elife00632.xml

<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.1d1 20130915//EN"  "JATS-archivearticle1.dtd"><article article-type="research-article" dtd-version="1.1d1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><front><journal-meta><journal-id journal-id-type="nlm-ta">elife</journal-id><journal-id journal-id-type="hwp">eLife</journal-id><journal-id journal-id-type="publisher-id">eLife</journal-id><journal-title-group><journal-title>eLife</journal-title></journal-title-group><issn publication-format="electronic">2050-084X</issn><publisher><publisher-name>eLife Sciences Publications, Ltd</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">00632</article-id><article-id pub-id-type="doi">10.7554/eLife.00632</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research article</subject></subj-group><subj-group subj-group-type="heading"><subject>Biophysics and structural biology</subject></subj-group></article-categories><title-group><article-title>Viral genome structures are optimal for capsid assembly</article-title></title-group><contrib-group><contrib contrib-type="author" id="author-4011"><name><surname>Perlmutter</surname><given-names>Jason D</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-4094"><name><surname>Qiao</surname><given-names>Cong</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-4033"><name><surname>Hagan</surname><given-names>Michael F</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="other" rid="par-1"/><xref ref-type="other" rid="par-2"/><xref ref-type="fn" rid="con3"/><xref ref-type="fn" rid="conf1"/></contrib><aff id="aff1"><institution content-type="dept">Martin A Fisher School of Physics</institution>, <institution>Brandeis University</institution>, <addr-line><named-content content-type="city">Waltham</named-content></addr-line>, <country>United States</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Roux</surname><given-names>Benoit</given-names></name><role>Reviewing editor</role><aff><institution>University of Chicago</institution>, <country>United States</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>hagan@brandeis.edu</email></corresp></author-notes><pub-date date-type="pub" publication-format="electronic"><day>14</day><month>06</month><year>2013</year></pub-date><pub-date pub-type="collection"><year>2013</year></pub-date><volume>2</volume><elocation-id>e00632</elocation-id><history><date date-type="received"><day>12</day><month>02</month><year>2013</year></date><date date-type="accepted"><day>14</day><month>05</month><year>2013</year></date></history><permissions><copyright-statement>© 2013, Perlmutter et al</copyright-statement><copyright-year>2013</copyright-year><copyright-holder>Perlmutter et al</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/3.0/"><license-p>This article is distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use and redistribution provided that the original author and source are credited.</license-p></license></permissions><self-uri content-type="pdf" xlink:href="elife00632.pdf"/><abstract><object-id pub-id-type="doi">10.7554/eLife.00632.001</object-id><p>Understanding how virus capsids assemble around their nucleic acid (NA) genomes could promote efforts to block viral propagation or to reengineer capsids for gene therapy applications. We develop a coarse-grained model of capsid proteins and NAs with which we investigate assembly dynamics and thermodynamics. In contrast to recent theoretical models, we find that capsids spontaneously ‘overcharge’; that is, the negative charge of the NA exceeds the positive charge on capsid. When applied to specific viruses, the optimal NA lengths closely correspond to the natural genome lengths. Calculations based on linear polyelectrolytes rather than base-paired NAs underpredict the optimal length, demonstrating the importance of NA structure to capsid assembly. These results suggest that electrostatics, excluded volume, and NA tertiary structure are sufficient to predict assembly thermodynamics and that the ability of viruses to selectively encapsidate their genomic NAs can be explained, at least in part, on a thermodynamic basis.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.001">http://dx.doi.org/10.7554/eLife.00632.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><object-id pub-id-type="doi">10.7554/eLife.00632.002</object-id><title>eLife digest</title><p>Viruses are infectious agents made up of proteins and a genome made of DNA or RNA. Upon infecting a host cell, viruses hijack the cell’s gene expression machinery and force it to produce copies of the viral genome and proteins, which then assemble into new viruses that can eventually infect other host cells. Because assembly is an essential step in the viral life cycle, understanding how this process occurs could significantly advance the fight against viral diseases.</p><p>In many viral families, a protein shell called a capsid forms around the viral genome during the assembly process. However, capsids can also assemble around nucleic acids in solution, indicating that a host cell is not required for their formation. Since capsid proteins are positively charged, and nucleic acids are negatively charged, electrostatic interactions between the two are thought to have an important role in capsid assembly. However, it is unclear how structural features of the viral genome affect assembly, and why the negative charge on viral genomes is actually far greater than the positive charge on capsids. These questions are difficult to address experimentally because most of the intermediates that form during virus assembly are too short-lived to be imaged.</p><p>Here, Perlmutter et al. have used state of the art computational methods and advances in graphical processing units (GPUs) to produce the most realistic model of capsid assembly to date. They showed that the stability of the complex formed between the nucleic acid and the capsid depends on the length of the viral genome. Yield was highest for genomes within a certain range of lengths, and capsids that assembled around longer or shorter genomes tended to be malformed.</p><p>Perlmutter et al. also explored how structural features of the virus—including base-pairing between viral nucleic acids, and the size and charge of the capsid—determine the optimal length of the viral genome. When they included structural data from real viruses in their simulations and predicted the optimal lengths for the viral genome, the results were very similar to those seen in existing viruses. This indicates that the structure of the viral genome has been optimized to promote packaging into capsids. Understanding this relationship between structure and packaging will make it easier to develop antiviral agents that thwart or misdirect virus assembly, and could aid the redesign of viruses for use in gene therapy and drug delivery.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.002">http://dx.doi.org/10.7554/eLife.00632.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>virus capsid</kwd><kwd>self assembly</kwd><kwd>RNA Packaging</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>Viruses</kwd></kwd-group><funding-group><award-group id="par-1"><funding-source><institution-wrap><institution>National Institutes of Health, National Institute of Allergy and Infectious Diseases</institution></institution-wrap></funding-source><award-id>R01AI080791</award-id><principal-award-recipient><name><surname>Hagan</surname><given-names>Michael F</given-names></name></principal-award-recipient></award-group><award-group id="par-2"><funding-source><institution-wrap><institution>National Science Foundation XSEDE (Keeneland, Longhorn, and Condor)</institution></institution-wrap></funding-source><award-id>TG-MCB090163</award-id><principal-award-recipient><name><surname>Hagan</surname><given-names>Michael F</given-names></name></principal-award-recipient></award-group><funding-statement>The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.</funding-statement></funding-group><custom-meta-group><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>Computer simulations reveal that viral nucleic acids have an ideal structure for being packaged into outer protein shells called capsids.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="s1" sec-type="intro"><title>Introduction</title><p>For many viruses the spontaneous assembly of a protein shell, or capsid, around the viral nucleic acid (NA) is an essential step in the viral lifecycle. Identifying the factors which enable capsids to efficiently and selectively assemble around the viral genome could identify targets for new antiviral drugs that block or derail the formation of infectious virions. Conversely, understanding how assembly depends on the NA and protein structure would guide efforts to reengineer capsid proteins and human NAs for gene therapy applications. From a fundamental perspective, high-order complexes that assemble from protein and/or NAs abound in biology. Learning how the properties of viral components determine their co-assembly can shed light on assembly mechanisms of a broad array of structures and the associated selective pressures on their components. In this article, we use GPU computing (<xref ref-type="bibr" rid="bib12">Anderson et al., 2008</xref>; <xref ref-type="bibr" rid="bib47">Nguyen et al., 2011</xref>; <xref ref-type="bibr" rid="bib40">LeBard et al., 2012</xref>) and a simplified, but quantitatively testable, model to elucidate the effects of electrostatics, capsid geometry, and NA tertiary structure on assembly.</p><p>Assembly around NAs is predominately driven by electrostatic interactions between NA phosphate groups and basic amino acids, often located in flexible tails known as arginine rich motifs (ARMs) (e.g., <xref ref-type="bibr" rid="bib54">Schneemann, 2006</xref>). There is a correlation between the net charge of these protein motifs and the genome length for many ssRNA viruses (<xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>; <xref ref-type="bibr" rid="bib29">Hu et al., 2008</xref>), with a ‘charge ratio’ of negative charge on NAs to positive charge on proteins typically of order 2:1 (i.e., viruses are ‘overcharged’). Electrophoresis measurements confirm that viral particles are negatively charged (e.g., [<xref ref-type="bibr" rid="bib57">Serwer et al., 1995</xref>; <xref ref-type="bibr" rid="bib56">Serwer and Griess, 1999</xref>; <xref ref-type="bibr" rid="bib49">Porterfield et al., 2010</xref>]), though these measurements include contributions from the capsid exteriors (<xref ref-type="bibr" rid="bib9">Bozic et al., 2012</xref>; <xref ref-type="bibr" rid="bib76">Zlotnick et al., 2013</xref>). Based on these observations, it has been proposed that viral genome lengths are thermodynamically optimal for assembly, meaning that their lengths minimize the free energy of the assembled nucleocapsids. However, while estimates of optimal lengths have varied (<xref ref-type="bibr" rid="bib2">van der Schoot and Bruinsma, 2005</xref>; <xref ref-type="bibr" rid="bib3">Angelescu et al., 2006</xref>; <xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>; <xref ref-type="bibr" rid="bib29">Hu et al., 2008</xref>; <xref ref-type="bibr" rid="bib58">Siber and Podgornik, 2008</xref>; <xref ref-type="bibr" rid="bib10">Ting et al., 2011</xref>; <xref ref-type="bibr" rid="bib48">Ni et al., 2012</xref>; <xref ref-type="bibr" rid="bib59">Siber et al., 2012</xref>), recent theoretical models based on linear polyelectrolytes (<xref ref-type="bibr" rid="bib58">Siber and Podgornik, 2008</xref>; <xref ref-type="bibr" rid="bib10">Ting et al., 2011</xref>; <xref ref-type="bibr" rid="bib48">Ni et al., 2012</xref>) have consistently predicted that optimal NA lengths correspond to ‘undercharging’ (fewer NA charges than positive capsid charges). These results lead to the conclusion that capsid assembly around genomic (overcharged) NAs requires an external driving force such as a Donnan potential (<xref ref-type="bibr" rid="bib10">Ting et al., 2011</xref>). Yet, viruses preferentially assemble around genomic length RNAs even in vitro (<xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>), in the absence of such a driving force.</p><p>The effect of NA structural features other than charge remains unclear. In some cases, genomic NAs are preferentially packaged over others with equivalent charge (<xref ref-type="bibr" rid="bib8">Borodavka et al., 2012</xref>) due to virus-specific packaging sequences (<xref ref-type="bibr" rid="bib15">Bunka et al., 2011</xref>; <xref ref-type="bibr" rid="bib8">Borodavka et al., 2012</xref>). However, experiments on other viruses have demonstrated a striking lack of virus-specific interactions (<xref ref-type="bibr" rid="bib49">Porterfield et al., 2010</xref>; <xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>). For example, cowpea chlorotic mottle virus (CCMV) proteins preferentially encapsidate BMV RNA over the genomic CCMV RNA (<xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>). Since the two NAs are of similar length, the authors propose that other structural features, such as NA tertiary structure (<xref ref-type="bibr" rid="bib73">Yoffe et al., 2008</xref>), may drive this preferential encapsidation. However, the relationship between NA structure and assembly has not been explored.</p><p>To clarify this relationship, we use a computational model to investigate capsid assembly dynamics and thermodynamics as functions of NA and capsid charge, solution ionic strength, capsid geometry, and NA size (resulting from tertiary structure). We first test the proposed link between the thermodynamic optimum length, <inline-formula><mml:math id="inf1"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and assembly, finding that the yield of assembled nucleocapsids at relevant timescales is maximal near <inline-formula><mml:math id="inf2"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. Longer-than-optimal NAs lead to non-functional structures, indicating that the thermodynamic optimum <inline-formula><mml:math id="inf3"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> corresponds to an upper bound for the genome size for capsids which spontaneously assemble and package their genome. We then explore how <inline-formula><mml:math id="inf4"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> depends on solution conditions and the structures of capsids and NAs. We find that overcharging occurs spontaneously, requiring no external driving force. When base-pairing is accounted for, predicted optimal NA lengths are consistent with the genome size for a number of viruses, suggesting that electrostatics and NA tertiary structure are important factors in the formation and stability of viral particles. Our predictions can be tested quantitatively in in vitro packaging experiments (e.g., [<xref ref-type="bibr" rid="bib49">Porterfield et al., 2010</xref>; <xref ref-type="bibr" rid="bib1">Cadena-Nava et al., 2012</xref>; <xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>]).</p></sec><sec id="s2"><title>Model</title><p>Our coarse-grained capsid model (<xref ref-type="fig" rid="fig1">Figure 1</xref>) is motivated by the recent observation (<xref ref-type="bibr" rid="bib38">Kler et al., 2012</xref>) that purified simian virus 40 (SV40) capsid proteins assemble around ssRNA molecules in vitro to form capsids comprising 12 homopentamer subunits. We model the capsid as a dodecahedron, composed of 12 pentagonal subunits (each of which represents a rapidly forming and stable pentameric intermediate, which then more slowly assembles into the complete capsid, as is the case for SV40 [<xref ref-type="bibr" rid="bib41">Li et al., 2002</xref>]). Our model extends those of <xref ref-type="bibr" rid="bib68">Wales (2005)</xref>, <xref ref-type="bibr" rid="bib21">Fejer et al. (2009)</xref>, <xref ref-type="bibr" rid="bib33">Johnston et al. (2010)</xref>, with subunits attracted to each other via attractive pseudoatoms at the vertices (type ‘A’) and driven toward a preferred subunit–subunit angle by repulsive ‘Top’ pseudoatoms (type ‘T’) and ‘Bottom’ pseudoatoms (type ‘B’) (see <xref ref-type="fig" rid="fig1">Figure 1</xref> and the ‘Methods’). In contrast to previous models for polyelectrolyte encapsidation (<xref ref-type="bibr" rid="bib3">Angelescu et al., 2006</xref>; <xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>; <xref ref-type="bibr" rid="bib37">Kivenson and Hagan, 2010</xref>; <xref ref-type="bibr" rid="bib44">Mahalik and Muthukumar, 2012</xref>), the proteins contain positive charges located in flexible polymeric tails, representing the ARM (arginine-rich motif) NA binding domains typical of positive-sense ssRNA virus capsid proteins.<fig id="fig1" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.003</object-id><label>Figure 1.</label><caption><p>Schematics and representative images of model systems. (<bold>A</bold>), (<bold>B</bold>) Model schematic for (<bold>A</bold>) a single subunit, and (<bold>B</bold>) two interacting subunits, showing positions of the attractor (‘A’), Top (‘T’), and Bottom (‘B’) pseudoatoms, which are defined in the ‘Model’ section and in the ‘Methods’. (<bold>C</bold>) (left) The pentameric SV40 capsid protein subunit, which motivates our model. The globular portions of proteins are shown in blue and the beginning of the NA binding motifs (ARMs) in yellow, though much of the ARMs are not resolved in the crystal structure (<xref ref-type="bibr" rid="bib62">Stehle et al., 1996</xref>). Space-filling model of the basic subunit model (middle) and a pentamer from the PC2 model (right). (<bold>D</bold>) A cutaway view of complete CCMV and PC2 capsids (with respective biological charge ratios of 1.8 and 1.32). Beads are colored as follows: blue = excluders, green = attractors, yellow = positive ARM bead, gray = neutral ARM bead, red = polyelectrolyte.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.003">http://dx.doi.org/10.7554/eLife.00632.003</ext-link></p></caption><graphic xlink:href="elife00632f001"/></fig></p><p>To investigate the effect of NA properties on assembly we consider two models for the packaged polymer: (1) a linear flexible polyelectrolyte and (2) a NA with predefined secondary and tertiary structure (i.e., static base-pairs) that captures the size, shape, and rigidity of NAs. Single-stranded regions are modeled as flexible polymers with one bead per nucleotide (<xref ref-type="bibr" rid="bib75">Zhang and Glotzer, 2004</xref>; <xref ref-type="bibr" rid="bib20">ElSawy et al., 2011</xref>), with charge <inline-formula><mml:math id="inf5"><mml:mrow><mml:mo>−</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></inline-formula>. Double-stranded regions of NAs comprise two adjoined semiflexible strands with the net persistence length of dsDNA (<inline-formula><mml:math id="inf6"><mml:mrow><mml:mo>≈</mml:mo><mml:mn>50</mml:mn></mml:mrow></mml:math></inline-formula> nm), and base-paired nucleotides are connected by harmonic bonds. Electrostatics are modeled using Debye–Huckel interactions to account for screening, except where these are tested against simulations with Coulomb interactions and explicit salt ions (see <xref ref-type="fig" rid="fig3">Figure 3D</xref> below).</p><p>In addition to representing the secondary structures of specific ssRNA genomes, we are able to tune statistical measures of base-pairing, such as the fraction of nucleotides that are base-paired, the relative frequency of hairpins and higher-order junctions (<xref ref-type="fig" rid="fig6">Figure 6</xref>), and the maximum ladder distance (MLD), which measures the extension in graph space of a NA secondary structure (<xref ref-type="bibr" rid="bib73">Yoffe et al., 2008</xref>). As shown in <xref ref-type="fig" rid="fig6">Figure 6</xref>, the radius of gyration <inline-formula><mml:math id="inf7"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> of the model NAs depends on MLD as <inline-formula><mml:math id="inf8"><mml:mrow><mml:mn>1.7</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mtext>MLD</mml:mtext></mml:mrow><mml:mrow><mml:mn>0.43</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, which has a slightly smaller exponent than a theory in which only base-paired segments were accounted for (<xref ref-type="bibr" rid="bib73">Yoffe et al., 2008</xref>). Further model details and parameters are presented in the ‘Methods’.</p></sec><sec id="s3" sec-type="results"><title>Results</title><sec id="s3-1"><title>Capsid assembly leads to spontaneous overcharging</title><p>We begin by presenting the results of simulations on our simplest capsid and cargo models. Our model capsid has a dodecahedron inradius (defined as the distance from the capsid center to a face center) of <inline-formula><mml:math id="inf9"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula> nm, to give an interior volume consistent with that of the smallest icosahedral viruses, and contains 60 ARMs (i.e., a <inline-formula><mml:math id="inf10"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> capsid, where <italic>T</italic> is the triangulation number [<xref ref-type="bibr" rid="bib14">Caspar and Klug, 1962</xref>]) each containing five positively charged residues. The cargo is a linear polyelectrolyte. While we systematically alter both the cargo and capsid below to include more biological detail, the simple model demonstrates two important results (that are consistent with results from more complex models): (1) Viral particles spontaneously overcharge during assembly, and (2) The thermodynamic optimal polyelectrolyte length closely correlates with the length for which dynamical assembly leads to the highest yield of complete viral particles.</p><sec id="s3-1-1"><title>Dynamical simulations</title><p>The results of Brownian dynamics simulations of capsid assembly around a linear polyelectrolyte at physiological salt concentration (Debye screening length <inline-formula><mml:math id="inf11"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> nm) are shown in <xref ref-type="fig" rid="fig2">Figure 2</xref>. Consistent with most ssRNA virus proteins, the polymer is essential for assembly under the simulated conditions, since the subunit–subunit interactions are too weak for formation of empty capsids (see below). <xref ref-type="fig" rid="fig2">Figure 2A</xref> presents representative snapshots of the assembly process for a polyelectrolyte with 600 segments (see also <xref ref-type="other" rid="video1">Video 1</xref>). The subunits first adsorb onto the polymer in a disordered fashion, with on average about eight subunits adsorbing before first formation of a critical nucleus (a complex comprising five subunits, <xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1</xref>). Once a critical nucleus forms, additional subunits add to it sequentially and reversibly until the final subunit closes around the polymer.<fig-group><fig id="fig2" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.004</object-id><label>Figure 2.</label><caption><p>Capsid assembly around a linear polyelectrolyte. (<bold>A</bold>) Snapshots illustrating assembly of subunits with ARM length = 5 around a linear polyelectrolyte with 600 segments. Beads are colored as in <xref ref-type="fig" rid="fig1">Figure 1</xref>. (<bold>B</bold>) Fraction of trajectories leading to a complete capsid as a function of polymer length (top axis) or charge ratio (bottom axis). The dashed line indicates the thermodynamic optimum charge ratio or length <inline-formula><mml:math id="inf12"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>from equilibrium calculations. Snapshots of typical outcomes above and below the optimal length are shown. (Far right) A typical assembly outcome for polymer length 1200 (twice <inline-formula><mml:math id="inf13"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>) is compared to an EM image of CCMV proteins assembled around an RNA which is twice the CCMV genome length (image extracted from panel C of Figure 5 in <xref ref-type="bibr" rid="bib1">Cadena-Nava et al., 2012</xref>).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.004">http://dx.doi.org/10.7554/eLife.00632.004</ext-link></p></caption><graphic xlink:href="elife00632f002"/></fig><fig id="fig2s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.005</object-id><label>Figure 2—figure supplement 1.</label><caption><title>Estimation of the critical nucleus size.</title><p>As in <xref ref-type="bibr" rid="bib37">Kivenson and Hagan (2010)</xref>, we define the critical nucleus size <inline-formula><mml:math id="inf224"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>nuc</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>as the smallest cluster of subunits for which more than 50% of trajectories proceed to complete assembly before complete disassembly (defined as reaching a state of <inline-formula><mml:math id="inf225"><mml:mrow><mml:mi>n</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula>). This plot, which is for trajectories with a linear polymer of length 575, indicates <inline-formula><mml:math id="inf226"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>nuc</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.005">http://dx.doi.org/10.7554/eLife.00632.005</ext-link></p></caption><graphic xlink:href="elife00632fs001"/></fig><fig id="fig2s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.006</object-id><label>Figure 2—figure supplement 2.</label><caption><title>The residual chemical potential <inline-formula><mml:math id="inf227"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mtext>r</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>calculated by the Widom test-particle insertion method.</title><p>Here, the residual chemical potential is shown for a linear polyelectrolyte, isolated in solution (red squares) and encapsidated in the simple capsid model (<xref ref-type="fig" rid="fig1">Figure 1</xref>) (black circles). Results from replica exchange (REX) simulations on the encapsidated polymer are also shown (blue triangles).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.006">http://dx.doi.org/10.7554/eLife.00632.006</ext-link></p></caption><graphic xlink:href="elife00632fs002"/></fig></fig-group><media content-type="glencoe play-in-place height-250 width-310" id="video1" mime-subtype="mp4" mimetype="video" xlink:href="elife00632v001.mov"><object-id pub-id-type="doi">10.7554/eLife.00632.007</object-id><label>Video 1.</label><caption><title>Capsid Assembly.</title><p>Movie illustrating assembly of subunits with ARM length=5 around a linear polyelectrolyte with 600 segments. Beads are colored as in <xref ref-type="fig" rid="fig1">Figure 1</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.007">http://dx.doi.org/10.7554/eLife.00632.007</ext-link></p></caption></media><fig-group><fig id="fig3" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.008</object-id><label>Figure 3.</label><caption><title>Effect of control parameters on the thermodynamic optimal length and charge ratio.</title><p>(<bold>A</bold>) Effect of increasing capsid charge, with capsid <inline-formula><mml:math id="inf14"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula> nm. (<bold>B</bold>) Effect of increasing capsid size for fixed ARM length = 5. (<bold>C</bold>) Effect of base-pairing, with <inline-formula><mml:math id="inf15"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula> base-paired nucleotides and varying maximum ladder distance (MLD), for capsid <inline-formula><mml:math id="inf16"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula> nm and ARM Length = 5. Snapshots of our model NA structures with small and large MLD’s are shown (prior to encapsidation), with double-stranded regions in red and single-stranded regions in blue. The result for no base-pairing (linear) is shown as a dashed line. (<bold>D</bold>) Effect of ionic strength and comparison between Debye–Huckel interactions and explicit ions. The thermodynamic optimum lengths <inline-formula><mml:math id="inf17"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and corresponding optimal charge ratios are shown as functions of the ionic strength (Debye screening length), calculated with simulations using Debye–Huckel (DH) interactions (red circles) or Coulomb interactions with explicit ions, 1:1 salt and no divalent cations (blue diamonds), 1% 2:1 salt (blue triangles), or 5% 2:1 salt (green triangles). An additional system with monovalent free ions and divalent cations irreversibly bound to the polyelectrolyte is also presented (yellow diamonds, see ‘Model potentials and parameters’). Calculations were performed using the simple capsid model (<xref ref-type="fig" rid="fig1">Figure 1</xref>) and a linear polyelectrolyte.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.008">http://dx.doi.org/10.7554/eLife.00632.008</ext-link></p></caption><graphic xlink:href="elife00632f003"/></fig><fig id="fig3s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.009</object-id><label>Figure 3—figure supplement 1.</label><caption><title>The thermodynamic optimum lengths <inline-formula><mml:math id="inf230"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and charge ratios monotonically decrease with increasing persistence length for a linear, semiflexible polyelectrolyte.</title><p>While this observation could be anticipated on intuitive grounds, the quantitative decrease is substantial—a 32 % decrease in optimal charge ratio between our most flexible polymer (<inline-formula><mml:math id="inf231"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>2.1</mml:mn></mml:mrow></mml:math></inline-formula> nm) and our stiffest polymer (<inline-formula><mml:math id="inf232"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>53.4</mml:mn></mml:mrow></mml:math></inline-formula> nm). The persistence length is obtained by simulating the polymer unencapsidated in solution and fitting the segmental autocorrelation function to an exponential decay, where the persistence length is the decay constant. The simulations were performed using the simple capsid model (<xref ref-type="fig" rid="fig1">Figure 1</xref>, with dodecahedron inradius <inline-formula><mml:math id="inf233"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula>, and an ARM length of 5 positive charges). Representative snapshots of the encapsidated polymer (taken from simulations at the optimal length) are shown for several persistence lengths. The capsid and ARMs are rendered invisible in these snapshots to enable visibility of the polyelectrolyte.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.009">http://dx.doi.org/10.7554/eLife.00632.009</ext-link></p></caption><graphic xlink:href="elife00632fs003"/></fig><fig id="fig3s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.010</object-id><label>Figure 3—figure supplement 2.</label><caption><title>Effect of varying the ion radius.</title><p>Explicit ions (blue circles) are varied in size at 100 mM monovalent ions, compared with the DH result (dashed red line).</p><p>The explicit ion results approach those of the Debye–Huckel model simulations at physiological salt concentrations (100 mM) as the explicit ion radius is decreased, since ion excluded-volume is reduced, with the two methods agreeing to within 10% for the most realistic ion radius (0.125 nm). Note, in <xref ref-type="fig" rid="fig3">Figure 3D</xref> all ion radii were set to 0.125 nm.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.010">http://dx.doi.org/10.7554/eLife.00632.010</ext-link></p></caption><graphic xlink:href="elife00632fs004"/></fig><fig id="fig3s3" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.011</object-id><label>Figure 3—figure supplement 3.</label><caption><title>Effects of counterion condensation on <inline-formula><mml:math id="inf234"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</title><p>In a previous work (<xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>) it was noted that the high charge densities of RNA and peptide tails will give rise to counterion condensation, where the linear charge density of a polyelectrolyte is renormalized by condensed counterions to an effective linear charge density of <inline-formula><mml:math id="inf235"><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>B</mml:mtext></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>charges/nm, with <inline-formula><mml:math id="inf236"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.714</mml:mn></mml:mrow></mml:math></inline-formula>nm as the Bjerrum length. We chose not to charge renormalize in our simulations which used the Debye–Huckel model because association of RNA or an anionic linear polyelectrolyte with the oppositely charged peptide tails will lead to dissociation of condensed counterions. To test the validity of this choice (and to further test the validity of the Debye–Huckel model), we calculated optimal lengths <inline-formula><mml:math id="inf237"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>as a function of the linear charge density for a linear polyelectrolyte using both the Debye–Huckel (with no assumed counterion condensation red circles) and Coulomb interactions with explicit counterions (blue diamonds). In the latter simulations counterion condensation arises naturally and responds to local charge densities with no approximations. The linear charge density was varied by adjusting the equilibrium bond lengths between neighboring beads in the polyelectrolyte; all other parameters were unchanged. The results are compared to the results of simulations with the Debye–Huckel model and irreversible counterion condensation (black line). To obtain this result, we performed a single simulation using the Debye–Huckel model with a linear charge density of 1 <inline-formula><mml:math id="inf238"><mml:mrow><mml:mrow><mml:mrow><mml:mtext>charge</mml:mtext></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>B</mml:mtext></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, and then assumed all charge densities exceeding this value are renormalized, so that the optimal charge ratio increases proportionally with the bare linear charge density. That is, at a charge density of 2 <inline-formula><mml:math id="inf239"><mml:mrow><mml:mrow><mml:mrow><mml:mtext>charges</mml:mtext></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>B</mml:mtext></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, only half the polymer is effectively charged and the optimal charge ratio (calculated as a ratio of bare charge on the RNA to bare charge on the peptide arms) doubles. The simulations used the simple capsid model (<inline-formula><mml:math id="inf240"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula>, ARM Length = 5) at a Deybe length of <inline-formula><mml:math id="inf241"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>nm or a salt concentration of 100 mM, respectively.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.011">http://dx.doi.org/10.7554/eLife.00632.011</ext-link></p></caption><graphic xlink:href="elife00632fs005"/></fig></fig-group><media content-type="glencoe play-in-place height-250 width-310" id="video2" mime-subtype="mp4" mimetype="video" xlink:href="elife00632v002.mov"><object-id pub-id-type="doi">10.7554/eLife.00632.012</object-id><label>Video 2.</label><caption><title>Base-Paired Capsid Assembly.</title><p>Movie illustrating assembly of subunits with ARM length=5 around a base-paired molecule with a charge ratio of 2.5 and normalized MLD of 0.67. Beads are colored as in <xref ref-type="fig" rid="fig6">Figure 6</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.012">http://dx.doi.org/10.7554/eLife.00632.012</ext-link></p></caption></media></p><p>The assembly outcome depends on polymer length, with successful capsid formation occurring when there is overcharging, meaning that the negative charge on the polymer exceeds the net positive charge on an assembled capsid (<inline-formula><mml:math id="inf18"><mml:mrow><mml:mn>300</mml:mn><mml:mi>e</mml:mi></mml:mrow></mml:math></inline-formula> for this model). <xref ref-type="fig" rid="fig2">Figure 2B</xref> shows the yield of well-formed capsids at <inline-formula><mml:math id="inf19"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mi>t</mml:mi><mml:mtext>u</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> (<inline-formula><mml:math id="inf20"><mml:mrow><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> time steps), at which point the fraction assembled has approximately plateaued for most parameter values. Here a well-formed capsid is defined as a structure comprising 12 subunits that each interact with five neighboring subunits and together completely encapsulate the polymer. Assembly is robust (yield <inline-formula><mml:math id="inf21"><mml:mrow><mml:mo>∼</mml:mo><mml:mn>0.9</mml:mn></mml:mrow></mml:math></inline-formula>) near an optimal polymer length of <inline-formula><mml:math id="inf22"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>dyn</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:mn>575</mml:mn></mml:mrow></mml:math></inline-formula> segments, corresponding to a ‘charge ratio’ of <inline-formula><mml:math id="inf23"><mml:mrow><mml:mrow><mml:mrow><mml:mn>575</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>300</mml:mn></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>1.9</mml:mn></mml:mrow></mml:math></inline-formula>. Above the optimal length, the polymer is typically not fully incorporated when capsid assembly nears completion. For sufficiently long polymers (e.g., 2 <inline-formula><mml:math id="inf24"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>dyn</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, <xref ref-type="fig" rid="fig2">Figure 2B</xref> right) multiple capsids assemble on the same polymer. These multiplet structures resemble configurations seen in a previous simulation study which did not explicitly consider electrostatics (<xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>) and observed in experiments in which CCMV proteins assembled around RNAs longer than the CCMV genome length (<xref ref-type="bibr" rid="bib1">Cadena-Nava et al., 2012</xref>). For polymer lengths well below <inline-formula><mml:math id="inf25"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>dyn</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> the polymer is completely encapsulated before assembly completes, and addition of the remaining subunits slows substantially. Although capsids which are incomplete at the conclusion of these simulation might eventually reach completion, the low yield of assembled capsids at our finite measurement time reflects the fact that assembly at these parameters is less efficient than for polymer lengths near <inline-formula><mml:math id="inf26"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>dyn</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p></sec><sec id="s3-1-2"><title>Equilibrium calculations</title><p>We calculated the thermodynamic optimal polymer length <inline-formula><mml:math id="inf27"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, or the length of encapsulated polymer that minimizes the free energy of the polymer–capsid complex, with two different methods. First, we performed Brownian dynamics simulations of a long polymer and a preassembled capsid with one subunit made permeable to the polymer, so that the length of encapsulated polymer is free to equilibrate. Second, we calculated the residual chemical potential difference between the encapsidated polymer and a polymer free in solution (<xref ref-type="bibr" rid="bib69">Widom, 1963</xref>; <xref ref-type="bibr" rid="bib39">Kumar et al., 1991</xref>; <xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>). The first method predicts an optimal polymer length of <inline-formula><mml:math id="inf28"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:mn>574</mml:mn></mml:mrow></mml:math></inline-formula> while the latter suggests <inline-formula><mml:math id="inf29"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>550</mml:mn><mml:mo>−</mml:mo><mml:mn>575</mml:mn></mml:mrow></mml:math></inline-formula>, indistinguishable from the optimal length found in the finite-time dynamical assembly simulations (<xref ref-type="fig" rid="fig2">Figure 2B</xref>). The observation that the yield of encapsulated polymers from dynamical assembly trajectories diminishes above <inline-formula><mml:math id="inf30"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, together with the observation that many viruses with single-stranded genomes assemble and package their nucleic acid spontaneously, suggests that this equilibrium value may set an upper bound on the size of a viral genome.</p></sec></sec><sec id="s3-2"><title>The effect of control parameters on packaged lengths</title><sec id="s3-2-1"><title>Capsid structure affects packaged lengths</title><p>Since our simulations show that <inline-formula><mml:math id="inf31"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf32"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>dyn</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> are closely correlated, we performed a series of equilibrium calculations in which ionic strength, capsid structural parameters, and the NA model were systematically varied, to determine the effect of each parameter on <inline-formula><mml:math id="inf33"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. To determine how <inline-formula><mml:math id="inf34"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and the optimal charge ratio depend on the number of positive charges in the capsid, we first varied the length of the ARMs, keeping all ARM residues positively charged. As shown in <xref ref-type="fig" rid="fig3">Figure 3A</xref> (inset), <inline-formula><mml:math id="inf35"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> increases sub-linearly with capsid charge, meaning that each additional ARM charge increases the equilibrium polymer packaging length by a smaller amount, leading to a diminishing charge ratio. We obtained a similar result when, instead of modeling flexible ARMs, we placed charges in rigid patches on the inner capsid surface (e.g., corresponding to MS2 [<xref ref-type="bibr" rid="bib66">Valegard et al., 1997</xref>]). However, we find that charges on the surface lead to a lower optimal charge ratio than the equivalent number of charges located in flexible ARMs (<xref ref-type="fig" rid="fig3">Figure 3A</xref>), since the ARM flexibility increases the volume of configuration space available for NA–ARM interactions. These observations demonstrate that, while electrostatics is an important factor, excluded-volume and the lengths of polyelectrolyte segments that bridge between ARMs (discussed below) also affect the length of packaged polyelectrolyte. However, in the biologically relevant range of 5–20 positive charges per protein monomer (<xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>; <xref ref-type="bibr" rid="bib29">Hu et al., 2008</xref>), the optimal length appears roughly linear with capsid charge (but with a positive intercept).</p><p>To understand how capsid size influences <inline-formula><mml:math id="inf36"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, we varied the model capsid radius while holding the number of capsid charges fixed. As shown in <xref ref-type="fig" rid="fig3">Figure 3B</xref>, <inline-formula><mml:math id="inf37"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and hence the optimal charge ratio increase dramatically with capsid size, scaling roughly with capsid radius as <inline-formula><mml:math id="inf38"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow><mml:mrow><mml:mn>1.6</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>. The non-integer exponent is intriguing, as it rules out scaling with capsid volume, surface area, or a linear path length, which would respectively result in <inline-formula><mml:math id="inf39"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow><mml:mn>3</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="inf40"><mml:mrow><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula>, or <inline-formula><mml:math id="inf41"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. Projecting the density of packaged polymer segments onto angular coordinates (<xref ref-type="fig" rid="fig5s2">Figure 5—figure supplement 2</xref>) reveals that the polymer is not homogenously distributed throughout the capsid surface, but instead has enriched density at the vertices and edges relative to the subunit faces. This result is consistent with experimental observations that nucleic acids form dodecahedral cages in viral particles (<xref ref-type="bibr" rid="bib60">Speir and Johnson, 2012</xref>), and our model may describe scaling of the optimal charge ratio with volume for these capsids. For model capsids with <inline-formula><mml:math id="inf42"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mn>12.5</mml:mn></mml:mrow></mml:math></inline-formula> nm, the amount of polymer segments directly interacting with ARM charges becomes independent of capsid size, and the dependence of optimal length on volume can be attributed to the lengths of polymer between ARMs (see ‘Discussion’).</p></sec><sec id="s3-2-2"><title>Base-pairing increases packaged lengths</title><p>To understand how the geometric effects of base-pairing contribute to packaging, we performed dynamical assembly simulations and equilibrium calculations of <inline-formula><mml:math id="inf43"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for a wide range of base-pairing patterns and fraction of base-paired nucleotides (see section ‘Base-paired polymer’). The key result is that for all simulated base-pairing patterns, increasing the fraction of base-paired nucleotides (up to the biological fraction of 50%) increases <inline-formula><mml:math id="inf44"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> (<xref ref-type="fig" rid="fig3 fig6">Figures 3C and 6D</xref>). The increase in optimal length can be as large as 200–250 nucleotides for our small <inline-formula><mml:math id="inf45"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> capsid, indicating that base-pairing can contribute significantly to the amount of polymer that can be packaged. This effect can be explained by the fact that nucleotide–nucleotide interactions which drive NA structure formation effectively cancel some NA charge–charge repulsions and result in NA structures that are compact in comparison to linear polymers with the same lengths. Thus encapsulated NAs incur smaller excluded–volume interactions, electrostatic repulsions, and conformational entropy penalties during assembly.</p><p>However, the connection between the size of a molecule in solution and <inline-formula><mml:math id="inf46"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is surprisingly subtle. As described in the section ‘Base-paired polymer’, we have quantified base-pairing patterns by their maximum ladder distance (MLD), which counts the maximum number of base-pairs along any non-repeating path across the NA and thus describes the extent of the molecule in the secondary structure graph space. As shown in <xref ref-type="fig" rid="fig6">Figure 6</xref>, for a NA with 1000 segments and 50% base-pairing, the solution radius of gyration varies with MLD as <inline-formula><mml:math id="inf47"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mtext>MLD</mml:mtext></mml:mrow><mml:mrow><mml:mn>0.43</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> to yield <inline-formula><mml:math id="inf48"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>≈</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math></inline-formula> nm to <inline-formula><mml:math id="inf49"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>≈</mml:mo><mml:mn>20</mml:mn></mml:mrow></mml:math></inline-formula> nm, in comparison the linear model <inline-formula><mml:math id="inf50"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>25.5</mml:mn></mml:mrow></mml:math></inline-formula> nm. As shown in <xref ref-type="fig" rid="fig3">Figure 3C</xref> the inclusion of base-pairing has a large effect on <inline-formula><mml:math id="inf51"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, but changes in MLD have only a minor effect. Though over this range of MLDs the solution <inline-formula><mml:math id="inf52"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> more than doubles, <inline-formula><mml:math id="inf53"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>*</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> changes by only about 10%, with an even smaller variation over the range of MLDs that we estimate for biological RNA molecules <inline-formula><mml:math id="inf54"><mml:mrow><mml:mrow><mml:mrow><mml:mtext>MLD</mml:mtext></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mtext>Max</mml:mtext><mml:mo> </mml:mo><mml:mtext>MLD</mml:mtext></mml:mrow></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0.25</mml:mn><mml:mo>,</mml:mo><mml:mn>0.55</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> based on <xref ref-type="bibr" rid="bib22">Gopal et al. (2012)</xref> (see section ‘Base-paired polymer’ for additional detail).</p></sec><sec id="s3-2-3"><title>Semiflexible polymer</title><p>The effect of persistence length without tertiary structure (i.e., dsDNA) is shown in <xref ref-type="fig" rid="fig3s1">Figure 3—figure supplement 1</xref>.</p></sec><sec id="s3-2-4"><title>Effect of salt concentration</title><p>To understand the dependence of <inline-formula><mml:math id="inf55"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> on ionic strength and to evaluate the effect of the approximations made in the Debye–Huckel treatment of electrostatics, we performed a number of simulations using the primitive model representation of electrostatics and explicit ions to represent neutralizing counterions and added salt (the ‘Model potentials and parameters’ section). Ions are modeled as repulsive spheres (<xref ref-type="disp-formula" rid="equ6">Equation 6</xref> below) and electrostatics are calculated according to Coulomb interactions (<xref ref-type="disp-formula" rid="equ12">Equation 12</xref> below) with the relative permittivity set to 80.</p><p>As shown in <xref ref-type="fig" rid="fig4">Figure 4</xref>, the optimal length <inline-formula><mml:math id="inf56"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and charge ratio increase with increasing ionic strength (i.e., decreasing Debye length <inline-formula><mml:math id="inf57"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>). This effect can be explained by the fact that a smaller fraction of NA charges interact with positive capsid charges as the screening length decreases (see the ‘Discussion’ section). Importantly, the simulations predict overcharging at all salt concentrations investigated <inline-formula><mml:math id="inf58"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo> </mml:mo><mml:mtext>mM</mml:mtext><mml:mo>≤</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mtext>salt</mml:mtext></mml:mrow></mml:msub><mml:mi mathvariant="normal">≲</mml:mi><mml:mn>400</mml:mn><mml:mo> </mml:mo><mml:mtext>mM</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. Over this range, we see that optimal lengths predicted by simulations using explicit ions or Debye–Huckel interactions agree to within about 10% (<xref ref-type="fig" rid="fig4">Figure 4</xref>). The Debye–Huckel simulations slightly overpredict the optimal length at high salt concentrations because they neglect counterion excluded-volume, while they underpredict the optimal length at low ionic strength because they neglect ion–ion correlations. We also present the results of the limiting case where only neutralizing counterions are used (resulting in <inline-formula><mml:math id="inf59"><mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> mM cations and 0 anions, for a total ionic strength of <inline-formula><mml:math id="inf60"><mml:mrow><mml:mo>∼</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula> mM). Further simulations exploring the effect of divalent cations show only a slight increase in <inline-formula><mml:math id="inf61"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> at physiologically relevant divalent cation concentrations (1 mM) (<xref ref-type="fig" rid="fig4">Figure 4</xref>). Results of additional simulations examining the effect of ion size and charge renormalization are shown in <xref ref-type="fig" rid="fig3s2">Figure 3—figure supplement 2</xref> and <xref ref-type="fig" rid="fig3s3">Figure 3—figure supplement 3</xref>. We focus on <inline-formula><mml:math id="inf62"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mtext>salt</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></inline-formula> mM 1:1 salt for all other results in this article.<fig-group><fig id="fig4" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.013</object-id><label>Figure 4.</label><caption><p>Correspondance between viral genomes and model calculations. Comparison between viral genome lengths and calculated thermodynamic optimal lengths (<bold>A</bold>) and charge ratios (<bold>B</bold>) for models based on the indicated viral capsid structures (see <xref ref-type="table" rid="tbl1">Table 1</xref>). Predicted optimal lengths are shown for linear polyelectrolytes (red circles) and model NAs (blue triangles) with 50% base-pairing. Viral genome lengths are shown with green pentagons symbols. Error bars fall within the symbol sizes.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.013">http://dx.doi.org/10.7554/eLife.00632.013</ext-link></p></caption><graphic xlink:href="elife00632f004"/></fig><fig id="fig4s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.014</object-id><label>Figure 4—figure supplement 1.</label><caption><title>Our capsid model can be modified to describe specific viral capsids by altering the capsid radius and ARM sequence.</title><p>Atomic-resolution structures of capsids are available for PC2, STNV, STMV, SPMV, PaV, BMV, and CCMV (<xref ref-type="bibr" rid="bib34">Jones and Liljas, 1984</xref>; <xref ref-type="bibr" rid="bib4">Ban and McPherson, 1995</xref>; <xref ref-type="bibr" rid="bib61">Speir et al., 1995</xref>; <xref ref-type="bibr" rid="bib63">Larson et al., 1998</xref>; <xref ref-type="bibr" rid="bib65">Tang et al., 2001</xref>; <xref ref-type="bibr" rid="bib42">Lucas et al., 2002</xref>; <xref ref-type="bibr" rid="bib36">Khayat et al., 2011</xref>). For each capsid structure, we estimated the radius by fitting the radial density of capsid protein (C, N, S, O atoms), as plotted here, to a Gaussian. For <inline-formula><mml:math id="inf242"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>capsids, we scaled the inradius of our dodecahedral model capsid (<xref ref-type="fig" rid="fig1">Figure 1</xref>) until its interior volume was equal to the volume of a sphere with the radius of the biological capsid. The ARMs were anchored as shown in <xref ref-type="fig" rid="fig1">Figure 1</xref>, midway across the pentagonal radius (we found that changing the locations of anchor points did not substantially affect <inline-formula><mml:math id="inf243"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>), and the sequence of positive, negative, and neutral beads was set to match the amino acid sequence of the capsid protein for the virus being modeled. For <inline-formula><mml:math id="inf244"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula>capsids, an icosahedrally symmetric capsid was designed with the excluders and ARMs placed based on the crystal structure of the Brome Mosaic Virus (<xref ref-type="bibr" rid="bib42">Lucas et al., 2002</xref>). For other <inline-formula><mml:math id="inf245"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula>viruses the ARM sequence and capsid radius were adjusted. For the satellite viruses, there are basic residues located on the capsid inner surface (in addition to those found in the ARM); for each such residue a positive charge was rigidly fixed to the inner surface of the model capsid. No atomic-resolution structures for capsids of viruses in the Nanoviridae family are available, so the capsid radius for BBT was based on electron microscopy (<xref ref-type="bibr" rid="bib27">Harding et al., 1991</xref>).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.014">http://dx.doi.org/10.7554/eLife.00632.014</ext-link></p></caption><graphic xlink:href="elife00632fs006"/></fig><fig id="fig4s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.015</object-id><label>Figure 4—figure supplement 2.</label><caption><title>Optimal lengths are sensitive to multiple factors.</title><p>Values of the thermodynamic optimum length <inline-formula><mml:math id="inf246"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>for the capsids considered in <xref ref-type="fig" rid="fig4">Figure 4</xref> plotted against (<bold>A</bold>) total capsid charge, (<bold>B</bold>) capsid inradius, (<bold>C</bold>) ARM volume fraction. Values are shown for a linear polyelectrolyte, the model base-paired NA, and the actual genome length for each virus. In panel (<bold>A</bold>) we find that a linear fit yields a slope similar to that previously observed (1.75), which we present as a comparison (<xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.015">http://dx.doi.org/10.7554/eLife.00632.015</ext-link></p></caption><graphic xlink:href="elife00632fs007"/></fig></fig-group></p></sec></sec><sec id="s3-3"><title>Predictions for specific viral capsid structures</title><p>To evaluate the significance of the trends identified above for packaging in a biological context, we performed equilibrium calculations in which the structural parameters discussed above (capsid volume, ARM length, charge, and NA base-pairing) were based on specific <inline-formula><mml:math id="inf64"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf65"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> viruses (whose capsids are assembled from 60 and 180 protein copies respectively). For each investigated virus, the capsid radius was fit to protein densities in capsid crystal structures (<xref ref-type="bibr" rid="bib11">Carrillo-Tripp et al., 2009</xref>), the ARM length was determined from the structure, and charges in the ARM and on the capsid inner surface were assigned based on amino acid sequence (see <xref ref-type="table" rid="tbl1">Table 1</xref>). NAs were modeled with 50% base-pairing and <inline-formula><mml:math id="inf66"><mml:mrow><mml:mrow><mml:mrow><mml:mtext>MLD</mml:mtext></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mtext>Max</mml:mtext><mml:mo> </mml:mo><mml:mtext>MLD</mml:mtext></mml:mrow></mml:mrow><mml:mo>≈</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula>. Visualizations of <inline-formula><mml:math id="inf67"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf68"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> viruses (PC2 and CCMV) are presented in <xref ref-type="fig" rid="fig1">Figure 1D</xref> and further details details are provided in <xref ref-type="fig" rid="fig4s2">Figure 4—figure supplement 2</xref>.<table-wrap id="tbl1" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.016</object-id><label>Table 1.</label><caption><p>Details for the models of biological capsids studied in this article. The capsid inradius (distance from capsid center to face center), number of residues in the arginine rich motif (ARM), and net charge of the ARM and inner surface are features used to build these models. The viral genome length is then presented for comparison to the value of <inline-formula><mml:math id="inf69"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> predicted for the base-paired model. Finally, the fraction of occupied volume within the capsid is given for the base-paired model at the optimal length</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.016">http://dx.doi.org/10.7554/eLife.00632.016</ext-link></p></caption><table frame="hsides" rules="groups"><thead><tr><th>Virus</th><th>Inradius (nm)</th><th>ARM Length/Net charge</th><th>Genome length</th><th><inline-formula><mml:math id="inf70"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula></th><th>Occupied volume fraction</th></tr></thead><tbody><tr><td>PaV</td><td align="char" char=".">13.0</td><td>48/+13</td><td align="char" char=".">4322</td><td align="char" char=".">4766</td><td align="char" char=".">0.074</td></tr><tr><td>CCMV</td><td align="char" char=".">11.5</td><td>48/+10</td><td align="char" char=".">3233</td><td align="char" char=".">3136</td><td align="char" char=".">0.099</td></tr><tr><td>BMV</td><td align="char" char=".">11.5</td><td>44/+9</td><td align="char" char=".">3233</td><td align="char" char=".">3087</td><td align="char" char=".">0.093</td></tr><tr><td>PC2</td><td align="char" char=".">8.0</td><td>43/+22</td><td align="char" char=".">1767</td><td align="char" char=".">1672</td><td align="char" char=".">0.265</td></tr><tr><td>STNV</td><td align="char" char=".">7.7</td><td>28/+16</td><td align="char" char=".">1239</td><td align="char" char=".">1242</td><td align="char" char=".">0.240</td></tr><tr><td>BBT</td><td align="char" char=".">7.5</td><td>27/+12</td><td align="char" char=".">1066</td><td align="char" char=".">1058</td><td align="char" char=".">0.209</td></tr><tr><td>STMV</td><td align="char" char=".">7.2</td><td>19/+11</td><td align="char" char=".">1058</td><td align="char" char=".">922</td><td align="char" char=".">0.232</td></tr><tr><td>SPMV</td><td align="char" char=".">6.8</td><td>20/+13</td><td align="char" char=".">826</td><td align="char" char=".">918</td><td align="char" char=".">0.276</td></tr></tbody></table></table-wrap></p><p>The predicted values of <inline-formula><mml:math id="inf71"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for linear polyelectrolytes and base-paired NAs are compared to the actual viral genome lengths in <xref ref-type="fig" rid="fig4">Figure 4</xref>. We see that overcharging (charge ratios larger than 1, <xref ref-type="fig" rid="fig4">Figure 4B</xref>) is predicted for all structures. Furthermore, while the values of <inline-formula><mml:math id="inf72"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> predicted for linear polyelectrolytes fall short of the viral genome lengths for all investigated structures except for SPMV (whose virion has an unusually low charge ratio), <inline-formula><mml:math id="inf73"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for the NA models are relatively close to the viral genome lengths for most structures. We emphasize that the optimal length is sensitive to all of the control parameters; for example, <inline-formula><mml:math id="inf74"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is correlated not just with the capsid charge, but also with capsid volume and ARM packing fraction (see <xref ref-type="fig" rid="fig4s2">Figure 4—figure supplement 2</xref>). Recalling that <inline-formula><mml:math id="inf75"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> sets an upper bound on length of a polymer that can be efficiently packaged during assembly (<xref ref-type="fig" rid="fig2">Figure 2B</xref>), this result suggests that the geometric effects of base-pairing contribute to spontaneous packaging of viral genomes. The largest difference between <inline-formula><mml:math id="inf76"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and genome length occurs for STMV. This discrepancy may reflect the fact that we used a NA base-pairing fraction of <inline-formula><mml:math id="inf77"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula> whereas 57% of nucleotides participate in secondary structure elements in the STMV crystal structure (<xref ref-type="bibr" rid="bib63">Larson et al., 1998</xref>; <xref ref-type="bibr" rid="bib74">Zeng et al., 2012</xref>) (lower fractions of nucleotides are resolved in other virion structures, suggesting lower values of <inline-formula><mml:math id="inf78"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>).</p></sec></sec><sec id="s4" sec-type="discussion"><title>Discussion</title><p>We have shown that assembly simulations and equilibrium calculations based on our coarse-grained model predict optimal NA lengths which are overcharged and relatively close to actual genome sizes for a number of viruses. This finding contrasts with earlier continuum models solved under an assumption of spherical symmetry, which required either a Donnan potential (<xref ref-type="bibr" rid="bib10">Ting et al., 2011</xref>; <xref ref-type="bibr" rid="bib48">Ni et al., 2012</xref>) or irreversible charge renormalization of the NA (<xref ref-type="bibr" rid="bib6">Belyi and Muthukumar, 2006</xref>; see <xref ref-type="fig" rid="fig3s3">Figure 3—figure supplement 3</xref>) to account for overcharging. Our results (<xref ref-type="fig" rid="fig2 fig3 fig4">Figures 2, 3, 4</xref>) show that the optimal genome length is determined by a complex interplay between capsid charge, capsid size, excluded-volume, and RNA structure.</p><sec id="s4-1"><title>The origins of overcharging</title><p>Analysis of conformations of encapsulated polymers in our simulations shows that overcharging arises as a consequence of geometry and electrostatic screening. The presence of discrete positive charges located in ARMs (or on the capsid surface) combined with nm-scale screening of electrostatics limits the number of direct NA–protein electrostatic interactions; the remaining nucleotides are found in segments which bridge the gaps between positive charges. These interconnecting (bridging) segments are the primary origin of overcharging. Earlier approaches which assumed spherical symmetry could not capture these bridging segments and thus did not predict overcharging. The significance of bridging segments to overcharging is clearly revealed by the dependence of optimal length on capsid size under constant ARM length (<xref ref-type="fig" rid="fig3">Figures 3B</xref>). For <inline-formula><mml:math id="inf79"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mn>12.5</mml:mn></mml:mrow></mml:math></inline-formula> nm, the amount of NA interacting with the ARMs is constant, while bridging lengths increase with capsid radius (<xref ref-type="fig" rid="fig5s3">Figure 5—figure supplement 3</xref>) due to the increased typical distance between charges on different ARMs. The increase in <inline-formula><mml:math id="inf80"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> with capsid radius in these calculations can be attributed entirely to increased bridge lengths.</p><p>Although the amounts of bridging segments in the biological capsid models depend on many control parameters (e.g., charge, volume, packing fraction, RNA structure), we also confirmed the significance of bridging segments to overcharging in these calculations. <xref ref-type="fig" rid="fig5">Figure 5</xref> breaks down the <inline-formula><mml:math id="inf81"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> into the number of segments which interact with positive ARM charges and the number of segments which are bridging. If one counts only the NA segments that directly interact with capsid charges, the resulting charge ratio is slightly less than one for each of these capsids. However, when the bridging segments are included, all the capsids are overcharged. Interestingly, more bridging segments are found in the larger <inline-formula><mml:math id="inf82"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> capsids (56% bridging) than in <inline-formula><mml:math id="inf83"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> capsids (25% bridging), contributing to the larger predicted charge ratios for <inline-formula><mml:math id="inf84"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> capsids (<xref ref-type="fig" rid="fig4">Figure 4B</xref>). Though the fraction of nucleotides closely interacting with protein in capsids is difficult to measure experimentally, it might be estimated from the amounts of RNA resolved in crystallographic or EM structures. In a recent summary, Larsson et al. found that for 10 <inline-formula><mml:math id="inf85"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> crystal structures an average of 16% of NA were resolved. For <inline-formula><mml:math id="inf86"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> structures a wider range of values was obtained, where some had a large fraction of NA resolved (STMV = 62%, STNV = 34%), but other ssDNA viruses had resolved fractions similar to <inline-formula><mml:math id="inf87"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> viruses. An additional piece of evidence comes from low resolution neutron diffraction, where 72% of RNA was observed to be in the first layer of density along the inner capsid surface of the <inline-formula><mml:math id="inf88"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> STNV, again suggesting that much of the <inline-formula><mml:math id="inf89"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> viral genome is closely interacting with the protein (<xref ref-type="bibr" rid="bib7">Bentley et al., 1987</xref>). We present additional data describing the conformation of the polymer within the capsid, including the radial and angular densities as <xref ref-type="fig" rid="fig5s1 fig5s2 fig5s3 fig5s4">Figure 5—figure supplements 1, 2, 3, 4</xref>.<fig-group><fig id="fig5" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.017</object-id><label>Figure 5.</label><caption><title>Bridging in biological capsids. Number of NA segments that directly interact with positively charged ARM segments (interaction energy <inline-formula><mml:math id="inf90"><mml:mrow><mml:mo>≤</mml:mo><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>, blue squares) and bridging segments (interaction energy <inline-formula><mml:math id="inf92"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>, purple circles).</title><p>The numbers are calculated at the optimal length <inline-formula><mml:math id="inf94"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for each capsid shown in <xref ref-type="fig" rid="fig4">Figure 4</xref> using the base-paired model. For visual reference, the dashed line indicates a 1:1 correspondence between capsid charge and number of nucleotides.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.017">http://dx.doi.org/10.7554/eLife.00632.017</ext-link></p></caption><graphic xlink:href="elife00632f005"/></fig><fig id="fig5s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.018</object-id><label>Figure 5—figure supplement 1.</label><caption><title>Radial density for linear polymer and ARM segments in the simple capsid (A) and CCMV (B).</title><p>The sharp peak in ARM density is due to the first ARM segment, which is rigidly attached to the capsid shell. In the simple capsid the polymer segments are concentrated within a few nm of the capsid shell, with lower densities in the capsid center. For CCMV, the longer arms result in a more diffuse distribution of positive charges within the capsid interior as compared to the basic capsid model. While there is some co-localization of positively charged ARM and combined neutral and negatively charged polymer segments, their densities peak at slightly different radii. The CCMV ARM sequence contains 48 segments, with 11 positive segments and 1 negative segment. Though the charges are not homogenously distributed throughout the sequence (9 occur within a 19 segment stretch), the degree of separation observed was unexpected.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.018">http://dx.doi.org/10.7554/eLife.00632.018</ext-link></p></caption><graphic xlink:href="elife00632fs008"/></fig><fig id="fig5s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.019</object-id><label>Figure 5—figure supplement 2.</label><caption><title>Angular density of linear polymer segments (heat map) in the basic capsid model (A) and CCMV (B).</title><p>Green squares indicate the first ARM segment. Segment densities are averaged over radial distances of 5–6.25 nm (<bold>A</bold>) and 8.75–10 nm (<bold>B</bold>), as a function of the spherical angles, without angular averaging. For the simple capsid, the polymer more frequently resides in the vertices between subunits (between the clusters of 3 ARMs) as well as along the dodecahedral edges, and resides less frequently in the center of the subunit faces. The angular density is heterogeneous in CCMV, though to a lesser extent than found for the simple capsid.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.019">http://dx.doi.org/10.7554/eLife.00632.019</ext-link></p></caption><graphic xlink:href="elife00632fs009"/></fig><fig id="fig5s3" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.020</object-id><label>Figure 5—figure supplement 3.</label><caption><title>Capsid radius and polymer bridging.</title><p>Number of polymer segments interacting with positive capsid charges (red inverted triangles), and number of polymer segments not interacting with positive charges (bridging segments, blue diamonds), using threshold interaction distance of 0.74 nm, which corresponds to a screened electrostatic interaction of <inline-formula><mml:math id="inf248"><mml:mrow><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>. The numbers are calculated at the optimal polymer length <inline-formula><mml:math id="inf249"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> as a function of capsid inradius <inline-formula><mml:math id="inf250"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for the simple capsid with constant ARM length (<xref ref-type="fig" rid="fig3">Figure 3B</xref>). The number of polymer segments strongly interacting with ARM charges is constant for <inline-formula><mml:math id="inf251"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mn>12.5</mml:mn></mml:mrow></mml:math></inline-formula> nm, while the number of bridging segments increases to span the distances between arms. Hence, for capsids with <inline-formula><mml:math id="inf252"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mtext>in</mml:mtext></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mn>12.5</mml:mn></mml:mrow></mml:math></inline-formula> nm, the observed dependence of <inline-formula><mml:math id="inf253"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> on capsid size arises entirely due to bridging segments. For smaller capsids, there is a weak increase in the number of interacting segments with size as more conformational space around the ARMs becomes available.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.020">http://dx.doi.org/10.7554/eLife.00632.020</ext-link></p></caption><graphic xlink:href="elife00632fs010"/></fig><fig id="fig5s4" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.00632.021</object-id><label>Figure 5—figure supplement 4.</label><caption><title>Number of NA segments that directly interact with positively charged ARM segments and bridging segments, for both the linear and base-paired model.</title><p>For visual reference, the dashed line indicates a 1:1 correspondence between capsid charge and number of nucleotides. This data shows that while the base-paired polymer increases the charge ratio it does so by increasing both the number of segments which are tightly bound and bridging.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.021">http://dx.doi.org/10.7554/eLife.00632.021</ext-link></p></caption><graphic xlink:href="elife00632fs011"/></fig></fig-group><fig id="fig6" position="float"><object-id pub-id-type="doi">10.7554/eLife.00632.022</object-id><label>Figure 6.</label><caption><p>Base-paired polymer setup and analysis. (<bold>A</bold>) Schematic illustrations of the algorithm we used to obtain a wide range of base-paired structures. From left to right, double-stranded (ds) segments are first randomly assigned. These segments are then base-paired together, starting from one end. If base-paired segments are widely separated (i.e., <inline-formula><mml:math id="inf95"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>is large) then subsequent nested base-pairs lead to an extended structure. Conversely, if <inline-formula><mml:math id="inf96"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is small, less extended structures form. The right-most panel indicates a psuedoknot, a structural motif we have prevented from occurring in this model, by setting <inline-formula><mml:math id="inf97"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> to the last unpaired segment. (<bold>B</bold>) Radius of gyration <inline-formula><mml:math id="inf98"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for model NAs isolated in solution as a function of maximum ladder distance (MLD) normalized by the maximum possible MLD. The nucleic acid has 1000 nt, 50% of which are base-paired. (<bold>C</bold>) The frequency of junction numbers can be altered by varying <italic>λ</italic> in <xref ref-type="disp-formula" rid="equ2">Equation 2</xref>, with large values of <italic>λ</italic> leading to large values of <inline-formula><mml:math id="inf99"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. The symbols indicate the relative frequency of junction numbers for biological RNAs with indicated lengths, obtained from Ref. (<xref ref-type="bibr" rid="bib22">Gopal et al., 2012</xref>), and the lines are best fits to these distributions generated by varying <italic>λ</italic>. The inset illustrates several different junction orders. (<bold>D</bold>) The thermodynamic optimum length measured for the simple model capsid as a function of the fraction of base-paired nucleotides <inline-formula><mml:math id="inf100"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for a simplified ‘hairpins only’ model (red squares). (<bold>E</bold>) Snapshots illustrating assembly around a NA. Beads are colored as follows: blue = excluders, yellow = ARM bead, red = single-stranded NA, cyan = double-stranded NA. ‘Top’, ‘Bottom’, and ‘Attractor’ beads removed for clarity.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.00632.022">http://dx.doi.org/10.7554/eLife.00632.022</ext-link></p></caption><graphic xlink:href="elife00632f006"/></fig></p><p>We emphasize that our coarse-grained model is designed to incorporate the minimal set of features required to explain the thermodynamic stability of viral particles, and thus neglects some factors that contribute to packaging specific NAs. The in vivo experiments in <xref ref-type="bibr" rid="bib48">Ni et al. (2012)</xref> on brome mosaic virus (BMV) showed that even charge-conserving mutations to ARM residues could affect the amounts and types of packaged RNA, possibly by interfering with coordination of RNA replication and encapsidation (<xref ref-type="bibr" rid="bib50">Rao, 2006</xref>). Similarly, packaging signals, or regions of RNA that have sequence-specific interactions with the capsid protein, are known for some viruses (e.g., HIV [<xref ref-type="bibr" rid="bib16">D’Souza and Summers, 2005</xref>] or MS2 and satellite tobacco necrosis virus [STNV] [<xref ref-type="bibr" rid="bib15">Bunka et al., 2011</xref>; <xref ref-type="bibr" rid="bib8">Borodavka et al., 2012</xref>]). Packaging signals could be added to our model to investigate how they favor selective assembly around the viral genome through kinetic (<xref ref-type="bibr" rid="bib8">Borodavka et al., 2012</xref>) or thermodynamic effects. The fact that our model predicts <inline-formula><mml:math id="inf102"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for STNV close to its genome length without accounting for sequence specificity may suggest that packaging signals have only a small effect on the thermodynamic <inline-formula><mml:math id="inf103"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p><p>In conclusion, our results elucidate the connection between structure and assembly for viral capsids. Firstly, our simulations show that ‘overcharged’ capsids are favored thermodynamically and kinetically, even in the absence of cellular factors or other external effects. Secondly, our results delineate how the genome length which is most favorable for assembly depends on virus-specific quantities such as capsid charge, capsid volume, and genomic tertiary structure. While the correlation between predicted optimal lengths and viral genome sizes suggests that our results have biological relevance, the physical foundations of our model can be tested via controlled in vitro experiments. As noted above, several existing experimental observations agree with our results. A positive correlation between protein charge and amounts of packaged RNA has been demonstrated through experiments in which the charge on capsid protein ARMs was altered by mutagenesis (e.g., [<xref ref-type="bibr" rid="bib18">Dong et al., 1998</xref>; <xref ref-type="bibr" rid="bib35">Kaplan et al., 1998</xref>; <xref ref-type="bibr" rid="bib67">Venter et al., 2009</xref>]). Competition assays (<xref ref-type="bibr" rid="bib49">Porterfield et al., 2010</xref>; <xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>), in which two species of NAs or other polymers compete for packaging by a limiting concentration of capsid proteins, offer a quantitative estimate of <inline-formula><mml:math id="inf104"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. For example, our prediction that <inline-formula><mml:math id="inf105"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> for CCMV is roughly consistent with the genome length (<xref ref-type="fig" rid="fig4">Figure 4</xref>) agrees with the observation that CCMV proteins preferentially package longer RNAs, up to the wildtype genome length, over shorter RNAs in competition assays (<xref ref-type="bibr" rid="bib13">Comas-Garcia et al., 2012</xref>). Now, it is possible to quantitatively test the predictions of our model for the dependence of <inline-formula><mml:math id="inf106"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> on protein charge and salt concentration through similar competition assays in which NA length preferences are observed for proteins with charge altered by mutagenesis under different ionic strengths. Similarly, our prediction that base-pairing increases <inline-formula><mml:math id="inf107"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> can be evaluated by comparison of assembly experiments on RNA and synthetic polyelectrolytes (e.g., polystyrene sulfonate) or RNA with base-pairing inhibited through chemical modification (e.g., etheno-RNA [<xref ref-type="bibr" rid="bib17">Dhason et al., 2012</xref>]). Our simulations predict that above the optimal length for a linear polyelectrolyte, only base-paired RNA will be packaged in high yields of well-formed capsids.</p></sec></sec><sec id="s5" sec-type="materials|methods"><title>Methods</title><sec id="s5-1"><title>Model description</title><p>We have extended a model for empty capsid assembly (<xref ref-type="bibr" rid="bib68">Wales, 2005</xref>; <xref ref-type="bibr" rid="bib21">Fejer et al., 2009</xref>; <xref ref-type="bibr" rid="bib33">Johnston et al., 2010</xref>) to describe assembly around NAs. A complete listing of the interaction potentials is provided below; here we present a concise description of our model. The pseudoatoms in the capsid subunit model are illustrated in <xref ref-type="fig" rid="fig1">Figure 1</xref>. Subunit assembly is mediated through an attractive Morse potential between Attractor (‘A’) pseudoatoms located at each subunit vertex. The Top (‘T’) pseudoatoms interact with other ‘T’ psuedoatoms through a potential consisting of the repulsive term of the LJ potential, the radius of which is chosen to favor a subunit-subunit angle consistent with a dodecahedron (116°). The Bottom (‘B’) pseudoatom has a repulsive LJ interaction with ‘T’ pseudoatoms, intended to prevent ‘upside-down’ assembly. The ‘T’, ‘B’, and ‘A’ pseudoatoms form a rigid body (<xref ref-type="bibr" rid="bib68">Wales, 2005</xref>; <xref ref-type="bibr" rid="bib21">Fejer et al., 2009</xref>; <xref ref-type="bibr" rid="bib33">Johnston et al., 2010</xref>). See <xref ref-type="bibr" rid="bib55">Schwartz et al. (1998)</xref>, <xref ref-type="bibr" rid="bib51">Rapaport et al. (1999)</xref>, <xref ref-type="bibr" rid="bib52">Rapaport (2004</xref>, <xref ref-type="bibr" rid="bib53">2008</xref>), <xref ref-type="bibr" rid="bib23">Hagan and Chandler (2006)</xref>, <xref ref-type="bibr" rid="bib28">Hicks and Henley (2006)</xref>, <xref ref-type="bibr" rid="bib46">Nguyen et al. (2007)</xref>, <xref ref-type="bibr" rid="bib70">Wilber et al., (2007</xref>, <xref ref-type="bibr" rid="bib71">2009a</xref>, <xref ref-type="bibr" rid="bib72">2009b)</xref>, <xref ref-type="bibr" rid="bib25">Hagan (2008)</xref>, <xref ref-type="bibr" rid="bib45">Nguyen and Brooks (2008)</xref>, <xref ref-type="bibr" rid="bib31">Nguyen et al. (2009)</xref>, <xref ref-type="bibr" rid="bib19">Elrad and Hagan (2010)</xref>, <xref ref-type="bibr" rid="bib33">Johnston et al. (2010)</xref>, <xref ref-type="bibr" rid="bib24">Hagan et al. (2011)</xref>, <xref ref-type="bibr" rid="bib44">Mahalik and Muthukumar (2012)</xref>, <xref ref-type="bibr" rid="bib26">Hagan (2013)</xref> for related models.</p><p>To model electrostatic interaction with a negatively charged NA or polyelectrolyte we extend the model as follows. Firstly, to better represent the capsid shell we add a layer of ‘Excluder’ pseudoatoms which have a repulsive LJ interaction with the polyelectrolyte and the ARMs. Each ARM is modeled as a bead–spring polymer, with one bead per amino acid. The ‘Excluders’ and first ARM segment are part of the subunit rigid body. ARM beads interact through repulsive Lennard–Jones interactions and, if charged, electrostatic interactions modeled by a Debye–Huckel potential. Comparison to Coulomb interactions with explicit counterions is shown in <xref ref-type="fig" rid="fig3">Figure 3D</xref>. We also show that the results do not change significantly when experimentally relevant concentrations of divalent cations are added to the system (<xref ref-type="fig" rid="fig3">Figure 3D</xref>). The ability of the Debye–Huckel model to provide a reasonable representation of electrostatics in the system can be understood based on the relatively low packing fractions (see <xref ref-type="table" rid="tbl1">Table 1</xref>) within the assembled capsids and the fact that the relevant experimental and physiological conditions correspond to moderate to high salt concentrations.</p><p>Simulations were performed with the Brownian Dynamics algorithm of HOOMD, which uses the Langevin equation to evolve positions and rigid body orientations in time (<xref ref-type="bibr" rid="bib12">Anderson et al., 2008</xref>; <xref ref-type="bibr" rid="bib47">Nguyen et al., 2011</xref>; <xref ref-type="bibr" rid="bib40">LeBard et al., 2012</xref>). Simulations were run using a set of fundamental units. The fundamental energy unit is selected to be <inline-formula><mml:math id="inf108"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mtext>u</mml:mtext></mml:msub><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>. The unit of length <inline-formula><mml:math id="inf109"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>u</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is set to the circumradius of a pentagonal subunit, which is taken to be <inline-formula><mml:math id="inf110"><mml:mrow><mml:mn>1</mml:mn><mml:msub><mml:mi>D</mml:mi><mml:mtext>u</mml:mtext></mml:msub><mml:mo>≡</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula> nm so that the dodecahedron inradius of <inline-formula><mml:math id="inf111"><mml:mrow><mml:mn>1.46</mml:mn><mml:msub><mml:mi>D</mml:mi><mml:mtext>u</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula> nm gives an interior volume consistent with that of the smallest <inline-formula><mml:math id="inf112"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> capsids. Assembly simulations were run at least 10 times for each set of parameters, each of which were concluded at <inline-formula><mml:math id="inf113"><mml:mrow><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> time steps. The following parameters values were used in all of our dynamical assembly simulations: <inline-formula><mml:math id="inf114"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> nm, box size = 200 × 200 × 200 nm, subunit concentration = 12<italic>μ</italic>M. During calculation of the thermodynamic optimal polymer length <inline-formula><mml:math id="inf117"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, calculations were run at least <inline-formula><mml:math id="inf118"><mml:mrow><mml:mn>1</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>7</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> timesteps, with equilibrium assessed after convergence. Standard error was obtained based on averages of multiple <inline-formula><mml:math id="inf119"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> independent simulations. Separate calculations of <inline-formula><mml:math id="inf120"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> were also performed using using the Widom test-particle method (<xref ref-type="bibr" rid="bib69">Widom, 1963</xref>) as extended to calculate polymer residual chemical potentials (<xref ref-type="bibr" rid="bib39">Kumar et al., 1991</xref>; <xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>) (described in more detail below). Snapshots from simulations were visualized using VMD (<xref ref-type="bibr" rid="bib30">Humphrey et al., 1996</xref>).</p></sec><sec id="s5-2"><title>Base-paired polymer</title><p>To obtain base-paired polymers with a wide and tunable range of structures (i.e., maximum ladder distances), we implement the following strategy. Firstly, the polymer contour length <inline-formula><mml:math id="inf121"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mtext>C</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, length of the base-paired segments <inline-formula><mml:math id="inf122"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and fraction of nucleotides in base-pairs <inline-formula><mml:math id="inf123"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> are free parameters which we specify (typical values are <inline-formula><mml:math id="inf124"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mtext>C</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:mrow></mml:math></inline-formula> nucleotides, <inline-formula><mml:math id="inf125"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula> nucleotides per segment, and <inline-formula><mml:math id="inf126"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula>). Secondly, we iterate over the linear sequence of the polymer, randomly choosing segments which will undergo base-pairing to form double-stranded (ds) segments. Each segment consists of <inline-formula><mml:math id="inf127"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> consecutive nucleotides. Segments are numbered sequentially to facilitate pairing (i.e., the first ds segment in the sequence is 1, the second is 2, and so on). Thirdly, these ds segments are then paired together. In the case of the hairpin model, each ds strand is paired with the next ds segment in the sequence (i.e., the first segment with the second, third with fourth, and so on). In the general base-pairing model, pairs are assigned stochastically according to an algorithm which allows us to simultaneously tune the distribution of junction orders and the maximum ladder distance (MLD). The algorithm is described in <xref ref-type="fig" rid="fig6">Figure 6A</xref> defined as follows:</p><p>The first step in assigning a base-pair is to obtain a random separation <inline-formula><mml:math id="inf128"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> from an exponential distribution where <italic>λ</italic> is the inverse of the mean:<disp-formula id="equ1"><label>(1)</label><mml:math id="m1"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>l</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>To prevent pseudoknots this <inline-formula><mml:math id="inf129"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is then subtracted from the maximal available separation <inline-formula><mml:math id="inf130"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> to yield <inline-formula><mml:math id="inf131"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>:<disp-formula id="equ2"><label>(2)</label><mml:math id="m2"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>The obtained <inline-formula><mml:math id="inf132"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> defines the number of segments separating the current segment from its base-pairing partner. With this algorithm, the single control parameter parameter <italic>λ</italic> is used to control both the base-pairing pattern, and thus MLD and the distribution of junction types, that is, the number of double stranded segments emerging from a single stranded intersection (see <xref ref-type="fig" rid="fig6">Figure 6C</xref>). When <italic>λ</italic> is large, we are more likely to obtain small values of <inline-formula><mml:math id="inf133"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and thus large values of <inline-formula><mml:math id="inf134"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. Large <inline-formula><mml:math id="inf135"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> values lead to more extensive structures (i.e., larger MLDs and a larger fraction of two-junctions). When <italic>λ</italic> is lower, we have a broader distribution of <inline-formula><mml:math id="inf136"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>random</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> values, and thus obtain smaller values of <inline-formula><mml:math id="inf137"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. If <inline-formula><mml:math id="inf138"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>pair</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is small, it creates higher-order junctions and regions which are not part of the MLD.</p><p>To describe the structures of the polymers generated by this algorithm, we make use of two structural parameters: the maximum ladder distance (MLD) and radius of gyration <inline-formula><mml:math id="inf139"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. As in (<xref ref-type="bibr" rid="bib73">Yoffe et al., 2008</xref>), we define the MLD as the largest number of base-pairs in any single path across the molecule’s secondary structure. <xref ref-type="fig" rid="fig6">Figure 6B</xref> describes the polymer radius of gyration <inline-formula><mml:math id="inf140"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> as a function of MLD, normalized by the maximal possible MLD (i.e., if all base-pairs were along a single path), for polymers of length 1000 with fraction base-pairing <inline-formula><mml:math id="inf141"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mtext>bp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula>. All of the base-paired polymers are compressed relative to the linear polymer (<inline-formula><mml:math id="inf142"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>25.5</mml:mn></mml:mrow></mml:math></inline-formula> nm), but they differ amongst themselves significantly. We observe <inline-formula><mml:math id="inf143"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> to vary with MLD as <inline-formula><mml:math id="inf144"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mtext>MLD</mml:mtext></mml:mrow><mml:mrow><mml:mn>0.43</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> to yield sizes in the range <inline-formula><mml:math id="inf145"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>≈</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math></inline-formula> nm to <inline-formula><mml:math id="inf146"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>≈</mml:mo><mml:mn>20</mml:mn></mml:mrow></mml:math></inline-formula>.</p><sec id="s5-2-1"><title>Effect of MLD on optimal charge ratio</title><p>In order to estimate biological MLD values, we fit the histogram of junction numbers generated by our algorithm with different values of <italic>λ</italic> and against the distribution of junction numbers obtained for biological ssRNA molecules in <xref ref-type="bibr" rid="bib22">Gopal et al. (2012)</xref> (<xref ref-type="fig" rid="fig6">Figure 6C</xref>). For the two cellular, noncoding ssRNA segments, we obtain normalized MLDs of 0.55 and 0.36, and for a viral segment (RNA2 of CCMV) we obtain 0.25. As shown in <xref ref-type="fig" rid="fig6">Figure 6B</xref> the radii of gyration for RNAs with lengths of <inline-formula><mml:math id="inf147"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mtext>C</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:mrow></mml:math></inline-formula> nt and the normalized MLDs of the cellular RNAs of 0.55 and 0.36 are respectively 14.1 nm and 11.8 nm. A 1000 nt RNA with the viral normalized MLD of 0.25 has <inline-formula><mml:math id="inf148"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>G</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>10.1</mml:mn></mml:mrow></mml:math></inline-formula> nm; that is, the viral-like RNA is compressed by 14–29% in solution. However, as shown in <xref ref-type="fig" rid="fig3">Figure 3C</xref>, the optimal charge ratios for these RNAs in the simple capsid model are within the large statistical error (we obtain 2.70, 2.75, and 2.78 respectively from a linear fit to the data). An example assembly simulation is shown in <xref ref-type="fig" rid="fig6">Figure 6E</xref> and <xref ref-type="other" rid="video1">Video 2</xref>.</p></sec></sec><sec id="s5-3"><title>Subunit–subunit binding free energy estimates</title><p>Our method of calculating the subunit–subunit binding free energy is similar to that presented in our previous simulations (<xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>; <xref ref-type="bibr" rid="bib24">Hagan et al., 2011</xref>). Briefly, subunits were modified such that only one edge formed attractive bonds, limiting complex formation to dimers. We measured the relative concentration of dimers and monomers for a range of attraction strengths (<italic>ε</italic>). The free energy of binding along that interface is then <inline-formula><mml:math id="inf149"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi><mml:mo> </mml:mo><mml:mtext>ln</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mtext>ss</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mtext>d</mml:mtext></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> with standard state concentration <inline-formula><mml:math id="inf150"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mtext>ss</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> M and <inline-formula><mml:math id="inf151"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mtext>d</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> in molar units. This free energy is well fit by the linear expression <inline-formula><mml:math id="inf152"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>1.5</mml:mn><mml:mi>ε</mml:mi><mml:mo>−</mml:mo><mml:mi>T</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mtext>b</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> where <inline-formula><mml:math id="inf153"><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mtext>b</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>9</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>. We can then correct for the multiplicity of dimer conformations, by adding in the additional term <inline-formula><mml:math id="inf154"><mml:mrow><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mtext>Δ</mml:mtext><mml:msub><mml:mi>s</mml:mi><mml:mtext>c</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mtext>ln</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn>25</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, where the five pentagonal edges are assumed to be distinguishable, but complex orientations which differ only through global rotation are not. Our assembly simulations are run at <inline-formula><mml:math id="inf155"><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>, for which we observe only transient subunit–subunit associations except in the presence of an anionic polyelectrolyte. Our free energy calculations agree with this observation, suggesting that for this value of <italic>ε</italic> binding is very weak: <inline-formula><mml:math id="inf156"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mtext>d</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.33</mml:mn><mml:mi>M</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf157"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1.1</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>. Note that formation of additional bonds in a capsid structure will give rise to substantially more negative binding free energies. As shown in <xref ref-type="bibr" rid="bib23">Hagan and Chandler (2006)</xref> much of the binding entropy penalty associated with adding a subunit to a capsid is incurred during the formation of the first bond, with smaller decreases in entropy associated with forming additional bonds. A similar set of calculations for capsids with the ARMs removed decreased the binding free energy to <inline-formula><mml:math id="inf158"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1.84</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>, indicating that ARM–ARM interactions increase the free energy by about <inline-formula><mml:math id="inf159"><mml:mrow><mml:mn>0.74</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mtext>B</mml:mtext></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula> along each interface at <inline-formula><mml:math id="inf160"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mtext>salt</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></inline-formula> mM.</p></sec><sec id="s5-4"><title>Equilibrium encapsidation</title><p>The free energy as a function of encapsidated polymer length was obtained by two different methods. In the first, we placed a very long polymer in or near a preassembled capsid, with one of the capsid subunits made permeable to the polymer. We then performed unbiased Brownian dynamics. Once the amount of packaged polymer reached equilibrium, the thermodynamic optimum length <inline-formula><mml:math id="inf161"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and the distribution of fluctuations around it were measured.</p><p>In the second approach we used the Widom test-particle method (<xref ref-type="bibr" rid="bib69">Widom, 1963</xref>) as extended to calculate polymer residual chemical potentials (<xref ref-type="bibr" rid="bib39">Kumar et al., 1991</xref>; <xref ref-type="bibr" rid="bib19">Elrad and Hagan, 2010</xref>). We performed independent sets of simulations for a free and an encapsidated polymer in which we calculated the residual chemical potential <inline-formula><mml:math id="inf162"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mtext>r</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> according to:<disp-formula id="equ3"><label>(3)</label><mml:math id="m3"><mml:mtable><mml:mtr><mml:mtd><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:msub><mml:mi>μ</mml:mi><mml:mtext>r</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mtext>chain</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mtext>chain</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>=</mml:mo><mml:mtext>log</mml:mtext><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:mtext>exp</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mtext>I</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>〉</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>where <inline-formula><mml:math id="inf163"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the number of segments in the polymer and <inline-formula><mml:math id="inf164"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mtext>I</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the interaction energy experienced by a test segment inserted onto either end of the polymer. Importance sampling was used to make the calculation feasible, where the bond length of inserted segments was chosen from a normal distribution matching the equilibrium distribution of bond lengths, truncated at <inline-formula><mml:math id="inf165"><mml:mrow><mml:mo>±</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> standard deviations. The effect of using this biased insertion was removed a posteriori according to standard non-Boltzmann sampling. Between incrementing <inline-formula><mml:math id="inf166"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="inf167"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> steps of dynamics were run, and <inline-formula><mml:math id="inf168"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> insertions were attempted for each value of <inline-formula><mml:math id="inf169"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> in 100 independent runs. The results of these calculations are presented in <xref ref-type="fig" rid="fig2s2">Figure 2—figure supplement 2</xref>, and based on the point of intersection between the encapsidated and unencapsidated chemical potentials, we estimate the optimal length <inline-formula><mml:math id="inf170"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> to be between 550–575 segments (or a charge ratio of <inline-formula><mml:math id="inf171"><mml:mrow><mml:mn>1.83</mml:mn><mml:mo>−</mml:mo><mml:mn>1.92</mml:mn></mml:mrow></mml:math></inline-formula>), which is close agreement with the preassembled dynamics calculations (574 segments or a charge ratio of 1.91). If we integrate the difference in chemical potential between the encapsidated and unencapsidated polymers between 0 and 575, we obtain <inline-formula><mml:math id="inf172"><mml:mrow><mml:mo>−</mml:mo><mml:mn>500</mml:mn><mml:mi>k</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula> as an estimate for the free energy of polymer encapsidation due to both polymer–ARM and polymer–polymer interactions in the simple capsid model with ARM length = 5. Since the primary motivation for the Widom test-particle method calculations was to provide an independent test of optimal lengths calculated using the semipermeable capsid, we only considered the Debye–Huckel model for electrostatics in test-particle method calculations.</p><p>To further assess the convergence and sampling of both approaches for calculating the <inline-formula><mml:math id="inf173"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>eq</mml:mtext></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, we performed additional replica exchange (REX) simulations (<xref ref-type="bibr" rid="bib64">Sugita and Okamoto, 1999</xref>). In replica exchange, replicas of the system are simulated in parallel at different temperatures. Periodically, structures are exchanged between temperatures based on the Metropolis Criterion. In our systems, 12 replicas were run, with temperatures distributed exponentially between 1 kT and 1.5 kT. This resulted in a satisfactory exchange frequency of 30–40%. We present the results for REX simulations in <xref ref-type="fig" rid="fig2s2">Figure 2—figure supplement 2</xref>, but in this case and all other cases, the REX results quantitatively agree with the results of our simulations run at a single temperature.</p></sec><sec id="s5-5"><title>Model potentials and parameters</title><p>In our model, all potentials can be decomposed into pairwise interactions. Potentials involving capsomer subunits further decompose into pairwise interactions between their constituent building blocks—the excluders, attractors, ‘Top’ and ‘Bottom’, and ARM pseudoatoms. It is convenient to write the total energy of the system as the sum of 6 terms: a capsomer-capsomer <inline-formula><mml:math id="inf174"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> part (which does not include interactions between ARM pseudoatoms), capsomer-polymer <inline-formula><mml:math id="inf175"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, capsomer-ARM <inline-formula><mml:math id="inf176"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>ca</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, polymer-polymer <inline-formula><mml:math id="inf177"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pp</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, polymer-ARM <inline-formula><mml:math id="inf178"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pa</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and ARM-ARM <inline-formula><mml:math id="inf179"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>aa</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> parts. Each is summed over all pairs of the appropriate type:<disp-formula id="equ4"><label>(4)</label><mml:math id="m4"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>cap </mml:mtext><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mtext> </mml:mtext><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>cap </mml:mtext><mml:mi>j</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>cap </mml:mtext><mml:mi>i</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:munder><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>poly </mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cp</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>cap </mml:mtext><mml:mi>i</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:munder><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>ARM </mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>ca</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>poly </mml:mtext><mml:mi>i</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:munder><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>poly </mml:mtext><mml:mi>j</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pp</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>poly </mml:mtext><mml:mi>i</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:munder><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>ARM </mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pa</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>ARM </mml:mtext><mml:mi>i</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:munder><mml:munder><mml:mstyle displaystyle="true" mathsize="140%"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mtext>ARM </mml:mtext><mml:mi>j</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>aa</mml:mtext></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>where <inline-formula><mml:math id="inf180"><mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>cap</mml:mtext><mml:mo> </mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>cap</mml:mtext><mml:mo> </mml:mo><mml:mi>j</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mtext>​</mml:mtext></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula> is the sum over all distinct pairs of capsomers in the system, <inline-formula><mml:math id="inf181"><mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>cap</mml:mtext><mml:mo> </mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>poly</mml:mtext><mml:mo> </mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mtext>​</mml:mtext></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula> is the sum over all capsomer-polymer pairs, etc.</p><p>The capsomer-capsomer potential <inline-formula><mml:math id="inf182"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is the sum of the attractive interactions between complementary attractors, and geometry guiding repulsive interactions between ‘Top’–‘Top’ pairs and ‘Top’–‘Bottom’ pairs. There are no interactions between members of the same rigid body, but ARMs are not rigid and thus there are intra-subunit ARM–ARM interactions. Thus, for notational clarity, we index rigid bodies and non-rigid pseudoatoms in Roman, while the pseudoatoms comprising a particular rigid body are indexed in Greek. For example, for capsomer <italic>i</italic> we denote its attractor positions as <inline-formula><mml:math id="inf183"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> with the set comprising all attractors <italic>α</italic>, its ‘Top’ positions <inline-formula><mml:math id="inf184"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, and its ‘Bottom’ positions <inline-formula><mml:math id="inf185"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. The capsomer–capsomer interaction potential between two capsomers <italic>i</italic> and <italic>j</italic> is then defined as:<disp-formula id="equ5"><label>(5)</label><mml:math id="m5"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cc</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">a</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>t</mml:mtext></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mi>ε</mml:mi><mml:mi mathvariant="script">L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>t</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>b</mml:mtext></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mtext>t</mml:mtext></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mi>ε</mml:mi><mml:mi mathvariant="script">L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>b</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>+</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>a</mml:mtext></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mi>ε</mml:mi><mml:mi mathvariant="script">M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">a</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant="italic">ϱ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula>where <italic>ε</italic> is an adjustable parameter which both sets the strength of the capsomer–capsomer attraction at each attractor site and scales the repulsive interactions which enforce the dodecahedral geometry. <inline-formula><mml:math id="inf186"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>t</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="inf187"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>b</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math id="inf188"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>a</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> are the number of ‘Top’, ‘Bottom’, and attractor pseudoatoms respectively in one capsomer, <inline-formula><mml:math id="inf189"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>t</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf190"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>b</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> are the effective diameters of the ‘Top’–‘Top’ interaction and ‘Bottom’–‘Top’ interactions, which are set to 10.5 nm and 9.0 nm respectively throughout this work, <inline-formula><mml:math id="inf191"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is the minimum energy attractor distance, set to 1 nm, ϱ is a parameter determining the width of the attractive interaction, set to 2.5, and <inline-formula><mml:math id="inf192"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is the cutoff distance for the attractor potential, set to 10.0 nm.</p><p>The function <inline-formula><mml:math id="inf193"><mml:mi mathvariant="normal">ℒ</mml:mi></mml:math></inline-formula> is defined as the repulsive component of the Lennard–Jones potential shifted to zero at the interaction diameter:<disp-formula id="equ6"><label>(6)</label><mml:math id="m6"><mml:mrow><mml:mi mathvariant="normal">ℒ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>σ</mml:mi><mml:mi>x</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>:</mml:mo><mml:mtext>otherwise</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>The function <inline-formula><mml:math id="inf194"><mml:mi mathvariant="normal">ℳ</mml:mi></mml:math></inline-formula> is a Morse potential:<disp-formula id="equ7"><label>(7)</label><mml:math id="m7"><mml:mrow><mml:mi mathvariant="normal">ℳ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant="italic">ϱ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant="italic">ϱ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant="italic">ϱ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>:</mml:mo><mml:mtext>otherwise</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>The capsomer–polymer interaction is a short-range repulsion that accounts for excluded-volume. For capsomer <italic>i</italic> with excluder positions <inline-formula><mml:math id="inf195"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> and polymer subunit <italic>j</italic> with position <inline-formula><mml:math id="inf196"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, the potential is:<disp-formula id="equ8"><label>(8)</label><mml:math id="m8"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cp</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>x</mml:mtext></mml:msub></mml:mrow></mml:munderover><mml:mrow><mml:mi mathvariant="normal">ℒ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>xp</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf197"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>x</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the number of excluders on a capsomer and <inline-formula><mml:math id="inf198"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>xp</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>x</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is the effective diameter of the excluder–polymer repulsion. The diameter of the polymer bead is <inline-formula><mml:math id="inf199"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula> nm and the diameter for the excluder beads is <inline-formula><mml:math id="inf200"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>x</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>3.0</mml:mn></mml:mrow></mml:math></inline-formula> nm for the <inline-formula><mml:math id="inf201"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> model and <inline-formula><mml:math id="inf202"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>x</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>5.25</mml:mn></mml:mrow></mml:math></inline-formula> nm for the <inline-formula><mml:math id="inf203"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> model.</p><p>The capsomer–ARM interaction is a short-range repulsion that accounts for excluded-volume. For capsomer <italic>i</italic> with excluder positions <inline-formula><mml:math id="inf204"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> and ARM subunit <italic>j</italic> with position <inline-formula><mml:math id="inf205"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, the potential is:<disp-formula id="equ9"><label>(9)</label><mml:math id="m9"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>cA</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mtext>x</mml:mtext></mml:msub></mml:mrow></mml:munderover><mml:mrow><mml:mi mathvariant="normal">ℒ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>xA</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math></disp-formula>with <inline-formula><mml:math id="inf206"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>xA</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>x</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>A</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> as the effective diameter of the excluder–ARM repulsion with <inline-formula><mml:math id="inf207"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mtext>A</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></inline-formula> nm the diameter of an ARM bead.</p><sec id="s5-5-1"><title>Electrostatic interactions among polymer, ARM, and ion beads</title><p>The polymer–polymer non-bonded interaction is composed of electrostatic repulsions and short-ranged excluded-volume interactions. These polymers also contain bonded interactions which are only evaluated for segments occupying adjacent positions along the polymer chain and angular interactions which are only evaluated for three sequential polymer segments. As noted in the main text, electrostatics are represented either by Debye–Huckel interactions or by Coulomb interactions with explicit salt ions. For the case of Debye–Huckel interactions,<disp-formula id="equ10"><label>(10)</label><mml:math id="m10"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pp</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant="script">K</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mtext> bonded</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant="script">K</mml:mi><mml:mrow><mml:mtext>angle</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>angle</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mtext> angle</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>p</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mtext>nonbonded</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf208"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is the center-to-center distance between the polymer subunits, <inline-formula><mml:math id="inf209"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>p</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> is the valence of charge on each polymer segment, and <inline-formula><mml:math id="inf210"><mml:mrow><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is a Debye–Huckel potential smoothly shifted to zero at the cutoff:<disp-formula id="equ11"><label>(11)</label><mml:math id="m11"><mml:mrow><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>l</mml:mi><mml:mtext>b</mml:mtext></mml:msub><mml:mo> </mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>σ</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>r</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mi>r</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>l</mml:mi><mml:mtext>b</mml:mtext></mml:msub><mml:mo> </mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>σ</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>r</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mi>r</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub><mml:mo>&lt;</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mtext>otherwise</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula></p><p><inline-formula><mml:math id="inf211"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mtext>D</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the Debye length, <inline-formula><mml:math id="inf212"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mtext>b</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the Bjerrum length, and <inline-formula><mml:math id="inf213"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf214"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> are the valences of the interacting charges. For the cases using explicit electrostatics the <inline-formula><mml:math id="inf215"><mml:mrow><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> potential is replaced by a Coulomb potential:<disp-formula id="equ12"><label>(12)</label><mml:math id="m13"><mml:mrow><mml:mi mathvariant="script">C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>ε</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mn>1</mml:mn><mml:mi>r</mml:mi></mml:mfrac></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf216"><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>ε</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is the term for the permittivity of free space and <inline-formula><mml:math id="inf217"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the relative permittivity of the solution, set to 80. Above a cutoff distance (<inline-formula><mml:math id="inf218"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> = 5 nm) the electrostatics are calculated using the particle-particle particle-mesh (PPPM) Ewald summation (<xref ref-type="bibr" rid="bib40">LeBard et al., 2012</xref>). Explicit ions are included in these simulations to represent both neutralizing counterions and added salt. Ions interact with other charged beads in the solution according to the Coulomb potential (<xref ref-type="disp-formula" rid="equ12">Equation 12</xref>) and interact with all beads through the repulsive shifted LJ interaction (<xref ref-type="disp-formula" rid="equ6">Equation 6</xref>). Except for the results in <xref ref-type="fig" rid="fig3s2">Figure 3—figure supplement 2</xref>, ionic radii were set to 0.125 nm (i.e., <inline-formula><mml:math id="inf219"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.25</mml:mn></mml:mrow></mml:math></inline-formula> nm in <xref ref-type="disp-formula" rid="equ6">Equation 6</xref> below), which is roughly equal to the radii of Na<sup>+</sup> and CL<sup>−</sup> ions in the CHARMM force field (<xref ref-type="bibr" rid="bib5">Beglov and Roux, 1994</xref>; <xref ref-type="bibr" rid="bib32">MacKerell et al., 1998</xref>; <xref ref-type="bibr" rid="bib43">Mackerell, 2004</xref>).</p><p>Specific binding between Mg<sup>2+</sup> ions and RNA is known to affect RNA structure. To test the effect of such stably bound divalent cations on optimal length, we constructed a polyelectrolyte with a divalent cation irreversible bound (through a harmonic potential, see <xref ref-type="disp-formula" rid="equ13">Equation 13</xref>) to every 100th NA segment, in a solution containing 100 mM 1:1 salt. While this model does not capture the structural effects of specific Mg<sup>2+</sup> binding to RNA, it does represent the fact that these bound cations effectively cancel some NA charges.</p><p>Bonds are represented by a harmonic potential:<disp-formula id="equ13"><label>(13)</label><mml:math id="m14"><mml:mrow><mml:msub><mml:mi mathvariant="script">K</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Angles are also represented by a harmonic potential:<disp-formula id="equ14"><label>(14)</label><mml:math id="m15"><mml:mrow><mml:msub><mml:mi mathvariant="script">K</mml:mi><mml:mrow><mml:mtext>angle</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>angle</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>angle</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>ϑ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf220"><mml:mrow><mml:msub><mml:mi>ϑ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is the angle formed by the sequential polymer units <inline-formula><mml:math id="inf221"><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></inline-formula>.</p><p>The ARM–ARM interaction is similar to the polymer–polymer interaction, consisting of non-bonded interactions composed of electrostatic repulsions and short-ranged excluded-volume interactions. These ARMs also contain bonded interactions which are only evaluated for segments occupying adjacent positions along the polymer chain:<disp-formula id="equ15"><label>(15)</label><mml:math id="m16"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>aa</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant="script">K</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>a</mml:mtext></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mtext>bond</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mtext>bonded</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>a</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mtext>a</mml:mtext></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>:</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mtext>nonbonded</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf222"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is the center-to-center distance between the ARM subunits and <inline-formula><mml:math id="inf223"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the valence of charge on ARM segment <italic>i</italic>.</p><p>Finally, the ARM–Polymer interaction is the sum of short-ranged excluded-volume interactions and electrostatic interactions:<disp-formula id="equ16"><label>(16)</label><mml:math id="m17"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mtext>pa</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>ℒ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>ap</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mrow><mml:mtext>DH</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mtext>ap</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p></sec></sec></sec></body><back><ack id="ack"><title>Acknowledgements</title><p>We gratefully acknowledge Chuck Knobler, William Gelbart, and Adam Zlotnick for insightful discussions and critical reads of the manuscript.</p></ack><sec sec-type="additional-information"><title>Additional information</title><fn-group content-type="competing-interest"><title>Competing interests</title><fn fn-type="conflict" id="conf1"><p>The authors declare that 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letter</article-title></title-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Roux</surname><given-names>Benoit</given-names></name><role>Reviewing editor</role><aff><institution>University of Chicago</institution>, <country>United States</country></aff></contrib></contrib-group></front-stub><body><boxed-text><p>eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see <ext-link ext-link-type="uri" xlink:href="http://www.elifesciences.org/the-journal/review-process">review process</ext-link>). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.</p></boxed-text><p>Thank you for sending your work entitled “Viral genome structures are optimal for capsid assembly” for consideration at <italic>eLife</italic>. Your article has been favorably evaluated by a Senior editor and 2 reviewers, one of whom is a member of our Board of Reviewing Editors.</p><p>The Reviewing editor and the other reviewer discussed their comments before we reached this decision, and the Reviewing editor has assembled the following comments to help you prepare a revised submission.</p><p>While both reviewers agree that the study is of very high interest, there is a need to provide further clarification and justification for the treatment of electrostatic screening in the model and its consequences on the robustness of the overall conclusions. The manuscript is clearly and cleanly written, logical, beautifully carried out, and of significance for the understanding of viral assembly and function. The authors first show dynamical assembly of capsids around NAs occurs optimally for parameters that thermodynamically optimize assembly. This allows for more controlled thermodynamic simulations that disentangle the role of charge, excluded volume, and tertiary structure have in the assembly process. Significantly, the authors find that assembled capsids spontaneously overcharge in the absence of eternal fields or influences. In all, one finds no flaws with the manuscript, either stylistically or in terms of the science. The study adds significantly to the field of virus biophysics. However, because the simulations are based on a simplified model of viral capsid assembly, there could be serious concerns about all the approximations that are made. Some clarification by the authors would be welcome to further strengthen the study.</p><p>1) Part of the appeal of the study is the simplicity of the underlying model, which confers great clarity to the analysis and the description of the results. The nicest and most impressive results are probably summarized in <xref ref-type="fig" rid="fig4">Figure 4</xref>. However, upon more scrutiny, one does also start to worry about the significance of the approximations that are being made, and how much confidence one has in the robustness of the conclusions. For example, it would be nice to convey visually how the viruses differ in <xref ref-type="fig" rid="fig4">Figure 4</xref>. The match between computations and length of genome is nice, but it is not clear what is the underlying basis for the agreement. Is the size of the base-paired length packed merely reflecting the accessible volume inside the capsid? The model is simple enough that one should be able to summarize what are the principal differences between the different viruses shown in <xref ref-type="fig" rid="fig4">Figure 4</xref> (size of full capsid, charge of the arms, etc.).</p><p>2) Much is made of the observation that the viruses are overcharged, that isi.e., the ratio between the negative (nucleic acid) over the positive (protein arms) charges is larger than 1. Essentially, this means that the assembled capsid with its packed nucleic acid chain is overall negative. The text is emphatic: “our simulations show that ‘overcharged’ capsids are favored thermodynamically and kinetically, even in the absence of cellular factors or other external effects.” But while one does not dispute the outcome of the simulations, it is disappointing that no simple argument is provided nor sketched to explain <italic>why</italic> this is so (a back-of-the-envelope type of explanation). This would add much clarification on the underlying reasons for this outcome of the model. More importantly, it isn't clear that the charge ratio predicted by the model is correct or even reasonable. For instance, what about the likely presence of magnesium ions, which would presumably cancel out the charge of the excess of negative charges? The ionic screening model used here is Debye-Huckel and the model does not incorporate the possible effect of divalent binding. Furthermore, is it experimentally known that these capsids are negatively charged? This ought to be detectable through electrophoretic mobility measurements (zeta potential). A figure supplement provides a comparison of a calculation with explicit ions presented as a validation of the Debye-Huckel approximation (<xref ref-type="fig" rid="fig3s3">Figure 3–figure supplement 3</xref>). Not much detail is offered about this test; presumably this is for a 1:1 electrolyte in a primitive model representation of ionic solution (continuum dielectric solvent and hard-sphere ions), but no ion radii are given so it is hard to see how realistic this model is. And again, what about these divalent ions binding to the DNA?</p><p>3) Additional cause for concern about the treatment of ionic screening arise from <xref ref-type="fig" rid="fig2s2">Figure 2–figure supplement 2</xref>, which shows the residual excess chemical potential for adding one segment to the nucleic acid chain when it is in bulk solution (red line) or when it is in a capsid (blue line). What is noteworthy here is that the residual excess chemical potential is positive for both cases. Normally, insertion of one charged particle in a continuum solvent with an electrolyte is a negative number because the interaction of the charged particle with the surroundings, which comprises solvent and mobile counterions, is favorable. This excess free energy to add one chain segment in solution would probably be negative, even within the context of Debye-Huckel. We believe that the numbers in <xref ref-type="fig" rid="fig2s2">Figure 2–figure supplement 2</xref> are positive because this interaction between the segment of the salt solution is not included in the present model. The latter probably only treats the Debye-Huckel screening of the interactions of the charged segment with other segments (which is positive and unfavorable) but the self-energy is neglected. Please confirm.</p><p>If this is so, the problem is that the favorable self-energy is probably much reduced once the segment is inserted inside the capsid because there is much less room to have counterions in the densely packed capsid. To take an analogy, this is a bit like an implicit solvent that changes the bare Coulomb's law q1*q1/r12 to q1*q2/(eps*r12), but which ignores the Born self-energy of the charges. Ignoring the self-term amounts to assume that it cancels out. This is okay if the environment is assumed to remain roughly the same. Is there enough free space left inside the capsid for the implicit solvent to account for the same Debye-Huckel treatment as in the bulk solution? What is the total volume of the interior of the capsid? What is the total fraction of volume occupied by the nucleic acid? How much free space is left for ionic solution (needed to justify the Debye-Huckel screening treatment)? Please clarify this issue and explain why the implicit treatment of counterions is sufficiently accurate here.</p></body></sub-article><sub-article article-type="reply" id="SA2"><front-stub><article-id pub-id-type="doi">10.7554/eLife.00632.024</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><p><italic>1) Part of the appeal of the study is the simplicity of the underlying model, which confers great clarity to the analysis and the description of the results. The nicest and most impressive results are probably summarized in <xref ref-type="fig" rid="fig4">Figure 4</xref>. However, upon more scrutiny, one does also start to worry about the significance of the approximations that are being made, and how much confidence one has in the robustness of the conclusions. For example, it would be nice to convey visually how the viruses differ in <xref ref-type="fig" rid="fig4">Figure 4</xref></italic>.</p><p>We have modified the text (under “Predictions for specific viral capsid structures”) to direct the reader to <xref ref-type="fig" rid="fig1">Figure 1D</xref> where visualizations of the models are presented. We have only included images of one T=1 capsid and one T=3 capsid because the complex interior of the capsids makes it difficult to visually resolve factors other than capsid volume.</p><p><italic>The match between computations and length of genome is nice, but it is not clear what is the underlying basis for the agreement. Is the size of the base-paired length packed merely reflecting the accessible volume inside the capsid</italic>?</p><p>As described later in the answer to point 3, the volume of the capsid is only one of several factors; we find that there is still a significant amount of free volume within the capsid when at the optimal packaging length. In order to further clarify how the structural features of the capsid lead to observed packaging, we have presented additional data as <xref ref-type="fig" rid="fig4s2">Figure 4–figure supplement 2</xref>. Please see the response to point 3 below for additional details.</p><p><italic>The model is simple enough that one should be able to summarize what are the principal differences between the different viruses shown in <xref ref-type="fig" rid="fig4">Figure 4</xref> (size of full capsid, charge of the arms, etc.)</italic>.</p><p>The differences between the capsid structures were summarized in a supplementary file. To better draw attention to this table, we have moved it to the main text (now <xref ref-type="table" rid="tbl1">Table 1</xref>). We have also added additional information (“Genome Length, Model Optimal Length, and Occupied Volume Fraction”).</p><p><italic>2) Much is made of the observation that the viruses are overcharged, i.e., the ratio between the negative (nucleic acid) over the positive (protein arms) charges is larger than 1. Essentially, this means that the assembled capsid with its packed nucleic acid chain is overall negative. The text is emphatic: “our simulations show that ‘overcharged’ capsids are favored thermodynamically and kinetically, even in the absence of cellular factors or other external effects.” But while one does not dispute the outcome of the simulations, it is disappointing that no simple argument is provided nor sketched to explain why this is so (a back-of-the-envelope type of explanation). This would add much clarification on the underlying reasons for this outcome of the model</italic>.</p><p>The simulations show that overcharging is a consequence of two factors – the electrostatic screening and geometry. Namely, it is geometrically not possible for every charge on the encapsulated NA to approach within a Debye length of an opposite charge on the capsid. It is primarily the presence of these charges not able to interact with capsid charges that lead to overcharging. Earlier models were not able to capture this because the locations of charges were smeared by the assumption of spherical symmetry. Based on the reviewers’ question, we have modified the beginning of the Discussion to more clearly explain the origins of overcharging. We also present a new figure (<xref ref-type="fig" rid="fig5">Figure 5</xref>) to clarify this observation: if one accounts only for the NA charges that directly interact with capsid charges the system appears slightly undercharged (as predicted by the earlier continuum models with an assumption of spherical symmetry). The overcharging arises due to the presence of NA, which forms a path from capsid charge to capsid charge.</p><p><italic>More importantly, it isn’t clear that the charge ratio predicted by the model is correct or even reasonable. For instance, what about the likely presence of magnesium ions, which would presumably cancel out the charge of the excess of negative charges? The ionic screening model used here is Debye-Huckel and the model does not incorporate the possible effect of divalent binding. […] And again, what about these divalent ions binding to the DNA</italic>?</p><p><xref ref-type="fig" rid="fig3">Figure 3D</xref> and <xref ref-type="fig" rid="fig3s2">Figure 3–figure supplement 2</xref> have been modified to include multiple sets of additional explicit ion simulations. The first set contains divalent cations as part of the salt solution. The second set of simulations includes divalent cations that are irreversibly bound to the NA (to represent specifically bound Mg<sup>2+</sup> ions). These simulations indicate only a small effect due to divalent cations at physiological concentrations.</p><p><italic>Furthermore, is it experimentally known that these capsids are negatively charged? This ought to be detectable through electrophoretic mobility measurements (zeta potential)</italic>.</p><p>The genome lengths of viruses assembled in vivo are known to high precision, as are the structures for the capsids we have considered. Based on these results it is incontrovertible that the genome length exceeds the positive capsid ARM charge. Given the complexity of the cellular environment, it could be argued that other multivalent cationic biomolecules might be present in these capsids. However, <italic>in vitro</italic> experiments mentioned in the text (e.g., Self-assembly of viral capsid protein and RNA molecules of different sizes: Requirement for a specific high protein/RNA mass ratio, J. Virol., 86(6):3318–3326, 2012; RNA encapsidation by SV40-derived nanoparticles follows a rapid two-state mechanism. J. Am. Chem. Soc., 134(21):8823–8830, 2012) have clearly shown that capsids spontaneously assemble into overcharged states under conditions in which the ionic composition is carefully controlled. Most notably, Cadena-Nava et al (J. Virol., 86(6):3318–3326, 2012) performed competition experiments, which showed that CCMV capsid proteins preferentially assemble around genomic length RNA rather than shorter fragments.</p><p>Zeta potential measurements do in fact show that capsids are negatively charged. However, these measurements do not evaluate overcharging of the capsid interior; rather, they are sensitive to the charge on the capsid exterior, which is separated by at least 5 nm from the interior. The capsid exterior is negatively charged because of acidic residues found there. (Since these negative charges on the capsid exterior are separated by at least 5 nm from the capsid interior, they have negligible effect on the thermodynamics of genome encapsulation. Therefore, we did not include them in our model.)</p><p><italic>A figure supplement provides a comparison of a calculation with explicit ions presented as a validation of the Debye-Huckel approximation (<xref ref-type="fig" rid="fig3s3">Figure 3–figure supplement 3</xref>). Not much detail is offered about this test; presumably this is for a 1:1 electrolyte in a primitive model representation of ionic solution (continuum dielectric solvent and hard-sphere ions), but no ion radii are given so it is hard to see how realistic this model is</italic>.</p><p>We now completely describe the explicit ion (primitive model) simulations in the Methods section “Electrostatic interactions among polymer, ARM, and ion beads”. Furthermore, to fully understand the effect of excluded volume on our results, we have performed additional explicit ion simulations at various ion radii. The predicted dependence of optimal length on the ion radius is now shown in <xref ref-type="fig" rid="fig3s2">Figure 3–figure supplement 2</xref>. These simulations show that the optimal length predicted by the Debye-Huckel model is within 10% of that predicted by the primitive model with an ion radius of 0.125 nm (roughly the radius for Na<sup>+</sup> and Cl<sup>-</sup> ions in the CHARMM force field) at physiological salt concentration. We appreciate the reviewers for inspiring us to consider this further, and we have re-performed all the necessary explicit ion simulations using this realistic ion size, rather than the larger size (0.25 nm) we had used originally.</p><p><italic>3) Additional cause for concern about the treatment of ionic screening arise from <xref ref-type="fig" rid="fig2s2">Figure 2–figure supplement 2</xref>, which shows the residual excess chemical potential for adding one segment to the nucleic acid chain when it is in bulk solution (red line) or when it is in a capsid (blue line). What is noteworthy here is that the residual excess chemical potential is positive for both cases. Normally, insertion of one charged particle in a continuum solvent with an electrolyte is a negative number because the interaction of the charged particle with the surroundings, which comprises solvent and mobile counterions, is favorable. This excess free energy to add one chain segment in solution would probably be negative, even within the context of Debye-Huckel. We believe that the numbers in <xref ref-type="fig" rid="fig2s2">Figure 2–figure supplement 2</xref> are positive because this interaction between the segment of the salt solution is not included in the present model. The latter probably only treats the Debye-Huckel screening of the interactions of the charged segment with other segments (which is positive and unfavorable) but the self-energy is neglected. Please confirm</italic>.</p><p><italic>If this is so, the problem is that the favorable self-energy is probably much reduced once the segment is inserted inside the capsid because there is much less room to have counterions in the densely packed capsid. To take an analogy, this is a bit like an implicit solvent that changes the bare Coulomb’s law q1*q1/r12 to q1*q2/(eps*r12), but which ignores the Born self-energy of the charges. Ignoring the self-term amounts to assume that it cancels out. This is okay if the environment is assumed to remain roughly the same. Is there enough free space left inside the capsid for the implicit solvent to account for the same Debye-Huckel treatment as in the bulk solution? What is the total volume of the interior of the capsid? What is the total fraction of volume occupied by the nucleic acid? How much free space is left for ionic solution (needed to justify the Debye-Huckel screening treatment)? Please clarify this issue and explain why the implicit treatment of counterions is sufficiently accurate here</italic>.</p><p>The excess chemical potential calculations were indeed performed using the Debye-Huckel model. We chose this model for these calculations because we had already found that simulations with the Debye-Huckel agree reasonably with those using the primitive model (e.g., <xref ref-type="fig" rid="fig3">Figure 3D</xref>). The purpose of the chemical potential computations was to confirm the results of the calculations in which optimal length was estimated by rendering part of the capsid permeable to the encapsulated polymer. (While the thermodynamic justification for these latter calculations is straightforward, it is a somewhat novel approach to calculating the optimal length and thus worth testing by independent calculations.) We now clearly state that the excess chemical potential simulations were performed using the Debye-Huckel model in the Methods section “Equilibrium encapsidation”.</p><p>There is a significant amount of free space for counterions within capsids (both within our model and in actual capsids with single-stranded genomes). The most densely packed systems that we simulated are the T=1 biological capsids, which contain relatively long ARMs within a small capsid volume. For these models the fraction of <italic>free volume</italic> within the capsid is between 0.7–0.8. For the T=3 biological capsid models and for the simple capsid with ARM charge = +5, the free volume fraction is &gt;0.9. These numbers indeed explain why the Debye- Huckel treatment is a reasonable starting point for modeling this system. We were remiss in not providing this justification for our model in the original manuscript; we now provide it in <xref ref-type="table" rid="tbl1">Table 1</xref> and in the text of the Methods section.</p></body></sub-article></article>