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| <?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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" dtd-version="1.1d1"><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">02043</article-id><article-id pub-id-type="doi">10.7554/eLife.02043</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research article</subject></subj-group><subj-group subj-group-type="heading"><subject>Cell biology</subject></subj-group><subj-group subj-group-type="heading"><subject>Ecology</subject></subj-group></article-categories><title-group><article-title>Systems analysis of the CO<sub>2</sub> concentrating mechanism in cyanobacteria</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes" id="author-9591"><name><surname>Mangan</surname><given-names>Niall M</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="fn" rid="pa1">†</xref><xref ref-type="other" rid="par-1"/><xref ref-type="other" rid="par-2"/><xref ref-type="other" rid="par-3"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-3071"><name><surname>Brenner</surname><given-names>Michael P</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="corresp" rid="cor2">*</xref><xref ref-type="other" rid="par-1"/><xref ref-type="other" rid="par-2"/><xref ref-type="other" rid="par-3"/><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf1"/></contrib><aff id="aff1"><label>1</label><institution content-type="dept">School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology</institution>, <institution>Harvard University</institution>, <addr-line><named-content content-type="city">Cambridge</named-content></addr-line>, <country>United States</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor" id="author-9992"><name><surname>Milo</surname><given-names>Ron</given-names></name><role>Reviewing editor</role><aff><institution>Weizmann Institute for Science</institution>, <country>Israel</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>niallmm@gmail.com</email> (NMM);</corresp><corresp id="cor2"><email>brenner@seas.harvard.edu</email> (MPB)</corresp><fn fn-type="present-address" id="pa1"><label>†</label><p>Mechanical Engineering/LMP, Massachusetts Institute of Technology, Cambridge, MA, United States</p></fn></author-notes><pub-date publication-format="electronic" date-type="pub"><day>29</day><month>04</month><year>2014</year></pub-date><pub-date pub-type="collection"><year>2014</year></pub-date><volume>3</volume><elocation-id>e02043</elocation-id><history><date date-type="received"><day>11</day><month>12</month><year>2013</year></date><date date-type="accepted"><day>12</day><month>04</month><year>2014</year></date></history><permissions><copyright-statement>© 2014, Mangan and Brenner</copyright-statement><copyright-year>2014</copyright-year><copyright-holder>Mangan and Brenner</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="elife02043.pdf"/><abstract><object-id pub-id-type="doi">10.7554/eLife.02043.001</object-id><p>Cyanobacteria are photosynthetic bacteria with a unique CO<sub>2</sub> concentrating mechanism (CCM), enhancing carbon fixation. Understanding the CCM requires a systems level perspective of how molecular components work together to enhance CO<sub>2</sub> fixation. We present a mathematical model of the cyanobacterial CCM, giving the parameter regime (expression levels, catalytic rates, permeability of carboxysome shell) for efficient carbon fixation. Efficiency requires saturating the RuBisCO reaction, staying below saturation for carbonic anhydrase, and avoiding wasteful oxygenation reactions. We find selectivity at the carboxysome shell is not necessary; there is an optimal non-specific carboxysome shell permeability. We compare the efficacy of facilitated CO<sub>2</sub> uptake, CO<sub>2</sub> scavenging, and <inline-formula><mml:math id="inf1"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport with varying external pH. At the optimal carboxysome permeability, contributions from CO<sub>2</sub> scavenging at the cell membrane are small. We examine the cumulative benefits of CCM spatial organization strategies: enzyme co-localization and compartmentalization.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.001">http://dx.doi.org/10.7554/eLife.02043.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><object-id pub-id-type="doi">10.7554/eLife.02043.002</object-id><title>eLife digest</title><p>Cyanobacteria are microorganisms that live in water and, like plants, they capture energy from the sun to convert carbon dioxide into sugars and other useful compounds. This process—called photosynthesis—releases oxygen as a by-product. Cyanobacteria were crucial in making the atmosphere of the early Earth habitable for other organisms, and they created the vast carbon-rich deposits that now supply us with fossil fuels. Modern cyanobacteria continue to sustain life on Earth by providing oxygen and food for other organisms, and researchers are trying to bioengineer cyanobacteria to produce alternative, cleaner, fuels.</p><p>Understanding how cyanobacteria can be as efficient as possible at harnessing sunlight to ‘fix’ carbon dioxide into carbon-rich molecules is an important step in this endeavor. Carbon dioxide can readily pass through cell membranes, so instead cyanobacteria accumulate molecules of bicarbonate inside their cells. This molecule is then converted back into carbon dioxide by an enzyme found in specials compartments within cells called carboxysomes. The enzyme that fixes the carbon is also found in the carboxysomes. However, several important details in this process are not fully understood.</p><p>Here, Mangan and Brenner further extend a mathematical model of the mechanism that cyanobacteria use to concentrate carbon dioxide in order to explore the factors that optimize carbon fixation by these microorganisms. Carbon fixation is deemed efficient when there is more carbon dioxide in the carboxysome than the carbon-fixing enzyme can immediately use (which also avoids wasteful side-reactions that use oxygen instead of carbon dioxide). However, there should not be too much bicarbonate, otherwise the enzyme that converts it to carbon dioxide is overwhelmed and cannot take advantage of the extra bicarbonate.</p><p>Mangan and Brenner's model based the rates that carbon dioxide and bicarbonate could move in and out of the cell, and the rates that the two enzymes work, on previously published experiments. The model varied the location of the enzymes (either free in the cell or inside a carboxysome), and the rate at which carbon dioxide and bicarbonate could diffuse in and out of the carboxysome (the carboxysome's permeability). Mangan and Brenner found that containing the enzymes within a carboxysome increased the concentration of carbon dioxide inside the cell by an order of magnitude. The model also revealed the optimal permeability for the carboxysome outer-shell that would maximize carbon fixation.</p><p>In addition to being of interest to researchers working on biofuels, if the model can be adapted to work for plant photosynthesis, it may help efforts to boost crop production to feed the world’s growing population.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.002">http://dx.doi.org/10.7554/eLife.02043.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>cyanobacteria</kwd><kwd>carboxysomes</kwd><kwd>carbon fixation</kwd><kwd>systems modeling</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>other</kwd></kwd-group><funding-group><award-group id="par-1"><funding-source><institution-wrap><institution content-type="university">NSF Harvard Materials Research Science and Engineering Center</institution></institution-wrap></funding-source><award-id>DMR-0820484</award-id><principal-award-recipient><name><surname>Mangan</surname><given-names>Niall M</given-names></name><name><surname>Brenner</surname><given-names>Michael P</given-names></name></principal-award-recipient></award-group><award-group id="par-2"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000121</institution-id><institution>NSF Division of Mathematical Sciences</institution></institution-wrap></funding-source><award-id>DMS-0907985</award-id><principal-award-recipient><name><surname>Mangan</surname><given-names>Niall M</given-names></name><name><surname>Brenner</surname><given-names>Michael P</given-names></name></principal-award-recipient></award-group><award-group id="par-3"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000057</institution-id><institution content-type="university">National Institute of General Medical Sciences</institution></institution-wrap></funding-source><award-id>Grant GM068763 for National Centers of Systems Biology </award-id><principal-award-recipient><name><surname>Mangan</surname><given-names>Niall M</given-names></name><name><surname>Brenner</surname><given-names>Michael P</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><meta-name>elife-xml-version</meta-name><meta-value>2.0</meta-value></custom-meta><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>Systems level modeling of cyanobacterial mechanism for concentrating carbon dioxide shows optimal organization and enzymatic activity for enhanced carbon fixation.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec sec-type="intro" id="s1"><title>Introduction</title><p>Intracellular compartments are critical for directing and protecting biochemical reactions. One of the simplest and most striking known examples of compartmentalization are the carboxysomes <xref ref-type="bibr" rid="bib6">(Cannon et al., 2001</xref>; <xref ref-type="bibr" rid="bib52">Yeates et al., 2008)</xref> of cyanobacteria and other autotrophic proteobacteria <xref ref-type="bibr" rid="bib40">(Savage et al., 2010</xref>; <xref ref-type="bibr" rid="bib39">Rosgaard et al., 2012)</xref>. These small, 100–200 nm compartments separate the principal reaction of the Calvin cycle, the RuBisCO catalyzed fixation of carbon dioxide (CO<sub>2</sub>) into 3-phosphoglycerate, from the rest of the cell (<xref ref-type="bibr" rid="bib5">Cannon et al., 1991</xref>). CO<sub>2</sub> and oxygen (O<sub>2</sub>) competitively bind as substrates of RuBisCO, and the reaction with O<sub>2</sub> produces phosphoglycolate, a waste product which must be recycled by the cell <xref ref-type="bibr" rid="bib18">(Jordan & Ogren, 1981</xref>; <xref ref-type="bibr" rid="bib45">Tcherkez et al., 2006</xref>; <xref ref-type="bibr" rid="bib41">Savir et al., 2010)</xref>. To maximize carboxylation and minimize oxygenation, the carboxysome is believed to act as a diffusion barrier to CO<sub>2</sub> <xref ref-type="bibr" rid="bib37">(Reinhold et al., 1989</xref>; <xref ref-type="bibr" rid="bib11">Dou et al., 2008)</xref>. There is much interest in the design and function of such compartments and whether they can be used to enhance carbon fixation in other organisms such as plants or to improve reaction rates in other metabolic systems <xref ref-type="bibr" rid="bib12">(Ducat & Silver, 2012</xref>; <xref ref-type="bibr" rid="bib1">Agapakis et al., 2012</xref>; <xref ref-type="bibr" rid="bib29">Papapostolou & Howorka, 2009</xref>; <xref ref-type="bibr" rid="bib13">Frank et al., 2013)</xref>. Increased efficiency of biochemical reactions will lead to better yield in bioengineered bacterial systems, creating new possibilities for production of high-value products such as biofuels. Enhancing carbon fixation in plants or other organisms could lead to increased carbon sequestration, or crop yield.</p><p>The concentrating mechanism in cyanobacteria relies on the interaction of a number of well characterized components, as shown in <xref ref-type="fig" rid="fig1">Figure 1</xref>, transferring inorganic carbon from outside the cell into cytosol and carboxysomes <xref ref-type="bibr" rid="bib2">(Allen, 1984</xref>; <xref ref-type="bibr" rid="bib3">Badger & Price, 2003</xref>; <xref ref-type="bibr" rid="bib19">Kaplan & Reinhold, 1999</xref>; <xref ref-type="bibr" rid="bib34">Price et al., 2007)</xref>. Due to this mechanism, inorganic carbon concentration is elevated well above 200–300 <italic>μ</italic>M, the CO<sub>2</sub> concentration required for saturating the RuBisCO. Additionally a high CO<sub>2</sub> concentration increases the ratio of CO<sub>2</sub> to O<sub>2</sub> so that carboxylation dominates over oxygenation. Concentrations of 20–40 mM inorganic carbon, up to 4000-fold higher than external levels, have been observed <xref ref-type="bibr" rid="bib19">(Kaplan & Reinhold, 1999</xref>; <xref ref-type="bibr" rid="bib50">Woodger et al., 2005</xref>; <xref ref-type="bibr" rid="bib44">Sultemeyer et al., 1995</xref>; <xref ref-type="bibr" rid="bib32">Price et al., 1998)</xref>. The bilipid outer and cell membranes are highly permeable to small uncharged molecules such as CO<sub>2</sub> <xref ref-type="bibr" rid="bib27">(Missner et al., 2008</xref>; <xref ref-type="bibr" rid="bib15">Gutknecht et al., 1977)</xref>, so instead the cell primarily accumulates the charged and less membrane soluble bicarbonate (<inline-formula><mml:math id="inf2"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>) <xref ref-type="bibr" rid="bib48">(Volokita et al., 1984</xref>; <xref ref-type="bibr" rid="bib31">Price & Badger, 1989)</xref>. Active transporters, both constitutive and inducible, bring <inline-formula><mml:math id="inf3"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> into the cell <xref ref-type="bibr" rid="bib35">(Price et al., 2008</xref>, <xref ref-type="bibr" rid="bib33">2004</xref>; <xref ref-type="bibr" rid="bib28">Omata et al., 1999</xref>), and mechanisms exist to actively convert CO<sub>2</sub> to <inline-formula><mml:math id="inf4"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> at the thylakoid and cell membrane <xref ref-type="bibr" rid="bib35">(Price et al., 2008</xref>; <xref ref-type="bibr" rid="bib23">Maeda et al., 2002</xref>; <xref ref-type="bibr" rid="bib43">Shibata et al., 2001)</xref>. Once it passively diffuses into the carboxysome, <inline-formula><mml:math id="inf5"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is rapidly brought into equilibrium with CO<sub>2</sub> by the enzyme carbonic anhydrase, resulting in the production of CO<sub>2</sub> near RuBisCO. Carbonic anhydrase is known to be localized on the interior side of the carboxysome shell <xref ref-type="bibr" rid="bib52">(Yeates et al., 2008</xref>; <xref ref-type="bibr" rid="bib5">Cannon et al., 1991</xref>; <xref ref-type="bibr" rid="bib9">Cot et al., 2008</xref>; <xref ref-type="bibr" rid="bib22">Long et al., 2008)</xref>. The carboxysome shell must be permeable enough to allow <inline-formula><mml:math id="inf6"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and 3-phosphoglycerate to readily diffuse in and out. The function of this system and its ability to concentrate inorganic carbon depends on the interplay between these various molecular components. Without a model, flux measurements cannot determine the components relative roles in enhancing the CO<sub>2</sub> concentration in the carboxysome. To date, it has not been possible to directly measure the partitioning of the internal carbon concentration in the cytosol and carboxysomes. We wish to characterize the distribution of internal carbon. Visualizations of the location of the carboxysomes with fluorescent microscopy in <italic>Synechococcus elongatus</italic> PCC7942 demonstrated that the carboxysomes are evenly spaced along the centerline of the cell, (<xref ref-type="bibr" rid="bib40">Savage et al., 2010</xref>), raising the question of how spatial organization, beyond simple partitioning, changes the efficacy of the system.<fig id="fig1" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.003</object-id><label>Figure 1.</label><caption><title>Schematic of the CCM in cyanobacteria.</title><p>Outer and cell membranes (in black), as well as, thylakoid membranes where the light reactions take place (in green) are treated together. Carboxysomes are shown as four hexagons evenly spaced along the centerline of the cell. The model treats a spherically symmetric cell, with one carboxysome at the center. Active <inline-formula><mml:math id="inf7"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport into the cell is indicated (in light blue), as well as active conversion from CO<sub>2</sub> to <inline-formula><mml:math id="inf8"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, sometimes called ‘facilitated uptake’ or ‘scavenging’, at membranes (in orange). Both CO<sub>2</sub> and <inline-formula><mml:math id="inf9"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> can leak in and out of the cell, with CO<sub>2</sub> leaking out much more readily. Both species passively diffuse across the carboxysome shell. Carbonic anhydrase (red) and RuBisCO (blue) are contained in the carboxysomes and facilitate reactions as shown.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.003">http://dx.doi.org/10.7554/eLife.02043.003</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f001"/></fig></p><p>The goal of this study is to further develop a mathematical model of the CCM <xref ref-type="bibr" rid="bib37">(Reinhold et al., 1989</xref>; <xref ref-type="bibr" rid="bib14">Fridlyand et al., 1996</xref>; <xref ref-type="bibr" rid="bib38">Reinhold et al., 1991)</xref> that uses recent experimental progress on the CCM to untangle the relative roles of the different molecular components, and predict the region of parameter space where efficient carbon fixation occurs. We are considering conditions where CO<sub>2</sub> is limiting (15<italic>μM</italic> external inorganic carbon) and, for the moment, ignore other biological pressures. In this context, efficient carbon fixation requires two conditions: First, the CO<sub>2</sub> concentration must be high enough that RuBisCO is saturated, and the competitive reaction with O<sub>2</sub> is negligible. We emphasize that for the oxygenation reaction to be negligible the CO<sub>2</sub> concentration should be higher than needed to merely saturate RuBisCO. Secondly, the carbonic anhydrase within the carboxysome must be unsaturated, so that extra energy isn't wasted transporting unused <inline-formula><mml:math id="inf10"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> into the cell.</p><p>Examination of the system performance with varying expression levels of <inline-formula><mml:math id="inf11"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transporters, carboxysome permeability, and conversion from CO<sub>2</sub> to <inline-formula><mml:math id="inf12"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, reveals a parameter window where these conditions are simultaneously satisfied. We comment on the relation of this window to measured carbon pools, carbon fixation rates, and <inline-formula><mml:math id="inf13"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transporters. We find that the <inline-formula><mml:math id="inf14"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration in the cytosol is constant across the cell, set by the <inline-formula><mml:math id="inf15"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport and leakage rates, and depends very little on the carboxysome permeability. The carboxysome permeability does, however, set how the CO<sub>2</sub> is partitioned between the carboxysome and cytosol. At optimal carboxysome permeability, <inline-formula><mml:math id="inf16"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> diffusion into the carboxysome is fast enough to supply inorganic carbon for fixation, but the rate of CO<sub>2</sub> leakage out of the carboxysome is low. We explore the fluxes from CO<sub>2</sub> facilitated uptake and scavenging with varying ratios of external CO<sub>2</sub> and <inline-formula><mml:math id="inf17"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. Finally we discuss the proportion the carbon concentration comes from different methods of spatial organization such as co-localization, encapsulation, and spatial location of carboxysomes. Concentration of carbonic anhydrase increases the maximum rate of reaction for carbonic anhydrase per volume, causing carbonic anhydrase to saturate at a higher level of <inline-formula><mml:math id="inf18"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, and achieve an order of magnitude higher local CO<sub>2</sub> concentrations. Encapsulation of the reactions in an optimally permeable carboxysome shell achieves another order of magnitude of CO<sub>2</sub> concentration.</p><sec id="s1-1"><title>Reaction diffusion model</title><p>We present our mathematical model, which captures all aspects of the CCM as described above. This model is an expansion of previously developed models <xref ref-type="bibr" rid="bib37">(Reinhold et al., 1989</xref>; <xref ref-type="bibr" rid="bib14">Fridlyand et al., 1996</xref>; <xref ref-type="bibr" rid="bib38">Reinhold et al., 1991)</xref>. Our three dimensional model of the CCM solves for both the <inline-formula><mml:math id="inf19"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub> concentration throughout a spherical cell. We solve this model numerically and analytically at steady state for three different spatial organizations of carbonic anhydrase and RuBisCO in the cell (See Figure 6): enzymes distributed evenly throughout the cell, enzymes localized to the center of the cell but not encapsulated (as they would be on a scaffold), enzymes encapsulated in a carboxysome. We compare the effects of these scenarios in the discussion section, and for now consider a spherical cell of radius R<sub>b</sub> = 0.5 <italic>μm</italic> with a single spherical carboxysome of radius R<sub>c</sub> = 50 <italic>nm</italic> containing RuBisCO and carbonic anhydrase. Numerical computations are carried out with finite difference methods in MATLAB. The details of analytic solutions are given in the <xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref>.</p><p>We include the effects of diffusion, active transport and leakage through the cell membrane, and reactions with carbonic anhydrase and RuBisCO. In the carboxysome (r < <italic>R</italic><sub><italic>c</italic></sub>), the equations governing the <inline-formula><mml:math id="inf22"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub>, here H and C respectively, are<disp-formula id="equ1"><label>(1)</label><mml:math id="m1"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mo>∇</mml:mo> <mml:mn>2</mml:mn></mml:msup><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>R</mml:mi><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula><disp-formula id="equ2"><label>(2)</label><mml:math id="m2"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mi>H</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mo>,</mml:mo></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula>where here <italic>D</italic> is the diffusion constant, and R<sub>CA</sub> is the carbonic anhydrase reaction, and R<sub>Rub</sub> is the RuBisCO reaction. The carbonic anhydrase reaction follows reversible Michaelis–Menten kinetics <xref ref-type="bibr" rid="bib19">(Kaplan & Reinhold, 1999</xref>; <xref ref-type="bibr" rid="bib34">Price et al., 2007)</xref>,<disp-formula id="equ3"><label>(3)</label><mml:math id="m3"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>C</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mo>+</mml:mo><mml:mi>K</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>V</italic><sub><italic>ca</italic></sub> and <italic>V</italic><sub><italic>ba</italic></sub> are hydration and dehydration rates, proportional to the local carbonic anhydrase concentration. <italic>K</italic><sub><italic>ca</italic></sub> and <italic>K</italic><sub><italic>ba</italic></sub> are the concentration at which hydration and dehydration are half maximum. The RuBisCO reaction follows Michaelis–Menten kinetics with competitive binding with O<sub>2</sub>, <inline-formula><mml:math id="inf23"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>R</mml:mi><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow></mml:msub><mml:mi>C</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="inf24"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. Here <italic>V</italic><sub><italic>max</italic></sub> is the maximum rate of carbon fixation and K<sub>m</sub> is the apparent half maximum concentration value, which has been modified to include competitive binding with O<sub>2</sub>, <italic>O</italic>. K<sub>i</sub> is the dissociation constant of <italic>O</italic><sub>2</sub> with the RuBisCO and <inline-formula><mml:math id="inf25"><mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is the half maximum concentration with no <italic>O</italic><sub>2</sub> present. RuBisCO also requires ribulose-1,5-bisphosphate, the substrate which <italic>CO</italic><sub>2</sub> reacts with to produce 3-phosphoglycolate. Under CO<sub>2</sub> limiting conditions it has been shown that there is sufficient ribulose-1,5-bisphosphate to saturate all RuBisCO active sites, and the reaction rates are independent of ribulose-1,5-bisphosphate concentrations <xref ref-type="bibr" rid="bib25">(Mayo et al., 1989</xref>; <xref ref-type="bibr" rid="bib49">Whitehead et al., 2014)</xref>.</p><p>In the cytosol there is no carbonic anhydrase or RuBisCO activity, so R<sub>CA</sub> = 0 and <inline-formula><mml:math id="inf26"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>R</mml:mi><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>, and there is only diffusion of CO<sub>2</sub> and <inline-formula><mml:math id="inf27"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. We do not include the natural, but slow, interconversion of CO<sub>2</sub> and <inline-formula><mml:math id="inf28"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> in the cytosol. This assumption is a good one given that the <inline-formula><mml:math id="inf29"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration is known to be held out of equilibrium in the cell <xref ref-type="bibr" rid="bib48">(Volokita et al., 1984</xref>; <xref ref-type="bibr" rid="bib31">Price & Badger, 1989)</xref>. In agreement with this experimental observation, we find that all the other processes effecting the concentration of <inline-formula><mml:math id="inf30"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> in the cytosol happen much faster than the natural interconversion.</p><p>Boundary conditions prescribe the inorganic carbon fluxes into the cell and the diffusion across the carboxysome boundary. We treat the inorganic carbon fluxes at cell and thylakoid membranes together. At this cell boundary, there is passive leakage of both CO<sub>2</sub> and <inline-formula><mml:math id="inf31"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>: the velocity of CO<sub>2</sub> across the cell membrane, <inline-formula><mml:math id="inf32"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>c</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula> is about 1000-fold higher than that of <inline-formula><mml:math id="inf33"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="inf34"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula>, due to the lower permeability of the membrane to charged molecules. To account for active import of <inline-formula><mml:math id="inf35"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, we combine the total <inline-formula><mml:math id="inf36"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> flux, <italic>j</italic><sub><italic>c</italic></sub>, from all <inline-formula><mml:math id="inf37"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transporters. These transporters include BCT1 (encoded by cpm), which is thought to be powered by ATP; and BicA and SbtA which are thought to be symporters between <inline-formula><mml:math id="inf38"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and Na<sup>+</sup>, driven by the highly controlled electrochemical gradient for Na<sup>+</sup> <xref ref-type="bibr" rid="bib35">(Price et al., 2008</xref>, <xref ref-type="bibr" rid="bib33">2004</xref>; <xref ref-type="bibr" rid="bib28">Omata et al., 1999</xref>). Additionally, there are two complexes NDH-1<sub>3</sub> and NDH-1<sub>4</sub> responsible for converting CO<sub>2</sub> to <inline-formula><mml:math id="inf39"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. This conversion is thought to either decrease CO<sub>2</sub>, creating a gradient across the membranes and ‘facilitating uptake’ of CO<sub>2</sub>, or ‘scavenge’ CO<sub>2</sub> which has escaped from the carboxysome. These are localized to the thylakoid and possibly the plasma membrane. They have been linked to the photosynthetic linear and cyclic electron transport chain <xref ref-type="bibr" rid="bib35">(Price et al., 2008</xref>; <xref ref-type="bibr" rid="bib23">Maeda et al., 2002</xref>; <xref ref-type="bibr" rid="bib43">Shibata et al., 2001)</xref>. It has been proposed that electron transport drives the formation of local alkaline pockets where CO<sub>2</sub> more rapidly converts to <inline-formula><mml:math id="inf40"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. We more simply describe the conversion with a maximal reaction rate <italic>α</italic>, and concentration of half maximal activity of <italic>K</italic><sub><italic>α</italic></sub>. Combining these effects, the boundary condition setting diffusive flux of <inline-formula><mml:math id="inf41"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub> at the cell membrane is<disp-formula id="equ4"><label>(4)</label><mml:math id="m4"><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>C</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula><disp-formula id="equ5"><label>(5)</label><mml:math id="m5"><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:msub><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula>where the subscript <italic>cytosol</italic> and <italic>out</italic> indicate we are taking the concentration immediately inside and outside the cell boundary respectively. The diffusion constant times partial derivatives with respect to the radial coordinate, <italic>r</italic>, define the diffusive flux at the membrane.</p><p>The carboxysome shell is composed of proteins with a radius R<sub>c</sub> ≈ 50 nm. As of yet, there have been no direct measurements of the carboxysome permeability to small molecules. Using the carboxysome geometry, we can calculate an upper bound for the diffusive velocity across the carboxysome shell, which is directly related to the carboxysome permeability. Crystal structures <xref ref-type="bibr" rid="bib52">(Yeates et al., 2008</xref>; <xref ref-type="bibr" rid="bib8">Cheng et al., 2008</xref>; <xref ref-type="bibr" rid="bib51">yeates et al., 2007)</xref> show the surface has approximately N<sub>pores</sub> = 4800 small pores with radius <inline-formula><mml:math id="inf42"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.35</mml:mn></mml:mrow></mml:math></inline-formula> nm, and thickness <italic>l</italic> = 1.8 nm. If <italic>k</italic><sub><italic>c</italic></sub> is the characteristic velocity that small molecules pass through the shell, these numbers imply the upper bound for diffusive transport <inline-formula><mml:math id="inf43"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo><</mml:mo><mml:mfrac><mml:mrow><mml:mi>π</mml:mi><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msubsup><mml:mi>R</mml:mi><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>0.02</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. This calculation is done by taking the probability that a molecule will encounter a pore on the carboxysome shell <inline-formula><mml:math id="inf44"><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:mtext>pore surface area</mml:mtext></mml:mrow><mml:mrow><mml:mtext>carboxysome surface area</mml:mtext></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and multipling it by the speed a small molecule will diffuse through the length of the pore (<italic>D/l</italic>). Since it does not take into account any charge effects, which would add an additional energy barrier, it is an upper bound. Although there has been much speculation that the positively charged pores might enhance diffusion of negatively charged <inline-formula><mml:math id="inf45"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> <xref ref-type="bibr" rid="bib52">(Yeates et al., 2008</xref>; <xref ref-type="bibr" rid="bib11">Dou et al., 2008</xref>; <xref ref-type="bibr" rid="bib8">Cheng et al., 2008)</xref>, here we explore the simplest assumption, that both <inline-formula><mml:math id="inf46"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub> have the same permeability. Namely, the boundary conditions at the carboxysome shell are<disp-formula id="equ6"><label>(6)</label><mml:math id="m6"><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula><disp-formula id="equ7"><label>(7)</label><mml:math id="m7"><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:msub><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>).</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>We will vary <italic>k</italic><sub><italic>c</italic></sub> (henceforth called carboxysome permeability, although it is a velocity) within our model and see that there is a range of <italic>k</italic><sub><italic>c</italic></sub> where the CCM is effective even with <italic>k</italic><sub><italic>c</italic></sub> identical for CO<sub>2</sub> and <inline-formula><mml:math id="inf47"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p></sec></sec><sec sec-type="results" id="s2"><title>Results</title><sec id="s2-1"><title>Analysis of model: Finding functional parameter space</title><p>Now that we have defined our model, we wish to find the range of parameters where efficient carbon fixation occurs. In what follows, we fix the enzymatic rates, cell membrane permeability, and diffusion constant as reported in the literature <xref ref-type="bibr" rid="bib18">(Jordan & Ogren, 1981</xref>; <xref ref-type="bibr" rid="bib27">Missner et al., 2008</xref>; <xref ref-type="bibr" rid="bib15">Gutknecht et al., 1977</xref>; <xref ref-type="bibr" rid="bib17">Heinhorst et al., 2006)</xref> (see <xref ref-type="table" rid="tbl1">Table 1</xref> and <xref ref-type="table" rid="tbl2">Table 2</xref>). Note that full analytic solutions are available in <xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref> sections S3 and S4, so the effect of varying other parameters can be analyzed. We consider the efficacy of the CCM as a function of <italic>j</italic><sub><italic>c</italic></sub>, the flux of <inline-formula><mml:math id="inf48"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> into the cell, <italic>k</italic><sub><italic>c</italic></sub>, the carboxysome permeability, and the parameters (α, <italic>K</italic><sub>α</sub>) governing the <italic>CO</italic><sub><italic>2</italic></sub> facilitated uptake mechanism. Both <italic>α</italic> and <italic>j</italic><sub><italic>c</italic></sub> can be regulated by the organism and vary depending on environmental conditions, whereas the carboxysome permeability, <italic>k</italic><sub><italic>c</italic></sub>, is the parameter with the largest uncertainty and debate <xref ref-type="bibr" rid="bib6">(Cannon et al., 2001</xref>; <xref ref-type="bibr" rid="bib52">Yeates et al., 2008</xref>; <xref ref-type="bibr" rid="bib8">Cheng et al., 2008)</xref>.<table-wrap id="tbl1" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.005</object-id><label>Table 1.</label><caption><p>Parameter values chosen for main set of simulations, unless otherwise indicated</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.005">http://dx.doi.org/10.7554/eLife.02043.005</ext-link></p></caption><table frame="hsides" rules="groups"><thead><tr><th>Parameter</th><th>Definition</th><th>Value</th><th>Reference</th></tr></thead><tbody><tr><td><italic>H</italic><sub><italic>out</italic></sub></td><td>concentration of bicarbonate outside the cell</td><td>14 <italic>μM</italic>*</td><td>(<xref ref-type="bibr" rid="bib35">Price et al., 2008</xref>)</td></tr><tr><td><italic>C</italic><sub><italic>out</italic></sub></td><td>concentration of carbon dioxide outside of cell</td><td>0.14 <italic>μM</italic>*</td><td>(<xref ref-type="bibr" rid="bib35">Price et al., 2008</xref>)</td></tr><tr><td>D</td><td>diffusion constant of small molecules, CO<sub>2</sub> and <inline-formula><mml:math id="inf49"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula></td><td>10<sup>−5</sup> <inline-formula><mml:math id="inf50"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td>(<xref ref-type="bibr" rid="bib14">Fridlyand et al., 1996</xref>)</td></tr><tr><td><inline-formula><mml:math id="inf51"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>c</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula></td><td>permeability of cell membrane to CO<sub>2</sub></td><td>0.3 <inline-formula><mml:math id="inf52"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td><xref ref-type="bibr" rid="bib27">(Missner et al., 2008</xref>; <xref ref-type="bibr" rid="bib15">Gutknecht et al., 1977)</xref></td></tr><tr><td><inline-formula><mml:math id="inf53"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula></td><td>permeability of cell membrane to <inline-formula><mml:math id="inf54"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula></td><td><inline-formula><mml:math id="inf55"><mml:mrow><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td><xref ref-type="bibr" rid="bib27">(Missner et al., 2008</xref>; <xref ref-type="bibr" rid="bib15">Gutknecht et al., 1977)</xref></td></tr><tr><td><italic>R</italic><sub><italic>c</italic></sub></td><td>radius of carboxysome</td><td>5×10<sup>−6</sup> cm</td><td><xref ref-type="bibr" rid="bib8">(Cheng et al., 2008</xref>; <xref ref-type="bibr" rid="bib42">Schmid et al., 2006)</xref></td></tr><tr><td><italic>R</italic><sub><italic>b</italic></sub></td><td>radius of bacteria</td><td>5 × 10<sup>−5</sup> cm</td><td>(<xref ref-type="bibr" rid="bib40">Savage et al., 2010</xref>)</td></tr><tr><td><italic>j</italic><sub><italic>c</italic></sub></td><td><inline-formula><mml:math id="inf56"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport rate resulting in 30mM cytosolic <inline-formula><mml:math id="inf57"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> pool</td><td>0.6 <inline-formula><mml:math id="inf58"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>*</td><td>calculated here</td></tr><tr><td><inline-formula><mml:math id="inf59"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula></td><td>optimal carboxysome permeability</td><td>10<sup>−3</sup> <inline-formula><mml:math id="inf60"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>*</td><td>calculated here</td></tr><tr><td><mml:math id="inf250"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></td><td>cell volume</td><td><inline-formula><mml:math id="inf61"><mml:mrow><mml:mn>5.2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:msup><mml:mi>μ</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:math></inline-formula></td><td>calculated</td></tr><tr><td><inline-formula><mml:math id="inf62"><mml:mrow><mml:mi>S</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula></td><td>cell surface area</td><td><inline-formula><mml:math id="inf63"><mml:mrow><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:msup><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula></td><td>calculated</td></tr></tbody></table><table-wrap-foot><fn id="tblfn1"><label>*</label><p>these parameters are varied in the text, but these values are use unless noted otherwise.</p></fn></table-wrap-foot></table-wrap><table-wrap id="tbl2" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.006</object-id><label>Table 2.</label><caption><p>Table comparing enzymatic rates <xref ref-type="bibr" rid="bib50">(Woodger et al., 2005</xref>; <xref ref-type="bibr" rid="bib44">Sultemeyer et al., 1995</xref>; <xref ref-type="bibr" rid="bib17">Heinhorst et al., 2006)</xref></p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.006">http://dx.doi.org/10.7554/eLife.02043.006</ext-link></p></caption><table frame="hsides" rules="groups"><thead><tr><th>Enzyme reaction</th><th>active sites</th><th><inline-formula><mml:math id="inf65"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="inf66"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>s</mml:mi></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th><th><inline-formula><mml:math id="inf67"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> in ‘cell’ <inline-formula><mml:math id="inf68"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th><th><inline-formula><mml:math id="inf69"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> in carboxysome <inline-formula><mml:math id="inf70"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th><th><inline-formula><mml:math id="inf71"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="inf72"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>μ</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th></tr></thead><tbody><tr><td>carbonic anhydrase hydration</td><td>80</td><td>8 × 10<sup>4</sup></td><td>8.8 × 10<sup>3</sup></td><td>1.5 × 10<sup>7</sup></td><td>3.2 × 10<sup>3</sup></td></tr><tr><td>carbonic anhydrase dehydration</td><td>80</td><td>4.6 × 10<sup>4</sup></td><td>1.5 × 10<sup>4</sup></td><td>8.8 × 10<sup>6</sup></td><td>9.3 × 10<sup>3</sup></td></tr><tr><td>RuBisCO carboxylation</td><td>2160</td><td>26</td><td>178</td><td>1.8 × 10<sup>5</sup></td><td>270</td></tr></tbody></table><table-wrap-foot><fn><p><inline-formula><mml:math id="inf64"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> in cell and carboxysome refer to the volumetric reaction rate assuming the enzymes are distributed throughout the entire cell or only carboxysome. <italic>V</italic><sub><italic>ba</italic></sub> (<italic>V</italic><sub><italic>max</italic></sub> for carbonic anhydrase dehydration) is estimated by assuming <italic>K</italic><sub><italic>eq</italic></sub> = 5 and using parameters found in (<xref ref-type="bibr" rid="bib17">Heinhorst et al., 2006</xref>). <italic>V</italic><sub><italic>ca</italic></sub> is <italic>V</italic><sub><italic>max</italic></sub> for carbonic anhydrase hydration. Similarly, <italic>K</italic><sub><italic>ba</italic></sub>, and <italic>K</italic><sub><italic>ca</italic></sub> are <inline-formula><mml:math id="inf73"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for dehydration and hydration respectively.</p></fn></table-wrap-foot></table-wrap></p><p>For any given pair of <italic>k</italic><sub><italic>c</italic></sub> and <italic>j</italic><sub><italic>c</italic></sub>, we ask whether the CO<sub>2</sub> concentrating mechanism is effective, using the criteria of saturating RuBisCO, reducing oxidation reactions, and not increasing the <inline-formula><mml:math id="inf74"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration beyond carbonic anhydrase saturation. Our central result is presented in <xref ref-type="fig" rid="fig2">Figure 2</xref>, which shows the range of <italic>k</italic><sub><italic>c</italic></sub> and <italic>j</italic><sub><italic>c</italic></sub> where these conditions are met, assuming that there is no facilitated uptake, <italic>α</italic> = 0. The blue shaded region shows where RuBisCO is unsaturated, and the red shaded region shows where carbonic anhydrase is saturated. There is a crescent shaped region between these regions, where the CCM is effective according to our criteria. In the white region oxygenation reactions happen at a rate of greater than 1%. In the green shaded region oxygenation reactions occur at a rate of less than 1%. Within the white and green regions the CO<sub>2</sub> concentration in the carboxysome varies greatly.<fig-group><fig id="fig2" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.007</object-id><label>Figure 2.</label><caption><title>Phase space for <inline-formula><mml:math id="inf75a"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport, <italic>j</italic><sub><italic>c</italic></sub>, and carboxysome permeability <italic>k</italic><sub>c</sub>.</title><p>Plotted are the parameter values at which the CO<sub>2</sub> concentration reaches some critical value. The left most line (dark blue) indicates for what values of <italic>j</italic><sub><italic>c</italic></sub> and <italic>k</italic><sub><italic>c</italic></sub> the CO<sub>2</sub> concentration in the carboxysome would half-saturate RuBisCO (<italic>K</italic><sub><italic>m</italic></sub>). The middle line (light blue) indicates the parameter values which would result in a CO<sub>2</sub> concentration where 99% of all RuBisCO reactions are carboxylation reactions and only 1% are oxygenation reactions when O<sub>2</sub> concentration is 260 <italic>μM</italic>. The right most (red) line indicates the parameter values which result in carbonic anyhdrase saturating. Here α = 0, so there is no CO<sub>2</sub> scavenging or facilitated uptake. The dotted line (grey) shows the <italic>k</italic><sub><italic>c</italic></sub> and <italic>j</italic><sub><italic>c</italic></sub> values, where the <inline-formula><mml:math id="inf76"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration in the cytosol is 30 <italic>mM</italic>. The <inline-formula><mml:math id="inf77"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration in the cytosol does not vary appreciably with <italic>k</italic><sub><italic>c</italic></sub> in this parameter regime, and reaches 30 <italic>mM</italic> at <inline-formula><mml:math id="inf78"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.6</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. All other parameters, such as reaction rates are held fixed and the value can be found in the <xref ref-type="table" rid="tbl1">Table 1</xref> and <xref ref-type="table" rid="tbl2">Table 2</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.007">http://dx.doi.org/10.7554/eLife.02043.007</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f002"/></fig><fig id="fig2s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.02043.008</object-id><label>Figure 2—figure supplement 1.</label><caption><title>Phase space for <inline-formula><mml:math id="inf75"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport and carboxysome permeability.</title><p>Solid lines show lines of constant CO<sub>2</sub> concentration in the carboxysome for <inline-formula ><mml:math id="inf211"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mi>e</mml:mi><mml:mo>−</mml:mo><mml:mn>5</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, or the diffusion constant of small molecule in water. Dashed lines show the same lines of constant CO<sub>2</sub> concentration, bur for <inline-formula ><mml:math id="inf212"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mi>e</mml:mi><mml:mo>−</mml:mo><mml:mn>7</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, or the diffusion constant of a small molecule in a 60% sucrose solution.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.008">http://dx.doi.org/10.7554/eLife.02043.008</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043fs001"/></fig><fig id="fig2s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.02043.009</object-id><label>Figure 2—figure supplement 2.</label><caption><title>Phase space for <inline-formula><mml:math id="inf213"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport, <italic>j</italic><sub><italic>c</italic></sub>, and carboxysome permeability, <italic>k</italic><sub><italic>c</italic></sub>.</title><p>Plotted are the parameter values at which CO<sub>2</sub> concentration reaches some critical value. The left most line (dark blue) indicates for what values of <italic>j</italic><sub><italic>c</italic></sub> and <italic>k</italic><sub>c</sub> the CO<sub>2</sub> concentration in the carboxysome would saturate RuBisCO. The middle line (light blue) indicates the parameter values which would result in a CO<sub>2</sub> concentration where 99% of all RuBisCO reactions are carboxylation reactions and only 1% are oxygenation reactions when O<sub>2</sub> concentration is 260 <italic>µM</italic>. The right most (red) line indicates the parameter values which result in carbonic anyhdrase saturating. Here <inline-formula ><mml:math id="inf214"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (solid lines) and <inline-formula ><mml:math id="inf215"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (dashed line), showing the effect of CO<sub>2</sub> scavenging or facilitated uptake on the phase space. All other parameters, such as reaction rates are held fixed and the value can be found <xref ref-type="table" rid="tbl1">Table 1</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.009">http://dx.doi.org/10.7554/eLife.02043.009</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043fs002"/></fig></fig-group></p><p>The lines dividing the regions in <xref ref-type="fig" rid="fig2">Figure 2</xref> are lines of constant carboxysomal CO<sub>2</sub> concentration in <italic>j</italic><sub><italic>c</italic></sub> and <italic>k</italic><sub><italic>c</italic></sub> parameter space. The dark blue line is where CO<sub>2</sub> = K<sub>m</sub>, the CO<sub>2</sub> concentration for half-maximum RuBisCO reactions. The light blue line indicates parameter values resulting in the CO<sub>2</sub> concentration (C<sub>99%</sub>) where rate of oxygenation reactions is 1% for O<sub>2</sub> concentration of 260 <italic>μ</italic>M. Varying carboxysome permeability, <italic>k</italic><sub><italic>c</italic></sub> values, require more or less HCO<sub>3</sub> transport, <italic>j</italic><sub><italic>c</italic></sub>, to achieve the same carboxysomal CO<sub>2</sub> concentration.</p><p>We can calculate an amplification factor for the C<sub>99%</sub> carboxysomal CO<sub>2</sub> concentration as <inline-formula><mml:math id="inf79"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>133</mml:mn></mml:mrow></mml:math></inline-formula>. Any combination of <italic>j</italic><sub><italic>c</italic></sub> and <italic>k</italic><sub><italic>c</italic></sub> which produce C = C<sub>99%</sub>, make 133 times more CO<sub>2</sub> available in the carboxysome than there is total inorganic carbon outside the cell. Generally, increasing <inline-formula><mml:math id="inf80"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport, below the carbonic anhydrase saturation point results in higher CO<sub>2</sub> concentration in the carboxysome.</p></sec><sec id="s2-2"><title>Varying <inline-formula><mml:math id="inf81"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport saturates enzymes</title><p>The basic physics of the phase diagram <xref ref-type="fig" rid="fig2">Figure 2</xref> follows from examining how CO<sub>2</sub> and <inline-formula><mml:math id="inf82"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> in the carboxysome change as <italic>j</italic><sub><italic>c</italic></sub> is varied. <xref ref-type="fig" rid="fig3">Figure 3</xref> shows the response to varying <italic>j</italic><sub><italic>c</italic></sub>, with <inline-formula><mml:math id="inf83"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (the optimal value in <xref ref-type="fig" rid="fig4">Figure 4</xref>).<fig-group><fig id="fig3" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.010</object-id><label>Figure 3.</label><caption><title>Numerical solution (diamonds and circles) and analytic solutions (carbonic anhydrase unsaturated, solid lines, and saturated, dashed lines) correspond well.</title><p><inline-formula><mml:math id="inf84"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport is varied, and all other system parameters are held constant. The CO<sub>2</sub> concentration above which RuBisCO is saturated is <italic>K</italic><sub><italic>m</italic></sub> (grey dashed line). The CO<sub>2</sub> concentration where the oxygen reaction error rate will be 1% is C<sub>99%</sub> (grey dash-dotted line). The transition between carbonic anyhdrase being unsatruated and saturated happens where the two analytic solutions meet (where the dashed and solid red lines meet). Below a critical value of transport, <inline-formula><mml:math id="inf85"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> the level of transport is lower than the <inline-formula><mml:math id="inf86"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> leaking through the cell membrane. A value of <inline-formula><mml:math id="inf87"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mtext> cm</mml:mtext><mml:mo>/</mml:mo><mml:mtext>s</mml:mtext></mml:mrow></mml:math></inline-formula> for the carboxysome permeability was used for these calculations.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.010">http://dx.doi.org/10.7554/eLife.02043.010</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f003"/></fig><fig id="fig3s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.02043.011</object-id><label>Figure 3—figure supplement 1.</label><caption><title>No effect of localizing carbonic anhydrase to the shell of the carboxysome.</title><p>We assume the same amount of carbonic anhydrase and RuBisCO activity for each simulation and compare the case with the enzymes evenly distributed throughout the carboxysome to the case where the carbonic anhydrase is localized to the inner carboxysome shell. The (-.-) lines are for no organization and (x) for localization with <inline-formula ><mml:math id="inf216"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. The (…) lines are for no organization and (o) are for localization with <inline-formula ><mml:math id="inf217"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.011">http://dx.doi.org/10.7554/eLife.02043.011</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043fs003"/></fig></fig-group><fig id="fig4" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.012</object-id><label>Figure 4.</label><caption><title>Concentration of CO<sub>2</sub> in the carboxysome with varying carboxysome permeability (<bold>A</bold>).</title><p>Numerical solution (diamonds and circles) and analytic solutions (carbonic anhydrase unsaturated, solid lines, and saturated, dashed lines) correspond well. On all plots CO<sub>2</sub> (red circle) < <inline-formula><mml:math id="inf88"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> (blue diamond). Concentration in the cell along the radius, <italic>r</italic>, with carboxysome permeability <inline-formula><mml:math id="inf89"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (<bold>B</bold>), <inline-formula><mml:math id="inf90"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (<bold>C</bold>), <inline-formula><mml:math id="inf91"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> (<bold>D</bold>). Grey dotted lines in (<bold>B</bold>), (<bold>C</bold>), (<bold>D</bold>) indicate location of the carboxysome shell boundary. The transition from low CO<sub>2</sub> at high permeability (<bold>D</bold>) to maximum CO<sub>2</sub> concentration at optimal permeability (<bold>C</bold>) occurs at <inline-formula><mml:math id="inf92"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>D</mml:mi><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. At low carboxysome permeability (<bold>B</bold>) <inline-formula><mml:math id="inf93"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> diffusion into the carboxysome is slower than consumption. For all subplots <inline-formula><mml:math id="inf94"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf95"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.6</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. Qualitative results remain the same with varying <italic>j</italic><sub><italic>c</italic></sub>, increasing <italic>α</italic> will increase the gradient of CO<sub>2</sub> across the cell as CO<sub>2</sub> is converted to <inline-formula><mml:math id="inf96"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> at the cell membrane.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.012">http://dx.doi.org/10.7554/eLife.02043.012</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f004"/></fig></p><p>When <italic>j</italic><sub><italic>c</italic></sub> is low, the ratio of CO<sub>2</sub> and <inline-formula><mml:math id="inf97"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is constant, set by the chemical equilibrium at a given pH. In this case the rate of the carbonic anhydrase reaction is much faster than diffusion within the carboxysome, so that <inline-formula><mml:math id="inf98"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi></mml:mrow></mml:math></inline-formula>. Unlike the uncatalyzed interconversion of CO<sub>2</sub> and <inline-formula><mml:math id="inf99"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> in the cytosol, carbonic anhydrase brings the concentrations in the carboxysome to equilibrium very quickly. The chemical equilibrium is <inline-formula><mml:math id="inf100"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo>/</mml:mo><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)/</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula>, for pH around 7 [<xref ref-type="bibr" rid="bib17">Heinhorst et al., 2006</xref>; <xref ref-type="bibr" rid="bib10">DeVoe & Kistiakowsky, 1961)</xref>], so that <inline-formula><mml:math id="inf101"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> > CO<sub>2</sub> in the carboxysome. Increased pH would increase <italic>K</italic><sub><italic>eq</italic></sub> and the proportion of <inline-formula><mml:math id="inf102"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, while decreased pH would decrease <italic>K</italic><sub><italic>eq</italic></sub> and the proportion of <inline-formula><mml:math id="inf103"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>. Such variations do not substantially effect the subsequent discussion and mechanisms, although they will change the absolute values of CO<sub>2</sub> concentration in the carboxysome.</p><p>The <italic>K</italic><sub><italic>m</italic></sub> dashed line in <xref ref-type="fig" rid="fig3">Figure 3</xref> shows the CO<sub>2</sub> level above which RuBisCO reaction is saturated: this gives the RuBisCO saturated (blue) boundary in <xref ref-type="fig" rid="fig2">Figure 2</xref>. We have similarly marked the concentration C<sub>99%</sub> where there is a 1% oxygen reaction error rate with a dash-doted line.</p><p>At higher levels, the CO<sub>2</sub> concentration no longer increases with increasing <italic>j</italic><sub><italic>c</italic></sub>, because the carbonic anhydrase is saturated. The saturated regime occurs in <xref ref-type="fig" rid="fig3">Figure 3</xref> when <inline-formula><mml:math id="inf104"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mo>></mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, so that increasing <inline-formula><mml:math id="inf105"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> (controlled directly by <italic>j</italic><sub><italic>c</italic></sub>) no longer increases the rate of production of <inline-formula><mml:math id="inf106"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. This transition from unsaturated to saturated carbonic anhydrase defines the line for the carbonic anhydrase saturated region in <xref ref-type="fig" rid="fig2">Figure 2</xref>.</p></sec><sec id="s2-3"><title>Carboxysome permeability has optimal value</title><p>For each line of constant concentration in <xref ref-type="fig" rid="fig2">Figure 2</xref> there is an optimal permeability value, where the least <inline-formula><mml:math id="inf107"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport is required to achieve the same CO<sub>2</sub> concentration. The optimal permeability value shifts downward with increasing CO<sub>2</sub> concentration (compare light and dark blue curves). For <italic>C</italic><sub><italic>99%</italic></sub> the optimal permeability is <inline-formula><mml:math id="inf108"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, below the calculated upper bound: <inline-formula><mml:math id="inf109"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo><</mml:mo><mml:mn>0.02</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> obtained above from the carboxysome structure. To further understand the effect of permeability, we examine the CO<sub>2</sub> concentration in the carboxysome for varying carboxysome permeabilities and a fixed <inline-formula><mml:math id="inf110"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport rate in <xref ref-type="fig" rid="fig4">Figure 4</xref>. <xref ref-type="fig" rid="fig4">Figure 4A</xref>, shows that there is a broad range of <italic>k</italic><sub><italic>c</italic></sub> where the CCM has maximal efficacy. <xref ref-type="fig" rid="fig4">Figure 4</xref> shows the distribution of inorganic carbon throughout the cell when the permeability is low (<bold>B</bold>), optimal (<bold>C</bold>), and high (<bold>D</bold>). At high permeability, the CO<sub>2</sub> produced in the carboxysome rapidly leaks out of the carboxysome, and the CO<sub>2</sub> concentration in the cytosol, shown in <xref ref-type="fig" rid="fig4">Figure 4D</xref>, is relatively high. When the carboxysome permeability decreases to near the optimal value, <xref ref-type="fig" rid="fig4">Figure 4C</xref>, the carboxysome traps CO<sub>2</sub>, and the cytosolic levels are lower, decreasing leakage out of the cell. This transition occurs when diffusion across the cell (and carboxysome) takes a shorter time than diffusion through the carboxysome shell; or the CO<sub>2</sub> in the carboxysome is effectively partitioned from the CO<sub>2</sub> in the cell.</p><p>If the carboxysome permeability is below optimal, diffusion of <inline-formula><mml:math id="inf111"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> into the carboxysome cannot keep up with consumption from RuBisCO, <xref ref-type="fig" rid="fig4">Figure 4B</xref>. The existence of an optima requires RuBisCO consumption to be low enough that there is a <italic>k</italic><sub><italic>c</italic></sub> where the cytosol and carboxysome are partitioned, but <inline-formula><mml:math id="inf112"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> diffusion in can keep up. When such an optima exists, the carboxysome permeability can improve the CO<sub>2</sub> concentration in the carboxysome without any special selectivity between <inline-formula><mml:math id="inf113"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub>. The location and concentrating power of the optimal regime, is dependent on the size of the cell and the membrane permeabilities to CO<sub>2</sub> and <inline-formula><mml:math id="inf114"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p></sec></sec><sec sec-type="discussion" id="s3"><title>Discussion</title><sec id="s3-1"><title>Are the fluxes and concentrations reasonable?</title><p>While we have solved our model to describe a vast parameter space it is instructive to compare the fluxes and concentrations we find within our optimal parameter space (the green region in <xref ref-type="fig" rid="fig2">Figure 2</xref>) to actual numbers. At low external inorganic carbon conditions, internal inorganic carbon pools due to CCM activity are regularly measured as high as C<sub>i</sub> = 30 <italic>mM</italic>. The inorganic carbon is predominantly in the form of <inline-formula><mml:math id="inf115"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>, and measurements do not distinguish between the cytosol and carboxysome <xref ref-type="bibr" rid="bib19">(Kaplan & Reinhold, 1999</xref>; <xref ref-type="bibr" rid="bib50">Woodger et al., 2005</xref>; <xref ref-type="bibr" rid="bib44">Sultemeyer et al., 1995</xref>; <xref ref-type="bibr" rid="bib32">Price et al., 1998</xref>, <xref ref-type="bibr" rid="bib35">2008</xref>). In our model, we find that the cytosolic <inline-formula><mml:math id="inf116"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration is 30 mM when <inline-formula><mml:math id="inf117"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.6</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, over a wide range of the carboxysome permeability (indicated as the dashed grey line in <xref ref-type="fig" rid="fig2">Figure 2</xref>). From <xref ref-type="fig" rid="fig4">Figure 4</xref> we can also see that the cytosolic <inline-formula><mml:math id="inf118"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration is the dominate form of inorganic carbon in the cell at <inline-formula><mml:math id="inf119"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.6</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. We examine the fate of the <inline-formula><mml:math id="inf120"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transported into the cell in terms of the <inline-formula><mml:math id="inf121"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> leaking out, CO<sub>2</sub> leaking out, CO<sub>2</sub> fixation or carboxylation, and O<sub>2</sub> fixation or oxygenation (<xref ref-type="table" rid="tbl3">Table 3</xref>).<table-wrap id="tbl3" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.013</object-id><label>Table 3.</label><caption><p>Fate of carbon brought into the cell for <italic>j</italic><sub><italic>c</italic></sub> = 0.6<italic>cm/s</italic> and <italic>k</italic><sub><italic>c</italic></sub> = 10<sup>–3</sup><italic>cm/s</italic></p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.013">http://dx.doi.org/10.7554/eLife.02043.013</ext-link></p></caption><table frame="hsides" rules="groups"><thead><tr><th/><th>formula</th><th><inline-formula><mml:math id="inf122"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mi>i</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mtext> </mml:mtext><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th><th>% of flux</th></tr></thead><tbody><tr><td><inline-formula><mml:math id="inf123"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport</td><td><italic>j</italic><sub><italic>c</italic></sub><italic>H</italic><sub><italic>out</italic></sub></td><td>3.26 × 10<sup>−4</sup></td><td/></tr><tr><td><inline-formula><mml:math id="inf124"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> leakage</td><td><inline-formula><mml:math id="inf125"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td>3.2 × 10<sup>−4</sup></td><td>98.6%</td></tr><tr><td>CO<sub>2</sub> leakage</td><td><inline-formula><mml:math id="inf126"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>C</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td>4.5 × 10<sup>−4</sup></td><td>1.4 %</td></tr><tr><td>carboxylation</td><td><inline-formula><mml:math id="inf127"><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mi>O</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>O</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td>8.2 × 10<sup>−8</sup></td><td>0.03 %</td></tr><tr><td>oxygenation</td><td><inline-formula><mml:math id="inf128"><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow><mml:mi>O</mml:mi></mml:msubsup><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>m</mml:mi><mml:mi>O</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mi>C</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td>6.7 × 10<sup>−10</sup></td><td>2 × 10<sup>−4</sup> %</td></tr></tbody></table></table-wrap></p><p>For cells grown under low inorganic carbon conditions net <inline-formula><mml:math id="inf129"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> fluxes (transport - leakage) are measured <inline-formula><mml:math id="inf130"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:mtext>pmol</mml:mtext></mml:mrow><mml:mrow><mml:mtext>mgChls</mml:mtext></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula>, with CO<sub>2</sub> net flux being slightly lower but the same order of magnitude <xref ref-type="bibr" rid="bib49">(Whitehead et al., 2014</xref>; <xref ref-type="bibr" rid="bib4">Badger et al., 1994)</xref>. High external inorganic carbon conditions produce slightly higher net <inline-formula><mml:math id="inf131"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> rates (<xref ref-type="bibr" rid="bib46">Tchernov et al., 1997</xref>). Assuming chlorophyll per cell volume of around <inline-formula><mml:math id="inf132"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>11</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mtext>mgChl</mml:mtext></mml:mrow><mml:mrow><mml:mtext>cell</mml:mtext></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula> for cells of our size we can convert this into a flux of <inline-formula><mml:math id="inf133"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>(c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mi>s)</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula> <xref ref-type="bibr" rid="bib49">(Whitehead et al., 2014</xref>; <xref ref-type="bibr" rid="bib21">Keren et al., 2004</xref>, <xref ref-type="bibr" rid="bib20">2002</xref>). The net <inline-formula><mml:math id="inf134"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> flux for our model cell is <inline-formula><mml:math id="inf135"><mml:mrow><mml:mn>6</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>(c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mi>s)</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula>, so we are about an order of magnitude too high. If we choose a <inline-formula><mml:math id="inf136"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport value one order of magnitude smaller, we will get net fluxes of the same order of magnitude as the measurements at the cost of slightly lower carboxylation rates and higher oxygenation rates (<xref ref-type="table" rid="tbl4">Table 4</xref>). This would also mean a lower internal <inline-formula><mml:math id="inf137"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> pool. Alternatively, the same internal <inline-formula><mml:math id="inf138"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> could be reached at a lower flux rate, if the external <inline-formula><mml:math id="inf139"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is higher. Since the majority of the <inline-formula><mml:math id="inf140"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport is balanced by <inline-formula><mml:math id="inf141"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> leakage, we can find the transport rate needed to sustain a particular amplification by the simple formula: <inline-formula><mml:math id="inf142"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>.<table-wrap id="tbl4" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.014</object-id><label>Table 4.</label><caption><p>Fate of carbon brought into the cell for <italic>j</italic><sub><italic>c</italic></sub> = 0.06 <italic>cm/s</italic> and <italic>k</italic><sub><italic>c</italic></sub> = 10<sup>–3</sup><italic>cm/s</italic></p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.014">http://dx.doi.org/10.7554/eLife.02043.014</ext-link></p></caption><table frame="hsides" rules="groups"><thead><tr><th/><th>formula</th><th><inline-formula><mml:math id="inf143"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mi>i</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mtext> </mml:mtext><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></th><th>% of flux</th></tr></thead><tbody><tr><td><inline-formula><mml:math id="inf144"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport</td><td><italic>j</italic><sub><italic>c</italic></sub><italic>H</italic><sub><italic>out</italic></sub></td><td>2.8 × 10<sup>−5</sup></td><td/></tr><tr><td><inline-formula><mml:math id="inf145"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> leakage</td><td><inline-formula><mml:math id="inf146"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>H</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td>2.7 × 10<sup>−5</sup></td><td>96.6 %</td></tr><tr><td>CO<sub>2</sub> leakage</td><td><inline-formula><mml:math id="inf147"><mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>m</mml:mi><mml:mi>C</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td>8.8 × 10<sup>−7</sup></td><td>3.2 %</td></tr><tr><td>carboxylation</td><td><inline-formula><mml:math id="inf148"><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mi>O</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>O</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td>5.4 × 10<sup>−8</sup></td><td>0.2 %</td></tr><tr><td>oxygenation</td><td><inline-formula><mml:math id="inf149"><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mi>V</mml:mi><mml:mrow><mml:mi>max</mml:mi></mml:mrow><mml:mi>O</mml:mi></mml:msubsup><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>m</mml:mi><mml:mi>O</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mi>C</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula></td><td>2.3 × 10<sup>−9</sup></td><td>8 × 10<sup>−3</sup> %</td></tr></tbody></table></table-wrap></p><p>While we can compare the net fluxes, we have not found direct experimental evidence for the absolute <inline-formula><mml:math id="inf150"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> uptake rate. To determine whether this <inline-formula><mml:math id="inf151"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport rate is reasonable we perform a back of the envelope calculation. Our simulated cell has a flux of 2 × 10<sup>8</sup> molecules/s. Assuming the rate of transport per transporter of <inline-formula><mml:math id="inf152"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:mtext>molecules</mml:mtext></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> and our cell's surface area this requires about <inline-formula><mml:math id="inf153"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>4</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:mtext>transporters</mml:mtext></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula>. This is about an order of magnitude higher than the number of ATP synthase complexes on the thylakoid membrane in spinach, <inline-formula><mml:math id="inf154"><mml:mrow><mml:mn>700</mml:mn><mml:mtext> </mml:mtext><mml:mfrac><mml:mrow><mml:mtext>complexes</mml:mtext></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mtext>m</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula> (<xref ref-type="bibr" rid="bib26">Miller and Staehelin, 1979</xref>).</p><p>According to our calculation only around 1 % of the carbon transported into the cell is fixed into 3-phosophoglycerate. Even in this highly CO<sub>2</sub> concentrating regime, 5 × 10<sup>4</sup> 2-phosophoglycolate produced per second. Cyanobacteria have been shown to have multiple pathways for recycling 2-phosophoglycolate (<xref ref-type="bibr" rid="bib16">Hackenberg et al., 2009</xref>). Our system fixes CO<sub>2</sub> at a rate of 0.14 pg/hour. Given the volume of our cell, and the fact that between 115–300 fg/<italic>μ</italic>m<sup>3</sup> of carbon are needed to produce a new cyanobacterial cell (<xref ref-type="bibr" rid="bib24">Mahlmann et al., 2008</xref>) we need between 0.1 and 0.3 picograms of carbon per cell. At the higher flux rate (<xref ref-type="table" rid="tbl3">Table 3</xref>) this means that a cell could replicate every 7–21 hr and the lower flux rate (<xref ref-type="table" rid="tbl4">Table 4</xref>) allows replication every 11–35 hr. Both are consistent with the division times of cyanobacteria.</p></sec><sec id="s3-2"><title>Concentration profiles of CO<sub>2</sub> and <inline-formula><mml:math id="inf155"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> across the cell</title><p>At <inline-formula><mml:math id="inf156"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.6</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, varying the carboxysome permeability changes how the available inorganic carbon is partitioned between the carboxysome and cytosol, thereby setting the carboxysomal CO<sub>2</sub> concentration as shown in <xref ref-type="fig" rid="fig4">Figure 4</xref>. Strikingly, the <inline-formula><mml:math id="inf157"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration is constant across the cytosol. This is because the cell membranes have low permeability to <inline-formula><mml:math id="inf158"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>; thus, the rate of escape is slow and <inline-formula><mml:math id="inf159"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> equilibrates across the cell. A consequence of this flat <inline-formula><mml:math id="inf160"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> profile is that the carboxysome experiences the same <inline-formula><mml:math id="inf161"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration, independent of its position in the cell. This means the incoming inorganic carbon source for the carboxysome system is invariant with the position of the carboxysome in the cell.</p><p>In contrast, there is a gradient in CO<sub>2</sub> concentration across the cell when the carboxysome permeability is at or above the optimum (<xref ref-type="fig" rid="fig4">Figure 4C,D</xref>). The cell membrane is more permeable to CO<sub>2</sub>. The gradient means that the CO<sub>2</sub> leakage out of the cell affects the CO<sub>2</sub> leakage out of the carboxysome. Moving the carboxysome close to the cell membrane increases the leakage rate of CO<sub>2</sub> out of the carboxysome. Notably, in <italic>S. elongatus</italic> the carboxysomes are located along the center line of the cell, away from the cell membranes (<xref ref-type="bibr" rid="bib40">Savage et al., 2010</xref>). The spatial profiles of <inline-formula><mml:math id="inf162"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> and CO<sub>2</sub> give no hint as to why the carboxysomes are spaced apart from one another. Since the gradient in <inline-formula><mml:math id="inf163"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is flat, there is no competition between the carboxysomes for <inline-formula><mml:math id="inf164"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> (the main incoming source of inorganic carbon). In fact, since the local concentration of CO<sub>2</sub> is higher near a carboxysome, nearby carboxysomes would ‘feed’ each other CO<sub>2</sub>. As has been shown, such clumping would reduce the probability of distributing carboxysomes equitably to daughter cells, possibly counteracting any benefit (<xref ref-type="bibr" rid="bib40">Savage et al., 2010</xref>).</p><p>The concentration across the carboxysome is basically constant, because the carboxysome is so small that diffusion across it takes very little time. A consequence of this is that the organization of the reactions in the carboxysome does not effect the CO<sub>2</sub> concentration in the carboxysome (<xref ref-type="fig" rid="fig3s1">Figure 3—figure supplement 1</xref>). Therefore, the localization of the carbonic anhydrase to the inner carboxysome shell seems to have no effect on the CCM. It has been suggested that diffusion in the carboxysome should be slower, since the carboxysome is packed with RuBisCO. One proposed consequence of slower diffusion in the carboxysome is that it could trap CO<sub>2</sub>, making a low carboxysome permeability unnecessary. We have tested this hypothesis (<xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1</xref>), and find that assuming the diffusion constant one would expect for small molecules in a 60% sucrose solution (<inline-formula><mml:math id="inf165"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>), does reduce the optimal carboxysome permeability. However, for any carboxysome permeability a higher <inline-formula><mml:math id="inf166"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport rate is needed to achieve the same carboxsomal CO<sub>2</sub> concentration. So if the diffusion is indeed slower in the carboxysome it does not aid the CCM. Even at this slower diffusion, the CO<sub>2</sub> concentration across the carboxysome is flat.</p></sec><sec id="s3-3"><title>Benefit of CO<sub>2</sub> to <inline-formula><mml:math id="inf167"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> conversion: facilitated uptake or scavenging of CO<sub>2</sub></title><p>We investigate the effect of CO<sub>2</sub> to <inline-formula><mml:math id="inf168"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> conversion at the thylakoid and cell membranes (combined in the model). Increasing conversion, <italic>α</italic> > 0, can facilitate uptake of CO<sub>2</sub> from outside the cell and scavenge CO<sub>2</sub> escaped from the carboxysome. Facilitated uptake results in saturating both carbonic anhydrase and RuBisCO at a lower level of <inline-formula><mml:math id="inf169"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport. Scavenging broadens the range of carboxysome permeability which will effectively separate the inorganic carbon pools in the carboxysome and outside. Scavenging decreases the concentration of CO<sub>2</sub> in the cytosol, so a more permeable carboxysome can still result in a low leakage rate of inorganic carbon out of the cell (more of the inorganic carbon in the cytosol is in the form of <inline-formula><mml:math id="inf170"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> which leaks out less readily). However, neither of these effects is particularly strong in our current range of reaction rates, and cell membrane permeability (<xref ref-type="fig" rid="fig2s2">Figure 2—figure supplement 2</xref>).</p><p>The relative effect of these two mechanisms depends on the external CO<sub>2</sub> and <inline-formula><mml:math id="inf171"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentrations. In saltwater environments the pH is near 8 and <inline-formula><mml:math id="inf172"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> is the predominant inorganic carbon source. While external pH is not explicitly treated in our model, we can account for changes to pH through the external inorganic carbon concentration. To be consistent with oceanic environment, thus far we have shown results for low external inorganic carbon concentrations of [CO<sub>2</sub>] = 0.1 <italic>μM</italic> and [<inline-formula><mml:math id="inf173"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>] = 14.9 <italic>μM</italic>. The effect of facilitated uptake, under these assumptions, is very small. In freshwater or under conditions of ocean acidification, where the pH could fall to 6 or lower, there can be a much larger proportion of CO<sub>2</sub> (>50%). <xref ref-type="fig" rid="fig5">Figure 5</xref> shows the absolute contribution of <inline-formula><mml:math id="inf174"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport, facilitated CO<sub>2</sub> uptake, and CO<sub>2</sub> scavenging for varying proportions of external CO<sub>2</sub>. Even though we assume the same velocity of facilitated uptake and <inline-formula><mml:math id="inf175"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport (<inline-formula><mml:math id="inf176"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>), facilitated uptake contributes less because it is limited by CO<sub>2</sub> diffusion across the membrane. At the same rates of transport the facilitated uptake mechanism only contributes more than active <inline-formula><mml:math id="inf177"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> if the CO<sub>2</sub> concentration is greater than 80% of external inorganic carbon. This is consistent with observations that oceanic cyanobacteria such as Prochlorococcus only seem to possess gene homologs for <inline-formula><mml:math id="inf178"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport systems, while other freshwater and estuarine cyanobacteria have gene homologs for both constitutive (NDH-1<sub>4</sub>) and inducible (NDH-1<sub>3</sub>) CO<sub>2</sub> uptake systems as well as inducible <inline-formula><mml:math id="inf179"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport systems (BicA, SbtA, and BCT1) (<xref ref-type="bibr" rid="bib36">Price, 2011</xref>).<fig id="fig5" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.015</object-id><label>Figure 5.</label><caption><title>Size of the <inline-formula><mml:math id="inf180"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> flux in one cell from varying sources, as the proportion of <inline-formula><mml:math id="inf181"><mml:mrow><mml:msub><mml:mrow><mml:mtext>CO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math id="inf182"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> outside the cell changes changes.</title><p>We show results for three carboxysome permeabilities, <inline-formula><mml:math id="inf183"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and only the scavenging is effected. Total external inorganic carbon is <inline-formula><mml:math id="inf184"><mml:mrow><mml:mn>15</mml:mn><mml:mi>μ</mml:mi><mml:mi>M</mml:mi></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="inf185"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf186"><mml:mrow><mml:mfrac><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>. Scavenging is negligibly small for all values of <inline-formula><mml:math id="inf187"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> shown. Unless there is very little <inline-formula><mml:math id="inf188"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> in the environment, <inline-formula><mml:math id="inf189"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport seems to be more efficient than <inline-formula><mml:math id="inf190"><mml:mrow><mml:msub><mml:mrow><mml:mtext>CO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> facilitated uptake.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.015">http://dx.doi.org/10.7554/eLife.02043.015</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f005"/></fig></p><p>Scavenging is negligibly small for all values of <inline-formula><mml:math id="inf191"><mml:mrow><mml:msub><mml:mtext>k</mml:mtext><mml:mtext>c</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> shown. There is very little <inline-formula><mml:math id="inf192"><mml:mrow><mml:msub><mml:mrow><mml:mtext>CO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> in the cytosol, so there is very little <inline-formula><mml:math id="inf193"><mml:mrow><mml:msub><mml:mrow><mml:mtext>CO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to scavenge, <xref ref-type="fig" rid="fig5">Figure 5</xref>. The effect of scavenging is dependent on the cell membrane permeability to <inline-formula><mml:math id="inf194"><mml:mrow><mml:msub><mml:mrow><mml:mtext>CO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf195"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p><p>Given that scavenging has no obvious affect on <inline-formula><mml:math id="inf196"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentrations, it is reasonable to wonder why this mechanism exists at all. One might assume that scavenging prevents leakage, but if the energy to bring a ‘new’ <inline-formula><mml:math id="inf197"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> molecule from outside the cell is the same as the energy required to save an ‘old’ CO<sub>2</sub> molecule from leaking out, there is no obvious advantage of preventing the leakage. It is possible that since the scavenging mechanism is associated with the electron transport chain of the light reactions of photosynthesis scavenging can be ramped up more easily when there is excess light energy. If this were the case, a comparison of <inline-formula><mml:math id="inf198"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="inf199"><mml:mrow><mml:mfrac><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula> is deceiving and <inline-formula><mml:math id="inf200"><mml:mrow><mml:mfrac><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula> could be much larger. Indeed it has been suggested that the cell uses this mechanism as a way to dissipate excess light energy <xref ref-type="bibr" rid="bib46">(Tchernov et al., 1997</xref>, <xref ref-type="bibr" rid="bib47">2003</xref>).</p></sec><sec id="s3-4"><title>Cellular organization</title><p>The most striking aspect of the CCM is the way that spatial organization is used to increase the efficacy of the reactions. <xref ref-type="fig" rid="fig6">Figure 6</xref> compares the effect of different enzymatic reaction organizations. Concentrating carbonic anhydrase and RuBisCO to a small region in the center of the cell, on a scaffold for example, leads to an order of magnitude increase in the concentration of CO<sub>2</sub>. Localizing the carbonic anhydrase to a small volume concentrates it, increasing the maximum reaction rate per volume, <italic>V</italic><sub><italic>ca</italic></sub> and <italic>V</italic><sub><italic>ba</italic></sub>. A larger <italic>V</italic><sub><italic>ba</italic></sub> increases the <inline-formula><mml:math id="inf201"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration at which carbonic anhydrase is saturated allowing the mechanism to take advantage of a larger <inline-formula><mml:math id="inf202"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> flux, <italic>j</italic><sub><italic>c</italic></sub>. A small increase can be gained from encapsulating the enzymes in a permeable carboxysome shell and another order of magnitude is gained at the optimal permeability. At optimal carboxysome permeability, the CO<sub>2</sub> is effectively partitioned into the carboxysome and conversion can act only as facilitated uptake as shown in <xref ref-type="fig" rid="fig5">Figure 5</xref>.<fig id="fig6" position="float"><object-id pub-id-type="doi">10.7554/eLife.02043.004</object-id><label>Figure 6.</label><caption><title>Concentration of CO<sub>2</sub> achieved through various cellular organizations of enzymes, where we have selected the <inline-formula><mml:math id="inf20"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport level such that the <inline-formula><mml:math id="inf21"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentration in the cytosol is 30 <italic>mM</italic>.</title><p>O<sub>2</sub> concentration is 260 <italic>μM</italic>. The oxygenation error rates, as a percent of total RuBisCO reactions are indicated on the concentration bars. The cellular organizations investigated are RuBisCO and carbonic anhydrase distributed throughout the entire cytosol, co-localizing RuBisCO and carbonic anhydrase on a scaffold at the center of the cell without a carboxysome shell, RuBisCO and carbonic anhydrase encapsulated in a carboxysome with high permeability at the center of the cell, and RuBisCO and carbonic anhydrase encapsulated in a carboxysome with optimal permeability at the center of the cell.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.004">http://dx.doi.org/10.7554/eLife.02043.004</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife02043f006"/></fig></p><p>Another advantage of localizing the enzymes in a small region at the center of the cell is separating carbonic anhydrase from the <italic>α</italic> (CO<sub>2</sub> to <inline-formula><mml:math id="inf203"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula>) conversion mechanism, preventing a futile cycle. The futile cycle is most detrimental when the enzymes are distributed through out the cytosol, and increases the oxygenation error rate (data not shown). Concentrating the enzymes away from the cell and thylakoid membranes, where conversion happens, removes this effect. On a scaffold the oxygenation rate is almost exactly the same with and without the <italic>α</italic> conversion mechanism. This is consistent with the previously shown detrimental effect of having active carbonic anhydrase free within the cytosol (<xref ref-type="bibr" rid="bib31">Price & Badger, 1989</xref>). It would be impossible to keep the cytosol completely free from carbonic anhydrase enzyme, so there must be a way of activating it within the carboxysome only. Carbonic anhydrase is inactivated under reducing conditions (<xref ref-type="bibr" rid="bib30">Peña et al., 2010</xref>). Recently it was shown that carboxysomes oxidize after assembly, providing a way to keep carbonic anhydrase inactive until fully enclosed in a carboxysome (<xref ref-type="bibr" rid="bib7">Chen et al., 2013</xref>).</p></sec><sec id="s3-5"><title>Effects of pH</title><p>Cyanobacteria must regulate pH as almost all biochemical reactions are pH sensitive. We do not attempt to model this regulation or potential pH variation within the cell, however pH may be included implicitly in a couple ways. We have already explored the effect of varying external pH, and the effects of pH on carbonic anhydrase. Cytosolic pH would have little direct effect on the CO<sub>2</sub> and <inline-formula><mml:math id="inf204"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> levels since the non-enzymatic interconversion is very slow as previously discussed. The effect of internal pH could also be explored by varying the reaction rate of RuBisCO, which is pH sensitive. Varying the reaction rate of RuBisCO greatly could change the range where a non-specific carboxysome permeability can increase the concentration of CO<sub>2</sub> in the carboxysome. It would be unexpected that the RuBisCO rate be much faster than we assume, as we have assumed a rate on the high end. A lower RuBisCO rate would increase the range of effective carboxysome permeabilities. As previously mentioned the CO<sub>2</sub> facilitated uptake mechanism functions by creating local alkaline pockets. Diffusion of hydrogen ions across the cell would be very fast, so such pockets would require a massive reduction from the light reactions to maintain local alkalinity. Whether such pH gradients are possible, is certainly a subject of future interest.</p></sec><sec id="s3-6"><title>Conclusions</title><p>We have described and analyzed a model for the CO<sub>2</sub> concentrating mechanism in cyanobacteria. There exists a broad range of <inline-formula><mml:math id="inf205"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport and carboxysome permeability values which result in effective CO<sub>2</sub> concentration in the carboxysome. This effective concentration parameter space is defined by CO<sub>2</sub> levels high enough to saturate RuBisCO and produce a favorable ratio of carboxylation to oxygenation reactions, but not so high as to saturate carbonic anhydrase (after which increasing <inline-formula><mml:math id="inf206"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> transport will not increase the CO<sub>2</sub> concentration). An optimal carboxysome permeability exists, where <inline-formula><mml:math id="inf207"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> diffusion into the carboxysome is not substantially inhibited, but CO<sub>2</sub> leakage is minimal. <inline-formula><mml:math id="inf208"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> concentrations across the cell are flat and are predominately set by the transport rate in, and leakage out. We quantitatively compare the transport rates and concentrations we predict in our optimal parameter space, and find them to be in good agreement with experiment. We also comment on the effects of external pH on CO<sub>2</sub> versus <inline-formula><mml:math id="inf209"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>HCO</mml:mtext></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo></mml:msubsup></mml:mrow></mml:math></inline-formula> uptake mechanisms. Finally we describe the cumulative benefits of co-localization, encapsulation, and optimal carboxysome permeability on the CCM.</p><p>Further comparison of this model to experimental flux measurements, especially to determine the quantitative contributions of different transporters under different physiological conditions would be very interesting. Current solutions are for steady state at constant external concentration, but most gas exchange measurements, by necessity, measure the fluxes as the inorganic carbon is depleted in the media. The model could be modified to solve the time dependent problem with varying external inorganic carbon. As of yet the carboxysome permeability has not been measured directly, and it would be quite interesting to see how close it is to our ‘optimal’ prediction.</p></sec></sec></body><back><ack id="ack"><title>Acknowledgements</title><p>We thank Colleen Cavanaugh, Jeremy Gunarawenda and Pam Silver for important conversations. This research was supported by the National Science Foundation through the Harvard Materials Research Science and Engineering Center (DMR-0820484) and the Division of Mathematical Sciences (DMS-0907985). MPB is an Investigator of the Simons Foundation.</p></ack><sec sec-type="additional-information" id="s4"><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 no competing interests exist.</p></fn></fn-group><fn-group content-type="author-contribution"><title>Author contributions</title><fn fn-type="con" id="con1"><p>NMM, preformed calculations, Conception and design, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con2"><p>MPB, oversaw and mentored research, Conception and design, Drafting or revising the article</p></fn></fn-group></sec><sec sec-type="supplementary-material" id="s5"><title>Additional files</title><supplementary-material id="SD1-data"><object-id pub-id-type="doi">10.7554/eLife.02043.016</object-id><label>Supplementary file 1.</label><caption><p>Mathematical derivation appendix. Mathematical derivations of analytic solutions for a spherical cell with reactions organized in a variety of ways. We present analytic solutions for the concentration of CO<sub>2</sub> and HCO<sub>3</sub><sup>−</sup> a carboxysome located at the center of the cell. We derive analytic solutions assuming a number of different cases for the enzymatic rates in the carboxysome: RuBisCO reaction rate negligible with carbonic anhydrase saturated and unsaturated, RuBisCO reaction rate non-negligible with carbonic anhydrase unsaturated. Additionally we derive analytic solutions for the enzymatic reactions throughout the cell and localized to a scaffold without a carboxysome shell.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02043.016">http://dx.doi.org/10.7554/eLife.02043.016</ext-link></p></caption><media xlink:href="elife02043s001.pdf" mimetype="application" mime-subtype="pdf"/></supplementary-material></sec><ref-list><title>References</title><ref id="bib1"><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Agapakis</surname><given-names>CM</given-names></name><name><surname>Boyle</surname><given-names>PM</given-names></name><name><surname>Silver</surname><given-names>PA</given-names></name></person-group><year>2012</year><article-title>Natural strategies for the spatial optimization of metabolism in synthetic biology</article-title><source>Nature Chemical 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