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new post on active modeling

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tlnagy committed Sep 7, 2017
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@@ -71,3 +71,72 @@ @incollection{boyd_area_2013
keywords = {Artificial Intelligence (incl. Robotics), Data Mining and Knowledge Discovery, Discrete Mathematics in Computer Science, Information Storage and Retrieval, pattern recognition, Probability and Statistics in Computer Science},
file = {boyd_et_al_2013_machine_learning_and_knowledge_discovery_in_databases.pdf:/Users/tamasnagy/Articles/storage/RZFP23UT/boyd_et_al_2013_machine_learning_and_knowledge_discovery_in_databases.pdf:application/pdf;Snapshot:/Users/tamasnagy/Articles/storage/JBF5H6ZX/10.html:text/html}
}
@article{Sumino2012-dc,
title = "Large-scale vortex lattice emerging from collectively moving
microtubules",
author = "Sumino, Yutaka and Nagai, Ken H and Shitaka, Yuji and Tanaka,
Dan and Yoshikawa, Kenichi and Chat{\'e}, Hugues and Oiwa,
Kazuhiro",
affiliation = "Aichi University of Education, Aichi 448-8542, Japan.",
abstract = "Spontaneous collective motion, as in some flocks of bird and
schools of fish, is an example of an emergent phenomenon. Such
phenomena are at present of great interest and physicists have
put forward a number of theoretical results that so far lack
experimental verification. In animal behaviour studies,
large-scale data collection is now technologically possible,
but data are still scarce and arise from observations rather
than controlled experiments. Multicellular biological systems,
such as bacterial colonies or tissues, allow more control, but
may have many hidden variables and interactions, hindering
proper tests of theoretical ideas. However, in systems on the
subcellular scale such tests may be possible, particularly in
in vitro experiments with only few purified components.
Motility assays, in which protein filaments are driven by
molecular motors grafted to a substrate in the presence of
ATP, can show collective motion for high densities of motors
and attached filaments. This was demonstrated recently for the
actomyosin system, but a complete understanding of the
mechanisms at work is still lacking. Here we report
experiments in which microtubules are propelled by
surface-bound dyneins. In this system it is possible to study
the local interaction: we find that colliding microtubules
align with each other with high probability. At high
densities, this alignment results in self-organization of the
microtubules, which are on average 15 µm long, into vortices
with diameters of around 400 µm. Inside the vortices, the
microtubules circulate both clockwise and anticlockwise. On
longer timescales, the vortices form a lattice structure. The
emergence of these structures, as verified by a mathematical
model, is the result of the smooth, reptation-like motion of
single microtubules in combination with local interactions
(the nematic alignment due to collisions)--there is no need
for long-range interactions. Apart from its potential
relevance to cortical arrays in plant cells and other
biological situations, our study provides evidence for the
existence of previously unsuspected universality classes of
collective motion phenomena.",
journal = "Nature",
volume = 483,
number = 7390,
pages = "448--452",
month = "21~" # mar,
year = 2012,
language = "en"
}
@article{Marchetti2013,
title = {Hydrodynamics of soft active matter},
author = {Marchetti, M. C. and Joanny, J. F. and Ramaswamy, S. and Liverpool, T. B. and Prost, J. and Rao, Madan and Simha, R. Aditi},
journal = {Rev. Mod. Phys.},
volume = {85},
issue = {3},
pages = {1143--1189},
numpages = {0},
year = {2013},
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/RevModPhys.85.1143},
url = {https://link.aps.org/doi/10.1103/RevModPhys.85.1143}
}
@@ -0,0 +1,36 @@
---
title: "Active matter modeling"
author: Tamas Nagy
layout: post
tags: [science, math]
image:
---
Yesterday was the systems biology module of the iPQB bootcamp here at UCSF. I was in charge of the modeling section and I decided to try to get the students to model the relatively simple active matter system from @Sumino2012-dc since it was relatively straightforward, very visual, and shows complex emergent properties from simple rules (See Figure 1). Active matter systems are ubiquitous nonequilibrium condensed systems whose "unifying characteristic is that they are **composed of self-driven units, active particles, each capable of converting stored or ambient free energy into systematic movement**" [@Marchetti2013]. An *in vitro* microtubule and dynein system, like the one we're investigating here, is much more controlled and reproducible compared to the flocking of birds or fish and thus serves as a nice model system in which to study active matter.
![Figure 1A-B from @Sumino2012-dc. Cy3 labeled microtubules form vortices when placed on dynein-c coated glass coverslips in the presence of ATP](/assets/images/2017-09-07-microtubule-vortices/0de43a55.png)
The authors proposed a relatively simple model that could recapitulate this phenomenon. They modeled the behavior of the microtubules as a biased Ornstein–Uhlenbeck process where the microtubules moved at a constant velocity in a direction, $\theta_i$, that was updated by an angular velocity, $\omega_i$, at each time step. They added some normal brownian noise to $\omega_i$ with mean $\xi_{\mu}$ and standard deviation $\xi_{\sigma}$. They noticed that the microtubules had a slight clockwise curvature preference and that after relaxation time $\lambda$, the angular velocity $\omega_i$ would approach the preferred angular velocity, $\omega_0$.
@Sumino2012-dc also noted that microtubules would almost always align or anti-align after collisions (they would sometimes stall or cross). They modeled this by taking the aggregate angle of all neighbors of a microtubule that are within a certain distance from the microtubule. This was then weighted by a parameter $A$ that controls the relative influence of other microtubules.
Mathematically, this can be expressed in the following non-dimensionalized form:
$$
\begin{align}
\frac{d\Omega_i}{dT} &= - \frac{1}{\lambda}\left(\Omega_i - \Omega_0\right) + \textrm{Normal}(\xi_{\mu}, \xi_{\sigma})\\
\frac{d\theta_i}{dT} &= \Omega_i + \frac{A}{N_i(T)} \sum_{j \sim i} \sin\left(2(\theta_j - \theta_i)\right) \\
\frac{d\mathbf{X_i}}{dT} &= \mathbf{e_x}\cos \theta_i + \mathbf{e_y} \sin \theta_i
\end{align}
$$
I implemented the code using Python 3, NumPy, SciPy, and Matplotlib and it is available as a Jupyter Notebook in this gist: <https://gist.github.com/tlnagy/cba938ffd5c98236e90bfd1dc3d23d11>. The students found a quite few parameter combinations that yielded interesting results, most of which were highly unrealistic. I suspect this is due to the much lower "concentration" of microtubules that we used in our simulation to achieve interactivity, $N=1500$ instead of $N=2621440$. We were still able to get vortices to form:
![Initially, microtubule movement looks pretty random.](/assets/images/2017-09-07-microtubule-vortices/mt_initial.gif)
![However, they start to coalesce into a grid of vortices](/assets/images/2017-09-07-microtubule-vortices/mt_vortices.gif)
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
## References

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