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<title>Spherical caps in cell polarization</title>
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<h1>Spherical caps in cell polarization</h1>
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<p><i>February 27, 2018 — Support my next blog post, <a href="https://www.paypal.com/donate/?hosted_button_id=UCLCSJLFL433E">buy me a coffee</a> ☕.</i></p>
<center>
<img src="/blog/sphericalcaps1.png" class="img-responsive">
</center>
<p>I have just submitted a <a href="https://www.cell.com/biophysj/fulltext/S0006-3495(18)30672-6">paper</a> about the modeling of
<a href="https://en.wikipedia.org/wiki/Embryogenesis">embryogenesis</a> with the
<a href="http://shvartsmanlab.com">Shvartsman group</a> at
<a href="http://www.princeton.edu">Princeton University</a>.
Let me review the main ideas of our work.</p>
<p>We proposed a new model for the formation of polarized patterns observed in early embryogenesis of
<a href="https://en.wikipedia.org/wiki/Caenorhabditis_elegans"> <i>C. elegans</i></a>
where <a href="http://en.wikipedia.org/wiki/Conservation_of_mass">mass conservation</a> is the driving force.
In our model, a species \(V\) moves freely in a <a href="http://en.wikipedia.org/wiki/Cell_(biology)">cell</a>
\(\Omega\) and reversibly adheres to the <a href="http://en.wikipedia.org/wiki/Cell_membrane">cell membrane</a>
\(\partial\Omega\).
The membrane-bound species \(U\) diffuses on the surface and recruits
more of itself to the membrane.
For a sphere of radius \(r\), \(U\) and \(V\) satisfy the following equations,
<small>
$$
\left\{
\begin{array}{l}
U_t = \frac{D_U}{r^2}\Delta U + k_b\big(U_0 + H(U - \Gamma)\big)V - k_d U
\;\text{on $\partial\Omega$}, \\[1.5pt]
V_t = D_V\Delta V\;\text{in $\Omega$}, \\[1.5pt]
D_V\nabla V\cdot n = -k_b\big(U_0 + H(U - \Gamma)V\big) + k_d U,
\end{array}
\right.
$$
</small>
for some constants \(D_U\), \(D_V\), \(k_b\), \(k_d\), \(\Gamma\) and \(U_0\).
Upon nondimensionalization by \(u=k_dU/k_bV_0\), we obtain a single PDE
$$
u_\tau = \delta^2\Delta u + \big(1 - \alpha\bar{u}\big)\big(\beta+H(u - \gamma)\big) - u.
$$
Using the algorithms presented in a previous blog post,
we showed that for arbitrary initial conditions, the only non-uniform long-term
behavior was a single axisymmetric spherical cap:</p>
<center>
<img src="/blog/sphericalcaps2.jpg" class="img-responsive">
</center>
<p>We also showed that, starting from a constant initial condition, transient convection is enough to start the polarization process:</p>
<center>
<img src="/blog/sphericalcaps3.jpg" class="img-responsive">
</center>
<p>These findings reinforce the idea that symmetry breaking is crucial for the development of all living organisms.</p>
<hr>
<h4>Blog posts about spectral methods</h4>
<p>2020 <a href="2020-05-19.html">Exponential integrators for stiff PDEs</a></p>
<p>2018 <a href="2018-12-05.html">Computer-assisted proofs for PDEs</a></p>
<p>2018 <a href="2018-02-27.html">Spherical caps in cell polarization</a></p>
<p>2018 <a href="2018-01-25.html">Solving nonlocal equations on the sphere</a></p>
<p>2018 <a href="2018-01-04.html">Gibbs phenomenon and Cesàro mean</a></p>
<p>2017 <a href="2017-10-26.html">Solving PDEs on the sphere</a></p>
<p>2017 <a href="2017-10-09.html">When planets dance</a></p>
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