### #77 Brownian Carnot engine-I 微觀卡諾引擎-I

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This is our 67th post in 3min Biophysics. However, this is our first post in English!! (Hurray!!) In our previous post, we have reviewed how E. coli accomplishes chemotaxis and we have written a simple simulation codes about it. Today we are going to discuss the chemotaxis of Dictyostelium discoideum, or slime mold. In these 2 episode, we are going to discuss the theoretical limits of the accuracy of gradient sensing by slime mold. While in the following episode, we are going to write another simple simulation code for it. Stay tuned!

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The chemotaxis of slime mold is fundamentally different from that of E. coli, which largely relies on temporal sensing. Bacteria have to "walk" far enough before its next tumbling episode to make sure the sensed difference in nutrients or repellents is caused by the gradient rather than the background fluctuations. A single cell of slime mold is large enough to have different nutrient concentration across its different part of cell membranes. In the original articles by Endres & Wingreen, they compare the limits of concentration sensing by a perfect absorbing sphere and a perfectly monitoring sphere. However, we will limit our discussion in the case of perfect absorbing sphere and we refer our readers to the original article for the perfectly monitoring sphere.

Figure 1. Model of gradient sensing by a perfectly absorbing sphere.

Let's see how the perfectly absorbing sphere works. In Figure 1, there is obviously a background concentration gradient along the x-axis. If the nutrient particles hit the cell membrane, it is absorbed and its location is recorded by the cell (possibly with the aids of receptors, transporters, and cytoskeleton). Due to the concentration gradient, most particles hit the cell from the left side rather than the right side. Let's simply call how particles hit the cell "the hitting pattern."

Here is the main tricks: because it is possible to calculate the probability of a hitting pattern given a known background concentration gradient, we could use the formalism of maximum likelihood to derive an estimated concentration gradient given an observed hitting pattern.

That is to say, given a hitting pattern looks like this:

It is far more possible that the background concentration gradient looks like this:

rather than looks like these two:

They look weird enough, right? That is what your basic instincts about maximum likelihood tried to tell you.

So I guess it is appropriate now to introduce the mathematical parts of all these concepts. It would take 2 episodes to finish it to keep our articles short enough to be finished in 3 min.

Let's first review the Fick's law of diffusion, where J is the particle flow, D is the diffusion coefficient, and c represents concentration.

We have to figure out the particle flow into a perfectly absorbing sphere without a background gradient before we discuss the case with background gradient. For a perfectly absorbing sphere, the concentration of particles along the cell border is 0. Therefore in steady state(∂c/∂t=0), it could be simplified into a boundary value problem looks like this:

Given the laplacian in a spherical coordinates:

The last 2 terms in laplacian could be neglected due to the spherical symmetry of our problem. Fundamental calculus tells us that the solution reads

and the particle flow across the surface of a perfectly absorbing sphere is

It is important to recognize the similarity between diffusion equations and electrostatics.That is

We will rely on this similarity to derive the particle flow across the surface in a gradient field. Stay tuned!

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