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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

這是本站第67篇文章，也是本站第一篇雙語的文章。在上一集我們回顧了大腸桿菌的梯度感應並寫了一個簡單的程式來模擬他的行為。今天我們要來到黏菌的梯度感應，也就是推薦文章作者所做的研究。這集會回顧理論的部分，下集中我們會寫另一個程式來模擬黏菌的梯度感應。

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The chemotaxis of slime mold is fundamentally different from that of

黏菌的梯度感應和大腸桿菌的梯度感應在本質上非常不同。大腸桿菌需要在改變方向之前移動足夠長的距離，才能確保他偵測到的粒子數差異來自於背景梯度而不是粒子數的隨機波動。但由於黏菌單一的細胞體足夠大，在他細胞膜的不同地方就足以感應到粒子數的差異，因此不需要移動就可以偵測到梯度的方向。在Endres & Wingreen的原始文章中，他們把重點放在比較perfect absorbing sphere和perfectly monitoring sphere在梯度感應的效率差異上。但在本站文章中，我們只會討論perfect absorbing sphere的情形，後者留給讀者自行閱讀原文。

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."

我們來看看 perfectly absorbing sphere是如何運作的。在圖一中，有一個很明顯的背景濃度梯度存在，如果養分分子撞擊到細胞膜之上，他就會被立刻吸收，而且他撞擊的位置會被細胞紀錄起來。由於濃度的差異，很顯然的撞擊到細胞左側的粒子數應該比撞擊到右側的粒子數來的多。我們姑且把粒子們撞擊到細胞膜上的情形稱為「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.

現在重點來了。如果我們已經知道背景濃度梯度長甚麼樣子，我們可以很容易的計算出特定 「hitting pattern」發生的機率有多大。因此如果我們利用最大似然估計理論(maximum likelihood)的話，我們就可以從觀測到的「hitting pattern」去回推到底怎樣的背景濃度梯度最可能產生出這樣的 「hitting pattern」。如果你聽不懂的話，我們來看看下面的例子。

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

簡單來說呢，如果我們的細胞記錄到撞擊到細胞表面的粒子位置長得像這樣：

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

很顯然的，我們會偷偷期待說背景濃度梯度應該要長得像這樣子：

rather than looks like these two:

而不是長得像這兩張圖這樣：

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

你心裡會強烈感受到後面兩張圖非常詭異，這就是最大似然估計理論在你的腦袋裡面尖叫說──它們長得真是怪透了，因為發生機率太低。

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.

所以我想差不多可以來引進這些邏輯背後的數學囉。我們至少要花兩集以上來講解才能控制每篇閱讀時間在三分鐘以內。(謎：有哪一次真的只要三分鐘閱讀了QwQ")

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.

我們先回顧一下Fick的擴散定律，中文版讀者已經看過不只一次囉，請回顧：(酵素的噴射引擎_中一：https://goo.gl/Rxa00Q 、酵素的噴射引擎_中二：https://goo.gl/NWfUQt 、酵素的噴射引擎_中三：https://goo.gl/Uo3oxu)。

We have to figure out the particle flow into a perfectly absorbing sphere

在我們能討論有背景梯度的情形之前，我們需要先解出沒有背景梯度的情形下，perfectly absorbing sphere感應到粒子流動的情形是如何。由於是一個perfectly absorbing sphere，在細胞膜的位置濃度是零。因此在穩定狀態假設

Given the laplacian in a spherical coordinates:

已知在球座標中的laplacian長這樣：

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

由於題目中有球狀對稱，laplacian後面兩個項可以丟掉，原本的擴散方程就變成一個只需要基本的微積分就能解的東西。答案長得像這樣子：

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

這並沒有甚麼難的，但是有一件很重要的事情是，我們必須注意到上面的擴散方程和靜電學中的方程式非常非常相似：

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

我們在下集中會利用這個相似性來推導出在有背景梯度的情形下，流過細胞膜表面的粒子流有多大。我們下次見！

/*---------Divider---------*/

Suggested reading and reference:

推薦閱讀與參考資料：

Endres, R. G. & Wingreen, N. S. (2008). Accuracy of direct gradient sensing by single cells.

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!這是本站第67篇文章，也是本站第一篇雙語的文章。在上一集我們回顧了大腸桿菌的梯度感應並寫了一個簡單的程式來模擬他的行為。今天我們要來到黏菌的梯度感應，也就是推薦文章作者所做的研究。這集會回顧理論的部分，下集中我們會寫另一個程式來模擬黏菌的梯度感應。

/*---------Divider---------*/

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.黏菌的梯度感應和大腸桿菌的梯度感應在本質上非常不同。大腸桿菌需要在改變方向之前移動足夠長的距離，才能確保他偵測到的粒子數差異來自於背景梯度而不是粒子數的隨機波動。但由於黏菌單一的細胞體足夠大，在他細胞膜的不同地方就足以感應到粒子數的差異，因此不需要移動就可以偵測到梯度的方向。在Endres & Wingreen的原始文章中，他們把重點放在比較perfect absorbing sphere和perfectly monitoring sphere在梯度感應的效率差異上。但在本站文章中，我們只會討論perfect absorbing sphere的情形，後者留給讀者自行閱讀原文。

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."

我們來看看 perfectly absorbing sphere是如何運作的。在圖一中，有一個很明顯的背景濃度梯度存在，如果養分分子撞擊到細胞膜之上，他就會被立刻吸收，而且他撞擊的位置會被細胞紀錄起來。由於濃度的差異，很顯然的撞擊到細胞左側的粒子數應該比撞擊到右側的粒子數來的多。我們姑且把粒子們撞擊到細胞膜上的情形稱為「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.

現在重點來了。如果我們已經知道背景濃度梯度長甚麼樣子，我們可以很容易的計算出特定 「hitting pattern」發生的機率有多大。因此如果我們利用最大似然估計理論(maximum likelihood)的話，我們就可以從觀測到的「hitting pattern」去回推到底怎樣的背景濃度梯度最可能產生出這樣的 「hitting pattern」。如果你聽不懂的話，我們來看看下面的例子。

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

簡單來說呢，如果我們的細胞記錄到撞擊到細胞表面的粒子位置長得像這樣：

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

很顯然的，我們會偷偷期待說背景濃度梯度應該要長得像這樣子：

rather than looks like these two:

而不是長得像這兩張圖這樣：

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

你心裡會強烈感受到後面兩張圖非常詭異，這就是最大似然估計理論在你的腦袋裡面尖叫說──它們長得真是怪透了，因為發生機率太低。

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.

所以我想差不多可以來引進這些邏輯背後的數學囉。我們至少要花兩集以上來講解才能控制每篇閱讀時間在三分鐘以內。(謎：有哪一次真的只要三分鐘閱讀了QwQ")

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.

我們先回顧一下Fick的擴散定律，中文版讀者已經看過不只一次囉，請回顧：(酵素的噴射引擎_中一：https://goo.gl/Rxa00Q 、酵素的噴射引擎_中二：https://goo.gl/NWfUQt 、酵素的噴射引擎_中三：https://goo.gl/Uo3oxu)。

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:在我們能討論有背景梯度的情形之前，我們需要先解出沒有背景梯度的情形下，perfectly absorbing sphere感應到粒子流動的情形是如何。由於是一個perfectly absorbing sphere，在細胞膜的位置濃度是零。因此在穩定狀態假設

**(**∂c/∂t=0**)**中，擴散問題可以被化簡成像下面這樣的boundary value problem：Given the laplacian in a spherical coordinates:

已知在球座標中的laplacian長這樣：

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

由於題目中有球狀對稱，laplacian後面兩個項可以丟掉，原本的擴散方程就變成一個只需要基本的微積分就能解的東西。答案長得像這樣子：

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

而流過細胞膜的粒子流總量就是

而流過細胞膜的粒子流總量就是

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

這並沒有甚麼難的，但是有一件很重要的事情是，我們必須注意到上面的擴散方程和靜電學中的方程式非常非常相似：

我們在下集中會利用這個相似性來推導出在有背景梯度的情形下，流過細胞膜表面的粒子流有多大。我們下次見！

/*---------Divider---------*/

Suggested reading and reference:

推薦閱讀與參考資料：

Endres, R. G. & Wingreen, N. S. (2008). Accuracy of direct gradient sensing by single cells.

*PNAS***105**(41): 15749-15754.
chemotaxis
computer simulation
diffusion
Fick's law
gradient sensing
laplacian
partial differential equation
slime mold

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