# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYS7635

## 物理代写|统计物理代写Statistical Physics of Matter代考|Cell Capture

One particularly interesting problem is reaction or capture of certain molecules on specific sites in cells. There are plentiful examples in biology such as ligand (signaling molecules) binding on receptors, and metabolites (such as oxygen and sucrose molecules) uptake by cells. The aqueous solution is regarded as a bath so large that the bulk concentration of the molecules is not affected by the local binding on the cells. At what rate can these molecules be delivered to a cell if the transport is controlled by the diffusion?

As a simple model, consider a spherical cell of radius $R$ immersed in an unbounded solution at a steady state in which the concentration of the molecules

infinitely away from the cell is maintained to be a constant. $n_{\infty}$ (Fig. 13.2a). For the time being, suppose that the entire surface of the cell absorbs the molecules whenever they arrive on the surface. In a steady state of our interest, $\partial n / \partial t=0$ and the distribution of the molecules is spherically symmetric, $n=n(r)$, so the diffusion equation is reduced to
$$D \nabla^2 n=D \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r} n\right)=0$$
The equation is satisfied by $r^2 \frac{\partial}{\partial r} n=A$, which is further integrated to yield
$$n(r)=-\frac{A}{r}+B$$
The two constants $A$ and $B$ are determined by the two boundary conditions $(\mathrm{BC})$. While the $\mathrm{BC}$ at $r=\infty$ is $n=n_{\infty}$, the $\mathrm{BC}$ at the cell surface $r=R$ is $n(R)=0$, the so-called adsorbing $\mathrm{BC}$ : once the molecules bind on the cell they disappear. The solution that meets the BCs can be written as
$$n(r)=n_{\infty}\left(1-\frac{R}{r}\right)$$
As the molecules approach the absorbing boundary, their concentration decreases until they vanish on surface.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Ionic Diffusion Through Membrane

Ionic transport through cell membranes is a critical process occurring ubiquitously in cells, for our every thought, perception, and movement. It costs the energy much higher than thermal energy for an ion to cross the membranes (6.8), making the lipid-bilayer highly impermeable. A structure of membrane proteins called the ion channel provides a conduction pathway for specific ions to transverse the membranes. Although the structures of ion channel are so complex (Doyle et al. 1998), for simplicity, we regard the transport of an ion as one dimensional diffusion process of crossing the free energy barriers caused by the structures and interactions (Lee and Sung 2002).

We are interested in the ionic distribution and current at a steady state in the channel across a membrane in the presence of a potential as well as an imbalance of the ionic concentrations on both sides of the membrane. Suppose that ions undergo the Brownian motion along one-dimension ( $x$-axis) subject to a potential $U(x)$ (Fig. 13.5). Its one-dimensional density $n(x)$ at steady state is described by
$$\frac{\partial}{\partial x} J(x)=0$$

where the flux (13.20) and (13.21) is
\begin{aligned} J(x) &=-D\left[\frac{\partial}{\partial x}+\beta \frac{\partial}{\partial x} U(x)\right] n(x) \ &=-D e^{-\beta U(x)} \frac{\partial}{\partial x} e^{\beta U(x)} n(x) \end{aligned}
Equation (13.39) assures us that the $J(x)$ is uniform throughout, that is, a constant $J$. Rewriting (13.40) as $J e^{\beta U(x)}=-D \partial / \partial x e^{\beta U(x)} n(x)$, which is integrated from the outer boundary $x_o$ to an arbitrary position $x$ in the channel, we have
$$J \int_{x_o}^x d x^{\prime} e^{\beta U\left(x^{\prime}\right)}=D\left[e^{\beta U\left(x_o\right)} n\left(x_o\right)-e^{\beta U(x)} n(x)\right]$$

## 物理代写|统计物理代写物质统计物理学代考|细胞捕获

.

.

$$D \nabla^2 n=D \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r} n\right)=0$$
，方程由$r^2 \frac{\partial}{\partial r} n=A$满足，进一步积分得到
$$n(r)=-\frac{A}{r}+B$$
，两个常数$A$和$B$由两个边界条件$(\mathrm{BC})$决定。而$r=\infty$上的$\mathrm{BC}$是$n=n_{\infty}$，细胞表面的\mathrm{BC}$$r=R是n(R)=0，所谓的吸附\mathrm{BC}:一旦分子在细胞上结合，它们就消失了。满足BCs的溶液可以写成$$ n(r)=n_{\infty}\left(1-\frac{R}{r}\right) $$当分子接近吸收边界时，它们的浓度下降，直到它们在表面消失。 ## 物理代写|统计物理代写Statistical Physics of Matter代考|离子膜扩散 通过细胞膜的离子运输是细胞中普遍存在的一个关键过程，与我们的每一个思想、知觉和运动有关。离子穿过膜所消耗的能量(6.8)要比热能高得多，这使得脂双分子层高度不渗透。一种叫做离子通道的膜蛋白结构为特定离子提供了穿过膜的传导途径。虽然离子通道的结构非常复杂(Doyle et al. 1998)，但为了简单起见，我们将离子的传输看作是穿过由结构和相互作用引起的自由能垒的一维扩散过程(Lee and Sung 2002) 我们感兴趣的离子分布和电流在恒定状态下的通道，在一个存在电位和离子浓度不平衡的膜两侧。假设离子沿一维(x -轴)进行布朗运动，受制于潜在的U(x)(图13.5)。稳态时的一维密度n(x)用$$ \frac{\partial}{\partial x} J(x)=0 $$来描述 ，其中通量(13.20)和(13.21)$$ \begin{aligned} J(x) &=-D\left[\frac{\partial}{\partial x}+\beta \frac{\partial}{\partial x} U(x)\right] n(x) \ &=-D e^{-\beta U(x)} \frac{\partial}{\partial x} e^{\beta U(x)} n(x) \end{aligned} $$公式(13.39)使我们确信J(x)始终是均匀的，即，一个常数J。将(13.40)重写为J e^{\beta U(x)}=-D \partial / \partial x e^{\beta U(x)} n(x)，它从外部边界x_o集成到通道中的任意位置x，我们得到$$ J \int_{x_o}^x d x^{\prime} e^{\beta U\left(x^{\prime}\right)}=D\left[e^{\beta U\left(x_o\right)} n\left(x_o\right)-e^{\beta U(x)} n(x)\right]\$

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