Chapter 13, A Statistical View of Biological Dynamics Video Solutions, Physical Biology of the Cell

Problem 1

Generate a series of plots like that shown in Figure 13.4 for all three choices of diffusion constant shown in Table 13.1 Justify these choices of diffusion coefficients by using the Stokes-Einstein relation $D=k_{\mathrm{B}} T / 6 \pi \eta a$.

Ozenc Gungor

Numerade Educator

00:59

Problem 2

The idea in this problem is to derive the solution to the one-dimensional diffusion equation for a point source, given by Equation $13.32 .$ The tools we invoke in this problem may seem heavy-handed on the first try, but illustrate a bevy of important ideas from mathematical physics.
(a) Take the Fourier transform of the diffusion equation by transforming in the spatial variables to obtain a new differential equation for $\tilde{c}(k, t).$
(b) Solve the resulting differential equation for $\tilde{c}(k, t) .$ Then compute the inverse Fourier transform to arrive at the solution in real space, $c(x, t).$
(c) Show that the solution for an arbitrary initial concentration distribution $c(x, t=0)$ can be written as an integral over the solution for a point source. In particular, consider an initial concentration profile of the form $c(x, 0)=c_{0}$ for $x<0$ and $c(x, 0)=0$ for $x>0$ and find the resulting diffusive profile.
(d) Formally derive the relation $\left\langle x^{2}\right\rangle=2 D t.$

Hast Aggarwal

Numerade Educator

02:10

Problem 3

In the chapter, we argued that diffusion is the result of nothing more than molecules flipping coins. In one dimension, this leads to a simple and beautiful "flux distribution" function. In this problem, consider two adjacent planes, one of which has $N_{1}$ molecules and the other of which has $N_{2}$ molecules, and derive the flux distribution function.

Lottie Adams

Numerade Educator

03:30

Problem 4

Figure $13.16(\mathrm{A})$ shows different snapshots of an $E .$ coli cell after it has been subjected to photobleaching. Use the solution for the FRAP problem of a one-dimensional bacterium (that is, Equation 13.47 ) to produce a plot of the difference between the initial concentration (that is, before photobleaching and the concentration at time $t$ as shown in Figure $13.16(\mathrm{B}) .$ Make a series of plots for different time points using a diffusion constant for GFP in $E .$ coli of $D=7 \mu \mathrm{m}^{2} / \mathrm{s} .$ The ambitious reader is encouraged to use a more realistic treatment of the $t=0$ concentration profile than the highly simplified uniform hole worked out in the chapter. Relevant data for this problem is provided on the book's website.

Nick Johnson

Numerade Educator

03:20

Problem 5

The goal of this problem is to generalize the one-dimensional treatment of FRAP given in the chapter. Consider a cell as a planar circle of radius $R$ uniformly covered with freely diffusing fluorescent proteins. Imagine that the laser photobleaches a hole of radius $a$ in the middle of the cell. Note that we ignore the presence of the nucleus. Work out the concentration of fluorescent proteins in the cell as a function of position and time in analogy with the one-dimensional treatment of the problem done in the chapter. Compute the number of molecules in the hole after photobleaching as a function of time.

Sana Riaz

Numerade Educator

07:38

Problem 6

(a) Consider a sphere of radius $R$ in water. Due to random collisions with the water molecules, the sphere will rotationally diffuse. The diffusion law in this case is analogous to the one obtained for translational motion,
\[\left\langle\Delta \theta^{2}\right\rangle=2 D_{\mathrm{r}} t\]
What are the units of the rotational diffusion coefficient $D_{r} ?$ Write down the formula for $D_{\mathrm{r}}$ using the Einstein relation and the rotational friction coefficient obtained in Problem $12.5,$ and convince yourself that the units are correct.
(b) Estimate how long it takes for an $E$. coli to diffuse over an angle equal to 1 radian. What is the distance traveled by the bacterium during that time?

Nicholas Sacco

Numerade Educator

01:15

Problem 7

In the chapter, we solved the problem of diffusion to capture using physical arguments to bypass explicitly solving the diffusion equation. In this problem, we do the math.
(a) Write the diffusion equation for the perfect-absorber case in spherical coordinates. Use the method of separation of variables and reproduce the solution given in the chapter.
(b) Use the flux to compute the number of molecules absorbed per unit time and find the corresponding $k_{\text {on }}$ implied by this solution. Plug in reasonable numbers to compute the diffusive speed limit for the case of oxygen binding to hemoglobin.

Khoobchandra Agrawal

Numerade Educator

02:00

Problem 8

There is strong evidence that chemoreceptors in $E .$ coli tend to cluster near one pole (see Kentner and Sourjik (2006) and Figure 13.23 ). One hypothesis about the role of such clustering is that it might increase the ability of a bacterium to better detect molecules in its environment. Determine if this is the most efficient strategy for counting (absorbing) molecules of chemoattractant. Approximate $E .$ coli as a sphere $a=1 \mu \mathrm{m}$ in radius and neglect its motion. Then compare the diffusive current to $N=1000$ receptors (absorbing patches of radius $s=1 \mathrm{nm}$ ) scattered over the surface of the cell with the diffusive current to the same receptors incorporated into a single patch with the same total area. Make use of the result that the diffusive current onto a sphere of radius $a$ with $N$ absorbing patches of radius $s$ spread uniformly over its surface is
$$I=\frac{4 \pi D a c_{\infty}}{1+\pi a / N s},$$
where $D$ is the diffusion constant of the molecules, while $c_{\infty}$ is their concentration far from the cell. (Adapted from a problem courtesy of H. C. Berg.)

Sana Riaz

Numerade Educator