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Physical Biology of the Cell

Rob Phillips, Jane Kondev, Julie Theriot

Chapter 14

Life in Crowded and Disordered Environments - all with Video Answers

Educators

Chapter Questions

03:10

Problem 1

In the chapter, we argued that the mean spacing between molecules in an in vitro biochemical experiment is roughly $100 \mathrm{nm}$ at $\mu \mathrm{M}$ concentrations while in the cell the spacings are a factor of 10 smaller. Justify these statements with simple estimates. The biochemical "standard state" is often taken as $1 \mathrm{M}$. Work out the mean spacing between molecules at this concentration.

Lottie Adams
Lottie Adams
Numerade Educator
03:27

Problem 2

The effect of crowding on the chemical potential of a molecular species in solution can be captured by the equation $\mu=\mu_{0}+k_{\mathrm{B}} T \ln \left(c \gamma / c_{0}\right),$ where the subscripts zero are for a reference state. The effective concentration is given by $c \gamma,$ where $c$ is the actual concentration that is present in the solution and $\gamma$ is called the activity coefficient. The simple lattice model of proteins in solution used repeatedly in the chapter implies a corresponding model for the activity coefficient. Work out this activity coefficient and compare your formula with the experimental results shown in Figure 14.25

Sana Riaz
Sana Riaz
Numerade Educator
06:40

Problem 3

Use the approximate formula for the pressure of a gas of hard spheres, Equation 14.11 , to extract an effective hard-sphere radius for hemoglobin from the data given in Figure $14.11 .$ How does this effective radius compare with the dimensions of the molecule obtained by X-ray scattering?
Relevant data for this problem are provided on the book's website.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:29

Problem 4

Compute the depletion force between a sphere of radius $R$ and a planar surface by carrying out the calculation indicated schematically in Figure $14.13 .$ The radius of the small spheres is $r$

Rabeya Zahid
Rabeya Zahid
Numerade Educator
03:20

Problem 5

In the chapter, we worked out a general statement of the free energy for two large objects in solution and in the presence of small depletant molecules, which was based upon the osmotic pressure of the molecules and the volume excluded by the objects when at distance $D$. Repeat that derivation leading up to the formula
$G(D)=\Pi_{\mathrm{o}} V_{\mathrm{ex}}(D)$
where $\Pi_{\mathrm{o}}=N k_{\mathrm{B}} T / V$ is the osmotic pressure of the depletant molecules. Estimate the force between two beads whose radius is $1 \mu \mathrm{m}$ in a concentrated protein solution, when the concentration is equal to the value characteristic of an $E .$ coli cell. Use $r=3$ nm for the typical radius of a protein.

Sana Riaz
Sana Riaz
Numerade Educator
03:29

Problem 6

Repeat the Flory calculation from the chapter (Section 14.2.4) for DNA confined to a two-dimensional surface.
(a) Find the scaling of the size of the polymer as a function of its length, incorporating self-avoidance.
(b) Estimate the DNA length for which self-avoidance becomes important. How does this compare with the length of genomic DNA from a $\lambda$ -phage?

Sana Riaz
Sana Riaz
Numerade Educator
00:30

Problem 7

In this problem, we extend the one-dimensional model of diffusion in the presence of crowding molecules to account for the difference in size between a tracer particle (considered to be present at low concentration) and the crowders. This situation is relevant for the data shown in Figure $14.24(\mathrm{A}) .$ The tracer particles are assumed to be undergoing random walk motion on the larger tracer lattice with lattice constant $b,$ while the crowders are hopping between adjacent sites of the smaller lattice, with lattice constant $a$ (see Figure 14.26 ). The two lattices are introduced to account for the difference in size between the two molecular species.
(a) Calculate the diffusion coefficient by considering the possible trajectories of the tracer particles and their probabilities. Note that the tracer can hop to an adjacent site of the tracer lattice only if there are no crowders present. Express your answer in terms of the diffusion coefficient $D_{0}$ of the tracer particles in the absence of crowders, the volume fraction of the crowders $\phi,$ and the ratio of the tracer and crowder sizes $r=b / a$
(b) Plot $\ln \left(D / D_{0}\right)$ as a function of the volume fraction for different values of $r .$ How well does this model explain the data shown in Figure $14.24(\mathrm{A}) ?$ To make this comparison, you will need to estimate the sizes of the molecules used in the experiment from their molecular masses and a typical protein density that is 1.3 times that of water. The data are provided on the book's website.

Sana Riaz
Sana Riaz
Numerade Educator