Physical Biology of the Cell

Rob Phillips, Jane Kondev, Julie Theriot

Chapter 6

Entropy Rules! - all with Video Answers

Educators

Chapter Questions

Problem 1

Disulfide bond formation
A protein, shown schematically in Figure $6.30,$ may contain several cysteines (represented as light colored balis), which may pair together to form disulfide bonds. For six cysteines, three disulfide bonds can form; the figure shows the pairing arrangement $1-6,2-5,3-4 .$ How many different disulfide pairing arrangements are possible? Derive the general formula for the number of different pairing arrangements when there are $n$ cysteines (and $n$ is even). (Adapted from problem 1.11 of $\mathrm{K}$. Dill and S. Bromberg, Molecular Driving Forces, 2 nd ed. Garland Science, 2011 .)

Sana Riaz

Sana Riaz

Numerade Educator

Problem 2

Statistical mechanics of an optical trap.
In the Computational Exploration in Chapter $5(p .207)$ we described the use of laser light to trap micron-sized beads. The dynamics of such beads can be
(a) Obtain the probability of binding as a function of the total concentration of ligand and receptor.
(b) Examine the limit where $[\mathrm{L}]_{\text {tot }} \gg[\mathrm{R}]_{\text {tot }}$. Notice that this is precisely the limit used in Section 6.1 .1 to calculate the probability of binding using a lattice model. Comment on the relation between the statistical mechanics and thermodynamic identifications made in Section 6.4 .1 and in Problem 6.3
(c) Maeda et al. (2000) measured binding curves of different $\sigma$ subunits to RNA polymerase core enzyme. Fit the data from their Figure 1 (A) (available on the book's website) to the model you derived in (a), assuming a total concentration of receptor $\left(\sigma^{70}\right)$ of $0.4 \mathrm{nM}$. Compare that with the results of fitting to the expression derived in (b).

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Numerade Educator

Problem 3

Polymerase binding to the promoter revisited
The probability of promoter occupancy can be computed using both statistical mechanics and thermodynamics (that is, using equilibrium constants). These two perspectives were already exploited for simple ligand-receptor binding in Sections 6.1 .1 and 6.4 .1
(a) Write an expression for the probability of finding RNA polymerase bound to the promoter as a function of the equilibrium constants for specific and nonspecific binding.
(b) In vitro, the dissociation constant of RNA polymerase binding to nonspecific DNA is approximately $10 \mu \mathrm{M}$ and the dissociation constants of RNA polymerase to the lac $P 1$ and T7A1 promoters are $550 \mathrm{nM}$ and $3 \mathrm{nM}$, respectively. Use these constants and the results from (a) to estimate the in vivo binding energies of RNA polymerase to $\operatorname{lac} P 1$ and $\mathrm{T} 7 \mathrm{Al}$
promoters.

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Problem 4

Free versus bound ligand
The expression found in Equation 6.111 gave the probability of binding of a ligand to a receptor as a function of the free concentration of ligand, [L]. However, quantities easier to tune experimentally are the total concentrations of ligand, [L]tot, and receptor, [R] tot.
(a) Obtain the probability of binding as a function of the total concentration of ligand and receptor.
(b) Examine the limit where [L] tot $\gg[\mathrm{R}]_{\mathrm{tot}} .$ Notice that this is precisely the limit used in Section 6.1 .1 to calculate the probability of binding using a lattice model. Comment on the relation between the statistical mechanics and thermodynamic identifications made in Section 6.4 .1 and in Problem 6.3
(c) Maeda et al. (2000) measured binding curves of different $\sigma$ subunits to RNA polymerase core enzyme. Fit the data from their Figure $1(\mathrm{A})$ (available on the book's website) to the model you derived in (a), assuming a total concentration of receptor $\left(\sigma^{70}\right)$ of $0.4 \mathrm{nM}$. Compare that with the results of fitting to the expression derived in (b).

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Numerade Educator

Problem 5

Distinguishable ligands Derive the probability that a receptor is occupied by a ligand using a model that treats the $L$ ligands in solution as distinguishable particles. Show that the expression is the same as obtained in the text (Equation 6.19 ), where the ligands were treated as indistinguishable.

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Problem 6

Derivation of the Boltzmann distribution
Derive the Boltzmann distribution using the counting argument from Section $6.1 .4 .$ Modify the argument given there such that every particle has at least an energy of $\varepsilon$

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Problem 7

Binding polynomials and polymerization
Many cellular processes involve polymerization, where a bunch of monomers bind together to form a polymer. Examples include transcription, translation, construction of the cytoskeleton, etc. Here we consider a simple model of polymerization where each monomer is added to the growing chain with the same equilibrium binding constant $K$ This situation is described by the following set of chemical equations:
The symbol $X_{n}$ denotes a polymer $n$ monomers in size. In equilibrium, there will be polymers of all different sizes. We use this model to compute the average polymer size, and how it depends on the concentration of monomers.
(a) Find an expression for the probability that a polymer is $n$ monomers in length in terms of $K$ and $x=\left[X_{1}\right],$ the concentration of free monomers. Use this result to get an expression for the average polymer size $(n)$
(b) Show that the average polymer size can be written as
$$(n)=\frac{d \ln Q}{d K x}$$
where $Q=1+K x+(K x)^{2}+(K x)^{3}+\dots$ is the binding polynomial.
(c) Plot $(n)$ as a function of $K x$. Show that $(n)$ diverges as $K x$ approaches 1 from below, What is the physical interpretation of this divergence? (In reality, there is no such thing as a polymer that is infinite in size.)

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Problem 8

6.8 Lattice model of the chemical potential
Intuitively, the chemical potential really tells us the free energy cost associated with changing the number of solute molecules in solution by 1 as
$$\mu_{\text {solute }}=G_{\text {tot }}\left(N_{\mathrm{s}}+1\right)-G_{\text {tot }}\left(N_{\mathrm{s}}\right)$$
Using the lattice model of a solution, compute the free energy and obtain an expression for the chemical potential.

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Problem 9

6.9 Osmotic pressure of a cell
In Section $6.2 .3,$ we derived the van't Hoff formula for the osmotic pressure and performed an estimate of the osmotic pressure experienced by a bacterium as a result of its impermeability to inorganic ions. Examine the contribution to the osmotic pressure of a bacterium coming from the presence of proteins within the cell.
How does this compare with the contribution to the osmotic pressure from inorganic ions described in the section? See Table 2.1 and Figure 2.4 for the relevant data.

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Problem 10

The die problem revisited
Work out the probability distribution for a dishonest die that has average values $(i)=2.5,(i)=3.5,$ and $(i)=4.5$ Report the values of the Lagrange multipliers in each case and make a plot (a bar plot) of the probability distribution for all three of these cases.

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Problem 11

The missing information revisited
For the single-receptor binding problem, show that the missing information for ligand concentrations $f K_{\mathrm{d}}$ and $K_{\mathrm{d}} / f$ are the same. This explains the symmetry of the curve shown in Figure 6.26

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Problem 12

6.12 Simple estimate of first passage times
The rate of a molecule passing over a free-energy barrier of height $B$ is proportional to the Boltzmann factor $e^{-B / k_{B} T}$ with $T$ the (absolute) temperature and $k_{\mathrm{B}}$ the Boltzmann constant. The exponential dependence on $B$ can be understood heuristically as follows: Thermal fluctuations typically give the molecule an energy of about $k_{\mathrm{B}} T .$ To go up in energy by another factor of $k_{\mathrm{B}} T$ is somewhat more unlikely than to go down. Consider the barrier of height $B$ as being a staircase of energy steps each of height $k_{\mathrm{B}} T .$ Next suppose that the probability that the next step of the particle is up is $q$ and the probability that the next step is down is given by $1-q$ with $q<1 / 2$. Make a crude estimate of the probability $p$ of getting to the top of the barrier in "one try." You will need to say what you mean by "one try." Compare your answer with the Boltzmann factor for $B \gg k_{\mathrm{B}} T .$ (Problem courtesy of Daniel Fisher.)

Sana Riaz

Numerade Educator