Chapter 26, Cooperativity: the Helix-Coil, Ising, & Landau Models Video Solutions, Molecular Driving Forces

Problem 1

A micellization model. You have developed a model for the formation of micelles, based on expressions for the chemical potential $\mu(\mathrm{mono})$ of the free monomeric molecules in solution and the chemical potential $\mu(\mathrm{mic})$ of the aggregated state of the monomers in micelles as a function of the mole fraction $x$ of monomers in solution:
$$
\mu(\text { mono })=0.5+0.1 x^{2}
$$
and
$$
\mu(\mathrm{mic})=1.3-25 x^{2}
$$
(a) What is the state of the system at low concentration?
(b) What is the state of the system at high concentration?
(c) What is the concentration at which the micelle transition occurs?

Kratika Bhadauria

Numerade Educator

03:02

Problem 2

Stabilities of droplets. Certain molecular aggregates (like surfactant micelles and oil droplets) shrink owing to surface tension and expand owing to volumetric forces. Suppose the free energy $g(r)$ as a function of the radius $r$ of the aggregate is $g(r)=2 r^{2}-r^{3}$. Such a system has two equilibrium radii.
(a) Calculate the two radii.
(b) Identify each radius as stable, unstable, neutral, or metastable.

Ajay Singhal

Numerade Educator

00:26

Problem 3

The energy of the Ising model. Derive an expression for the energy of the Ising model from the partition function.

Mayukh Banik

Numerade Educator

07:08

Problem 4

The Landau model for the critical exponent of $C_{V}(T)$. In the Landau model, show that $C_{V} \propto T$ as $T$ increases toward the critical temperature $T_{c}$.

Eduard Sanchez

Numerade Educator

00:33

Problem 5

Nucleation droplet size. What is the number of particles $n^{*}$ in a critical nucleus for phase separation?

Salamat Ali

Numerade Educator

03:22

Problem 6

Broad energy wells imply large fluctuations. A stable state of a thermodynamic system can be described by the free energy $G(x)$ as a function of the degree of freedom $x$. Suppose $G$ obeys a square law with spring constant $k_{s}: G(x) / k T=k_{s} x^{2}$. A small spring constant $k_{s}$ implies a broad shallow well (see Figure 26.19).
(a) Show that the fluctuations $\left\langle x^{2}\right\rangle$ are larger when $k_{s}$ is small.
(b) Suppose two phases in equilibrium have narrow energy wells, each with spring constant $k_{1}$, and that the critical point of the system has a broad well with spring constant $k_{2}=(1 / 4) k_{1}$. What is the ratio of the fluctuations, $\left(x_{2}^{2}\right) /\left\langle x_{1}^{2}\right\rangle ?$
Figure 26.19 Two different square-law free energies for Problem 6: one minimum is broad and the other is narrow.

George Dekermenjian

Numerade Educator

01:54

Problem 7

Critical exponents. What are the critical exponents for the following free energy functions? $A$ is a constant and $t=\left|\left(T-T_{c}\right) / T_{c}\right|$ is the reduced temperature.
(a) $g(t)=A t^{2} e^{-t}$.
(b) $g(t)=A t^{5 / 3}$.
(c) $g(t)=A$.

Seth Gerberding

Numerade Educator

01:45

Problem 8

Zimm-Bragg helix-coil theory for $N=4$ chain units.
(a) Write the Zimm-Bragg partition function $Q_{4}$ in terms of $\sigma$ and $s$ for a four-unit chain, where HHHH is the helical state.
(b) Write an expression for $f_{H}(s)$ for this transition.

Charles Machakwa

Numerade Educator

04:32

Problem 9

The Schellman helix-coil model. A helix-coil model developed by JA Schellman $[7]$ is simpler than the ZimmBragg model, and works well for short chains. Consider a chain having $N$ units.
(a) Write an expression for $\Omega_{k}$, the number of configurations of a chain that has all its $H$ units in a single helix $k$ units long, as a function of $N$ and $k$.
(b) If $\sigma$ is the parameter for nucleating a helix and $s$ is the propagation parameter, write an expression for the partition function $Q_{N}$ over all possible helix lengths $k$.
(c) Write an expression for $p_{k}(N)$, the probability of finding a $k$-unit helix in the $N$-mer.

Kevin Zaborsky

Numerade Educator

01:24

Problem 10

Overlapping transition curves as evidence for a two-state transition. Consider a two-state transition $A \rightarrow B . m$ is an order parameter: when $m=0$, the sys tem is fully in state $A$; when $m=1$, the system is in state $B$. Suppose some agent $x$ (which could be the temperature or the concentration of ligands or salts) shifts the equilibrium as indicated in Figure $26.20$. An experiment measures a property $g$ (which might be fluorescence, light scattering, circular dichroism, heat, etc.):
$$
g(m)=g_{B} m+g_{A}(1-m)
$$
where $g_{B}$ and $g_{A}$ are the measured values for pure $B$ and A, respectively. A different type of experiment measures another property $h$ :
$$
h(m)=h_{g} m+h_{A}(1-m)
$$
where $h_{B}$ and $h_{A}$ are also the measured values for the pure states. Show that if the baselines of the two types of experiment are superimposed $\left(g_{A}=h_{A}\right)$ and the amplitudes are scaled to be the same $\left(h_{B}-h_{A}=g_{B}-g_{A}\right)$, then the curve $h(x)$ must superimpose on $g(x)$.
Figure $26.20$ A transition curve for folding, binding, or conformational change, in which the order parameter $m$ is a function of some agent $x$ that shifts the equilibrium.

Manik Pulyani

Numerade Educator

01:47

Problem 11

Calorimetric evidence for a two-state transition: the relationship between $\Delta H_{\text {cal }}$ and $\Delta H_{\text {van }}$ ?notf* $\mathrm{A}$ calorimeter measures the heat capacity $C_{p}(T)$ as a function of temperature $T$ (see Figure 26.21). The area under the full curve describes an absorption of heat
$$
q_{\mathrm{o}}=\Delta H_{\mathrm{cal}}=\int_{T_{A}}^{T_{\mathrm{a}}} C_{p} d T .
$$
Suppose the absorption of heat results from a two-state process $A \stackrel{K}{\longrightarrow} B$, where $K=[B] /[A]$.
(a) Express the fraction $f=[B] /([A]+[B])$ in terms of $K$.
(b) Express $C_{p}(T)$ as a function of the heat $q_{0}$ and the equilibrium constant $K(T)$.
Figure 26.21 The heat capacity change in a two-state transition.
(c) Assume a two-state model (Equation (13.35)):
$$
\left(\frac{\partial \ln K}{\partial T}\right)=\frac{\Delta H_{\text {van }^{\prime} \text { thaff }}}{k T^{2}},
$$
and $K=1$ at $T=T_{\max }$, the point at which $C_{p}$ is a maximum.
Derive a relationship between $\Delta H_{\text {van thoff }}, \Delta H_{\text {all }}$, and $C_{p}\left(T_{\max }\right)$.

Ajay Singhal

Numerade Educator

06:50

Problem 12

Cooperativity in a three-state system. Perhaps the simplest statistical mechanical system having cooperativity is the three-level system shown in Figure $26.22$.
Energies
$2 \varepsilon$
2
Degeneracies
$y$
1
\begin{tabular}{ll}
\hline & 1 \\
\hline
\end{tabular}
Figure 26.22 Diagram of three energy levels and degeneracies.
(a) Write an expression for the partition function $q$ as a function of energy $\varepsilon$, degeneracy $y$, and temperature $T$.
(b) Write an expression for the average energy $(\varepsilon)$ verSus $T$.
(c) For $\varepsilon / k T=1$ and $y=1$, compute the populations, or probabilities $p_{1}, p_{2}$, and $p_{3}$, of the three energy levels.
(d) Now if $\varepsilon=2 \mathrm{kcal} \mathrm{mol}^{-1}$ and $y=1000$, find the tem perature $T_{0}$ at which $p_{1}=p_{3}$ -
(e) Under the condition of (d), compute $p_{1}, p_{2}$, and $p_{3}$ at temperature $T_{0} .$ In what sense is this system cooperative?

Stanley Enemuo

Numerade Educator