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Molecular Driving Forces

K.Dill and S.Bromberg

Chapter 28

Multi-site & Cooperative Ligand Binding - all with Video Answers

Educators

Chapter Questions

00:19

Problem 1

The entropy in the grand ensemble. Write an expres sion for the entropy in the grand canonical ensemble $(T, V, \mu)$.

Zachary Warner
Zachary Warner
Numerade Educator
10:20

Problem 2

Three-site binding. A ligand $X$ can bind to a macromolecule $P$ at three different binding sites with binding constants $K_{1}, K_{2}$, and $K_{3}$ :
$$
\begin{aligned}
&X+P \stackrel{K_{1}}{\longrightarrow} P X, \quad X+P X \stackrel{K_{2}}{\longrightarrow} P X_{2}, \\
&\text { and } X+P X_{2} \stackrel{K_{3}}{\longrightarrow} P X_{3} .
\end{aligned}
$$
(a) Write the binding polynomial, $Q$.
(b) Write an expression for the number of ligands $v$ bound per $P$ molecule.
(c) Compute $v$ for $x=[X]=0.05$, assuming $K_{1}=1$, $K_{2}=1$, and $K_{3}=1000$.
(d) Assume the same $K$ values as in (c). Below ligand concentration $x=x_{0}$, most of the macromolecular $P$ molecules have zero ligands bound. Above $x=x_{0}$, most of the $P$ molecules have three ligands bound. Compute $x_{0}$.
(e) For $x=x_{0}$ in (d), show the relative populations of the ligation states with zero, one, two, and three ligands bound.

Cadan Holden
Cadan Holden
Numerade Educator
14:11

Problem 3

Drug binding to a protein. A drug $D$ binds to a protein $P$ with two different equilibrium binding constants, $K_{1}$ and $K_{2}$ :
$$
D+P \stackrel{K_{1}}{\longrightarrow} P D_{1}, \quad \text { and } \quad D+P D_{1} \stackrel{K_{2}}{\longrightarrow} P D_{2},
$$
where $K_{1}=1000 \mathrm{M}^{-1}$, and $K_{2}=2000 \mathrm{M}^{-1}$ at $T=300 \mathrm{~K}$.
(a) Write an algebraic expression for the fraction of protein molecules that have two drug molecules bound as a function of drug concentration $x=[D]$.
(b) If the drug concentration is $x=10^{-3} \mathrm{M}$, what is the fraction of protein molecules that have one drug molecule bound?
(c) If the drug concentration is $x=10^{-3} \mathrm{M}$, what is the fraction of protein molecules that have zero drug molecules bound?
(d) If the drug concentration is $x=10^{-3} \mathrm{M}$, what is the fraction of protein molecules that have two drug molecules bound?

Md Mohibullah
Md Mohibullah
Auburn University Main Campus
01:50

Problem 4

Polymerization equilibrium. Consider the process of polymerization and assume the principle of equal reactivity, that each monomer adds with the same equilibrium constant as the previous one:
$$
\begin{gathered}
2 X_{1} \stackrel{K}{\longrightarrow} X_{2}, \\
3 X_{1} \stackrel{K^{2}}{\longrightarrow} X_{3}, \\
\vdots \\
n X_{1} \stackrel{K^{m-1}}{\longrightarrow} X_{n} .
\end{gathered}
$$
(a) Write the binding polynomial $Q$.
(b) Write the average chain length $\langle n\rangle$ in terms of $Q$.
(c) Plot $\langle n\rangle$ versus $x$ for $K=1$ and for $K x<1$.

Jorge Villanueva
Jorge Villanueva
Numerade Educator
00:46

Problem 5

Scatchard plot for an antibody. The plot in Figure $28.18$ shows the result of binding dinitrophenol (DNP) lig. and to an antibody (Ab). Compute the binding constant $K$ and the number of binding sites per Ab molecule $n$.
Figure $28.18 \quad v$ is the number of moles of DNP bound per mole of Ab and $x$ is the bulk concentration of DNP in solution in moles per liter. Source: J Damell, H Lodish, and D Baltimore, Molecular Cell Biology, 2nd edition, WH Freeman, San Francisco, 1990 .

Alexander Burbelo
Alexander Burbelo
Numerade Educator
04:08

Problem 6

Relating stoichiometric to site constants. Derive the relationship between stoichiometric-model binding constants $K_{1}$ and $K_{2}$ and site-model binding constants $K_{a}$ and $K_{b}$. Suppose the sites are independent, $K_{c}=K_{a} K_{b}$.

Eileen Sullivan
Eileen Sullivan
Numerade Educator
12:31

Problem 7

Scatchard plots don't work for multiple types of binding site. If a binding process involves $n_{1}$ sites with affinity $K_{1}$, a Scatchard plot of $v / x$ versus $v$ gives a straight line with $K_{1}$ as slope and $n_{1}$ as horizontal-axis intercept (see Figure 28.6). Show that if you have two types of sites having (number, affinity) $=\left(n_{1}, K_{1}\right)$ and $\left(n_{2}, K_{2}\right)$, a Scatchard plot doesn't give two straight lines from which you can get $\left(n_{2}, K_{2}\right)$. To fit multiple types of sites, you need a different model.

Bhumika Jayee
Bhumika Jayee
Numerade Educator
01:37

Problem 8

A model for Alzheimer's fibrillization. Consider a solution of protein molecules in three possible states of aggregation: in state $A_{1}$, the protein is a monomer in solution; in state $A_{10}$, the protein is hydrophobically clustered into oligomers having 10 molecules each; and in state $A_{100}, 100$ protein molecules are all aggregated into long fibrils. The equilibrium is given by
$$
\begin{gathered}
10 A_{1} \stackrel{K_{\text {olim }}}{\longrightarrow} A_{10} \\
100 A_{1} \stackrel{K_{\text {mpel }}}{\longrightarrow} A_{100}
\end{gathered}
$$
(a) Express the binding polynomial $Q$ for this association equilibrium in terms of $K_{\text {allgo, }} K_{\text {tharil }}$, and $x$, where $x$ is the concentration of protein $A$. (b) Write an expression for the average number $v$ of protein molecules per object (monomer, oligomer, or fibril).

Yifan Zhou
Yifan Zhou
Numerade Educator
11:57

Problem 9

Alzheimer's again: chain-length dependence. In a variant of the problem above, suppose the oligomer and fibril can be formed from any number of monomers, $M$. Suppose the oligomer is driven by hydrophobic clustering:
$$
K_{\text {olliga }}=K_{1}^{\mathrm{M}}
$$
where $K_{1}$ represents the equilibrium constant for adding a monomer to an oligomer.

Suppose the fibril is held together by hydrogen bonds: $K_{\text {thbrl }}=K_{2}^{M}$, where $K_{2}$ is the equilibrium constant for adding a monomer to a fibril.
(a) Now, express $v$ as a function of $x, K_{1}, K_{2}$, and $M$.
(b) If $x=1, K_{1}=2, K_{2}=1.1$, and $M=40$, what is the value of $v$ ?
(c) If $x=1, K_{1}=2, K_{2}=0.9$, and $M=40$, what is the value of $v$ ?

Shubham Kumar
Shubham Kumar
Numerade Educator
05:15

Problem 10

Cooperative assembly of membrane proteins. You have constructed an alpha-helical peptide. It partitions into lipid bilayer membranes and assembles into oligomeric complexes within the membrane. You measure the concentration $x$ of the peptide in the membrane and the fraction $\theta$ of helical peptide molecules that form oligomeric complexes in the membrane. At temperature $T=300 \mathrm{~K}$, the peptide concentrations $x$ and the ratios of oligomeric complexes $\theta /(1-\theta)$ are as given in Table 28.1.
Table 28.1
\begin{tabular}{lr}
\hline$x(\mu \mathrm{M})$ & $\theta /(1-\theta)$ \\
\hline 1 & 10 \\
2 & 160 \\
3 & 810 \\
4 & 2560 \\
5 & 6250 \\
\hline
\end{tabular}
(a) What are the association constant $K$ and the Hill coefficient $n$ ?
(b) Make a Hill plot.
(c) Plot $\theta$ versus $x$.
(d) At temperature $T=330 \mathrm{~K}$, you find that all the values of $\theta /(1-\theta)$ in the table are doubled. Compute the $\Delta H$ and $\Delta S$ of binding.

Joanna Quigley
Joanna Quigley
Numerade Educator
08:56

Problem 11

Misusing the Scatchard plot. Consider a binding problem in which you have a ligand $x$ and a protein $P$ that have $N_{1}$ independent sites with binding affinity $K_{1}$, and $N_{2}$ additional independent sites with binding affinity $K_{2}$.
(a) Write the binding polynomial for this system. (b) Write an expression for $v / x$, where $x$ is the liganc concentration in solution and $v$ is the average num ber of ligands bound per $P$ molecule.
(c) Now, consider the Scatchard plot shown in Figur $28.19$, which has been used in an attempt to finc the parameters $N_{1}, K_{1}, N_{2}$, and $K_{2}$ from the mode above:
$$
v / x=a_{0}-b_{0} v+a_{1}-b_{1} v
$$
Show that the parameters $a_{0}, b_{0}, a_{1}$, and $b_{1}$ that you obtain from this curve fitting are not able to give yot the quantities $N_{1}, K_{1}, N_{2}$, and $K_{2}$.

Cadan Holden
Cadan Holden
Numerade Educator