Chapter 29, Bio & Nano Machines Video Solutions, Molecular Driving Forces

Problem 1

A molecular machine. You have a complex binding process, a 'molecular machine,' in which three ligands $X$, $Y$, and $Z$ bind to a molecule $P$. The free concentrations of the ligands in the solution are $x, y$, and $z$, respectively. Molecule $X$ can bind with either 'foot' on $P$, with binding constant $K_{1}$, or with both feet on $P$, with binding constant $K_{1}^{2}$. Molecule $Y$ can bind only if $X$ is bound with both feet on $P$. Molecule $Z$ can bind to $P$ only if $X$ is doubly bound and $Y$ is bound. The binding equilibria are shown in Figure $29.16$. (a) Write the binding polynomial for this molecular machine.
(b) What is the fraction of $P$ molecules that have $Y$ and $Z$ and both feet of $X$ bound?
(c) What species dominate(s) at small $x$ ?

Khoobchandra Agrawal

Numerade Educator

01:44

Problem 2

Saturation of myoglobin. Suppose that $\mathrm{O}_{2}$ molecules bind to myoglobin with association constant $K=2 \mathrm{Torr}^{-1}$ at $25^{\circ} \mathrm{C}$ and $\mathrm{pH}$ 7.4.
(a) Show a table of the fractional saturation of myoglobin for pressures of $1,2,4,8$, and 16 Torr $\mathrm{O}_{2}$.
(b) Does the fractional saturation double for each doubling of the pressure?

Narayan Hari

Numerade Educator

01:47

Problem 3

Oxygen shifts the dimer/tetramer equilibrium in hemoglobin. Hemoglobin dissociates in two steps: tetramer $\rightarrow$ dimers $\rightarrow$ monomers. Oxygen binding alters the equilibrium between hemoglobin dimers and tetramers. The free energies shown in Figure $29.17$ have been measured.
(a) What is $\Delta G_{4}$ ?
(b) Does oxygen binding shift the equilibrium toward the dimers or toward the tetramer?

Ashley Brady

Numerade Educator

19:08

Problem 4

Helicases unwind DNA. Suppose DNA has two states: wound (helical) and unwound. In chromosomal DNA, the wound state is more stable than the unwound form of DNA by $-4.1 \mathrm{kcal} \mathrm{mol}^{-1}$ (per unit length).
Figure 29.17 Thermodynamic cycle for $\mathrm{O}_{2}$ binding and tetramer formation of hemoglobin. Source: FC Mills and GK Ackers, J Biol Chem 259, 2881-2887 (1979); GK Ackers, Biophys J 32, 331-346 (1980).
(a) Compute the equilibrium constant $K$ at $T=300 \mathrm{~K}$ for unwound DNA $\stackrel{K}{\longrightarrow}$ wOund DNA.
(b) Helicase is a protein that binds with equilibrium binding constant $B_{h}=10$ to wound DNA (see Figure 29.18). Using the thermodynamic cycle in Figure $29.18$, compute the binding constant $B_{u}$ for binding helicase to unwound DNA that would be required to unwind the DNA (half the DNA is unwound and half is helical when helicase is unbound and $K_{0}=1$ ).
$\begin{array}{rll}\text { Unwound } & \text { Wound } \\ \text { DNA } & \text { DNA } \\ \text { Helicase } & \text { Binds } \\ \text { Helicase } & \end{array}$
Figure 29.18 Thermodynamic cycle for binding helicase to DNA. $B_{u}$ and $B_{h}$ are the equilibrium binding constants for binding helicase protein to unwound and wound DNA, respectively. $K=[$ helical DNA $] /[$ unwound DNA $]$ is the equilibrium constant for helix formation in DNA in the absence of helicase and $K_{0}$ is that equilibrium constant in the presence of ligand.
(c) Write an expression for the fraction of DNA molecules $f_{u}$ that are unwound (including both $u$ and $u x$ ) as a function of the concentration $x$ of helicase in solution and of $K, B_{h}$, and $B_{u}$.
(d) For the values of $K$ and $B_{u}$ that you found, and for $B_{h}=0.1$, find the value of $x$ that gives $f_{u} \approx 1 / 2$.

Eric Goldman

Numerade Educator

03:44

Problem 5

5. Oxygen binding to hemoglobin and myoglobin. Figure $29.19$ compares the single-site binding of oxygen to myoglobin with the four-site cooperative binding to hemoglobin. What biological advantage of hemoglobin is evident from these curves?
Fraction of $\mathrm{O}_{2}$ Binding Sites Filled
Figure 29.19 The binding of $\mathrm{O}_{2}$ to hemoglobin and myoglobin. Source: J Darnell, H Lodish, and D Baltimore, Molecular Cell Biology, 2nd edition, WH Freeman, San Francisco, $1990 .$

Eric Goldman

Numerade Educator

01:50

Problem 6

6. Inhibitors of enzyme kinetics. The MichaelisMenten model of enzyme kinetics gives the reaction velocity $v$ in terms of a maximum rate $v_{\max }$ as
$$
v=v_{\max }\left(\frac{K x}{1+K x}\right),
$$
where $K$ is the binding constant $\left(K=1 / K_{M}\right.$, where $K_{M}$ is the Michaelis constant) and $x$ is the substrate concentration. A Lineweaver-Burk plot is the linearized version of Equation (29.40): a plot of $1 / v$ versus $1 / x$.
(a) Give the quantities (i), (ii), and (iii) shown in Figure $29.20$ in terms of $K$ and $v_{\max }$.
(b) Write the linearized form: $1 / v$ versus $1 / x$. Competitive inhibitors obey the expression
$$
v=v_{\max }\left(\frac{K_{x} x}{1+K_{x} x+K_{y} y}\right) \text {. }
$$
Noncompetitive inhibitors obey the expression
$$
v=v_{\max }\left(\frac{K_{x} x}{1+K_{x} x+K_{y} y+K_{x} K_{y} x y}\right) .
$$
Uncompetitive inhibitors obey the expression
$$
v=v_{\max }\left(\frac{K_{x} x}{1+K_{x} x+K_{x} K_{y} x y}\right) .
$$
(c) Sketch the plots of $1 / v$ versus $1 / x$ for competitive, noncompetitive, and uncompetitive inhibition. In each case, include the curve for the uninhibited rates.
Figure 29.20 A Lineweaver-Burk plot has a slope and two intercepts that can help determine the nature of an inhibitor.

Rabeya Zahid

Numerade Educator

04:26

Problem 7

DNA-binding protein cooperativity. Single-strand binding protein (SSB) is a tetrameric protein that binds single-stranded DNA oligomers. One DNA oligomer can bind to each protein monomer. A maximum of four oligomers can bind one tetrameric protein. The binding constant for one oligonucleotide is $K . x$ is the concentration of oligonucleotide. Binding is observed to be cooperative. Bujalowski and Lohman [14] have proposed a Pauling-like model of cooperativity. Think of the four protein subunits as being arranged in a square. A multiplicative 'cooperativity' factor $f$ applies whenever two ligands occupy adjacent sites on the protein square, but not when two ligands are diagonally across from each other.
(a) Write the binding partition function in terms of $K$, $f$, and $x$.
(b) Compute $v(x)$, the average number of DNA ligands bound per protein tetramer.

Josee Pacheco

Numerade Educator

02:38

Problem 8

Helicase nucleotide binding site. DNA B helicase is a hexametric protein that binds nucleotide ligands (see Figure 28.1). The binding fits a Pauling-like model involving six sites with binding constant $K$, arranged in a hexamer, and with interaction parameter $f$ whenever two ligands are on adjacent sites.
(a) Write the binding polynomial for this model.
(b) Write an expression for the average number of sites filled as a function of ligand concentration $x$.

Lara Gossage

Numerade Educator

04:26

Problem 9

Running a molecular transporter backwards. A molecular transporter undergoes the thermodynamic cycle shown in Figure $29.21$. Equilibrium and rate constants are given in the diagram. Assume that the concentration of the ligand on the left $\left(C_{L}\right)$ is smaller than that on the right $\left(C_{L} / C_{R}<1\right)$. Certain sets of values $T, K_{L}, K_{R}$, $k_{f}, k_{r}$ will allow for the transport of the ligand up a concentration gradient from a region of lower concentration $\left(C_{L}\right)$ to a higher concentration $\left(C_{R}\right)$.
Figure 29.21 The four states of a membrane transporter protein.
(a) Show that the flux around the cycle is
$$
\frac{k_{f} T K_{L} C_{L}-k_{r} K_{R} C_{R}}{\left(1+K_{L} C_{L}\right)+T\left(1+K_{R} C_{R}\right)}
$$
(b) If you want to pump uphill from left to right, for a given value of $C_{L} / C_{R}<1$, what ratio of the quantities $T, K_{L}, K_{R}, k_{f}, k_{r}$ is required?
(c) For $C_{1}=0$, write an expression for the cycle flux $J$ versus $C_{R}$ that gives a linearized plot. What is the slope and what is the intercept?

Josee Pacheco

Numerade Educator

05:59

Problem 10

A model transcriptional regulatory system. Consider the genetic regulatory system shown in Figure $29.22$. The action of the system depends on the concentrations $[\mathrm{DNA}],[P],[R]$, and $[A]$, where DNA is the operon DNA, $P$ is the RNA promoter protein, $R$ is the repressor protein, and $A$ is the activator protein.
Figure 29.22 Different binding sites of a transcriptional regulator.
(a) Write the binding polynomial $Q$ for this system.
(b) It is found that only state 6 leads to transcription of the lac genes. Write an expression for $f_{6}$, the population of molecules in state 6 , as a function of the species concentrations.
(c) Draw a qualitative plot of $f_{6}$ versus $[A]$, with all else constant.
(d) Draw a qualitative plot of $f_{6}$ versus $[R]$, all else constant.
(e) Draw a qualitative plot of $f_{6}$ versus $[P]$, all else constant.

Ashley Boni

Numerade Educator

03:42

Problem 11

Affinities of protein inhibitors. A receptor protein $P$ binds a metabolite molecule $M$ with affinity $K_{\text {assoc }}=$ $10^{6} \mathrm{M}^{-1}$.
(a) What ligand concentration in solution is required so that $75 \%$ of the receptor sites are occupied?
(b) A competitive inhibitor drug binds to the same protein with nanomolar affinity, i.e., $R_{\operatorname{assoc}}=10^{9} \mathrm{M}^{-1}$. If the inhibitor has concentration $y=10^{-7} \mathrm{M}$ in solution, in the situation described above, what is the fractional occupancy of $P$ now?

Rabeya Zahid

Numerade Educator

05:31

Problem 12

12. Cooperative protein binding to a DNA plasmid. Many different classes of proteins can bind to DNA. By binding to a promoter, these proteins can act to either increase or decrease the expression of a gene. The portion of the DNA sequence that a protein binds to is called a box.
There is a protein HU that binds to sites on a DNA plasmid, which helps the DNA to maintain an unkinked state. A second protein IHF binds to specific boxes and forces the plasmid to adopt a $90^{\circ}$ angle. The resulting kinked structure can cause a cascade of stress response events. For example, this kinked structure is observed in bacteria when activating a virulence plasmid.

Consider a simplified model of the process. There are only three IHF-binding sites on the plasmid. Each IHFbinding site can bind either one IHF protein or up to $n \mathrm{HU}$ proteins. When IHF binds, it excludes all $n$ binding sites in a mutually exclusive manner. An idealized schematic is shown in Figure $29.23 .$
Figure $29.23$ Two proteins, $\mathrm{HU}$ and $\mathrm{IHF}$, bind to a DNA plasmid.
For purposes of illustration, a possible binding configuration is shown in Figure $29.24 .$
Figure $29.24$ The bound-state complex.
In this case, there is an IHF bound to site 2 (excluding the possibility of an HU binding there), two HU's bound to site 3 , and one HU bound to site 1 .
(a) Express the binding polynomial for HU, assuming there is no IHF present, in terms of the binding constant $K_{H}$. Also express the binding polynomial for IHF, assuming there is no HU present, in terms of the binding constant $K_{I}$.
(b) Now, let's consider the effect of cooperativity. Using the Pauling model for cooperativity, express the binding polynomial for IHF, assuming there is no HU present. Use $\varepsilon$ as the interaction free energy for each pair of IHF molecules that are bound.
(c) For the model in (b), what is the average number of IHF ligands bound per plasmid?
(d) Express the binding polynomial, assuming both HU IHF are present. Assume there is no cooperativity among any of the sites.
(e) According to the model in (d), what is the average number of HU ligands bound to DNA? What happens in the limit of large HU? What happens in the limit of large IHF?

Sana Riaz

Numerade Educator

02:25

Problem 13

Activator or inhibitor? From binding experiments, you determine that a binding polynomial is
$$
Q=1+K x+R K x y+T R K x y z^{4},
$$
where $x, y$, and $z$ represent the concentrations of three different ligands, and $K, R$, and $T$ represent their binding affinities. Explain how ligand $z$ binds and whether it can be an activator and inhibitor.

Danielle Ashley

Numerade Educator

04:36

Problem 14

Binding polynomials. Here are two binding polynomials:
(i) $Q=1+K_{3} z+K_{3} K_{2} z y+K_{3} K_{2} K_{1} x y z$;
(ii) $Q=1+K_{3} z+K_{2} z y+K_{1} x y z$.
(a) Which describes the following process: $x$ can bind to $P$ (with affinity $K_{1}$ ) only if $y$ and $z$ are bound, $y$ can bind to $P$ (with affinity $K_{2}$ ) only if $z$ is bound, and $\mathrm{z}$ binds to $P$ with affinity $K_{3}$.
(b) For the $Q$ you chose, write an expression for the fraction of molecules $v_{y}$ that have $y$ bound.

Zach Steedman

Numerade Educator

01:16

Problem 15

Cooperative binding to an enzyme. An enzyme has three binding sites: $A, B$, and $C$. A ligand can bind to the enzyme at all three sites; however, site $C$ will not take up ligand until both sites $A$ and $B$ have been bound. And, sites $A$ and $B$ are independent of each other. This system is a single domain on the protein.
(a) Express the binding polynomial for this system.
(b) Write an expression for the average number of bound ligands.
(c) Express a different binding polynomial, assuming that the enzyme is uncompetitively inhibited by a different inhibitor at site $C$ and that the overall system consists of three independent domains.
(d) What is the average number of bound ligands in this latter case?

Joanna Quigley

Numerade Educator