Statistical mechanics of Monod-Wyman-Changeux (MWC) models

Abstract

The 50th anniversary of the classic Monod-Wyman-Changeux (MWC) model provides an opportunity to survey the broader conceptual and quantitative implications of this quintessential biophysical model. With the use of statistical mechanics, the mathematical implementation of the MWC concept links problems that seem otherwise to have no ostensible biological connection including ligand-receptor binding, ligand-gated ion channels, chemotaxis, chromatin structure and gene regulation. Hence, a thorough mathematical analysis of the MWC model can illuminate the performance limits of a number of unrelated biological systems in one stroke. The goal of our review is twofold. First, we describe in detail the general physical principles that are used to derive the activity of MWC molecules as a function of their regulatory ligands. Second, we illustrate the power of ideas from information theory and dynamical systems for quantifying how well the output of MWC molecules tracks their sensory input, giving a sense of the "design" constraints faced by these receptors.

Fig. 1

States-and-weights diagram of the one-site MWC molecule. (a) Each of the four states has an associated energy, part of which is due to the conformational degrees of freedom of the molecule and part of which reflects the free energy of the binding process. (b) p_active as given in Eq. (2) as a function of concentration in units of the inactive state's dissociation constant. The activity curve is shown on a log scale in the main plot and on a linear scale in the inset. (c) The four curves show the probabilities of each of the distinct states as a function of the ligand concentration. Each state is labeled by a pair of numbers. The first number of the pair is 1 if the receptor is active and 0 if the receptor is inactive; the second number of the pair is 1 if a ligand is bound and 0 if no ligand is bound. The parameter values used in the figure are Δε = ε_I − ε_A = −4 k_BT and $\Delta {\epsilon }_{\mathrm{b}}={\epsilon }_{\mathrm{b}}^{(\mathrm{A})}-{\epsilon }_{\mathrm{b}}^{(\mathrm{I})}=-5k_{\mathrm{B}}T$.

Fig. 2

Table of key quantities that can be computed within the MWC framework. (a) The activity curve on a linear scale for two MWC molecules: a one-site receptor with Δε = ε_I − ε_A = −4 k_BT, $K_{\mathrm{d}}^{(\mathrm{A})}=1\mu \mathrm{M}$ and $K_{\mathrm{d}}^{(\mathrm{I})}=148\mu \mathrm{M}$, giving a difference in binding energy of ${\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{A})}-{\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{I})}=\text{log}\frac{K_{\mathrm{d}}^{(\mathrm{A})}}{K_{\mathrm{d}}^{(\mathrm{I})}}=-4k_{\mathrm{B}}T$; and a two-site receptor with Δε = ε_I − ε_A = −4 k_BT, $K_{\mathrm{d}}^{(\mathrm{A})}=1\mu \mathrm{M}$, and $K_{\mathrm{d}}^{(\mathrm{I})}=12.2\mu \mathrm{M}$, giving a binding energy difference of ${\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{A})}-{\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{I})}=\text{log}\frac{K_{\mathrm{d}}^{(\mathrm{A})}}{K_{\mathrm{d}}^{(\mathrm{I})}}=-2k_{\mathrm{B}}T$. (b) The activity curves from (a) with concentrations on a log scale. The transition point concentration c* = 40.6 µM and effective Hill coefficient h_eff = 1 are shown with vertical and horizontal lines, respectively, for the one-site receptor. (c) This table gives formulas for some of the key parameters of interest in both statistical mechanics and thermodynamic language. Here, n is the total number of binding sites on the receptor, L = e^−βΔε is the conformational equilibrium constant where Δε = ε_I − ε_A is the difference in conformational energy between the inactive and active state, $K_{\mathrm{d}}^{(\mathrm{I})}=c_0{e}^{\mathrm{\beta }({\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{I})}-{\mathrm{\mu }}_0)}$ is the inactive state's dissociation constant for ligand binding, $K_{\mathrm{d}}^{(\mathrm{A})}=c_0{e}^{\mathrm{\beta }({\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{A})}-{\mathrm{\mu }}_0)}$ is the active state's dissociation constant for ligand binding and c is the ligand concentration.

Fig. 3

Toy model of a ligand-gated ion channel. (a) The model ion channel has two binding sites for the control ligand and can exist in four distinct states of occupancy (i.e., empty, 2 × single occupancy, double occupancy) for both the closed and open states. (b) Open probability for a cGMP-gated channel as a function of the cGMP concentration and fit to an MWC model with four binding sites.

Fig. 4

Bacterial chemotaxis. (a) A schematic showing the motion of a bacterium that consists of a series of runs and tumbles. (b) A chemoreceptor and the bacterial flagellar motor are shown in the same membrane region, although in real bacteria, they are often on opposite poles. In the presence of ligand, CheY is not phosphorylated and hence the motor is not induced to alter its rotation direction. (c) Activity of the chemoreceptor in the limits of low and high chemoattractant concentration with MWC parameters taken from Ref. .

Fig. 5

States-and-weights diagram for chemotaxis clusters. The various states shown in the figure correspond to different states of occupancy of the chemotactic receptors while in the inactive state. There is a corresponding set of diagrams (not shown) for the active state. The statistical weights of the different states reflect how many ligands have been drawn out of solution to bind the chemoreceptors. The degeneracies correspond to how many different ways there are of realizing a given state of binding. For example, in the second state shown in the figure, there is only one ligand bound on one of the n class 1 receptors. The class 1 receptors and class 2 receptors have conformational energy ε_(off) in the inactive state and ε_(on) in the active state. The n class 1 receptors have dissociation constant $K_{\mathrm{d}}^{(\text{off})}$ in the inactive state and $K_{\mathrm{d}}^{(\text{on})}$ in the active state; the m class 2 receptors have dissociation constants ${\kappa }_{\mathrm{d}}^{(\text{off})}$ in the inactive state and ${\kappa }_{\mathrm{d}}^{(\text{on})}$ in the active state.

Fig. 6

MWC model of nucleosome accessibility. States-and-weights diagram for a toy model of nucleosome accessibility that illustrates how transcription factors could alter the equilibrium of nucleosome-bound DNA. ε_c and ε_o refer to the conformational energies of the closed and open states, respectively, and $K_{\mathrm{d}}^{(\mathrm{C})}$ and $K_{\mathrm{d}}^{(\mathrm{O})}$ are the dissociation constants for transcription factor binding in those two states.

Fig. 7

Schematic description of MWC chromatin. (a) The genomic DNA exists in two classes of state, one of which is “off” and the other one of which is “on” and permits transcription. Transcription factor binding controls the relative probability of these different eventualities. (b) States and weights for the binding of two transcription factors, here denoted by A and B, which occupy the open and closed conformations with different affinity. The concentration of transcription factors A and B is given by c_A and c_B, respectively. The conformational energies of the closed and open states are given by ε_c and ε_o. The dissociation constant for A is $K_{\mathrm{A}}^{(\mathrm{C})}$ when chromatin is in the closed state and is $K_{\mathrm{A}}^{(\mathrm{O})}$ in the open state, and the dissociation constant for B is $K_{\mathrm{B}}^{(\mathrm{C})}$ when chromatin is in the closed state and is $K_{\mathrm{B}}^{(\mathrm{O})}$ when chromatin is in the open state.

Fig. 8

The Bohr effect and MWC models. (a) The Bohr effect and oxygen binding to hemoglobin as a function of pH. The hemoglobin binding curves are shown for five values of the pH: (a) 7.5, (b) 7.4, (c) 7.2, (d) 7.0 and (e) 6.8. The vertical lines indicate the partial pressures experienced in muscle and in the lungs. (b) The “Bohr effect” in the context of chromatin showing how the occupancy of a transcription factor on nucleosomal DNA changes as the histone–DNA affinity (for example) is changed, as described in Eq. (11). For the figure shown here, we have $K_{\mathrm{d}}^{(\mathrm{o})}={10}^{-9}\mathrm{M}$ and $K_{\mathrm{d}}^{(\mathrm{c})}=100K_{\mathrm{d}}^{(\mathrm{o})}$. The closed state energy has been chosen as the reference energy and is taken as zero.

Fig. 9

Information transmission through a two-site MWC molecule. The case of nACh receptors is illustrated for concreteness. (a) ACh (input) binds to ligand-gated ion channels (communication channel), thereby influencing the number of open ion channels (output). (b) The probability distribution of ACh concentration that maximizes the mutual information between input ([ACh]) and output (N_open) from Eq. (21). (c) The MWC ligand–receptor binding probabilities determine the conditional distribution of the total number of nACh receptors open as a function of ACh concentration, p(N_open|[ACh]) from Eq. (17), shown here as a heat map. (d) The probability distribution of N_open that maximizes the mutual information between input ([ACh]) and output (N_open). (b–d) Plots assume a total of 100 nACh receptors on the synaptic cleft for visualization purposes, although this underestimates the number of nACh receptors on a typical synaptic cleft and all plots use MWC parameters characteristic of nACh receptors.

Fig. 10

Sensor properties of an ensemble of independent MWC molecules with two binding sites. In plots (a), (b) and (d), the MWC parameters are characterized by − βε, the conformational energy difference between the open and closed states (in units of k_BT), and $\text{log}\frac{K_{\mathrm{d}}^{(\mathrm{o})}}{K_{\mathrm{d}}^{(\mathrm{c})}}=-\mathrm{\beta }({\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{o})}-{\mathrm{\epsilon }}_{\mathrm{b}}^{(\mathrm{c})})$, the difference in ligand binding energies between the open and closed states (in units of k_BT). (a) The effective Hill coefficient of a two-site MWC molecule plotted as a function of MWC parameters. (b) The dynamic range of 10⁵ independent two-site MWC molecules plotted as a function of MWC parameters. (c) The dynamic range of N two-site MWC molecule with MWC parameters characteristic of a nACh receptor, plotted as a function of the total number of receptors N. (d) The channel capacity of 10⁵ independent two-site MWC molecules plotted as a function of MWC parameters. (e) The channel capacity of N two-site MWC molecule with MWC parameters characteristic of a nACh receptor, plotted as a function of the total number of receptors N.

Fig. 11

Dynamics of an MWC ligand-gated ion channel with two binding sites. (a) Schematic showing states and rates of transition between MWC states in a simplified kinetic allostery model. (b) Probabilities of being in each state as a function of time, starting from a nonequilibrium configuration, as calculated using Eq. (33) with c = 0.5 µM and rate constants f_O = 1 µM⁻¹ ms⁻¹, b_O = 170 ms⁻¹, f_C = 1 µM⁻¹ ms⁻¹, b_C = 0.04 ms⁻¹, f_L = 1 µM⁻¹ ms⁻¹ and b_L = 8 × 10⁻⁴ ms⁻¹. The dotted lines show the equilibrium values of each of these probabilities.

Fig. 12

Characterization of the response properties of a toy model for a ligand-gated ion channel. (a) Step function increases in ligand concentration lead to smooth increases in the probability of an open channel. (b and c) Plots of the magnitude |G(ω)| and argument arg(G(ω)) of the frequency response function of the ligand-gated ion channel to small fluctuations in ligand concentration. These frequency responses are shown as a function of the mean ligand concentration c₀ about which the ligand concentration fluctuations. Rate constants: f_O = 1 µM⁻¹ ms⁻¹, b_O = 170 ms⁻¹, f_C = 1 µM⁻¹ ms⁻¹, b_C = 0.04 ms⁻¹, f_L = 1 µM⁻¹ ms⁻¹ and b_L = 8 × 10⁻⁴ ms⁻¹.

Fig. 13

Molecular cartoons showing the variety of different allowed states and subsets of states considered in different models.^, The states shaded in light blue correspond to the traditional MWC model. The states shaded in light pink correspond to a sequential model of the KNF form. The green box surrounds all of the states and generalizes the MWC scenario to include other intermediates. The version shown here is a slight variant on that presented in the excellent review by Hilser et al.