Transition in relaxation paths in allosteric molecules: Enzymatic kinetically constrained model

A hierarchy of timescales is ubiquitous in biological systems, where enzymatic reactions play an important role because they can hasten the relaxation to equilibrium. We introduced a statistical physics model of interacting spins that also incorporates enzymatic reactions to extend the classic model for allosteric regulation. Through Monte Carlo simulations, we found that the relaxation dynamics are much slower than the elementary reactions and are logarithmic in time with several plateaus, as is commonly observed for glasses. This is because of the kinetic constraints from the cooperativity via the competition for an enzyme, which has a different affinity for molecules with different structures. Our model showed symmetry breaking in the relaxation trajectories that led to inherently kinetic transitions without any correspondence to the equilibrium state. In this Rapid Communication, we discuss the relevance of these results for diverse responses in biology.

Figure 1

Schematics of the enzymatic MWC model. (a) All states that a single MWC molecule can take. $\sigma$ represents the T or R state and $m$, the number of modifications. (b) Energy level of each monomer. (c) Binding energies between the enzyme and the MWC molecules with different states. The binding energy depends on a molecular structure $\sigma$.

Figure 2

Time evolution of the average modification in the eMWC against the logarithmic Monte Carlo step. (a) Relaxation of the average modification to the equilibrium at various temperatures. Since ${\mathcal{U}}_{\mathrm{eq}}$ depends on the temperature, the normalized ratio $({\langle \mathcal{U}\rangle }_{\mathrm{ens}}-{\mathcal{U}}_{\mathrm{eq}})/(1-{\mathcal{U}}_{\mathrm{eq}})$ was plotted by setting $\langle n\rangle /N$ at 0.2. Different color lines indicate time courses with different values of $\beta$. (b) Relaxation of the average modification to the equilibrium for various values of $\langle n\rangle$. The time course of ${\langle \mathcal{U}\rangle }_{\mathrm{ens}}$ was plotted by setting $\beta$ at 1.75. The different line colors correspond to different values of $\langle n\rangle /N$. $({\mathcal{U}}_{\mathrm{eq}}$ is independent of $\langle n\rangle$.) Each line is an ensemble average of 1000 samples. (c) Logarithm of ${\langle \mathcal{U}\rangle }_{\mathrm{ens}}$ plotted for (b).

Figure 3

Distributions of the relaxation time and the variance of the logarithmic relaxation time. Probability distribution of the logarithmic normalized relaxation time at various $\langle n\rangle$ values with fixed temperature at (a) $\beta =1.75$, and at various temperatures with $\langle n\rangle /N$ fixed at (b) 0.8 and (c) 0.2, respectively. The logarithmic relaxation time was calculated as the base-10 logarithm of ${\tau }_{\mathrm{eq}}$. Plots are rescaled by ${\tau }_{\mathrm{eq}}{\langle n\rangle }^{-1}$ for (a) and rescaled by ${\tau }_{\mathrm{eq}}{e}^{-4\beta }$ for (b). Different color lines indicate the probability distributions under different parameters. (d) Averaged normalized relaxation time for different number of modification sites $M$ indicated by different symbols. The plot is rescaled in the similar way as in (a).

Figure 4

Schematic showing the energy landscape of the eKCM. Left: If the number of R-state molecules is lower than that of the enzyme, the modification reaction of T-state molecules (up and down arrows) can occur and progress until half of the modification sites are modified. Then, when more than half of the sites are modified, such molecules tend to be in the R state (lower right arrow). Right: When the number of R-state molecules exceeds that of the enzyme through modification, the R-state molecules monopolize the enzyme, and the modification reaction of the T-state molecules is kinetically inhibited (red crosses). Therefore, the structural changes from the T to R state in the less modified molecules (upper right arrow) are rate limiting.

Figure 5

Relaxation times at various temperatures and $\langle n\rangle$ values. (a) Dependence of the relaxation time on the temperature. The relaxation time is defined as the ensemble average of ${\tau }_{\mathrm{eq}}$ when $\mathcal{T}$ falls below the analytically calculated value ${\mathcal{T}}_{\mathrm{eq}}$ because $\mathcal{T}$ approaches equilibrium from above. The blue and green lines indicate lines proportional to $exp\left(\beta \right)$ and $exp\left(6\beta \right)$, respectively. $\langle n\rangle /N$ was set at 0.8. (b) Dependence of the relaxation time on $\langle n\rangle /N$. The green line indicates a line proportional to ${\langle n\rangle }^{-1}$. $\beta$ was set at 1.75.