Figure 5. A simple allosteric model for temperature activation of MthK.

(A) Four state binary elements model of MthK activation. The intrinsic equilibrium constant of pore opening is L₀ and the calcium-binding affinity is K. The state-dependent interactions between the pore and calcium sensor are represented by $L_0$, where X indicates the conformation of the calcium sensor (X = bound (B) or unbound (U)) and Y indicates the conformation of the Pore (Y = open (O) or closed (C)). Solid lines indicate the interactions between 'like' states and the dotted lines highlight interactions between 'unlike' states. (B) Using the model of MthK channel gating, described in A, we simulated the Hill-plot of MthK (i.e. ln [Po/(1-Po)] vs. Ca²⁺) at different temperatures for various model parameters. In each graph, all parameters, except the state-dependent interaction terms shown in each sub-panel, were kept constant across different temperatures. For all simulations, the values of the parameters used were: L₀ = 0.1; K = 20000 M⁻¹; $K=$ 8; ${\theta }_{UO}=$ 18; ${\theta }_{UC}=$ 150; ${\theta }_{BO}=$ 8. For temperature-dependent simulations of ${\theta }_{BC}=$, we use the following equation: ${\displaystyle {\theta }_{BC}\left(T\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\left(\mathrm{\Delta }{H}_{BC}-T\mathrm{\Delta }{S}_{BC}\right)/RT\right)}$, where ${\theta }_{BC}\left(T\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-(∆H_{BC}-T∆S_{BC})/RT)$ -71 kJ and $∆H_{BC}=$ -220 J/K. Similarly, $∆S_{BC}=$ (${\theta }_{BO}$ 81 kJ and $∆H_{BO}=$ 310 J/K), $∆S_{BO}=$ (${\theta }_{UC}$ -71 kJ and $∆H_{UC}=$ -220 J/K) and $∆S_{UC}=$ (${\theta }_{UO}$ 80 kJ and $∆H_{UO}=$ 300 J/K) were calculated.

Figure 5—figure supplement 1. Effect of temperature-dependent parameters on simulated Hill-plots of calcium-dependent gating of MthK.

Using the model of MthK channel gating, described in Figure 5A, we simulated the Hill-plot of MthK channel (i.e. ln [Po/(1-Po)] vs. Ca²⁺) at different temperatures. (A) All parameters, except $P_O=A+\left(B-A\right)*\frac{x^n}{K^n+x^n}$ were kept constant across the different temperatures. This represents the case where the pore opening is innately temperature sensitive. The simulations show that the Hill-plots symmetrically translate vertically with a change in temperature but there is no change in $\mathrm{\Delta }\chi$ values. (B) When the binding affinity of calcium is temperature-dependent, the Hill curves are displaced horizontally along with the calcium concentration, again with no change in $L_0$. For all the simulations, the values of the parameters used were: L₀ = 0.1; K = 20000 M⁻¹; $K=$ 8; ${\theta }_{UO}=$ 18; ${\theta }_{UC}=$ 150; ${\theta }_{BO}=$ 8. For temperature-dependent simulations of ${\theta }_{BC}=$ in panel A, H_L = 60 kJ and S_L = 200 J/K and the $∆S_L=$ at each temperature was directly calculated as: $L_0$. Similarly, for temperature-dependent simulations of $L_0\left(T\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-(∆H_L-T∆S_L)/RT)$ shown in panel B, H_K = 60 KJ and S_K = 220 J/K, respectively.

Figure 5—figure supplement 2. MthK activation involving multiple calcium binding sites with differing temperature-dependence.

(A) A schematic of MthK with two calcium binding sites, one of which is temperature-dependent, whereas the other is not. The intrinsic equilibrium constant for pore opening is $∆S_K=$, the binding affinity of calcium to the temperature-independent calcium sensor is $L_0$, and the intrinsic temperature-dependent constant of the calcium sensor is $K$. Each of the three binary elements are allosterically coupled to each other. The coupling constants are indicated as D, C, and E shown in the diagram. (B) Hill plots of model simulations at different temperatures for various values of parameters C and E, as indicated in the inset. The remaining parameters were kept constant: L₀ = 0.1, K = 200, and D = 100. All the parameters, except ${\displaystyle M}$, were assumed to be temperature independent. For the temperature sensor, H_M = 60 kJ and S_M = 200 J/K, respectively.

Figure 5—figure supplement 3. An alternate allosteric model of MthK activation involving independent calcium and temperature sensing domains.

(A) Schematic model depicting pore domain, calcium-binding domain, and temperature sensing domain each of which undergoes a change between active and passive forms. The intrinsic equilibrium constant for pore opening is $∆S_M=$, the binding affinity of calcium to the RCK domain is $L_0$, and the intrinsic temperature-dependent activation constant of the thermosensor is $K$. Each of the three binary elements are allosterically coupled to each other. D, C, and E indicate the coupling constants as shown in the diagram. This model is an adaptation of the nested MWC model of BK channel (Horrigan and Aldrich, 2002). (B) Hill plots of model simulations at different temperatures for various values of parameters C and E, as indicated in the inset. The remaining parameters were kept constant: L₀ = 0.1, K = 200, and D = 100. All the parameters, except ${\displaystyle M_0}$, were assumed to be temperature independent. For the temperature sensor, H_M = 60 kJ and S_M = 200 J/K, respectively.

Figure 5—figure supplement 4. Renormalization of state-dependent interaction parameters.

The model shown in Figure 5A can be described by a reduced set of parameters, as is common in classical representations of allosteric models. The reduced parameters correspond to experimentally determined from linkage analysis of the intact protein. The apparent equilibrium constants of pore opening and ligand binding ($∆S_M=$ and $L_0^{\text{'}}$) and the global coupling parameter, $K_0^{\text{'}}$, are now dependent on the state-dependent coupling parameters. This model and that in Figure 5A thus represent the same process, but the significance of the parameters is different. The intrinsic equilibrium constants $\theta$ and K can be extracted only and only if the pore activation and calcium binding to the calcium sensor is measured in isolation. Thus, these intrinsic parameters will be different from $L_0$ and $L_0^{\text{'}}$ terms obtained by allosteric analysis of intact channels if there is any state-dependent interaction between the two allosteric domains.