Homeostatic and adaptive energetics: Nonequilibrium fluctuations beyond detailed balance in voltage-gated ion channels

Stochastic thermodynamics has largely succeeded in characterizing both equilibrium and far-from-equilibrium phenomena. Yet many opportunities remain for application to mesoscopic complex systems—especially biological ones—whose effective dynamics often violate detailed balance and whose microscopic degrees of freedom are often unknown or intractable. After reviewing excess and housekeeping energetics—the adaptive and homeostatic components of a system's dissipation—we extend stochastic thermodynamics with a trajectory class fluctuation theorem for nonequilibrium steady-state, nondetailed-balanced complex systems. We then take up the neurobiological examples of voltage-gated sodium and potassium ion channels to apply and illustrate the theory, elucidating their nonequilibrium behavior under a biophysically plausible action potential drive. These results uncover challenges for future experiments and highlight the progress possible understanding the thermodynamics of complex systems—without exhaustive knowledge of every underlying degree of freedom.

Figure 1

Interaction between the stochastic process $X_{0:N}$, protocol ${\alpha }_{0:N}$, and realized (observed) trajectories $x_{0:N}$.

Figure 2

A thermal system and its interactions with various baths. While heat and work reservoirs (labeled with temperature $T$ and parameter $\alpha$, respectively) are ideal, by design we assume nothing about the auxiliary or “aux” reservoir, and so label its energetic contribution by $\mathrm{\Delta }{E}_{a}ux$ to avoid confusion with well-defined terms like heat and work. The labels and arrow directions indicate energetic fluxes to-from the system. Notably, we allow for nonequilibrium steady states and functionally split the total heat $Q$ into the excess heat $Q_{e}x$—corresponding to adaptive dissipation—and housekeeping heat $Q_{h}k$—referring to homeostatic dissipation.

Figure 3

Continuous-time Markov chain model of the ${\mathrm{K}}^{+}$ channel adapted from Fig. 5.12 of Ref. [15]. Self-transitions are implied. In the states labeled $An$, a number $n\in \{1,2,3,4\}$ activation gates close the channel. $\mathrm{O}$ labels the open channel state, the only one in which ${\mathrm{K}}^{+}$ current can flow through the channel. The rate parameters $a_n$ and $b_n$ are voltage-dependent; their functional forms are given in Eqs. (31). This channel model is fully detailed-balanced, in the sense that Eq. (33) vanishes for every allowed transition pair.

Figure 4

Continuous-time Markov chain model of the ${\mathrm{Na}}^{+}$ channel adapted from Fig. 5.13 of Ref. [15]. Self-transitions are implied. In the states labeled $An$, a number $n\in \{1,2,3\}$ activation gates close the channel. $\mathrm{O}$ labels the open channel state, in which ${\mathrm{Na}}^{+}$ current flows through the channel. Finally, I labels the channel's inactivation by its inactivation gate—its so-called ball and chain. The rate parameters $a_m, b_m$, and $a_h$ are all voltage dependent; their functional forms are given in Eqs. (29) and (30). The rate constants $k_1, k_2$, and $k_3$ are given by Eqs. (30). Unlike the ${\mathrm{K}}^{+}$ channel, this model of the ${\mathrm{Na}}^{+}$ channel features one-way transitions in the rate dynamic—states O to I and A2 to I. These transitions are maximally irreversible. These imply divergent infinitesimal housekeeping heat in the sense of Eq. (33). In addition to these, many of the other transition pairs do not satisfy detailed balance—Eq. (33) evaluates finite but nonzero.

Figure 5

The two channel models compared via their nonequilibrium excess work and housekeeping heat distributions, respectively, in response to the spike protocol drive. Nonzero values of $Q_{h}k$ indicate violations of the Crooks DFT Eq. (13), where the corrected NESS DFT Eq. (15) is needed. In addition, the axes here are of quantities with associated independent second laws; see Eqs. (20) and (23). And so, the labeled quadrants carry thermodynamic (and, in this case, biophysical) meaning as single-shot violations of each statistical second law.

Figure 6

${\mathrm{Na}}^{+}$ channel transition rates of housekeeping heat $Q_{h}k$ as functions of transmembrane potential. The total housekeeping heat along any stochastic trajectory, driven by any protocol, is the sum of these values with the associated state transition-protocol parameter pairs here.

Figure 7

Excess heats (solid lines) for both channels under the pulse protocol (dashed line). The ${\mathrm{K}}^{+}$ channel is less dissipative. Both expend energy as they relax to environmentally induced steady states.

Figure 8

Excess heats (solid lines) for both channels driven by the spike protocol (dashed line). Under this more biologically realistic protocol, the ${\mathrm{Na}}^{+}$ dissipates significantly more than the ${\mathrm{K}}^{+}$ channel and does so responding much more rapidly to changes in membrane voltage. This suggests a tradeoff between the speed of the channel's response and its dissipation, one not necessarily present in the more artificial pulse protocol.

Figure 9

Excess works (solid lines) for both channels under the pulse protocol (dashed line). Unlike excess heat, excess work is only done upon change in the driving parameter. Thus, we see changes only at the pulse's rise and fall. Much more excess work is done on the ${\mathrm{Na}}^{+}$ channel than on the ${\mathrm{K}}^{+}$ during the rise of the pulse, but these roles are reversed on its fall. Over the entire protocol, the ${\mathrm{Na}}^{+}$ channel produces (slightly) more excess environmental entropy.

Figure 10

Excess works (solid lines) for both channels under the spike protocol (dashed line). We see that more excess work is done on the ${\mathrm{Na}}^{+}$ channel across the board. This corresponds to larger environmental entropy production and indicates a greater potential for dissipated work in the ${\mathrm{Na}}^{+}$ channel. However, we also see that up until the peak and reset phases of the action potential spike, they track very closely before diverging.