Models have a habit of clinging to the boot-soles of ion channel biophysicists. Our primary electrophysiological recordings displayed in the Results section almost always presage a Discussion section replete with multistate schemes: gating models, conduction models, hidden Markov models, straightforward models that enlighten, opaque models that obfuscate, heuristic models with few parameters (oversimplified), complete models with many (overcomplexified) analytically tractable models for algebra lovers, gargantuan models for simulation lovers, models that spew out swarms of rate constants—well, reading the JGP can be exhausting sometimes. In this issue, Chowdhury and Chanda introduce an elegant analysis of voltage-dependent channel gating that yields a key parameter, the free energy of channel opening, in a virtually model-free way.
Voltage-gated ion channels have kept electrophysiologists off the streets for decades by challenging them with a fundamental and deeply appealing question to chew on: how does membrane voltage, which subjects any charged amino acid in the protein’s membrane-embedded core to enormous electric force fields tugging upon it (∼106 V/m), drive transitions between closed and open conformations of these proteins? This question has been attacked for over 60 years with increasingly sophisticated methods and elaborate models of the transmembrane movement of protein-associated charges that accompany pore opening. Because this charge movement is demanded by thermodynamics, it was confidently known to exist in voltage-gated channels long ago, even before its direct detection by Armstrong and Bezanilla (1974). Nowadays, this “gating current” is a readily observable, standard part of the electrophysiologist’s toolkit, and high-resolution structures of the charge-bearing voltage-sensitive domains (VSDs) of several Kv channels (Jiang et al., 2003; Long et al., 2007; Clayton et al., 2008), and of a putative Nav channel (Payandeh et al., 2011), have enriched the field with physical pictures—themselves food for current controversies—of how gating charge actually moves.
Gating current is detected by holding the membrane at a hyperpolarized voltage, where the channels are all closed and the VSDs are all in the “down” position (their charges exposed to the intracellular side of the membrane), and then stepping to a test voltage to observe, typically in the first few hundred microseconds after the voltage change, the transient blip of current that signals the VSD’s charged cargo moving outwards to the “up” position. Integrating that blip over time gives the net charge moved at the test voltage, and a plot of this “Q-V curve” will saturate at full charge movement, Qmax, when the test voltage is positive enough to have pushed all the VSDs in the system into the “up” position, where the channels are maximally open.
The simplest two-state model of voltage-dependent gating envisions a channel with a single VSD that can adopt either of two conformations, “down” (D) or “up” (U), the latter being uniquely associated with the open ion-conducting pore. The D↔U conformational equilibrium, and hence the observed charge movement, will depend on voltage according to what is commonly referred to as a Boltzmann function that describes the charge movement Q(V):
where z, the gating charge carried on the VSD, determines the steepness of the Q-V curve, and ΔGoc is the chemical part of the standard-state free energy of channel opening. This free energy term reflects the net result of all the differences in intrinsic chemical interactions within the protein in the U versus D conformations; it also sets the position of the Q-V curve on the voltage axis and is sensitive to the sorts of maneuvers that channel proteins suffer at the hands of biophysicists testing out mutants, or at the hands of evolution testing out isoforms, splice variants, and the like.
But real voltage-gated channels violate this two-state picture in ways that undermine its use down in the trenches of research. These channels carry four VSDs, a circumstance that means at minimum three intermediate conformational configurations are interposed between the fully closed and fully open channel, with all transitions among them involving charge movement. Once intermediates appear, we are forced into modeling the devilish details. Do the VSDs move independently of each other or cooperatively, such that after the first moves, the other three immediately follow? Or is there negative cooperativity, wherein the symmetry broken by the first VSD transition makes subsequent VSD movement require a more depolarized voltage range? How is the overall free energy partitioned among the many states involved? Is there a preordained order of VSD movement in the nonsymmetric Nav and Cav channels, or do the domains move randomly in response to voltage? Specific models embodying complications like these are required to understand channel behavior. These linked equilibria get very complicated very quickly as parameter piles upon parameter. In some cases, simplifying assumptions can collapse many parameters into just a few, as in the well-worn Monod-Wyman-Changeux formulation, but frequently such simplifications don’t work with the system at hand.
Chowdhury and Chanda’s paper does not cut through all these problems, but it does show how to extract a parameter of prime importance from experimental Q-V curves: the chemical component of the conformational free energy difference between the two extreme states, fully closed and fully open, the multistate analogue of ΔGoc in the two-state model. Their analysis pays tribute to Jeffries Wyman, the great physical chemist of hemoglobin who in the mid-1960s cut through this very same problem of intermediate states. Wyman showed that regardless of mechanistic detail, ligand-driven equilibria in a multisite protein may be characterized by an experimentally accessible parameter called the “median activity,” essentially a special kind of average ligand concentration read directly off the binding curve (Wyman, 1967; Wyman and Gill, 1990). The median activity allows a thermodynamically rigorous—and model-free—estimation of ΔGoc, the intrinsic free energy difference between all-sites-empty and all-sites-occupied conformations of the protein, analogous to the same parameter in Eq. 1 for the two-state channel model.
Chowdhury and Chanda (2012) apply similar logic to voltage-driven equilibria for VSD-type channels. They define on the Q-V curve a special “median voltage” Vm, such that the area under the curve from V = −∞ to Vm is equal to that between the curve and the Qmax line from V = Vm to ∞. They then do some Wymanesque mathematical handstands to show that Vm is a model-free measure of ΔGoc, the intrinsic free energy difference between the two conformational extremes: fully closed and fully open. This is a really useful maneuver, as in many cases, voltage-gated channels exhibit Q-V relations that look nothing like the two-state function of Eq. 1, and yet by simply measuring off areas, one can confidently extract ΔGoc from these otherwise model-dependent curves. The authors then go on to show, with experimental examples of Kv and Nav channels and with various simulated multistate models, that median-voltage analysis really works. And they also deal with a few of the qualifying complications that arise in this rarified thermodynamic treatment. Their examples illustrate in a palpable, dramatic way the erroneous ΔGoc estimates that can emerge from forcing “Boltzmann” fits to Q-V data, and the misleading effects of mutations on gating energetics inferred from them.
This paper gives us a good example of the power of thermodynamic reasoning, which is always true but often useless, as it necessarily eschews details of particular models. But sometimes, as is shown here, thermodynamics provides a path to circumvent the tyranny and heartbreak of model fitting. For me, reading this paper evoked the pleasure of engaging with a novel, elegant analysis, as well as the inevitable spite and envy at not having myself hit upon this strict analogy to Wyman’s work, which I’ve been teaching—proselytizing, even—to graduate students for over 20 years. So the authors have bestowed upon me, and I suspect will soon bestow upon other readers, a forehead-slapping, gosh-why-didn’t-I-think-of-that moment!
