The voltage dependence of charges in voltage-sensitive proteins, typically displayed as charge versus voltage (Q-V) curves, is often quantified by fitting it to a simple two-state Boltzmann function. This procedure overlooks the fact that the fitted parameters, including the total charge, may be incorrect if the charge is moving in multiple steps. We present here the derivation of a general formulation for Q-V curves from multistate sequential models, including the case of infinite number of states. We demonstrate that the commonly used method to estimate the charge per molecule using a simple Boltzmann fit is not only inadequate, but in most cases, it underestimates the moving charge times the fraction of the field.

Many ion channels, transporters, enzymes, receptors, and pumps are voltage dependent. This voltage dependence is the result of voltage-induced translocation of intrinsic charges that, in some way, affects the conformation of the molecule. The movement of such charges is manifested as a current that can be recorded under voltage clamp. The best-known examples of these currents are “gating” currents in voltage-gated channels and “sensing” currents in voltage-sensitive phosphatases. The time integral of the gating or sensing current as a function of voltage (V) is the displaced charge Q(V), normally called the Q-V curve.

It is important to estimate how much is the total amount of net charge per molecule (Qmax) that relocates within the electric field because it determines whether a small or a large change in voltage is necessary to affect the function of the protein. Most importantly, knowing Qmax is critical if one wishes to correlate charge movement with structural changes in the protein. The charge is the time integral of the current, and it corresponds to the product of the actual moving charge times the fraction of the field it traverses. Therefore, correlating charge movement with structure requires knowledge of where the charged groups are located and the electric field profile. In recent papers by Chowdhury and Chanda (2012) and Sigg (2013), it was demonstrated that the total energy of activating the voltage sensor is equal to Qmax VM, where VM is the median voltage of charge transfer, a value that is only equal to the half-point of activation V1/2 for symmetrical Q-V curves. VM is easily estimated from the Q-V curve, but Qmax must be obtained with other methods because, as we will show here, it is not directly derived from the Q-V curve in the general case.

The typical methods used to estimate charge per molecule Qmax include measurements of limiting slope (Almers, 1978) and the ratio of total charge divided by the number of molecules (Schoppa et al., 1992). The discussion on implementation, accuracy, and reliability of these methodologies has been addressed many times in the literature, and it will not be discussed here (see Sigg and Bezanilla, 1997). However, it is worth mentioning that these approaches tend to be technically demanding, thus driving researchers to seek alternative avenues toward estimating the total charge per molecule. Particularly, we will discuss here the use of a two-state Boltzmann distribution for this purpose. Our intention is to demonstrate that this commonly used method to estimate the charge per molecule is generally incorrect and likely to give a lower bound of the moving charge times the fraction of the field.

The two-state Boltzmann distribution describes a charged particle that can only be in one of two positions or states that we could call S1 and S2. When the particle with charge Qmax (in units of electronic charge) moves from S1 to S2, or vice versa, it does it in a single step. The average charge found in position S2, Q(V), will depend on the energy difference between S1 and S2, and the charge of the particle. The equation that describes Q(V) is:

 
Q(V)=Qmax1+exp[Qmax(VV1/2)kT],
(1)

where V1/2 is the potential at which the charge is equally distributed between S1 and S2, and k and T are the Boltzmann constant and absolute temperature, respectively. The Q(V) is typically normalized by dividing Eq. 1 by the total charge Qmax. The resulting function is frequently called a “single Boltzmann” in the literature and is used to fit normalized, experimentally obtained Q-V curves. The fit yields an apparent V1/2 (V1/2) and an apparent QMAX (Qmax), and this last value is then attributed to be the total charge moving Qmax. Indeed, this is correct but only for the case of a charge moving between two positions in a single step. However, the value of Qmax thus obtained does not represent the charge per molecule for the more general (and frequent) case when the charge moves in more than one step.

To demonstrate the above statement and also estimate the possible error in using the fitted Qmax from Eq. 1, let us consider the case when the gating charge moves in a series of n steps between n + 1 states, each step with a fractional charge zi (in units of electronic charge e0) that will add up to the total charge Qmax.

 
S1μ1S2μ2SiμiSi+1SnμnSn+1

The probability of being in each of the states Si is labeled as Pi, and the equilibrium constant of each step is given by

 
μi=exp[zi(VVi)kT],i=1n,

where zi is the charge (in units of e0) of step i, and Vi is the membrane potential that makes the equilibrium constant equal 1. In steady state, the solution of Pi can be obtained by combining

 
Pi+1Pi=μi,i=1nandi=1i=n+1Pi=1,

giving

 
Pi+1=m=1iμm1+j=1nk=1jμk,i=1nandP1=11+j=1nk=1jμk.

We define the reaction coordinate along the moved charged q as

 
qi=j=1izj,i=1n.

The Q-V curve is defined as

 
Q(V)=i=1nqiPi+1.

Then, replacing Pi yields

 
Q(V)=i=1n[j=1izj][m=1iμm]1+j=1nk=1jμk,

or written explicitly as a function of V:

 
Q(V)=i=1n[j=1izj][m=1iexp[zm(VVm)kT]]1+j=1nk=1jexp[zk(VVk)kT].
(2)

Eq. 2 is a general solution of a sequential model with n + 1 states with arbitrary valences and Vi’s for each transition. We can easily see that Eq. 2 has a very different form than Eq. 1, except when there is only a single transition (n = 1). In this latter case, Eq. 2 reduces to Eq. 1 because z1 and V1 are equal to Qmax and V1/2, respectively. For the more general situation where n > 1, if one fits the Q(V) relation obeying Eq. 2 with Eq. 1, the fitted Qmax value will not correspond to the sum of the zi values (see examples below and Fig. 1). A simple way to visualize the discrepancy between the predicted value of Eqs. 1 and 2 is to compute the maximum slope of the Q-V curve. This can be done analytically assuming that Vi = Vo for all transitions and that the total charge Qmax is evenly divided among those transitions. The limit of the first derivative of the Q(V) with respect to V evaluated at V = Vo is given by this equation:

 
dQ(V)dV|V=V0=Qmax(n+2)12nkT.
(3)

From Eq. 3, it can be seen that the slope of the Q-V curve decreases with the number of transitions being maximum and equal to Qmax /(4kT) when n = 1 (two states) and a minimum equal to Qmax /(12kT) when n goes to infinity, which is the continuous case (see next paragraph).

Infinite number of steps

Eq. 2 can be generalized to the case where the charge moves continuously, corresponding to an infinite number of steps. If we make

 
zi=Qmax/n,i=1n,Vi=Vo,i=1n,

then all µi = µ, and we can write Eq. 2 as the normalized Q(V) in the limit when n goes to infinity:

 
Qnor(V)=limni=1n[j=1iQmaxn]m=1iexp[Qmax(VVo)nkT]Qmax[1+i=1nj=1iexp[Qmax(VVo)nkT]]=[Qmax(VVo)kT]exp[Qmax(VVo)kT]+kTQmax(VVo)[exp[Qmax(VVo)kT]1].
(4)

Eq. 4 can also be written as

 
Qnor(V)=12[1+coth[Qmax(VVo)2kT]2kTQmax(VV0)],
(5)

which is of the same form of the classical equation of paramagnetism (see Kittel, 2005).

Examples

We will illustrate now that data generated by Eq. 2 can be fitted quite well by Eq. 1, thus leading to an incorrect estimate of the total charge moved. Typically, the experimental value of the charge plotted is normalized to its maximum because there is no knowledge of the absolute amount of charge per molecule and the number of molecules. The normalized Q-V curve, Qnor, is obtained by dividing Q(V) by the sum of all the partial charges.

Fig. 1 shows Qnor computed using Eq. 2 for one, two, three, four, and six transitions and for the continuous case using Eq. 5 (squares) with superimposed fits to a two-state Boltzmann distribution (Eq. 1, lines). The computations were done with equal charge in each step (for a total charge Qmax = 4e0) and also the same Vi = −25 mV value for all the steps. It is clear that fits are quite acceptable for cases up to four transitions, but the fit significantly deviates in the continuous case.

Considering that experimental data normally have significant scatter, it is then quite likely that the experimenter will accept the single-transition fit even for cases where there are six or more transitions (see Fig. 1). In general, the case up to four transitions will look as a very good fit, and the fitted Qmax value may be inaccurately taken and the total charge transported might be underestimated. To illustrate how bad the estimate can be for these cases, we have included as insets the fitted value of Qmax for the cases presented in Fig. 1. It is clear that the estimated value can be as low as a fourth of the real total charge. The estimated value of V1/2 is very close to the correct value for all cases, but we have only considered cases in which all Vi’s are the same.

It should be noted that if µi of the rightmost transition is heavily biased to the last state (Vi is very negative), then the Qmax estimated by fitting a two-state model is much closer to the total gating charge. In a three-state model, it can be shown that the fitted value is exact when V1→∞ and V2→−∞ because in that case, it converts into a two-state model. Although these values of V are unrealistic, the fitted value of Qmax can be very close to the total charge when V2 is much more negative than V1 (that is, V1 >> V2). On the other hand, If V1 << V2, the Q-V curve will exhibit a plateau region and, as the difference between V1 and V2 decreases, the plateau becomes less obvious and the curve looks monotonic. These cases have been discussed in detail for the two-transition model in Lacroix et al. (2012).

We conclude that it is not possible to estimate unequivocally the gating charge per sensor from a “single-Boltzmann” fit to a Q-V curve of a charge moving in multiple transitions. The estimated Qmax value will be a low estimate of the gating charge Qmax, except in the case of the two-state model or the case of a heavily biased late step, which are rare occurrences. It is then safer to call “apparent gating charge” the fitted Qmax value of the single-Boltzmann fit.

Addendum

The most general case in which transitions between states include loops, branches, and steps can be derived directly from the partition function and follows the general thermodynamic treatment by Sigg and Bezanilla (1997), Chowdhury and Chanda (2012), and Sigg (2013). The reaction coordinate is the charge moving in the general case where it evolves from q = 0 to q = Qmax by means of steps, loops, or branches. In that case, the partition function is given by

 
Z=iexp(qi(VVi)kT).
(6)

We can compute the mean gating charge, also called the Q-V curve, as

 
Q(V)=q=kTZZ=kTdlnZdV=iqiexp(qi(VVi)kT)iexp(qi(VVi)kT).
(7)

The slope of the Q-V is obtained by taking the derivative of 〈q〉 with respect to V:

 
dQ(V)dV=(kT)2d2lnZdV2.
(8)

Let us now consider the gating charge fluctuation. The charge fluctuation will depend on the number of possible conformations of the charge and is expected to be a maximum when there are only two possible charged states to dwell. As the number of intermediate states increases, the charge fluctuation decreases. Now, a measure of the charge fluctuation is given by the variance of the gating charge, which can be computed from the partition function as:

 
Δq2=q2q2=(kT)2(ZZ(ZZ)2)=(kT)2d2lnZdV2.
(9)

But the variance (Eq. 9) is identical to the slope of Q(V) (Eq. 8). This implies that the slope of the Q-V is maximum when there are only two states.

Acknowledgments

We thank Dr. Daniel Sigg for the contribution of the general derivation in the Addendum and all reviewers for pointing out several important improvements in the presentation. We also thank Dr. Rebecca Segal (Department of Mathematics at VCU) for her insightful comments.

This work was supported by National Institutes of Health grant GM030376 (to F. Bezanilla) and Clinical and Translational Science Award KL2TR000057 (to C.A. Villalba-Galea).

Kenton J. Swartz served as editor.

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