eq . 2 can be generalized to the case where the charge moves continuously , corresponding to an infinite number of steps . if we makezi = qmax / n ,  i=1  n ,  vi = vo ,  i=1  n , then all i =  , and we can write eq . 4 can also be written as(5)qnor(v)=12[1+coth[qmax(vvo)2kt]2ktqmax(vv0)],which is of the same form of the classical equation of paramagnetism ( see kittel , 2005 ) . 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 . 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 . examples of normalized q - v curves for a qmax = 4 computed with eq . 2 for the cases of one , two , three , four , and six transitions and the continuous case using eq . 5 ( squares ) . all the q - v curves were fitted with eq . 1 ( lines ) . the insets show the fitted valence ( qmax ) and half - point ( v1/2 ) . 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 . 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 i 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 . 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 . 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(6)z=iexp(qi(vvi)kt).we can compute the mean gating charge , also called the q - v curve , as(7)q(v)=q=ktzz = ktdlnzdv=iqiexp(qi(vvi)kt)iexp(qi(vvi)kt).the slope of the q - v is obtained by taking the derivative of q with respect to v:(8)dq(v)dv=(kt)2d2lnzdv2 . 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:(9)q2=q2q2=(kt)2(zz(zz)2)=(kt)2d2lnzdv2.but the variance ( eq . this implies that the slope of the q - v is maximum when there are only two states .
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 .