Time-dependent effects of bupivacaine on different Kv channels at + 60 mV at different concentrations have been analyzed in several studies, using voltage clamp technique (see Gonzalez et al., 2001; Nilsson et al., 2003). As can be inferred from these studies, a current peak can be observed at higher voltages. The voltage dependence of this induced peak, however, has been surprisingly little studied (but see Gonzalez et al., 2001).

We seek to find sufficient and necessary prerequisites for the existence of such a peak. We will first derive this for a three-state scheme, then use similar techniques to derive criteria for the four-state scheme, and in the Appendix for an n-state scheme. The techniques are generally those of dynamical systems and matrix algebra, which we (Gouwens et al., 2010; Zeberg et al., 2010, 2015; Sahlholm et al., 2016) and others (for an overview see Koch, 2004; Izhikevich, 2007) have used extensively for analyzing these systems. The differential equation for Scheme 1 written in matrix form is x′ = Ax where

with the general solution

and where V_i and r₁ are the eigenvectors and eigenvalues of the transition matrix A. c_i are constants dependent on the initial conditions. Experimentally, voltage clamp experiments are typically done with a strongly negative potential as an initial condition, meaning that all ion channels are in the (first) closed state, i.e.,

We will now use the Putzer algorithm to solve O(t). The Putzer Algorithm is a method for analytically evaluating matrix exponentials using only eigenvalues and components in the solution of a relatively simple linear system (Putzer, 1966). This approach might seem a little bit cumbersome for solving the three-state scheme but will later enable us to solve not just the four-state model but also the general case (complete proof given in the Appendix). By the Putzer algorithm, the solution to x′(t) = Ax(t), where A is a 3 × 3 matrix, can be written on the form

Where p_i(t) and M₁ are defined as follows. Define 3 × 3 matrices M₁, M₂, and M₃ by the formula

and let the functions p₁(t), p₂(t), and p₃(t) be given by solutions to the differential system

Note that here the eigenvalues can be in any given order, and we are not assuming that they are ordered in any particular way. We will now investigate p₃. Solving the equation system above reveals that

Since er3t is only to be found in p₃(t) we have that

To get the pre-exponential factor c₃V₃ we write

The result above will be a 3 × 1 vector. Since we are solving for O(t), we are only interested in the second element of this vector. After matrix multiplication, the pre-exponential factor to er3t in the expression for O(t) is

Since the sum of the eigenvalues of a matrix equals the trace, the expression be further reduced as in Equation (12). One of the eigenvalues is always zero so let r₁ = 0, and assume without loss of generality that r₃ > r₂. r₃ is then the slowest decaying term (note that all eigenvalues are non-positive). For some positive value of t the following inequalities must hold

why c₃V₃ will be the dominant term as t → ∞. Thus, a peak will exist for O(t) if the slowest decaying term r₃ has a positive pre-exponential factor, i.e.,

By the assumption that r₁ > r₃ > r₂ and that α is always positive it follows that this is true exactly when δ + r₃ < 0. Using Vieta's relationships in conjunction with the border condition, r₃ = −δ we can reduce δ + r₃ < 0 to a simpler form. This is due to the fact that there are three equations of the Vieta relationships for det(A−rI) = 0 if A is a 3 × 3 matrix. In conjunction with r₃ = −δ we have four equations and three eigenvalues, why it is possible to fully eliminate the eigenvalues. Some straight forward algebra reveals that

Thus a sufficient condition for a peak to exists in a three-state scheme is that the opening rate constant α must be greater than the dissociation rate constant δ.

For the four-state model the solution to p₄(t) is obtained by solving the natural extended version of Equation (8), and by the Putzer algorithm we have

To get the pre-exponential factor c₄V₄ we write

where U is some vector with a zero on the third row (due to the tri-diagonal nature of the matrix A, the number of non-zero diagonals increase with two for each multiplication and we can omit lower terms than A²). After matrix multiplication of Equation (17) and taking the third element of the resulting vector we obtain the pre-exponential factor to er4t in the expression for O(t),

Again, the expression is reduced using the relationship between the sum of eigenvalues and the trace. Now we can assume without loss of generality that r₄ > r₃ > r₂ and r₁ = 0. As for the three-state model, a peak will exist in the waveform if the pre-exponential factor to the slowest decaying eigenvalue is positive. This is true for a model with an arbitrary number of closed states, since all eigenvalues are real. Then by a sign analysis, the pre-exponential factor to er4t is positive if, and only if,

Again using the Vieta's relationships this can be reduced to

For the interested reader the solution to the general case is given in the Appendix. The technique used in the Appendix is the same as for the three and four state model. It transpires that the pre-exponential factor to ernt is

Again, let r_n > r_n−1 > ⋯ >r₂ and let r₁ = 0. A similar sign analysis can be performed, yielding that the pre-exponential factor to the slowest decaying eigenvalue is positive exactly when

If β = 0 this criterion is reduced to α > δ. We observe that the influence of β seems to increase with the number of closed states, whereas γ is not involved in the criterion for a peak. The appendix includes a proof that the association rate γ does not affect the criterion for a peak. Figure 1 shows regions associated with the existence of a peak in the β − δ plane, scaled for α.

To analyze the role of the rate constants in the height of the peak, we introduce a factor ψ such that the peak probability O_p can be expressed as

where ψ is dependent on the rate constants and o_ss is the steady state open probability in the presence of a blocking agent (see Nilsson et al., 2003). For the three-state model the peak will have its maximum at time

This formula was obtained by taking the time derivate of the solution for O(t) (see Appendix, A18) and equating this expression with zero. Taking O(t_p) yields

Using our analytical solution it is possible to analyze how extreme values of the rate constants affect the peak height. Extreme values are shown in Table 1.

In contrast to the criterion for the existence of a peak current, which involves only two rate constants (i.e., α and δ), the peak height depends on all rate constants of Equation (1) (i.e., α, β, γ and δ). To facilitate the use of the derived relationships, we can choose simplifying conditions for the system. Thus, assume that we investigate the blocking effect at high voltage steps (i.e., assuming β = 0 and α > γ + δ) and at a concentration equal to the K_d-value of the blocking agent (i.e., assuming γ = δ). Algebraically, the following formula is obtained

To analyze how the voltage affects the amplitude of the induced peak we used Monte Carlo simulations. Let α and β be described according to Eyring-Polanyi rate theory (Evans and Polanyi, 1935; Eyring, 1935) as

where k is the pre-exponential factor (or the characteristic frequency factor), V_1/2 is the potential for which α = β and s is a slope factor describing the influence of the potential. Again, let the concentration of blocking agent be at K_d-value (i.e., γ = δ).

Using Monte Carlo simulations in conjunction with analytical tools, we found three different regions in the δ−V plane: (i) One area, which we call A1, in which there is no open probability peak; (ii) a second area, A2, in which there is a peak and the relative amplitude (peak open probability relative to steady state with no blocking agent) decreases with potential and (iii) a third area, A3, in which there is a peak, but the relative amplitude increases with potential, approaching a value of one. Area A1 was defined using the criteria obtained above, i.e. α > δ. The border between A2 and A3 was obtained in the following way. Using custom written software in Mathematica 11.0 (Wolfram Inc.), we evenly assigned values using the built-in function RandomReal in parameter space of the right half of Figure 2 and investigated the voltage dependence (see Equations 26–28). The border between A2 and A3 were drawn using the built-in function ContourPlot. Figure 2 shows the three topological regions.

The findings in Figure 2 are to some extent congruent with the findings of the relatively few experimental investigations of local anesthetic effects on non-inactivating Kv channels (see e.g. the effect of bupivacaine on Kv1.5 channels, described by Gonzalez et al. (2001), where the relative peak amplitude goes from a potential region where there is no peak, i.e., corresponding to A1, and to a potential region where the relative peak increases with potential, i.e., corresponding to A3). Clearly this issue requires further experimental investigations to be clarified.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer AS and handling Editor declared their shared affiliation.