The Equivalent Two-State Model of Multi-State Ion Channels

2. The Open Probability and Stimuli Sensitivity of the Two-State Channel Model

As it was commented, multi-state channel models eventually represent a more complete physical description of ion channels, but they have a significant propensity to overparameterization and should be applied with care. The main features of multi-state models will be discussed later, and we center our attention on the simpler two-state model in the present section. Although the two-state Markov description constitutes a rather rough approximation, it captures the key phenomenological features of the gating dynamics of ion channels, i.e., their dichotomous nature and the stimulus-modulated open probability. Furthermore, by construction, the two-state model yields the right mean open/closed dwelling times, and the steady state probability for the open channel. So, such a simple model is still being frequently used, including systems with complex mechanistic [7,8,9,10].

The transition dynamics of a two-state channel model can be represented by the simple kinetic Scheme 1.

The corresponding rate equation for the open state, in combination with the normalization condition of the two states, is given by:

$${\displaystyle \frac{\partial P_O}{\partial t}}= k^{+}\left(1-P_O\right)-k^{-}{P}_O$$

(1)

Where $P_O$ is the open probability, while $k^{+}$ and $k^{-}$ are the stimuli-dependent rate constants for the close to open, and open to close state transitions, respectively. In the steady state (${\displaystyle \frac{\partial P_O}{\partial t}}=0$) the open probability adopts the following general form:

$$P_O={\displaystyle \frac{1}{1+\frac{k^{-}}{k^{+}}}}$$

(2)

The ratio $K={\displaystyle \frac{k^{+}}{k^{-}}}$ is known as the equilibrium constant of the reaction represented in the Scheme 1, and equation 4 can also be read as $P_O=1/(1+K^{-1})$. As it has been presented in previous papers, assuming the Eyring’s kinetic model, which obeys the Maxwell-Boltzmann distribution for the activation energy barriers determining the rate constants of the opening and closing dynamics [7,8,9,10,22,23,24,25], the equilibrium constant and the total free energy change correlate as $∆G_{Total}=-RTlnK$, and the open probability in the steady for the two-state channel model is given by:

$$P_O={\displaystyle \frac{1}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\Delta G_{Total}}{RT}\right)}}$$

(3)

$\Delta G_{Total}$ in equation (5) can be expressed as:

$${\Delta G_{Total}=∆G}^{‡}+{∆G}_{Ext}$$

(4)

Where ${∆G}^{‡}$ is the conformational free energy change, associated to the transition from the close to the open state at constant temperature, membrane potential, in the absence of ligands, membrane tension, radiation or any other external field forces. ${∆G}_{Ext}$ represents the additional free energy component to $\Delta G_{Total}$, under the action of external force fields. It is important to remark that in the particular case of ligands (L), or chemical species in general, their contribution as a ${∆G}_{Ext}$ component in $\Delta G_{Total}$ is given by ${∆G}_{Ligand}=k_{B}Tln({\displaystyle {\sum }_{i=0}^b}\left({\displaystyle \frac{b}{i}}\right){\left({\displaystyle \frac{\left[L\right]}{K_d}}\right)}^i)/({\displaystyle {\sum }_{i=0}^b}\left({\displaystyle \frac{b}{i}}\right){\left({\displaystyle \frac{\left[L\right]}{K_d}}\right)}^{i}exp\left({\displaystyle \frac{\frac{i}{4}∆∆H_{Max}}{RT}}\right))$, where [L] is the ligand concentration, b is the number of binding sites, K_d is the ligand affinity of a single agonist binding site when the channel is open and ΔΔH_Max is the maximal change in enthalpy in saturation conditions [26], and this expression is independent on the channel sensitivity to other nonchemical stimuli. So, equation (5) also includes these cases. External fields which reduce $\Delta G_{Total}$ with respect to ${∆G}^{‡}$ (${∆G}_{Ext}$ < 0) will favor the channel opening, while those increasing $\Delta G_{Total}$ (${∆G}_{Ext}$ > 0) will cause the opposite effect.

For the sake of simplicity, let’s define the parameter $\Delta \mathcal{G}\left(X\right)$:

$$\Delta \mathcal{G}\left(X\right)={\displaystyle \frac{\Delta G_{Total}\left(X\right)}{RT}}$$

(5)

which accounts for how many times $RT$ is nominally contained in $\Delta G_{Total}$ (the relative total free energy change). $X$ stands for the environmental or external stimulus ($X$ = V, $\gamma$, T, [L], etcetera; membrane potential, surface tension, temperature, and ligand concentration, respectively). Understanding that the relative total free energy change is a function of $X$, from now we will refer to this parameter simply as $\Delta \mathcal{G}$ instead of $\Delta \mathcal{G}\left(X\right)$, for the sake of simplicity (as it is with $P_O$). Thus, by deriving equation (3) the general form for the channel sensitivity to a stimulus $X$ (${\displaystyle \frac{\partial P_O}{\partial X}}$)can be written as:

$${\displaystyle \frac{\partial P_O}{\partial X}}=P_O^2\exp \left(\Delta \mathcal{G}\right)(-{\displaystyle \frac{\partial \Delta \mathcal{G}}{\partial X}})$$

(6)

This relation constitutes a generalized form of the ones obtained in a previous work [in preparation] for the cases of $X=T,V,\mathrm{a}\mathrm{n}\mathrm{d} \gamma$. The term ($-{\displaystyle \frac{\partial \Delta \mathcal{G}}{\partial X}}$) in equation (6) accounts for the sensitivity of the thermodynamic state of the channel (the relative total free energy) to be perturbated by the action of $X$. The corresponding minus sign indicates that if the relative total free energy of the channel is reduced by the action of the external field $X$ (${\displaystyle \frac{\partial \Delta \mathcal{G}}{\partial X}}<0$), favoring the channel opening, the corresponding sensitivity will be positive. Additionally, the term $-{\displaystyle \frac{\partial \Delta \mathcal{G}}{\partial X}}$ also contains the effect of the “coupling factor”, as it was defined and developed for the particular case of mechano-sensitivity in a previous paper [in preparation].

3. The Two-State Equivalent of Multi-State Channel Models

As it was mentioned above, although multi-state models can potentially offer a more complete description of the gating dynamics, they have a natural tendency of overparameterization, which not only can lead to numerically divergent kinetic parameters of the same system [21], but also limits physical interpretations (the overdetermination problem), or could even lack physical meaning. In this section we introduce some parameters which account for the pondered contribution of the kinetic rate constants for transitions between the different channel’s states (independently of their nature, conductivity, or any other condition), which gives rise to an equivalent two-state channel model. Thus, the results obtained above regarding the open probability and the sensitivity of the two-state model can be qualitatively extrapolated to the case of multi-state channel models, regardless of the number of states, circumventing the potential problem of overparameterization.

Let’s consider a single channel with n open states (O), m close states (C), and l inactive states (I). Each open state can have a different conductivity, while the close and inactive states have zero conductivity; see the schematic representation Scheme 2.

Then, the corresponding n rate equations for the open states have the following form (we only included here that of the k-th open state for representative purposes):

$${\displaystyle \frac{\partial P_{Ok}}{\partial t}}= {\displaystyle \sum _{j=1}^m}{k}_{Cj-Ok}{P}_{Cj}+{\displaystyle \sum _{j=1}^l}{k_{Ij-Ok}P}_{Ij}+{\displaystyle \sum _{j=1 (j\ne k)}^n}{k_{Oj-Ok}P}_{Oj}-\left[{\displaystyle \sum _{j=1}^m}{k}_{Ok-Cj}+{\displaystyle \sum _{j=1}^l}{k}_{Ok-Ij}+{\displaystyle \sum _{j=1 \left(j\ne k\right)}^n}{k}_{Ok-Oj}\right]P_{Ok}$$

(7)

Where P_Ok, P_Cj, and P_Ij, are the probabilities that the channel is in the open state k (Ok), the close state j (Cj), and the inactive state j (Ij), respectively. k_Cj-Ok, k_Ij-Ok, k_Oj-Ok, are the kinetic rate constants for Cj → Ok, Ij → Ok, and Oj → Ok transitions, respectively; while k_Ok-Cj, k_Ok-Ij, k_Ok-Oj are the kinetic rate constants corresponding to Ok → Cj, Ok → Ij, and Ok → Oj transitions, respectively. Note that in equation (7) the positive terms of the sum account for the total rate at which the Ok state is occupied from any other state, while the negative terms of the sum account for the total rate at which the Ok state transits to any other state. By summing the set of n equations represented in (7) we have:

$${\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)= {\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{k}_{Cj-Oi}{P}_{Cj}+{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^l}{k_{Ij-Oi}P}_{Ij}-\left[{\displaystyle \sum _{i=1}^n}{\left({\displaystyle \sum _{j=1}^m}{k}_{Oi-Cj}+{\displaystyle \sum _{j=1}^l}{k}_{Oi-Ij}\right)P}_{Oi}\right]+{\displaystyle \sum _{i=1}^n}\left({\displaystyle \sum _{j=1 \left(j\ne k\right)}^n}{k_{Oj-Oi}P}_{Oj}-{\displaystyle \sum _{j=1 \left(j\ne k\right)}^n}{k}_{Oi-Oj}{P}_{Oi}\right)$$

(8)

In equation (8), the two first terms of the sum account for the total rate at which any conductive state (O) is occupied from any other non-conductive state (C or I), the third term accounts for total rate at which any conductive state (O) transit to any other non-conductive state (C or I), while the fourth term accounts for the net transition rate between all the different open states (O). It is important to note that this last term is zero because the net rate at which the state Oi is occupied from the state Oj appears as a positive contribution in the equation of ${\displaystyle \frac{\partial P_{Oi}}{\partial t}}$, but exactly the same amount appears as negative contribution in the equation of ${\displaystyle \frac{\partial P_{Oj}}{\partial t}}$ (the rate at which Oj transits to Oi). Thus, multiplying and dividing the two first terms in equation (14) by ${\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij}$ (the probability of the occurrence of any C or I state), and considering that $1-{\displaystyle {\sum }_{i=1}^n}{P}_{Oi}={\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij}$ (from the normalization condition), and multiplying and dividing the third term by ${\displaystyle {\sum }_{j=1}^n}{P}_{Oj}$ (the probability of the occurrence of any O state) we have:

$${\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)= \left[{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{P_{Cj}}{({\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij})}}{k}_{Cj-Oi}+{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^l}{\displaystyle \frac{P_{Ij}}{({\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij})}}{k}_{Ij-Oi}\right]\left(1-{\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)-\left[{\displaystyle \sum _{i=1}^n}\left({\displaystyle \sum _{j=1}^m}{k}_{Oi-Cj}+{\displaystyle \sum _{j=1}^l}{k}_{Oi-Ij}\right){\displaystyle \frac{P_{Oi}}{{\displaystyle {\sum }_{i=1}^n}{P}_{Oi}}}\right]\left({\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)$$

(9)

Note that the bracket in first in the term is the statistically-pondered (or average) kinetic constant for the transitions from any non-conductive state (C or I) to any conductive state (O). So, the more populated a given non-conductive state is (and larger the corresponding transitions rate), the greater its weight in the average kinetic constant for non-conductive → conductive states transitions. Thus, the statistical weight of every state (in mechanical statistics terms [27]) ponders the contribution of the corresponding conformation to the transition rate dynamic of the channel. This idea can be paralleled, since it conserves the same mathematical form, to that of the more familiar concept of gravity center from Classical Mechanics. That is a kind of “kinetic rate center” of such transitions or equivalently, an effective kinetic rate constant. Analogously, the bracket in the second term in equation (9) is the pondered (or average) kinetic constant for the transitions from any conductive state (O) to any non-conductive state (C or I). The more populated a conductive state is (and larger the corresponding transitions rate), the bigger its statistical weight in the average kinetic constant for conductive → non-conductive states transitions. Then, the effective (pondered or “gravity center-like”) kinetic constant (which in general depend on time and on the stimuli) for non-conductive → conducting states transitions ($K_{eff.}^{+}$) and conductive → non-conductive states transitions ($K_{eff.}^{-}$) can be mathematically defined as:

$$K_{eff.}^{+} = {\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{P_{Cj}}{({\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij})}}{k}_{Cj-Oi}+{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^l}{\displaystyle \frac{P_{Ij}}{({\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij})}}{k}_{Ij-Oi}$$

(10)

$$K_{eff.}^{-}={\displaystyle \sum _{i=1}^n}\left({\displaystyle \sum _{j=1}^m}{k}_{Oi-Cj}+{\displaystyle \sum _{j=1}^l}{k}_{Oi-Ij}\right){\displaystyle \frac{P_{Oi}}{{\displaystyle {\sum }_{i=1}^n}{P}_{Oi}}}$$

(11)

From the typical assumption (experimentally supported) of Boltzmann statistics for the rate constants (k_Cj-Oi, k_Ij-Oi, k_Oi-Cj, k_Oi-Ij) of channel opening and closing in thermal equilibrium with the environment, such parameters can be modeled as exponentially-dependent on the stimuli $X$ (the Eyring model) [7,8,40]. So, the stimuli dependence of $K_{eff.}^{+}$ and $K_{eff.}^{-}$ in the steady state can be obtained by multiexponential fitting of the corresponding experimental data, note that the statistical weights ($w_i={\displaystyle \frac{P_{Yi}}{{\displaystyle {\sum }_{i=1}^n}{P}_{Yi}}}$; Y=C,I,O) in (10) and (11) are the coefficients of the exponentials while the stimuli $X$ are in their arguments (if only one thermodynamic sate were present in the conductive or non-conductive conformations, a single exponential would be observed). That is, the paradigm of the Eyring model implies that the effective kinetic constants ($K_{eff.}^{+/-}(X)={\sum }_{i=1}^{n,m or l}{w}_i{e}^{{\lambda }_{i}X}$) have the same mathematical form to that resulting from the Müntz-Szász theorem (in the context of Dirichlet series equivalence to Taylor series) which states that any smooth function can be represented in the form $f(x)={\sum }_{i=1}^n{A}_i{e}^{{\lambda }_{i}x}$. This fact means that the experimental $K_{eff.}^{+/-}(X) vs X$ curves can always be fitted by a multiexponential curve, with the proper number of exponentials (accounting for the different kinetically indistinguishable states).

The advantage of knowing (with a physical support) that multiexponential fitting is the key to obtain $K_{eff.}^{+}$ and $K_{eff.}^{-}$ from experiments is that it can allow to find the number of states, their statistical weights, and their ΔG_ij in a multi-state channel, without any initial assumption in this regard, as it is currently done. Substituting definitions (10) and (11) in equation (9), the last can be rewritten as:

$${\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)= K_{eff.}^{+}\left(1-{\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)-K_{eff.}^{-}\left({\displaystyle \sum _{i=1}^n}{P}_{Oi}\right)$$

(12)

As we mentioned above, the probability of the occurrence of any O state is $P_{cond.}={\displaystyle {\sum }_{i=1}^n}{P}_{Oi}$, thus:

$${\displaystyle \frac{\partial P_{cond.}}{\partial t}}= K_{eff.}^{+}\left(1-P_{cond.}\right)-K_{eff.}^{-}{P}_{cond.}$$

(13)

Note that equation (13) has the same form as that of the two-state channel model (1) with stimuli and time dependent kinetic constants. In other words, a multi-state channel behave like a two-state model (non-conductive ↔ conductive; Scheme 3), where the non-conductive state is a virtual one with an occurrence probability of $P_{non-cond.}={\displaystyle {\sum }_{j=1}^m}{P}_{Cj}+{\displaystyle {\sum }_{j=1}^l}{P}_{Ij}$ and a transition rate constant to the conductive state of $K_{eff.}^{+}$; while the conductive state is a virtual one characterized by an occurrence probability of $P_{cond.}={\displaystyle {\sum }_{j=1}^n}{P}_{Oj}$ and a transition rate constant to the non-conductive state of $K_{eff.}^{-}$.

In the steady state $K_{eff.}^{+}$ and $K_{eff.}^{-}$ are constant; accordingly, for a single channel with a non-zero average current in experimental terms (a channel which does not spent most of the time in non-conductive states, just to prevent the trivial solution), the solution of equation (13) is analogous to that of the general relation for the two-state channel model in (2).

$$P_{cond.}={\displaystyle \frac{1}{1+\frac{K_{eff.}^{-}}{K_{eff.}^{+}}}}$$

(14)

Thus, from the experimental steady state open probability of the single channel, the corresponding time-independent equilibrium constant ($K_{eff.}=K_{eff.}^{+}/K_{eff.}^{-}$) can be straightforwardly obtained. Note that for chemical reactions at the equilibrium (in this case, the two-state “reaction” at a given temperature and pressure which reached the steady state), under the “dilute conditions” approximation (the intrinsic physical properties of the channels are not interfered by the proximity of other channels), and from purely thermodynamic principles the general relation between the free energy change and the equilibrium constant is derived $\Delta G_{Total}=-{RTlnK}_{eff.}$ [28,29,30]. In the steady state, the results in the previous section 2 regarding the open probability and the correspondent sensitivity can be extrapolated here in the same way. Otherwise, the more general (time-dependent or not $K_{eff.}$) relation for the sensitivity holds:

$${\displaystyle \frac{\partial P_{cond.}}{\partial X}}=P_{cond.}^2\left(-{\displaystyle \frac{\partial K_{eff.}^{-1}}{\partial X}}\right)$$

(15)

Note that equation (15) has the same mathematical form to that of equation (6), so it is its generalized form in multi-state channels. As well, the term $\left(-{\displaystyle \frac{\partial K_{eff.}^{-1}}{\partial X}}\right)$ accounts for the stimuli sensitivity of the equilibrium shift (Scheme 3), which can be interpreted as and effective thermodynamic state (as discussed in the previous section).