An Expanded Markov State Model - Transition Path Theory Framework for Ion Conduction Reactive Pathways through Membrane Channels

Abstract

The combination of Markov State Models (MSMs), which approximate the dynamics of a system as stochastic transitions among a finite set of microstates, with transition path theory (TPT) enables a rigorous identification of the dominant reactive pathways controlling slow processes in complex systems. Although the MSM/TPT framework has been successfully applied to a wide range of problems, its use in analyzing unbound processes, such as ion conduction through channels, presents special conceptual challenges. The standard formulation of MSM/TPT, built on a picture of reactive transitions between two metastable basins, is incompatible with an unbound process like ion conduction. An expanded formulation of the MSM/TPT framework is introduced to enable a characterization of ion channels as well as a wide range of pumps, motors, and molecular machines.

Graphical Abstract

Kinetic models with discrete states have long been utilized to describe membrane transport processes.^1,2 Such models also provide a useful representation to analyze and interpret the results from detailed MD simulations.^3–5 Closely related, stochastic dynamics can be represented with a finite set of discrete states with a finite lag-time in a model referred to as a Markov State Model (MSM).⁶ The general procedure to map a molecular process onto a MSM consists in defining a finite set of discrete states associated with the configuration of the simulated system, which is then used to determine the transition probability over a finite lag-time between each pair of states using the information provided by MD trajectories. While there are challenges due to statistical convergence and numerical stability, the general procedure to map a molecular process onto a MSM is well-established. The outcome is a dynamical model with discrete states representing the dynamics of the system as a stochastic Markov process with a finite lag-time.^7–10 A powerful approach to analyze this stochastic dynamics is provided by transition path theory (TPT).^11,12 TPT presents a statistical description of the dynamics in terms of local spatial quantities.^13–15 A key ingredient in TPT is the committor probability, $q_i$, that the system will first reach the product state starting from state $i$. In the context of MSMs, TPT offers a rich framework to identify dominant reaction pathways and rate limiting steps,^11,12 that has been used to study a number of complex systems.^16–19

While the MSM/TPT framework is well established, there are specific conceptual challenges when trying to describe unbound processes such as the conduction of ions through a channel. In their traditional formulations, MSM assumes a finite set of microstates and TPT is built on the concept of reactive transitions between two metastable basins corresponding to reactant and product states. However, because an ion channel is an open system able to undergo unbound conduction events, a framework constructed with a finite set of microstates is inadequate. For example, in the case of ion conduction, an infinite number of pathways can lead to indistinguishable microstates of occupancy of the channel. Thus, standard procedures for monitoring and counting the transitions between microstates are inadequate to meaningfully capture the kinetic of system. Similar situations are encountered in the function of a wide range of molecule machines, such as membrane transporters and pumps, polymerases, microtubule growth, and motors processing along filaments.

To illustrate the central issue, let us consider an equilibrium MD simulations of ion channels embedded in a lipid membrane with periodic boundary conditions (PBC). While the system undergoes fluctuations over the course of a trajectory, with forward and backward ion conduction events, there is no net ionic flux across the channel (Figure 1). In principle, carrying an MSM analysis should begin with the identification of a set of discrete microstates associated with the configuration of the ions, water and protein atoms within the pore or in the neighborhood of the permeation pathway (Figure 2). It is relatively straightforward to define a finite set of microstates from the information of all the possible configurations of the pore region of a channel from a simulation, and any instantaneous configuration along the MD trajectory can be mapped to one of the discrete microstates.²⁰ From this, the number of transitions can be counted to try to determine the transition probability $T_{ij}$ between any pair of states $i$ and $j$. However, such a mapping is unable to correctly represent the process of ion conduction. For instance, any specific microstate of the system can be reached by an indefinite number of routes, with ions coming from either sides of the channel. Because of the PBC, ions that leave the channel from one side can re-enter from the opposite side at a later time. Except in the limit of a very short lag-time, it is not possible to obtain a meaningful transition matrix $T_{ij}$ for ion conduction through a channel on the basis of the standard microstate mapping. Regardless of the number of discrete microstates, simply recording the transitions $i\to j$ between discrete states is insufficient to monitor the dynamical evolution of the system. To resolve this issue we expand the standard microstate mapping by incorporating additional information about the overall unbound transport process in the representation of the system. In the following we formulate such an expanded MSM framework, which is then illustrated in the case of ion conduction through channels. The expanded MSM framework is general and could also serve to describe the kinetics of a wide range of molecular machines.

Figure 1:

Ion conduction events observed from an equilibrium MD simulation of the MthK channel embedded in a lipid membrane.

Figure 2:

Example of a transition from discrete state $i$ to state $j$. Without additional information, it is impossible to know if there was a conduction event toward one side or the other.

The present formulation of the expanded MSM is based on the following reasoning. The central issue arise from the incomplete information embedded in the transition matrix $T_{ij}$ in the case of an unbound process because there is an inherent degeneracy in the set of microstates. This is resolved by monitoring the true status of the unbound process as a function of time. Inspired by our previous analysis of ion conduction using a time-correlation function of the displacement charge,²¹ we introduce the time-dependent counting number $K\left(t\right)$ which acts as a global label accounting for the total number of crossing events in and out of the pore region. Because the system is simulated with PBC, the absolute value of the counting number $K\left(t\right)$ is not known, or rather, it is arbitrary. The global index $K$ does not change if the system re-arranges its configuration among the discrete states within the permeation pathway region. On the other hand, there is a net conduction in the forward or backward direction when an ion leaves the channel to enter the bulk region. While the absolute value of $K$ does not matter, it is possible to monitor changes in $K$ from a MD trajectory. For example, we can use the convention that $K$ increases by an increment of 1 when an ion leaves the channel toward the extracellular side, or when an ion enters the channel from the intracellular side. Correspondingly, $K$ decreases by an increment of 1 when an ion enters the channel from the extracellular side, or when an ion leaves the channel toward the intracellular side. The significance of the global index $K$ is akin to considering an “unwrapped” trajectory for a system simulated with PBC when one is trying to monitor the mean-square-displacement (MSD) of a molecule to calculate the diffusion constant; the absolute position does not matter but one must keep track of the displacement across periodic images to correctly monitor the diffusion. Importantly, for all we know, the ions may have crossed the channel an indefinite number of times forward or backward. In effect, the global counting index $K$ is not bounded, and as a consequence, the number of discrete states describing the status of the system is literally infinite.

These two indices, $i$ for the local state and $K$ as a global offset for counting the ion conduction events, are used to monitor the dynamic evolution of the system over a lag-time $\tau$. Let us assume that we record the transition from state $i\left(t\right)$ and offset $K\left(t\right)$ to state $j(t+\tau )$ with offset $K^{\prime }(t+\tau )$. Possible microscopic transitions are illustrated in Figure 3. Using this framework, we imagine that there is a transition matrix for the system with four indices, $T\left(i,K\to j,K^{\prime }\right)$. The trajectory generated with PBC defines an infinite transition matrix. If the ions only make a transition to a different configuration within the channel over the lag-time $\tau$, then there was no change in the global offset and $K^{\prime }=K$. This transition should be recorded in the diagonal block in the matrix $T(i,K\to j,K)$. If $K^{\prime }=K+1$, then a forward transport transition occurred, which should be recorded in the non-diagonal block in the matrix $T(i,K\to j,K+1)$. If $K^{\prime }=K-1$, then a backward transport transition occurred, which should be recorded in the non-diagonal block in the matrix $T(i,K\to j,K-1)$. And so on and so forth. The value of the index $K$ as a function of time can be reconstructed from the recorded increment $K^{\prime }-K$ at each lag-time. Since the absolute value of $K\left(t\right)$ is immaterial, we can arbitrarily set its initial value to 0 at $t=0$. The index $K$ for the MD trajectory of Figure 1 is plotted in Figure 4. In practice, it is important to parse the MD trajectory with a sufficiently small time intervale to extract the time-series $K\left(t\right)$ unambiguously. One should note that the net conduction of one ion across the channel requires a net change of +2 of the $K$ index.

Figure 3:

Possible transitions (indicated by double arrows) in the multi-ion configuration of a channel involving incremental changes in the global offset index $K$.

Figure 4:

(Top) Global counting index $K$ as a function of time from the simulation shown in Figure 1. The index $K$ was set to 0 at $t=0$. The lag-time is 100 ps. (Bottom) Displacement charge $Q_{\text{d}}\left(t\right)$ as a function of time from the simulation shown in Figure 1.²¹ The time-dependence of $K$ and $Q_{\text{d}}$ can be followed from the transitions in Figure 1. It is observed that an increment $\Delta Q_{\text{d}}=\pm 1$ is associated with an increment $\Delta K=\pm 2$.

The complete transition matrix can be written as $T\left(i,K\to j,K^{\prime }\right)$ in terms of four indices, and the time-dependent probability of the system arising from the the stochastic process obeys the Chapman-Kolmogorov,

$$P\left(K^{\prime },j;t+1\right)=\sum _{K=-\infty }^{\infty }\sum _{i=1}^{N}P(K,i;t)T\left(i,K\to j,K^{\prime }\right)$$ (1)

where $t$ is a discrete time label taking the integer values 1, 2, …. Formally, the matrix $\mathbf{T}$ is of infinite dimension, as the global index $K$ is not bounded. However, we know by translational symmetry that all the images of the system with PBC are equivalent and that the value of the global offset $K$ is arbitrary under equilibrium conditions. Therefore, we can write that $T\left(i,K\to j,K^{\prime }\right)=T\left(i,K+K_0\to j,K^{\prime }+K_0\right)$, where $K_0$ is some arbitrary integer. Thus, the invariance implies that one could always choose $K_0$ such that $K=0$ and $K^{\prime }=\Delta K$, where the increment is defined as $\Delta K=K^{\prime }-K$. For this reason, it is possible to reduce the transition matrix to a block-like form with three indices, $T_{\Delta K}(i\to j)$, where $\Delta K$ is an integer. The complete matrix $\mathbf{T}$ is defined by $\Delta K$-blocks corresponding to $n\times n$ submatrices, where $n$ is the number of microstates associated with the channel region. Those include a diagonal block, $T_{\Delta K=0}(i\to j)$, and non-diagonal blocks of increasing order, $T_{\Delta K=\pm 1}(i\to j)$, and $T_{\Delta K=\pm 2}(i\to j)$, etc. Because the blocks of a given order are formally equivalent, this representation of the matrix $\mathbf{T}$ is particularly convenient to parse the MD trajectory and accumulate all the relevant information about the transitions. The increment $\Delta K$ monitors the number of forward transition in the periodic system. For example, a transition with $\Delta K=2$ occurs when one ion leaves the pore region toward the extracellular side while another ion enters the pore region from the intracellular side over one lag-time period. The present analysis based on equilibrium simulations includes transitions up to second order with $-2\le \Delta K\le 2$. Analysis of the MD trajectories shows that there are no transitions with $\left|\Delta K\right|>2$ with the present lag-time of 100 ps (see Figure 4). If necessary, the analysis can be extended to account for higher order transitions occurring with a longer lag-time. A Chapman-Kolmogorov test comparing the magnitude of the matrix elements ${\mathbf{T}}^{\left(2\tau \right)}$ and ${\left({\mathbf{T}}^{\left(\tau \right)}\right)}^2$ was carried out to examine the validity of the Markov assumption with a lag-time $\tau$ of 100 ps for the AMBER force field. It is observed that the residuals $‖{\mathbf{T}}^{\left(2\tau \right)}-{\left({\mathbf{T}}^{\left(\tau \right)}\right)}^2‖$ are much smaller than the individual matrix elements (Supplemental Information).

The dynamical evolution of the expanded MSM is a stochastic jump process with two indices, $i$ and $K$ over the lag-time $\tau$. In the present study, the transition probability matrix was extracted from equilibrium MD simulations, and the expanded MSM does not yield a net unidirectional ion flux. The connection of the expanded MSM to channel conductance can be established on the basis of a linear response analysis.^21,22 In the limit of small membrane potential, the conductance of a channel is $g={𝒟}_Q/k_{\text{B}}T$, where ${𝒟}_Q={\text{lim}}_{t\to \infty }{⟨{\left(Q_{\text{d}}\left(t\right)-Q_{\text{d}}\left(0\right)\right)}^2⟩}_0/2t$ is the diffusion constant of the displacement charge $Q_{\text{d}}$ in the absence of membrane potential.^21,22 The latter can be expressed in terms of the microstates $(i,K)$ of the expanded MSM,^5,23

$$Q_{\text{d}}\left(i,K\right)=\sum _{\alpha =1}^{n_i}ef\left(z_{\alpha }\left(i\right)\right)+e\frac{\left(K-K_0\right)}{2}$$ (2)

where $K_0$ is some arbitrary integer, $n_i$ is the number of ions in the pore, $z_{\alpha }\left(i\right)$ is the position of ion $\alpha$ in the microstate $i$ along the pore axis, and $0\le f\left(z\right)\le 1$ is a dimensionless bounded function representing the coupling to a transmembrane potential, which can be calculated from continuum electrostatics,^24,25 or MD simulations.²³ The first term in Eq. (2) corresponds to the average charge displacement of the pore occupancy microstate $i$ and the second term is associated with the unbound global movements of the ions in the unbound system. The incremental change of the displacement charge over one random step $\left(i,K\to j,K^{\prime }\right)$ can be written as $\left(Q_{\text{d}}\left(j,K^{\prime }\right)-Q_{\text{d}}(i,K)\right)$. The dynamical evolution of $K$ is a random walk corresponding effectively to a diffusive process at long times. Accordingly, the diffusion coefficient of the displacement charge is

$${𝒟}_Q=\underset{t\to \infty }{\text{lim}}\frac{1}{2t}⟨{\left(Q_{\text{d}}\left(i\right(t),K(t\left)\right)-Q_{\text{d}}\left(i\right(t),K(0\left)\right)\right)}^2⟩=\frac{e^2}{4}{𝒟}_K$$ (3)

where ${𝒟}_K$ is the diffusion coefficient along the $K$ index, ${𝒟}_K={\text{lim}}_{t\to \infty }⟨\left(K\left(t\right)-K(0{)}^2\right.⟩/2t$. The contribution involving the function $f\left(z\right)$ in Eq. (3) become negligible over the long time limit. It follows that the channel conductance can be expressed as, $g=e^2{𝒟}_K/4k_{\text{B}}T$.

$$g=\frac{{𝒟}_Q}{k_{\text{B}}T}=300\text{pS}$$

$$k=\frac{1}{20\text{ns}}$$

$${𝒟}_Q=e^{2}k$$

$${𝒟}_z=2.45\times {10}^{-3}{Å}^2/\text{ps},6.1\times {10}^{-4}{Å}^2/\text{ps}$$

The most straightforward route to determine ${𝒟}_K$ is by direct stochastic simulations of $K\left(t\right)$ from the transition probability matrix $T\left(i,K\to j,K^{\prime }\right)$. Another route (see Supplemental Information) is to numerically solve the random walk along the $K$ index using the characteristic function.^26,27 Alternatively, one may consider the net reactive flux between two regions, $A$ and $B$ within the TPT framework,^11,12

$$J=\frac{1}{2\tau }\sum _{i,K}\sum _{j,K^{\prime }}\pi (i,K)T\left(i,K\to j,K^{\prime }\right){\left(q\left(j,K^{\prime }\right)-q(i,K)\right)}^2$$ (4)

where $q(i,K)$ is the committor probability to reach the region $B$ before $A$ starting from state $(K,i)$ defined by the backward Kolmogorov equation,

$$q(K,i)=\sum _{K^{\prime }=-\infty }^{\infty }\sum _{i=j}^{N}T\left(i,K\to j,K^{\prime }\right)q\left(K^{\prime },j;t\right)$$ (5)

where $q=0$ in the A region, and $q=1$ in the B region. To proceed further, it is necessary to define the intermediate region $I$ separating $A$ and $B$. Because the expanded MSM includes transitions with an increment $\left|\Delta K\right|\le 2$, the intermediate region must include at least two adjacent $K$-blocks. We choose to define the intermediate region $I$ as $K$ equal to 0 or 1. It follows that the region $A$ is defined as all $K<0$, and the region $B$ is defined as all $K>1$. The backward Kolmogorov equation $\mathbf{q}=\mathbf{T}\mathbf{q}$ for the committor probability is reduced to,

$$\left(\begin{array}{l}{\mathbf{q}}_0\\ {\mathbf{q}}_1\end{array}\right)=\left(\begin{array}{ll}{\mathbf{T}}_0& {\mathbf{T}}_1\\ {\mathbf{T}}_{-1}& {\mathbf{T}}_0\end{array}\right)\left(\begin{array}{l}{\mathbf{q}}_0\\ {\mathbf{q}}_1\end{array}\right)+\left(\begin{array}{l}{\mathbf{T}}_2\cdot 1\\ {\mathbf{T}}_1\cdot 1+{\mathbf{T}}_2\cdot 1\end{array}\right)$$

which can be solved in closed form (see Supplemental Information). In the diffusive limit, the diffusion coefficient along $K$ can be related to the TPT flux via the relation, $J={𝒟}_K\left(\sum _{i=1}^N\pi (i,K)\right)/\Delta K$, where $\Delta K$ is the length of the intermediate region $I$ separating the regions $A$ and $B$. In principle, ${𝒟}_K$ must be estimated in the limit in which the length of the intermediate region $\Delta K$ becomes large. In the present analysis, $\Delta K$ was set to 3. For convenience, we adopt a normalization convention such that $\sum _{i=1}^N\pi (i,K)=1$ for all values of $K$.

As a first illustration of the expanded MSM/TPT framework, equilibrium molecular dynamics (MD) trajectories of the MthK channel generated with the AMBER and Drude force fields were analyzed. Details about the simulations have been reported elsewhere.^20,21 Briefly, for the Drude simulation, fives trajectories of 600 ns were generated following a long equilibration with flat-bottom dihedral restraints on the backbone of the selectivity filter to prevent large excursions away from the experimental structure. For the AMBER simulation, a $3\mu \text{s}$ trajectory was generated without dihedral restraints following a long equilibration. Both simulations provide a meaningful picture of the ion conduction dynamics in the context of a selectivity filter undergoing natural fluctuations around this average structure. A lag-time of 100 ps was used for the purpose of the MSM/TPT analysis. This is comparable to the lag-time of 20 ps used in previous studies.²⁸

In Figure 5 we show a set of 37 microstates for the expanded MSM/TPT analysis combining all the data from the AMBER and Drude simulations. There is a fair amount of overlap between the two force fields, with a total of 20 microstates observed in both the AMBER and Drude simulations. There are 26 microstates associated with the AMBER simulation, with 6 microstates not observed in the Drude simulations. There are 31 microstates associated with the Drude simulations, with 11 microstates not observed in the AMBER simulation. Interestingly, the simulations from the polarizable Drude force field yield 6 microstates with 3 K⁺ ions in adjacent sites (27, 29, 30, 32, 33, and 36) and one microstate with 4 K⁺ ions in adjacent sites (37). Correspondingly, the simulation from the AMBER force field yield only 4 microstates with 3 K⁺ ions in adjacent sites (4, 8, 15, and 24). This could be indicative that including the effect of induced polarization decreases the effective ion-ion repulsion in the selectivity filter.

Figure 5:

Set of occupancy microstates from the AMBER and Drude trajectories. The microstates observed exclusively with AMBER are shown in red text, and those observed exclusively with Drude are shown in blue text. The microstates observed both with AMBER and Drude are shown in black text.

The channel conductance can be calculated from the expanded MSM/TPT framework the reactive TPT flux. For the AMBER force field, the total TPT reactive flux $J=1.892\times {10}^{-3}/\text{ns}$, the diffusion coefficient along the $K$ index, ${𝒟}_K=3J$, is 5.676 × 10⁻³)/ns, yielding a channel conductance 8.2 pS. Alternatively, a conductance of 5.3 pS is obtained by estimating the diffusion coefficient ${𝒟}_K$ directly from stochastic simulations of the expanded transition matrix. For the polarizable Drude force field, the total TPT reactive flux is 2.458 × 10⁻³)/ns, the diffusion coefficient ${𝒟}_K$ is 7.375 × 10⁻³)/ns, and the channel conductance is 10.7 pS. Alternatively, a conductance of 9.1 pS is obtained by estimating the diffusion coefficient ${𝒟}_K$ from stochastic simulations. These results can be compared with previous estimates obtained from Green-Kubo linear-response theory, which gave values of 9±4 pS and 5±5 pS for AMBER and Drude, respectively.²¹ While there are small quantitative differences, the channel conductances calculated from the MSM/TPT framework are in good agreement with the previous estimates.²¹ The discrepancies between may be attributed to a variety of factors, including incomplete sampling, non-Markovian dynamical effects. One additional source of discrepancy could be the length of the intermediate region $K_B-K_A$, which was set to 3 in the present analysis. In practice, $K_B-K_A$ must be sufficiently large to establish the diffusive regime along the $K$ index (see Supplementary Information). Nevertheless, the differences are small and one can be confident that the character of the reactive pathways from the TPT analysis is meaningful.

In Figure 6 are shown the reactive pathways contributing a large fraction of the total flux extracted from the TPT analysis of the expanded MSM based on the MD simulations generated with the AMBER and Drude force fields. For the AMBER trajectory, four pathways contribute about 31% of the total flux according to the TPT analysis. For the Drude trajectory, four pathways contribute about 24% of the total flux. Additional details about the paths are provided in the Supplemental Information. The reactive pathways shown for the AMBER and Drude force fields display both differences and similarities. At a fairly coarse level, it is clear that the dominant reactive pathways of both models involve several microstates where K⁺ ions occupy adjacent sites. In that sense, the ion conduction involves microstates with ion-ion contacts, according to the so-called “hard-knock” mechanism.²⁹ Similarly, the dominant reactive pathways determined from the TPT analysis bare some similarities with the results of a previous computational studies of the ion conduction mechanism in K⁺ channels, which showed transitions with K⁺ ions occupying adjacent sites S₁ and S₂ (CHARMM force field)^28,30 and sites S₂ and S₃ (AMBER force field)²⁸ along the selectivity filter. This view also appears to be consistent with a crystallographic study of ion conduction through the NaK2K channel under an explicit external electric field.³¹ However, closer examination reveals substantial differences among these studies. Furthermore, the present results based on the AMBER and Drude force fields also show clear discrepancies. Notably, the sequence of microstates yielding the dominant pathways contributing to the reactive flux are not the same for the two force fields. For the AMBER force field, all four dominant reactive pathways involve a sequence ending on microstate 1 or the AMBER-exclusive microstate 19. For the Drude force field, four of the dominant reactive pathways involve a sequence starting with microstates with three K⁺ ions. The dominant pathways for the AMBER force field tend to follow the sequence “ 3 bound K⁺ ions → 2 bound K⁺ ions → 3 bound K⁺ ions → 4 bound K⁺ ions”, whereas those of the Drude force field follow the sequence “ 3 bound K⁺ ions → 2 bound K⁺ ions → 3 bound K⁺ ions → 2 bound K⁺ ions”. Unlike AMBER, the Drude field allows for multiple K⁺ ions to bind in adjacent binding sites, and a transition to microstate 32 with four bound K⁺ ions. Such differences, together with the observation that exclusive microstates that are not shared between the AMBER and Drude force fields along the dominant reactive pathways, suggest that the ion conduction mechanism is sensitive to details and unlikely to be universal at the microscopic level.

Figure 6:

TPT analysis for the ion conduction through the MthK channel. At the top left are the results for the MD simulations of the MthK channel with the AMBER force field. The TPT analysis was carried out to include all pathways up to 35% of the total flux. At the top right are the results for the MD simulations of the MthK channel with the Drude force field. The TPT analysis was carried out to include all pathways up to 30% of the total flux. (Top) Schematic representation of the occupancy states laid out as a function of the committor $q(i,K)$ for the discrete microstates (with $i=1,N$ and $K=0,1$). (Bottom) Four paths highlighting the rate limiting steps, each contributing a 0.05–0.11 fraction of the total flux (cumulating to a total of 31% for AMBER and 24% for Drude). A more detailed analysis is provided in Supplemental Information.

It is worth clarifying how the present framework relates to previous studies of ion conduction through the K⁺ channel.^28,32–34 In the work of Lam et al²⁸ a large transmembrane potential of 300 mV was applied to elicit a net ion flux in a specific direction, for the detection and identification of “permeation cycles” corresponding to the net conduction of one ion. A large membrane potential (200 mV) was also present in the study of Catacuzzeno et al.³⁴ To resolve some of the inherent ambiguities of ion conducting transitions within traditional MSM and account for all transitions $i\to j$, the authors introduced a flux matrix $F_{ij}$ corresponding to the incremental conduction charge effectively carried along the transition $i\to j$. In effect, $F_{ij}$ is a composite average that is expressed recursively as a weighted sum over all the possible transition pathways.³⁴ Alternatively, the flux matrix $F_{ij}$ can also be extracted directly from the MD trajectory for any lag-time.³⁴ In this case, the equivalence with the present formulation can be established (Supplemental Information), although the situation is considerably simplified with the global counting index $K$ because the displacement charge $Q_{\text{d}}(i,K)$, can be attributed unambiguously via Eq. (2) to any microstate $(i,K)$, regardless of the lag-time. Furthermore, the formulation of the expanded MSM enables the application of TPT.

In conclusion, the classic MSM/TPT formulation was expanded to represent unbound processes such as ion conduction through a membrane channel. The present formulation, which includes the additional counting index $K$ to map the transitions among a set of microstates, has some similarities with the augmented TPT for sequences of history-dependent events introduced by Lorpaiboon et al.³⁵ While the new expanded MSM/TPT framework was exploited here to analyze ion conduction from equilibrium MD trajectories, it could also be used in the context of non-equilibrium trajectories where the condition of microscopic detailed balance is not satisfied. For instance the expanded MSM framework could be used to analyze the ion conduction under the influence of an external transmembrane potential. Previous advances to analyze ion conduction based on VAMP method could also be incorporated in this framework.³⁶ The expanded MSM/TPT formulation was illustrated with the multi-ion conduction process through the MthK K⁺ channel. This analysis allows also the identification of the TPT-consistent dominant reactive pathways and of the rate limiting steps controlling the process. It was found that only four reaction pathways already provide most, about 31 and 24% of the total reactive flux for AMBER and Drude, respectively. The key transitions implicate direct ions occupying adjacent binding sites along the selectivity filter, according to the hard-knock conduction mechanism.^29,37 However, differences between the two force fields suggest that the ion conduction mechanism is not universal at the microscopic level. Future work will be aimed at furthering our understanding the ion conduction mechanism.

The expanded MSM/TPT framework formulated here could provide a powerful tool to study the function of a wide range of open biomolecular systems in which a directional process or transport cycle can be repeated an indefinite number of times. In particular, the expanded MSM/TPT framework could be used to determine the rate-limiting step of molecular machines operating under nonequilibrium conditions if the free energy profile along the $N$ states can be represented as a pseudo-periodic sequence $G_i$ with an additional offset after each period, $G_{i+N}=G_i+\Delta G$.³⁸ This includes, for example, multi-state membrane transporters,^39,40 ATP-driven ion pumps,^38,41 ATP synthase,⁴² myosine motors,^43–46 DNA polymerases, F1F0-ATP synthase, and general molecular machines.^47–51

Supplementary Material

Additional mathematical developments for the expanded MSM, including formulation of the Chapman-Kolmogorov equation, Chapman-Kolmogorov test, relation of the expanded MSM to the theory of Catacuzzeno et al.,³⁴ characteristic function and general solution, average moments and mean-square-deviation of $K$ index, steady-state flux, and TPT analysis, followed by detailed results from the expanded MSM/TPT framework for ion conduction for the AMBER and Drude force fields.

Symbols used

$K$

Global index

$i$

internal states index

$T(i,K\to \left.j,K^{\prime }\right)$

transition matrix of the expanded MSM

$P(K,i;t)$

time-dependent probability

$T_{\Delta K}(i\to j)$

block submatrix

$g$

channel conductance

$Q_{\text{d}}$

displacement charge

${𝒟}_K$

diffusion coefficient of the $K$ index

${𝒟}_Q$

diffusion coefficient of the displacement charge

$J$

flux

$\pi (i,K)$

equilibrium probability

$q(i,K)$

committor probability