Invariant Manifolds in Markovian Ion Channel Dynamics

Transcription

1 Invariant Manifolds in Markovian Ion Channel Dynamics J. P. Keener Department of Mathematics University of Utah October 10, 2006 Ion channels are multiconformational proteins that open or close in a stochastic fashion, reflecting stochastic transitions between conformations. One popular way to model the dynamics of these transitions is as Markov jump processes. These Markov models are then used in conductance based models to study the dynamics of (for example) electrical activity in nerve cells, cardiac cells, muscle cells, etc. As Markovian models have become more detailed and complicated, the need for ways to reduce this complexity has also become apparent. One oft used technique to reduce model complexity is fast equilibrium or quasi-steady state analysis, in which the fastest transitions are taken to be in quasi-equilibrium. The reduced model then follows the dynamics on a slow manifold, which is of lower dimension than the full system. A different reduction is possible if the Markovian process has a stable invariant manifold. If there is such a stable invariant manifold then after transients have decayed (which, we argue below, is always the case), the dynamics of the system are restricted to the stable invariant manifold, which being of lower dimension than the full system, is therefore simpler to simulate. It seems to be not generally recognized that many ion channel models do in fact have stable invariant manifolds. The purpose of this paper, therefore, is to show that there are large classes of Markovian jump processes that possess stable invariant manifolds, and then to use this to reduce several wellknown ion channels models. Specifically, we will show reductions of models for potassium channels, sodium channels, and ryanodine (RYR) and IP 3 receptors. 1 Markovian Jump Processes We suppose that X is a discrete random variable which can take integer values j = 0., 1,, possibly infinite. We also suppose that the probability of jumping from state i to state j in the time interval between t and t + is k ij (t), in the limit that is small. Notice that this process need not be time independent. In particular, for ion channels, which is the application to which this discussion 1

2 applies, transition rates are often dependent on other time-varying variables such as voltage or ion concentration. Finally, we suppose that the reaction kinetics are irreducible, meaning that there is no random variable Y which takes on a subset of the values of X, which also forms a closed set. The master equations for this stochastic process are well known to be given by [1] dp j = k ij (t)p i k ji (t)p j, j = 0, 1, 2, (1) i j i j where p j (t) is the probability of being in state j at time t. Of necessity, p j (t) 0 and j p j(t) = 1. We can write the system (1) in matrix form recognizing that A = (a ij ) where dp = A(t)P, (2) a ij = k ji (t), i j, a ii = j i k ij (t). (3) An invariant manifold for this equation is a vector, say Q, parameterized by the variable q, having the feature that P = Q(q) is a solution of the differential equation (2) provided that q satisfies a differential equation of lower dimension = F (q, t). (4) It is easy to see that a sufficient condition for there to be an invariant manifold is that A(t)Q(p) = j Q q j F j (q, t). (5) The matrix A T has a null space spanned by the vector with all elements equal to one (hence an eigenvalue equal to zero), and all other eigenvalues have negative real part. This we can establish using the Perron-Frobenius theorem. Notice that the matrix A has all non-negative off-diagonal elements, and all negative diagonal elements. It follows that there is some number µ > 0 so that A T + µi is a non-negative matrix. It is also irreducible by virtue of our assumption that X contains no random variable Y which is a closed subset of X. Now, according to the Perron- Frobenius Theorem [?], A T + µi has a unique, positive eigenvector, with corresponding eigenvalue λ 0. Furthermore, all other eigenvalues are smaller than λ 0 in absolute value (so that λ 0 is the spectral radius of A T + µi). Since we know that one eigenvector of A T is the vector with all elements equal to one, this must be the unique positive eigenvector of A T + µi. The corresponding eigenvalue is λ 0 = µ. Since the eigenvalues of A T + µi are shifts of the eigenvalues of A T, it follows that all eigenvalues λ of A T lie in the disc λ µ < µ, and hence have negative real parts. Now we can show that an invariant manifold is stable. Suppose we let P = Q(q) + Y, where Y has components y i and i y i = 0. Then, dy = A(t)Y. (6) 2

3 Suppose that Φ is a matrix whose columns constitute an orthonormal basis for the range of A. By the Fredholm Alternative, Y is in the range of A, being orthogonal to the nullspace of A T. Hence there is a vector Z so that Y = ΦZ, (7) and dz = ΦT AΦ = MΦ. (8) Furthermore the eigenvalues of M are exactly the nonzero eigenvalues of A. Therefore, since all of the eigenvalues of M have negative real parts, Z is bounded by a decaying exponential (assuming the negative eigenvalues of A(t) are bounded away from zero). Hence lim t Y (t) = 0, so that the invariant manifold is globally stable. 2 Invariant Manifolds for Jump Processes Perhaps the easiest way to find invariant manifolds is to pick a probability distribution function and try to determine the master equation for which it is an invariant manifold. We begin with 2.1 Poisson Invariant Manifold Suppose we look for a solution of a master equation that is Poisson distributed. That is, we suppose that p k (q) = qk exp( q), (9) k! and seek a birth-death process for which this is a solution. We take a master equation for a general birth-death process of the form Upon substitution of (9) into (10) we find that dp k = a k 1p k 1 a k p k + b k+1 p k+1 b k p k. (10) q t = a k 1k a k q k q If this is to be independent of k, then it must be that in which case the equation governing the evolution of q is + q b k+1q b k (k + 1). (11) (k + 1)(k q) a k = α, b k = kβ, (12) q t = α βq, (13) even if α and β are time dependent. Notice that this birth-death process can be represented with a state diagram α S k β(k+1) S k+1, (14) 3

4 which can be understood as representing the chemical reaction A α β where the concentration of species A is held fixed. X, (15) 2.2 Binomial Invariant Manifold Suppose that the probability p k is given by the binomial distribution p k (q) = N! k!(n k)! qk (1 q) N k. (16) We ask if there is a single step jump process for which this is an invariant manifold. As we did with the Poisson distribution, we substitute (16) into (10) and find that ( k q N k 1 q ) = a k 1 which is independent of k provided k N k q q a k + b k+1 N k k + 1 q 1 q b k, (17) a k = α(n k), b k = βk. (18) With this choice, the parameter q is governed by the differential equation = α(1 q) βq. (19) It can also be shown that the smallest (in amplitude) negative eigenvalue of this system is (α+β). Notice that this Markov process can be represented by the transitions α(n k) S k S k+1. (20) β(k+1) which is also the transition associated with the simple chemical reaction X α β Y. (21) 2.3 Multinomial Invariant Manifold One can generalize this result for the binomial distribution to the multinomial distribution. Suppose we have N identical independent n-state random variables, and suppose that the single random variable probabilities are p 1 (t), p 2 (t),..., p n (t). Then, the master equation for the probability that there are i j random variables in state j, j = 1, 2,, n has an invariant manifold given by the multinomial distribution p i1,i 2,,i n = N! i 1!i 2!, i n! pi 1 1 p i 2 2 p in n, (22) 4

5 where the dynamics of the parameters (or probabilities) p 1 (t), p 2 (t),..., p n (t) are governed by the master equations for the single n-state Markov process (an n-dimensional system). The proof is by a direct calculation. One must write out the full master equation and then substitute (22) and somehow match up all the terms. To see how it works, we examine one of the possible transitions, between state j, and state k, say. We suppose that the transition rate for the n-state Markov process from state j to state k is k jk. Then, the full master equation must include the terms dp ij i k = (i j + 1)k jk p ij +1,i k 1 i j k jk p ij i k + (i k + 1)k kj p ij 1,i k +1 i k k kj p ij i k, (23) and we have suppressed all the other indices and transitions. Substitution of (22) into (23) gives (after dividing by the multinomial) i j dp j p j + i k dp k p k = k jk i k p j p k i j k jk + k kj i j p k p j i k k kj (24) = i k (k jk p j p k k kj ) + i j (k kj p k p j k jk ). (25) Because we want equations for p j and p k that are independent of i j and i k we take dp j = k kjp k k jk p j, dp k = k jkp j k kj p k, (26) which are terms in the master equation for the single n-state Markov process. In this way, we can see how the general case works, and the result is established. 2.4 The Negative Binomial Distribution We ask what is the master equation (if any) with an invariant manifold that is the negative binomial distribution with parameter q, p k (q) = (n + k 1)! q n (1 q) k. (27) (n 1)!k! Substituting this into the master equation (10) we find If we set ( n q k 1 q ) = a k 1 k 1 n + k 1 1 q a n + k k + b k+1 k + 1 (1 q) b k. (28) this reduces to the differential equation a k = α(n + k), b k = βk, (29) = q(β α βq), (30) which is a logistic equation. Notice that as before, α and β can be time dependent. This process corresponds to the transition diagram α(n+k) S k S k+1, (31) β(k+1) 5

6 which can also be interpreted as the chemical reaction A γ β X, X α 2X (32) with the concentration of A held fixed, where we make the identification γ[a] = αn. 2.5 Multiplicative Invariants With these basic invariant manifolds in hand, we can generate more invariant manifolds by multiplication. That is, if Q(q) is an invariant manifold for the stochastic process for random variable X, and R(r) is an invariant manifold for the stochastic process for the random variable Y, then the product P = Q T R is an invariant manifold for the random variable Z = X Y. We can also see how this works in reverse with a simple example. Suppose we have a random variable ( ) 0 Z = X. 1 (33) The states are represented by two indices, S j,k with j denoting a state for X and k being either 0 or 1. Now we suppose further that the transition rates between states with k = 0 and k = 1 are independent of j a S j,0 b S j,1. (34) If we denote by P the probability of being in states with k = 0 and by Q the probability of being in states with k = 1, then the master equations are the system of equations dp = AP ap + bq, dq = AQ + ap bq. (35) We can easily verify that this system has a multiplicative invariant manifold, with P = qs, Q = (1 q)s where ds = AS, = b(1 q) aq. (36) which has dimension half that of the original system. Again, all of these invariant manifolds are correct, even if the transition rates are time dependent. 3 Applications 3.1 Potassium Channels A simple model for the potassium channel found in the squid axon is that it is comprised of four independent subunits, each of which can be either open or closed, with voltage dependent transitions C α(v ) O. (37) β(v ) 6

7 The conducting state of the channel is that in which all subunits are open. If k denotes the number of open subunits, then the state diagram is represented by (20). It follows that if p k (t) is the probability of k subunits being open at time t, then the master equations governing p k (t) have a stable invariant manifold given by the binomial distribution with N = 4 and parameter q where = α(v )(1 q) β(v )q. (38) Furthermore the probability of being in the conducting state is P O = q 4. This is the open probability originally used by Hodgkin and Huxley [2], to model potassium conductance. More recent data on I Ks potassium channels have suggested that the independent subunits may have multiple closed states. For example, in the model of Silva and Rudy [5] the subunit has three states, C 2 α(v ) β(v ) γ(v ) C 1 O, (39) δ(v ) where C 1 and C 2 are closed states and O is the open state. The conducting state of the channel is when all subunits are in the open state, and there are four independent subunits. If i C1, i C2, and i O are the number of subunits in states C 1, C 2, and O, respectively, then there are a total of fifteen different configurations for the four subunits. Thus, the master equations consist of fifteen differential equations. However, because the subunits are independent, there is a two-dimensional invariant manifold, given by the multinomial distribution p ic1 i C2 i O = 4! i C1 i C2!i O! pi C 1 C 1 p i C 2 C 2 p i O O, (40) where the parameters p C1,p C2, and p O, are governed by the differential equations dp C2 dp C1 = βp C1 αp C2, (41) = αp C2 + δp O (β + γ)p C1, (42) and p C2 + p C1 + p O = 1 for all time. As before, this is an invariant manifold even if the transition rates are time dependent. Since the open state has i O = 4, the open probability is P O = p 4 O, (43) but now the dynamics of p O are governed by two differential equations, rather than one, as in the first potassium channel model. 3.2 Sodium Channels A simple model for the sodium channel of the squid axon is that it has three identical subunits that are activating (denoted by m) and one, also independent, that is inactivating (denoted by h). 7

8 The transition probabilities are again voltage dependent, with α j (V ) C j O j, j = m, h. (44) β j (V ) If we denote by p j,k the probability of j open m units and k open h units, (a total of eight possible states) it follows from our previous discussion that there is an invariant manifold for the master equations that is the product of two binomial distributions, p j,k = p 3 j (m)p1 k (h), (45) where by p N j (a) is meant a binomial distribution for N states with parameter a. The parameters m and h are governed by differential equations of the form = α q(1 q) β q q, q = m, h. (46) Furthermore, the probability of being in the conducting state is p 3,1 = m 3 h, which is also exactly the form originally used by Hodgkin and Huxley [2] to model sodium channel conductance. 3.3 Ryanodine Receptors One of the earliest models of the ryanodine receptor, due to Stern et al.[6], is shown in Fig. 1. This is a four state model with closed states R and RI, inactivated state I, and open state O. Transitions between these states are calcium dependent, reflecting binding of calcium to binding sites. A modification of the Stern et al. model in which the rate constants k 1 and k 2 were assumed to be dependent on the concentration of calcium in the sarcoplasmic reticulum was used by [?] The important observation here is that the in these models the transition rates between R and O are identical to those between RI and I, and transition rates between R and RI are identical to those between O and I. Thus, this can be viewed as the cross product of two two-state random variables. As discussed above, the master equations have a multiplicative stable invariant manifold of the form p R = pq, p RI = q(1 p), p O = q(1 p), p I = (1 p)(1 q), (47) and the parameters p and q satisfy the differential equations dp = k 1(1 p) k 1 c 2 p, = k 2(1 q) k 2 cq. (48) 3.4 IP 3 Receptors The Keizer-DeYoung model for IP 3 receptors [7] is shown in Fig. 2. This is an eight state model with states S ijk where i, j, and k can take on values 0 or 1. The index i represents binding (i = 1) or unbinding (i = 0) of IP 3, and the indices j and k represent binding or unbinding of calcium. In this model, the binding of calcium with index j is activating, and the binding of calcium with index 8

9 k 2 R RI k c 2 k 1 c 2 O k 1 k 2 k 2 c k c2 1 I k 1 Figure 1: Model of the RyR due to Stern et al. (1999). R and RI are closed states, O is the open state, and I is the inactivated state. k 1 p k 5 c k 1 p k 5 c S 110 S010 S000 S 100 k 1 k 5 k 1 k 5 k 2 k 2 c k 4 k 4 c k 4 k 4 c k 2 k 2 c k 2 k 2 c k 3 p k 5 c k 3 p k 5 c S111 S011 S001 S101 S 111 k 3 k 5 k 3 k 5 S 110 Figure 2: The binding diagram for the Keizer-DeYoung IP 3 receptor model. Here, c denotes Ca ++, and p denotes IP 3. 9

10 k is inactivating, so that the only conducting state is S 110. In general, the concentration dependent transition rates are time dependent. The observation that is relevant for this discussion is that there is a multiplicative structure, and hence an invariant manifold here. In particular, if we let X = , Y = , (49) (that is, we let X represent the states S i0k, and let Y represent the states S i1k ) then the master equations are of the form dp X = AP X k 5 cp X + k 5 P Y, dp Y = AP Y + k 5 P X k 5 P Y. (50) where P X and P Y are the probabilities of being in states X and Y, respectively. It follows that if we set P X = qp Z, P Y = (1 q)p Z, then dp Z = AP Z, = k 5(1 q) k 5 cq, (51) is a stable invariant manifold for this process. Thus, without approximation, the master equations can be reduced from a 7-dimensional to a 4-dimensional system. 3.5 Multiple Channels Now suppose we wish to determine the number of open channels in a cluster of channels. This might be relevant for ryanodine receptors or IP 3 receptors, which are known to occur in clusters (although it may be that RYR receptors are not independent, but are coupled by a binding protein FKBP12.6 [4, 3]). If the individual channel has n states, then the distribution for the number of channels in each of the states is governed by the multinomial distribution (22). However, if only one of the states, say j = 1 is a conducting state, then the probability distribution for the number of conducting states is the binomial p i1 (t) = N! i 1!(N i 1 )! pi 1 1 (1 p 1 ) N i 1. (52) It also follows that the expected value of number of conducting channels is E(i 1 ) = Np 1 (t). (53) This formula appears to be rather unremarkable, since it is exactly what one expects from standard probability theory for independent Bernoulli trials. What makes this interesting, however, is that it is correct even for time dependent Markov processes, once initial transients have decayed. 10

11 4 Discussion We have shown that a number of fundamentally important birth-death processes have stable invariant manifolds, even when the transition rates are time dependent. The significance of for simulations of ion channel behavior, for example, is that the size of the system that needs to be simulated can be reduced, sometimes quite significantly. This is possible because for any realistic experimental procedure, the initial data apply when the channel was originally cloned or the cell was originally cultured, not when the experiment (say, for example, a voltage clamp procedure) was begun. Further, by the time the experiment is performed, the initial transients have surely decayed. It is also interesting that it is common for many investigators to make a distinction between ion channel models that are of Hodgkin-Huxley type and Markovian ion channel models. However, we have shown here that the models of potassium and sodium ion channels used by Hodkin-Huxley were, in fact, the exact solutions of Markov models Acknowledgment: This research was supported in part by NSF Grant DMS Thanks to Berton Earnshaw for help in completing the proof that the invariant manifolds are stable. References [1] C. W. Gardiner. Handbook of Stochastic Methods. Springer, New York, 2nd edition, [2] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117: , [3] S. O. Marx, J. Gaburjakova, M. Gaburjakova, C. Henrikson, K. Ondrias, and A. R. Marks. Coupled gating between cardiac calcium release channels (ryanodine receptors). Circ. Res., 88: , [4] S. O. Marx, S. Reiken, Y. Hisamatsu, T. Jayaraman, D. Burkhoff, N. Rosemblit, and A. R. Marks. PKA phosphorylation dissociates FKBP12.6 from the calcium release channel (ryanodine receptor): Defective regulation in failing hearts. Cell, 101: , [5] J. Silva and Y. Rudy. Subunit interaction determines I Ks participation in cardiac repolarization and repolarization reserve. Circ., 121: , [6] M. D. Stern, L.-S. Song, H. Cheng, J. S.K. Sham, H. T. Yang, K. R. Boheler, and E. Rios. Local control models of cardiac excitation-contraction coupling: A possible role for allosteric interactions between ryanodine receptors. J. Gen. Phys., 113: , March [7] G. W. De Young and J. Keizer. A single pool [IP 3 ]-receptor based model for agonist stimulated [Ca ++ ] oscillations. Proc. Natl. Acad. Sci. USA, 89: ,