Stochastic Markovian modeling of electrophysiology of ion channels: Reconstruction of standard deviations in macroscopic currents

https://doi.org/10.1016/j.jtbi.2006.10.016Get rights and content

Abstract

Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.

Introduction

Ion channels are pore-forming proteins that permit and control the diffusion of ions across cellular membranes (Hille, 2001). They are the object of intensive research, both from the experimental as well as the mathematical perspective. As a consequence of this emphasis, various approaches exist to mathematically model ion channel behavior. Prior to the experimental verification of ion channel existence, models were already utilized to predict membrane response and the mechanisms underlying channel behavior. Such models are now ubiquitous in the ion channel research community.

Most mathematical electrophysiological models of ion channels fall into one of two categories: Markovian or Hodgkin–Huxley type. Markovian models describe ion channel behavior by assigning probabilities to potential channel states such as open, inactive, and closed states (Colquhoun and Hawkes, 1995a). Transitions between the states are governed by rate coefficients which are a function of physical parameters, e.g. temperature, transmembrane voltage and ion concentrations. Hodgkin–Huxley type models of ion channels follow a mathematical formalism introduced in the seminal work of Hodgkin and Huxley for describing the electrophysiology of membrane of a squid axon (Hodgkin and Huxley, 1952). Kinetics of channel properties are described by coefficients, which are dependent on time and physical parameters. The coefficients affect the current flow through channels in a multiplicative manner.

The traditional experimental approach for investigating and quantifying ion channel behavior involves the simultaneous application of voltage clamping protocols and measurement of the resulting electrical currents through either single or multiple membrane embedded ion channels. Ion channel modeling has proven to be a valuable addition to the experimental methodology and is capable of reconstructing experimental data and providing mechanistic insights into physiological and pathophysiological phenomena (Hodgkin and Huxley, 1952, McAllister et al., 1975, Luo and Rudy, 1991, Rudy, 2004, Sachse, 2004).

Noise in the current response of electrically active membranes can be categorized as either thermal or so-called excess noise (Verveen and DeFelice, 1974). Excess noise has been proposed to result from the granular ion current transport through channel pores, ion–ion interaction, and stochastic conformational changes of the channels such as their opening and closing. Further sources of fluctuations, particularly in macroscopic currents through ensembles of ion channels, may originate from the heterogeneity of kinetic properties and local variations in physical conditions in the channel vicinity.

Analysis of macroscopic current noise and single channel stochasticity provides significant biophysical insights and is an important tool for modeling (Colquhoun and Hawkes, 1995a, Colquhoun and Hawkes, 1995b; Colquhoun and Sigworth, 1995). However, ion channel models typically neglect noise and describe only the average behavior. Rather than computing a spread of outcomes resulting from randomly distributed input data and/or model parameters, these models produce a single result for deterministic input data and model parameters. While such simplifications are appropriate for certain systems, they render the model incapable of reconstructing heterogeneous behavior of channels. These restrictions can significantly reduce the insights to be gained from computational simulations.

In this paper, we assume Markovian models of ion channel electrophysiology and investigate the stochastic state dynamics and current response to randomly distributed rate coefficients. The approach aims at assessing current noise resulting from heterogeneity of kinetic channel properties and physical conditions in the channel vicinity.

A common method of investigating the effects of random parameters involves simulating the model multiple times, each with different values for the rate coefficients (varied individually or in combination). The current responses to each set of rate coefficients can then be compared to obtain an understanding of the effect of the rate coefficients. This is similar to taking specific realizations from a randomly distributed rate constant and using those values to run the deterministic model. From these discrete results, one can calculate an estimate of the response statistics, e.g. the mean and variance. As one increases the number of samples, the solution statistics converge. Such an approach falls under the category of sampling-based stochastic methods.

Requiring only a straightforward extension of the deterministic solver, Monte Carlo methods are the best known of these techniques due to their ease of implementation. However, such solutions are often computationally prohibitive even for systems of relatively low complexity, as they converge as 1/N where N is the number of realizations. Thus a large number of trials are necessary to obtain accurate statistics. Latin hypercube sampling (Loh, 1996), the quasi-Monte Carlo method (Niederreiter et al., 1998), and the Markov chain Monte Carlo method (Gamerman, 1997) all have accelerated convergence properties compared to the Monte Carlo method while maintaining ease of implementation. However, each method imposes certain restrictions on the process of interest which in turn limits their general applicability.

An alternative method to determine the effect of rate coefficient values upon a particular ion channel model is to assume a probability density function and directly calculate the current as a result of the (now) stochastic ion channel model. Such non-sampling methods avoid taking large samples of repetitive deterministic solvers and include perturbation methods (Kleiber and Hein, 1992) and second-moment analysis (Liu et al., 1986). Though more efficient than Monte Carlo under certain conditions, these methods have limited utility and robustness as they are only capable of resolving relatively small perturbations in both the random inputs and outputs. This is difficult to guarantee, especially for nonlinear systems where small perturbations in inputs can result in relatively large perturbations in the response.

In this paper, we adopt the generalized polynomial chaos–stochastic Galerkin (gPC–SG) method as an efficient computational means of obtaining solutions to complex stochastic differential systems. Generalized polynomial chaos (gPC) represents random processes via orthogonal polynomials (Xiu and Karniadakis, 2002a, Xiu and Karniadakis, 2002b). It is a generalization of the Wiener-Hermite polynomial chaos expansion (Wiener, 1938)1 which employs Hermite polynomials. The generalizations utilize sets of orthogonal polynomials to allow efficient representation of random processes with arbitrary probability distribution functions. Such expansions exhibit fast convergence rates when the stochastic response of the system is sufficiently smooth in the random space.

The traditional approach involves a Galerkin projection of the governing equations to the random polynomial basis functions defined by gPC, such that the mean square error of the residue is minimized. This is referred to as the stochastic Galerkin method, which efficiently reduces the stochastic governing equations to a system of deterministic equations that can be solved by conventional numerical techniques. Such a gPC–SG approach is capable of resolving systems with relatively large perturbations in both the inputs and responses and has been successfully applied to model uncertainty in complex stochastic solid and fluid dynamic problems (Ghanem and Spanos, 1991; Xiu and Karniadakis, 2002a, Xiu and Karniadakis, 2002b; Xiu and Karniadakis, 2003). However, the resulting deterministic equations can become very complicated and potentially intractable if the system of differential equations has nontrivial and/or nonlinear forms. Such difficulties are not typical of Markovian ion channel models which we study in this paper.

In this work we employ the gPC–SG method to assess the impact of rate coefficients on computational simulations of cardiac ion channel phenomena. We seek a quantitative understanding of the relationship between the distributions of rate coefficients and ion channel currents as a result of deterministic voltage clamping protocol. We first consider a simple Markovian model of ion channel behavior, and then apply our technique to a cardiac ion channel model of the slowly activating delayed rectifier potassium current, IKr (Iyer et al., 2004). While we only consider random rate coefficients in ion channel models, this methodology can be applied to other parameters, and, furthermore, a vast number of bio-mathematical models.

Section snippets

Generalized polynomial chaos expansion

We first briefly demonstrate the application of gPC to a general stochastic process with N independent parameters. To assess the impact of each of these parameters, we model them as random variables and assume that they are functions of N separate, independent and uncorrelated random variables, denoted (ξ)=(ξ1,ξ2,,ξN). The output of the system, f(ξ), is dependent on the parameters and is therefore also a random process. Utilizing gPC expansions, such a process is approximated by a linear

Simple Markovian model

We first present the simulated currents of the simple channel model for a given voltage clamping protocol (Table 4) with random rate coefficients of uniform distributions. The various computational results presented for this model assume the voltage clamping protocol shown in the insert of Fig. 2a. The mean current responses for all stochastic experiments presented for the simple model are nearly visually indistinguishable from the response to the deterministic model and are superimposed in

Summary and conclusions

In this work a stochastic numerical technique based on Galerkin projections is introduced to extend and study Markovian models of ion channel electrophysiology. The generalized polynomial chaos stochastic Galerkin (gPC–SG) method allows one to take stochastic model parameters into account. We have demonstrated that gPC–SG is an efficient method of determining the effects of stochastic model parameters on the output of models. This information is beneficial in sensitivity analysis which has a

Acknowledgments

This work was funded in part by NSF Career Award (Kirby) NSF-CCF0347791, the NIH NCRR Center for Bioelectric Field Modeling, Simulation and Visualization (www.sci.utah.edu/ncrr), NIH NCRR Grant No. 5P41RR012553-02, the Richard A. and Nora Eccles Fund for Cardiovascular Research and awards from the Nora Eccles Treadwell Foundation (Sachse). The authors also acknowledge the computational support and resources provided by the Scientific Computing and Imaging Institute.

References (24)

  • E. Hairer et al.

    Solving Ordinary Differential Equations I (2nd Revised ed.): Non-stiff Problems

    (1993)
  • B. Hille

    Ion Channels of Excitable Membranes

    (2001)
  • Cited by (12)

    • Generalized polynomial chaos-based uncertainty quantification and propagation in multi-scale modeling of cardiac electrophysiology

      2018, Computers in Biology and Medicine
      Citation Excerpt :

      As compared to MC, uncertainty propagation with gPC has been proved to be more efficient in terms of computational time in different modeling, control, and optimization problems [18,20–24]. Geneser et al. [16] introduced uncertainty in rate coefficients of ion channel model, and applied gPC for uncertainty propagation in ion channel gating. However, uncertainty was randomly assigned to model parameters and the quantification of uncertainty was only studied at the ion channel level, which cannot provide the information about the effect of uncertainty on higher organizational levels such as cell and tissue.

    • Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models?

      2016, Journal of Molecular and Cellular Cardiology
      Citation Excerpt :

      In both types of application it will be important to express a measure of confidence in the model outputs, given uncertainties and errors in the inputs. As a result, there has been growing interest and application of (VV)UQ in cardiac modelling [14], [10], [36], [37], [38]. In [38] Pathmanathan et al. quantified the natural variability in the steady-state inactivation of the canine fast sodium channel using a statistical framework known as Non-Linear Mixed Effects (NLME) modelling.

    • Uncertainty quantification of fast sodium current steady-state inactivation for multi-scale models of cardiac electrophysiology

      2015, Progress in Biophysics and Molecular Biology
      Citation Excerpt :

      UQ has been performed in a drug-based cardiac application, for parameters governing channel block of various compounds (Elkins et al., 2013). Also, in Geneser et al. (2007) a sophisticated framework for uncertainty propagation (the ‘stochastic Galerkin method’), was applied to determine the impact of uncertain rate coefficients in ion channel models. However, a simplified approach was used for the uncertainty characterisation stage, that was not based on experimental data.

    • Computer simulation of voltage sensitive calcium ion channels in a dendritic spine

      2013, Journal of Theoretical Biology
      Citation Excerpt :

      When detailed kinetic information becomes available for postsynaptic voltage-sensitive calcium channels, it will be a simple matter to substitute those kinetics for the ones used here. This kind of stochastic ion channel gating modeling has been done previously (Faber et al., 2007; Tanskanen et al., 2005; Geneser et al., 2007). Here, however, we study such stochastic ion channel gating in the context of electrodiffusion.

    • Functional role of the activity of ATP-sensitive potassium channels in electrical behavior of hippocampal neurons: Experimental and theoretical studies

      2011, Journal of Theoretical Biology
      Citation Excerpt :

      Single-channel data were modeled and transition rates were then obtained using the QUB suite (Qin et al., 2000; Wu et al., 2002). This type of kinetic analysis was based on the assumption that channel activity can be described by a finite-state Markovian model and that the channels in the patch are mutually independent (Geneser et al., 2007). Origin 8.0 (OriginLab, Northampton, MA) was used for fitting of experimental data and statistical analysis of the firing rate and interspike intervals in modeled neuron.

    • Uncertainty Quantification of the Effects of Segmentation Variability in ECGI

      2021, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    View all citing articles on Scopus
    View full text