The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Markov modeling of allosteric drug effects on ion channels, with particular reference to neuronal nicotinic acetylcholine receptors

Allosteric mechanisms have been suggested for an increasing number of ion channel drug interactions, but often these ideas are not examined quantitatively through use of Markov models that would allow statistical estimation of proposed coupling effects. In this paper we illustrate, using properties relevant to the neuronal nicotinic acetylcholine receptor, how these models can be used to provide insight into the behavior of cooperative systems. Such models would then provide the basis for inferential studies with experimental data aimed at quantifying the magnitude of drug-induced changes on particular channel parameters. It is shown that even small changes in agonist binding affinity or channel gating are sufficient to produce biphasic modulatory drug effects in an allosteric model of nicotinic receptor activity.     

Application of oscillating potentials to the Shaker potassium channel

Nonequilibrium response spectroscopy (NRS) has been proposed recently to complement standard electrophysiological techniques used to investigate ion channels. It involves application of rapidly oscillating potentials that drive the ion channel ensemble far from equilibrium. It is argued that new, so far undiscovered features of ion channel gating kinetics may become apparent under such nonequilibrium conditions. In this paper we explore the possibility of using regular, sinusoidal voltages with the NRS protocols to facilitate Markov model selection for ion channels. As a test case we consider the Shaker potassium channel for which various Markov models have been proposed recently. We concentrate on certain classes of such models and show that while some models might be virtually indistinguishable using standard methods, they show marked differences when driven with an oscillating voltage. Model currents are compared to experimental data obtained for the Shaker K+ channel expressed in mammalian cells (tsA 201).     

Gating kinetics of potassium channel in rat dorsal root ganglion neurons analyzed with fractal model

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and kinetic rate constants connecting these states are constant. To study the gating kinetics of voltage-dependent K(+) channel in rat dorsal root ganglion neurons, K(+) channel current were recorded using cell-attached patch-clamp technique. The K(+) channel characteristic of kinetics were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimension D for the closed times and for the open times depend on the pipette potential. For the open and closed times of kinetic setpoint, it was found dependent on the applied pipette potential, which indicated that the ion channel gating kinetics had nonlinear kinetic properties. Thus, the open and closed durations, which had the voltage dependence of the gating of this ion channel, were well described by the fractal model.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

Hidden Markov modeling for single channel kinetics with filtering and correlated noise

Hidden Markov modeling (HMM) can be applied to extract single channel kinetics at signal-to-noise ratios that are too low for conventional analysis. There are two general HMM approaches: traditional Baum's reestimation and direct optimization. The optimization approach has the advantage that it optimizes the rate constants directly. This allows setting constraints on the rate constants, fitting multiple data sets across different experimental conditions, and handling nonstationary channels where the starting probability of the channel depends on the unknown kinetics. We present here an extension of this approach that addresses the additional issues of low-pass filtering and correlated noise. The filtering is modeled using a finite impulse response (FIR) filter applied to the underlying signal, and the noise correlation is accounted for using an autoregressive (AR) process. In addition to correlated background noise, the algorithm allows for excess open channel noise that can be white or correlated. To maximize the efficiency of the algorithm, we derive the analytical derivatives of the likelihood function with respect to all unknown model parameters. The search of the likelihood space is performed using a variable metric method. Extension of the algorithm to data containing multiple channels is described. Examples are presented that demonstrate the applicability and effectiveness of the algorithm. Practical issues such as the selection of appropriate noise AR orders are also discussed through examples.     

Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments

Growth competition assays have been developed to quantify the relative fitness of HIV-1 mutants. In this article, we develop mathematical models to describe viral/cellular dynamic interactions in the assay system from which the competitive fitness indices or parameters are defined. In our previous HIV-viral fitness experiments, the concentration of uninfected target cells was assumed to be constant (Wu et al. 2006). But this may not be true in some experiments. In addition, dual infection may frequently occur in viral fitness experiments and may not be ignorable. Here, we relax these two assumptions and extend our earlier viral fitness model (Wu et al. 2006). The resulting models then become nonlinear ODE systems for which closed-form solutions are not achievable. In the new model, the viral relative fitness is a function of time since it depends on the target cell concentration. First, we studied the structure identifiability of the nonlinear ODE models. The identifiability analysis showed that all parameters in the proposed models are identifiable from the flow-cytometry-based experimental data that we collected. We then employed a global optimization approach (the differential evolution algorithm) to directly estimate the kinetic parameters as well as the relative fitness index in the nonlinear ODE models using nonlinear least square regression based on the experimental data. Practical identifiability was investigated via Monte Carlo simulations.     

A stochastic formulation of the gompertzian growth model for in vitro bactericidal kinetics: parameter estimation and extinction probability

Time-kill curves have frequently been employed to study the antimicrobial effects of antibiotics. The relevance of pharmacodynamic modeling to these investigations has been emphasized in many studies of bactericidal kinetics. Stochastic models are needed that take into account the randomness of the mechanisms of both bacterial growth and bacteria-drug interactions. However, most of the models currently used to describe antibiotic activity against microorganisms are deterministic. In this paper we examine a stochastic differential equation representing a stochastic version of a pharmacodynamic model of bacterial growth undergoing random fluctuations, and derive its solution, mean value and covariance structure. An explicit likelihood function is obtained both when the process is observed continuously over a period of time and when data is sampled at time points, as is the custom in these experimental conditions. Some asymptotic properties of the maximum likelihood estimators for the model parameters are discussed. The model is applied to analyze in vitro time-kill data and to estimate model parameters; the probability of the bacterial population size dropping below some critical threshold is also evaluated. Finally, the relationship between bacterial extinction probability and the pharmacodynamic parameters estimated is discussed.     

A direct optimization approach to hidden Markov modeling for single channel kinetics

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.     

A method for estimation of drug affinity constants to the open conformational state of calcium channels.

The affinity of D600 to calcium channels in the open state has been examined in isolated smooth muscle cells of the rabbit ear artery. Calcium channel currents were measured in high external barium solution by means of the patch-clamp technique. The current inhibition in various D600 concentrations (3-100 microM) on application of trains of short test pulses (20-80 ms) has been studied in nonmodified calcium channels and in cells where the calcium channels were modified by the agonist dihydropyridine (+) 202,791 (100 nM). The kinetics of the peak current decay has been analyzed with a mathematical model which is based on the experimental finding that D600 interacts primarily with calcium channels in the open conformational state. The model approach allows the estimation of drug affinity constants of D600 to the calcium channel in the open conformation. An association rate constant to the open conformational state of D600 of 6.16 x 10(4) M-1 s-1 was estimated. The association rate of the drug was not significantly changed after the calcium channels have been modified with 100 nM (+) 202,791. A method for correction of rate constants for possible drug trapping is discussed.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

The pore helix dipole has a minor role in inward rectifier channel function

Ion channels lower the energetic barrier for ion passage across cell membranes and enable the generation of bioelectricity. Electrostatic interactions between permeant ions and channel pore helix dipoles have been proposed as a general mechanism for facilitating ion passage. Here, using genetic selections to probe interactions of an exemplar potassium channel blocker, barium, with the inward rectifier Kir2.1, we identify mutants bearing positively charged residues in the potassium channel signature sequence at the pore helix C terminus. We show that these channels are functional, selective, resistant to barium block, and have minimally altered conductance properties. Both the experimental data and model calculations indicate that barium resistance originates from electrostatics. We demonstrate that potassium channel function is remarkably unperturbed when positive charges occur near the permeant ions at a location that should counteract pore helix electrostatic effects. Thus, contrary to accepted models, the pore helix dipole seems to be a minor factor in potassium channel permeation.     

Yeast "N"-hybrid systems for protein-protein and drug-protein interaction discovery

The majority of small molecule drugs act on protein targets to exert a therapeutic function. It has become apparent in recent years that many small molecule drugs act on more than one particular target and consequently, approaches which profile drugs to uncover their target binding spectrum have become increasingly important. Classical yeast two-hybrid systems have mainly been used to discover and characterize protein-protein interactions, but recent modifications and improvements have opened up new routes towards screening for small molecule-protein interactions. Such yeast "n"-hybrid systems hold great promise for the development of drugs which interfere with protein-protein interactions and for the discovery of drug-target interactions. In this review, we discuss several yeast two-hybrid based approaches with applications in drug discovery and describe a protocol for yeast three-hybrid screening of small molecules to identify their direct targets.     

Performance of the continuous glucose isomerase reactor system for the production of fructose syrup

Glucose isomerase in the form of heat-treated whole-cell enzyme prepared from Streptomyces phaeochromogenus follows the reversible single-substrate reaction kinetics in isomerization of glucose to fructose. Based on the Kinetic constants determined and the mathematical model of the reactor system developed, the preformance of a plug-flow-type continuous-enzyme reactor system was studied experimentally and also simulated with the aid of a computer for the ultimate objective of optimization of the glucose isomerase reactor system. The enzyme decay function for both the enzyme storage and during the use in the continuous reactor, was found to follow the first-order decay kinetics. When the enzyme decay function is taken into consideration, the ideal homogeneous enzyme reactor kinetics provided a satisfactory working model without further complicatin of the mathematical model, and the results of computer simulation were found to be in good agreement with the experimental results. Under a given set of constraints the performance of the continuous glucose isomerase reactor system can be predicted by using the computer simulation method described in this paper. The important parameters studied for the optimization of reactor operation were enzyme loading, mean space time of the reactor, substrate feed concentration, enzyme decay constants, and the fractional conversion, in addition to the kinetic constants. All these parameters have significant effect on the productivity. Some unique properties of the glucose isomerization reaction and its effects on the performance of the continuous glucose isomerase reactor system have been studied and discussed. The reaction kinetics of glucose isomerase and the effects of both the enzyme loading and the changes in reaction rate within a continuous reactor on the productivity are all found to be of particular importance to this enzyme reactor system.     

A physical model of sodium channel gating

Most current models of membrane ion channel gating are abstract compartmental models consisting of many undefined states connected by rate constants arbitrarily assigned to fit the known kinetics. In this paper is described a model with states that are defined in terms of physically plausible real systems which is capable of describing accurately most of the static and dynamic properties measured for the sodium channel of the squid axon. The model has two components. The Q-system consists of charges and dipoles that can move in response to an electric field applied across the membrane. It would contain and may compose the gating charge that is known to transfer prior to channel opening. The N-system consists of a charged group or dipole that is constrained to move only in the plane of the membrane and thus does not interact directly with the trans-membrane electric field but can interact electrostatically with the Q-system. The N-system has only two states, its resting state (channel closed) and its excited state (channel open) and its response time is very short in comparison with that of the Q-system. On depolarizing the membrane the the N-system will not make a transition to its open state until a critical amount of Q-charge transfer has occurred. Using only four adjustable parameters that are fully determined by fitting the equilibrium properties of the model to those of the sodium channel in the squid axon, the model is then able to describe with some accuracy the kinetics of channel opening and closing and includes the Cole and Moore delay. In addition to these predictions of the behaviour of assemblies of channels the model predicts some of the individual channel properties measured by patch clamp techniques.     

Ionic selectivity, saturation and block in gramicidin A channels: I. Theory for the electrical properties of ion selective channels having two pairs of binding sites and multiple conductance states.

A model for the gramicidin A channel is proposed which extends existing models by adding a specific cationic binding site at each entrance to the channel. The binding of ions to these outer channel sites is assumed to shift the energy levels of the inner sites and barriers and thereby alter the channel conductance. The resulting properties are analyzed theoretically for the simplest case of two inner sites and a single energy barrier. This for-site model (two outer and two inner) predicts that the membrane potential at zero current (Uo) should be a Goldman-Hodgkin-Katz equation with concentration-dependent permeability ratios. The coefficients of the concentration-dependent terms are shown to be related to the peak energy shifts of the barrier and to the binding constants of the outer sites. The thory also predicts the channel conductance in symmetrical solutions to exhibit three limiting behaviors, from which the properties of the outer and inner sites can be characterized. In two-cation symmetrical mixtures the conductance as a function of mole fraction is shown to have a minimum, and the related phenomenon of inhibition and block exerted by one ion on the other is explained explicitly by the theory. These various phenomena, having ion interactions in a multiply occupied channel as a common physical basis, are all related (by the theory) through a set of measurable parameters describing the properties of the system.     

Molecular variability elicits a tunable switch with discrete neuromodulatory response phenotypes

Conductance-based compartment modeling requires tuning of many parameters to fit the neuron model to target electrophysiological data. Automated parameter optimization via evolutionary algorithms (EAs) is a common approach to accomplish this task, using error functions to quantify differences between model and target. We present a three-stage EA optimization protocol for tuning ion channel conductances and kinetics in a generic neuron model with minimal manual intervention. We use the technique of Latin hypercube sampling in a new way, to choose weights for error functions automatically so that each function influences the parameter search to a similar degree. This protocol requires no specialized physiological data collection and is applicable to commonly-collected current clamp data and either single- or multi-objective optimization. We applied the protocol to two representative pyramidal neurons from layer 3 of the prefrontal cortex of rhesus monkeys, in which action potential firing rates are significantly higher in aged compared to young animals. Using an idealized dendritic topology and models with either 4 or 8 ion channels (10 or 23 free parameters respectively), we produced populations of parameter combinations fitting the target datasets in less than 80 hours of optimization each. Passive parameter differences between young and aged models were consistent with our prior results using simpler models and hand tuning. We analyzed parameter values among fits to a single neuron to facilitate refinement of the underlying model, and across fits to multiple neurons to show how our protocol will lead to predictions of parameter differences with aging in these neurons.     

Identification of structures for ion channel kinetic models

Markov models of ion channel dynamics have evolved as experimental advances have improved our understanding of channel function. Past studies have examined limited sets of various topologies for Markov models of channel dynamics. We present a systematic method for identification of all possible Markov model topologies using experimental data for two types of native voltage-gated ion channel currents: mouse atrial sodium currents and human left ventricular fast transient outward potassium currents. Successful models identified with this approach have certain characteristics in common, suggesting that aspects of the model topology are determined by the experimental data. Incorporating these channel models into cell and tissue simulations to assess model performance within protocols that were not used for training provided validation and further narrowing of the number of acceptable models. The success of this approach suggests a channel model creation pipeline may be feasible where the structure of the model is not specified a priori.