Estimating transition rates in aggregated Markov models of ion channel gating with loops and with nearly equal dwell times

A typical task in the application of aggregated Markov models to ion channel data is the estimation of the transition rates between the states. Realistic models for ion channel data often have one or more loops. We show that the transition rates of a model with loops are not identifiable if the model has either equal open or closed dwell times. This non-identifiability of the transition rates also has an effect on the estimation of the transition rates for models which are not subject to the constraint of either equal open or closed dwell times. If a model with loops has nearly equal dwell times, the Hessian matrix of its likelihood function will be ill-conditioned and the standard deviations of the estimated transition rates become extraordinarily large for a number of data points which are typically recorded in experiments.     

A model of the gating of ion channels

The gating of ion channels in biological membranes has usually been described in terms of Markov transitions between a few discrete open or closed states. Such models predict that the distributions of open and closed durations decay as a sum of exponential terms. Recent experimental data have indicated that certain channels are not easily described by these models. We show that distributions of open and closed times similar to those seen experimentally are predicted by a model that involves only one open and closed state but that assumes the activation energy of the gating process to be stochastic. This model involves only a few parameters and these have direct physical interpretations. Measurements of the correlation between the durations of successive open or closed events is shown to provide an experimental method for distinguishing between this and other models.     

Gating kinetics of batrachotoxin-modified Na+ channels in the squid giant axon. Voltage and temperature effects.

The gating kinetics of batrachotoxin-modified Na+ channels were studied in outside-out patches of axolemma from the squid giant axon by means of the cut-open axon technique. Single channel kinetics were characterized at different membrane voltages and temperatures. The probability of channel opening (Po) as a function of voltage was well described by a Boltzmann distribution with an equivalent number of gating particles of 3.58. The voltage at which the channel was open 50% of the time was a function of [Na+] and temperature. A decrease in the internal [Na+] induced a shift to the right of the Po vs. V curve, suggesting the presence of an integral negative fixed charge near the activation gate. An increase in temperature decreased Po, indicating a stabilization of the closed configuration of the channel and also a decrease in entropy upon channel opening. Probability density analysis of dwell times in the closed and open states of the channel at 0 degrees C revealed the presence of three closed and three open states. The slowest open kinetic component constituted only a small fraction of the total number of transitions and became negligible at voltages greater than -65 mV. Adjacent interval analysis showed that there is no correlation in the duration of successive open and closed events. Consistent with this analysis, maximum likelihood estimation of the rate constants for nine different single-channel models produced a preferred model (model 1) having a linear sequence of closed states and two open states emerging from the last closed state. The effect of temperature on the rate constants of model 1 was studied. An increase in temperature increased all rate constants; the shift in Po would be the result of an increase in the closing rates predominant over the change in the opening rates. The temperature study also provided the basis for building an energy diagram for the transitions between channel states.     

A quantitative description of KcsA gating II: single-channel currents

The kinetic transitions of proton-activated WT KcsA and the noninactivating E71A mutant were studied at the single-channel level in purified, liposome-reconstituted preparations. Single-channel currents were recorded using patch-clamp techniques under nonstationary and steady-state conditions. Maximum-likelihood analyses reveal that the key influence of acidic pH is to increase the frequency of bursting without an effect on the intraburst open and closed dwell times, consistent with the finding from macroscopic currents that protons promote activation without a significant effect on inactivation. However, in steady-conditions of pH, voltage not only alters the burst frequency but also affects their properties, such as the frequency of the flickers and the dwell times of the closed and open states. This is to be expected if voltage modulates pathways connecting open and inactivated states. Upon opening, KcsA can enter at least two closed states that are not part of the activation pathway. The frequency and duration of these closed states was found to be voltage dependent and therefore these are likely to represent short-lived inactivated states. Single-channel recordings of WT KcsA also show varying propensity for the presence of subconductance states. The probability of occurrence of these states did not show clear modulation by voltage or pH and their origin remains unclear and a focus for further investigation. A kinetic model is proposed to describe the gating events in KcsA that recapitulates its macroscopic and single-channel behavior. The model has been constrained by the single-channel analyses presented in this work along with data from macroscopic currents in the preceding paper.     

Gating kinetics of batrachotoxin-modified sodium channels in neuroblastoma cells determined from single-channel measurements

We have observed the opening and closing of single batrachotoxin (BTX)-modified sodium channels in neuroblastoma cells using the patch-clamp method. The conductance of a single BTX-modified channel is approximately 10 pS. At a given membrane potential, the channels are open longer than are normal sodium channels. As is the case for normal sodium channels, the open dwell times become longer as the membrane is depolarized. For membrane potentials more negative than about -70 mV, histograms of both open-state dwell times and closed-state dwell times could be fit by single exponentials. For more depolarized potentials, although the open-state histograms could still be fit by single exponentials, the closed-state histograms required two exponentials. This data together with macroscopic voltage clamp data on the same system could be accounted for by a three-state closed-closed-open model with transition rates between these states that are exponential functions of membrane potential. One of the implications of this model, in agreement with experiment, is that there are always some closed BTX-modified sodium channels, regardless of membrane potential.     

Shaker potassium channel gating. I: Transitions near the open state

Kinetics of single voltage-dependent Shaker potassium channels expressed in Xenopus oocytes were studied in the absence of fast N-type inactivation. Comparison of the single-channel first latency distribution and the time course of the ensemble average current showed that the activation time course and its voltage dependence are largely determined by the transitions before first opening. The open dwell time data are consistent with a single kinetically distinguishable open state. Once the channel opens, it can enter at least two closed states which are not traversed frequently during the activation process. The rate constants for the transitions among these closed states and the open state are nearly voltage-independent at depolarized voltages (> - 30 mV). During the deactivation process at more negative voltages, the channel can close directly to a closed state in the activation pathway in a voltage-dependent fashion.     

Estimating kinetic constants from single channel data.

The process underlying the opening and closing of ionic channels in biological or artificial lipid membranes can be modeled kinetically as a time-homogeneous Markov chain. The elements of the chain are kinetic states that can be either open or closed. A maximum likelihood procedure is described for estimating the transition rates between these states from single channel data. The method has been implemented for linear kinetic schemes of fewer than six states, and is suitable for nonstationary data in which one or more independent channels are functioning simultaneously. It also provides standard errors for all estimates of rate constants and permits testing of smoothly parameterized subhypotheses of a general model. We have illustrated our approach by analysis of single channel data simulated on a computer and have described a procedure for analysis of experimental data.     

Glutamate receptor-channel gating. Maximum likelihood analysis of gigaohm seal recordings from locust muscle.

Gigaohm recordings have been made from glutamate receptor channels in excised, outside-out patches of collagenase-treated locust muscle membrane. The channels in the excised patches exhibit the kinetic state switching first seen in megaohm recordings from intact muscle fibers. Analysis of channel dwell time distributions reveals that the gating mechanism contains at least four open states and at least four closed states. Dwell time autocorrelation function analysis shows that there are at least three gateways linking the open states of the channel with the closed states. A maximum likelihood procedure has been used to fit six different gating models to the single channel data. Of these models, a cooperative model yields the best fit, and accurately predicts most features of the observed channel gating kinetics.     

Model of multifractal gating of single ionic channels in biological membranes

A physico-mathematical model of the gating machinery of single ionic channels in biological membranes has been developed. In the paradigm of this model, gating particles are subjected to: (i) deterministic friction force responsible for interactions of gating particles with the surrounding solution; (ii) deterministic potential force depending on the structure and conformational state of the channel pore (the latter is controlled by the transmembrane voltage V and regulates the motion of particles overcoming potential barriers on going from the closed (open) to the open (closed) state of the channel); (iii) deterministic force responsible for interactions of water molecules with hydrophobic sites in the channel pore, and, finally, (iv) stochastic thermal fluctuation force. The model affords adequate approximation of experimental data.     

Gating of the squid sodium channel at positive potentials: II. Single channels reveal two open states.

Single-channel recordings from squid axon Na+ channels were made under conditions of reverse sodium gradient. In the range of potentials studied, +40-(+)120 mV, channels opened promptly after depolarization, closed and reopened several times during the pulse. In patches containing only one channel, the distributions of open dwell times showed two components showing the existence of a second open state. The ensemble average of single-channel records showed incomplete inactivation that became more pronounced at more positive potentials, showing that the maintained phase of the current is the result of only one type of sodium channel with two open states. Analysis of bursts indicated that the dwell times of the events at the onset of the depolarization are longer than those later in the pulse. The dwell open times of the first events could be fitted with a single exponential. This indicated that the channels open preferentially through the first open state, the access to the second open state happening subsequently. Maximum likelihood analysis was used to evaluate several possible kinetic schemes incorporating a second open state. The best model to fit the data from single channels, and consistent with the data from macroscopic and gating currents, has a second open state evolving from the inactivated state. A kinetic model is proposed that incorporates information obtained from dialyzed axons.     

Coupled and uncoupled Markov systems: a possible way to distinguish between them

The conditional distributions of openings and closings are computed for Markov schemes with two open and two closed states and with different pathways connecting the open and closed aggregates. The computation is performed for uncoupled schemes by directly applying the probability laws and by using a convolution algorithm for coupled schemes. The results show that, for coupled schemes, conditional distributions can be nonmonotonic functions of the dwell time duration. Simulations, illustrating how the difference between coupled and uncoupled models can be detected, are also reported.     

A biologically inspired hierarchical goal directed navigation model

We propose an extended version of our previous goal directed navigation model based on forward planning of trajectories in a network of head direction cells, persistent spiking cells, grid cells, and place cells. In our original work the animat incrementally creates a place cell map by random exploration of a novel environment. After the exploration phase, the animat decides on its next movement direction towards a goal by probing linear look-ahead trajectories in several candidate directions while stationary and picking the one activating place cells representing the goal location. In this work we present several improvements over our previous model. We improve the range of linear look-ahead probes significantly by imposing a hierarchical structure on the place cell map consistent with the experimental findings of differences in the firing field size and spacing of grid cells recorded at different positions along the dorsal to ventral axis of entorhinal cortex. The new model represents the environment at different scales by populations of simulated hippocampal place cells with different firing field sizes. Among other advantages this model allows simultaneous constant duration linear look-ahead probes at different scales while significantly extending each probe range. The extension of the linear look-ahead probe range while keeping its duration constant also limits the degrading effects of noise accumulation in the network. We show the extended model’s performance using an animat in a large open field environment.     

Optimal harvesting with exponential growth in an environment with random disasters and bonanzas

The optimal discounted present value of an exploited population under constant effort harvesting in an environment with random disasters and bonanzas is investigated. The deterministic component of growth is density independent (also called Malthusian or exponential). The disasters and bonanzas are random, occurring at the times of events of a Poisson process. The density independent properties of the model and the constant effort open loop policy lead to an exact solution for the expected present value. The optimal expected present value is compared with those in deterministic models with and without deterministic type jumps. Both deterministic and random jumps can have a significant influence on the optimal present value. However, the effort to achieve the optimal is not sensitive to variations in the total jump frequency or in the discount rate. The average random jump model is much easier to apply than the deterministic jump model. Bonanzas can have much more of an effect on the present value than disasters given similar jump rates.     

Some unexpected consequences of a simple physical mechanism for voltage-dependent gating in biological membranes

We consider a model for voltage-dependent gating of channels in which the gating charges are on the channel wall and move only a small distance. When this movement occurs across the closed gate, the charges move through the entire transmembrane potential, which is energetically equivalent to their moving across the entire membrane. The channel exists in two open states, O1 and O2, and two closed states, C1 and C2; each open and closed configuration is divided into two states because of the two possible positions of the gating charges. An unusual property of this model is that the electrical work in going from an open to a closed configuration (for example, in going from O1 to C2) is path dependent, and net work can result from going reversibly around a complete cycle. The model channel, like many biological channels, shows bursting activity. This flickering on and off of the channel enables the gate to sense the electric field and decide if it should be in the open or closed configuration. We prove here some general theorms concerning the electrical work associated with the movements of the walls of channels and the movements of charges on these walls.     

Two-dimensional kinetic analysis suggests nonsequential gating of mechanosensitive channels in Xenopus oocytes

Xenopus oocytes express mechanosensitive (MS(XO)) channels that can be studied in excised patches of membrane with the patch-clamp technique. This study examines the steady-state kinetic gating properties of MS(XO) channels using detailed single-channel analysis. The open and closed one-dimensional dwell-time distributions were described by the sums of 2-3 open and 5-7 closed exponential components, respectively, indicating that the channels enter at least 2-3 open and 5-7 closed kinetic states during gating. Dependency plots revealed that the durations of adjacent open and closed intervals were correlated, indicating two or more gateway states in the gating mechanism for MS channels. Maximum likelihood fitting of two-dimensional dwell-time distributions to both generic and specific models was used to examine gating mechanism and rank models. A kinetic scheme with five closed and five open states, in which each closed state could make a direct transition to an open state (two-tiered model) could account for the major features of the single-channel data. Two-tiered models that allowed direct transitions to subconductance open states in addition to the fully open state were also consistent with multiple gateway states. Thus, the gating mechanism of MS(XO) channels differs from the sequential (linear) gating mechanisms considered for MS channels in bacteria, chick skeletal muscle, and Necturus proximal tubule.     

The use of dwell time cross-correlation functions to study single-ion channel gating kinetics.

The derivation of cross-correlation functions from single-channel dwell (open and closed) times is described. Simulation of single-channel data for simple gating models, alongside theoretical treatment, is used to demonstrate the relationship of cross-correlation functions to underlying gating mechanisms. It is shown that time irreversibility of gating kinetics may be revealed in cross-correlation functions. Application of cross-correlation function analysis to data derived from the locust muscle glutamate receptor-channel provides evidence for multiple gateway states and time reversibility of gating. A model for the gating of this channel is used to show the effect of omission of brief channel events on cross-correlation functions.     

Deterministic epidemiological models at the individual level

In many fields of science including population dynamics, the vast state spaces inhabited by all but the very simplest of systems can preclude a deterministic analysis. Here, a class of approximate deterministic models is introduced into the field of epidemiology that reduces this state space to one that is numerically feasible. However, these reduced state space master equations do not in general form a closed set. To resolve this, the equations are approximated using closure approximations. This process results in a method for constructing deterministic differential equation models with a potentially large scope of application including dynamic directed contact networks and heterogeneous systems using time dependent parameters. The method is exemplified in the case of an SIR (susceptible-infectious-removed) epidemiological model and is numerically evaluated on a range of networks from spatially local to random. In the context of epidemics propagated on contact networks, this work assists in clarifying the link between stochastic simulation and traditional population level deterministic models.     

Carboxyl tail prevents yeast K(+) channel closure: proposal of an integrated model of TOK1 gating.

TOK1 encodes the channel responsible for the prominent outward K(+) current of the yeast plasma membrane. It can dwell in several impermeable states, including a rapidly transiting, K(+)-electromotive-force-dependent "R" (rectifying) state, a voltage-independent "IB" (interburst) state, and a set of [K(+)](ext) and voltage-dependent "C" (closed) states. Whereas evidence suggests that the C states result from the constriction of an inner gate at the cytosolic end of the pore, R is most likely an intrinsic gating property of the K(+) filter. Here, we present evidence that Tok1's carboxyl-tail domain also plays an intimate role in channel gating by dynamically preventing inner-gate closures. We present an integrated model of TOK1 gating in which the filter gate, inner gate, and carboxyl tail interact to produce the various phenomenological states. Both wild-type and tailless behaviors can be replicated using Monte Carlo computer simulations based on this model.     

Interactions of protons with single open L-type calcium channels. pH dependence of proton-induced current fluctuations with Cs+, K+, and Na+ as permeant ions

We studied the pH dependence of the proton-induced current fluctuations that appear in single open L-type Ca channels when monovalent ions are the charge carriers. We used different methods of analysis to obtain kinetic measurements even under conditions where the individual transitions were too fast to be resolved directly as discrete current steps between two conductance levels. The reciprocal of the dwell times at the high conductance level increased linearly with the pipette proton activity, with a slope that was similar for Cs, K, and Na as permeant ions. Contrary to the expectation for a simple model in which the high and low conductances represent the unprotonated and protonated states of the channel, respectively, the dwell times at the low conductance level were also pH dependent and lengthened with increasing proton activity. At all pH values the dwell times at the low conductance level were longest with Cs as permeant ion and shortened in the order Cs greater than K greater than Na. We introduce a more general model of the protonation cycle in which the channel is represented by four states and can be protonated and deprotonated both at the high and low conductance levels. The conductance change is represented by a conformational change of the channel protein. We discuss the validity of this model and its implications for the mechanism by which protons interact with ion permeation through L-type Ca channels.