Markov, fractal, diffusion, and related models of ion channel gating. A comparison with experimental data from two ion channels

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

Memory in Ion Channel Kinetics

Ion channels are transport proteins present in the lipid bilayers of biological membranes. They are involved in many physiological processes, such as the generation of nerve impulses, hormonal secretion, and heartbeat. Conformational changes in the ion channel-forming protein allow the opening or closing of pores to control the ionic flux through the cell membranes. The opening and closing of the ion channel have been classically treated as a random kinetic process, known as a Markov process. Here the time the channel remains in a given state is assumed to be independent of the condition it had in the previous state. More recently, however, several studies have shown that this process is not random but a deterministic one, where both the open and closed dwell-times and the ionic current flowing through the channel are history-dependent. This property is called long memory or long-range correlation. However, there is still much controversy regarding how this memory originates, which region of the channel is responsible for this property, and which models could best reproduce the memory effect. In this article, we provide a review of what is, where it is, its possible origin, and the mathematical methods used to analyze the long-term memory present in the kinetic process of ion channels.

Moving average filtering with deconvolution (MAD) for hidden Markov model with filtering and correlated noise

Ion channel data recorded using the patch clamp technique are low-pass filtered to remove high-frequency noise. Almanjahie et al. (Eur Biophys J 44:545-556, 2015) based statistical analysis of such data on a hidden Markov model (HMM) with a moving average adjustment for the filter but without correlated noise, and used the EM algorithm for parameter estimation. In this paper, we extend their model to include correlated noise, using signal processing methods and deconvolution to pre-whiten the noise. The resulting data can be modelled as a standard HMM and parameter estimates are again obtained using the EM algorithm. We evaluate this approach using simulated data and also apply it to real data obtained from the mechanosensitive channel of large conductance (MscL) in Escherichia coli. Estimates of mean conductances are comparable to literature values. The key advantages of this method are that it is much simpler and computationally considerably more efficient than currently used HMM methods that include filtering and correlated noise.

Hidden Quantum Processes, Quantum Ion Channels, and 1/ fθ-Type Noise

In this letter, we perform a complete and in-depth analysis of Lorentzian noises, such as those arising from [Formula: see text] and [Formula: see text] channel kinetics, in order to identify the source of [Formula: see text]-type noise in neurological membranes. We prove that the autocovariance of Lorentzian noise depends solely on the eigenvalues (time constants) of the kinetic matrix but that the Lorentzian weighting coefficients depend entirely on the eigenvectors of this matrix. We then show that there are rotations of the kinetic eigenvectors that send any initial weights to any target weights without altering the time constants. In particular, we show there are target weights for which the resulting Lorenztian noise has an approximately [Formula: see text]-type spectrum. We justify these kinetic rotations by introducing a quantum mechanical formulation of membrane stochastics, called hidden quantum activated-measurement models, and prove that these quantum models are probabilistically indistinguishable from the classical hidden Markov models typically used for ion channel stochastics. The quantum dividend obtained by replacing classical with quantum membranes is that rotations of the Lorentzian weights become simple readjustments of the quantum state without any change to the laboratory-determined kinetic and conductance parameters. Moreover, the quantum formalism allows us to model the activation energy of a membrane, and we show that maximizing entropy under constrained activation energy yields the previous [Formula: see text]-type Lorentzian weights, in which the spectral exponent [Formula: see text] is a Lagrange multiplier for the energy constraint. Thus, we provide a plausible neurophysical mechanism by which channel and membrane kinetics can give rise to [Formula: see text]-type noise (something that has been occasionally denied in the literature), as well as a realistic and experimentally testable explanation for the numerical values of the spectral exponents. We also discuss applications of quantum membranes beyond [Formula: see text]-type -noise, including applications to animal models and possible impact on quantum foundations.

Sensing Magnetic Fields with Magnetosensitive Ion Channels

[-15]Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k B T in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.

Memory in a fractional-order cardiomyocyte model alters properties of alternans and spontaneous activity

Cardiac memory is the dependence of electrical activity on the prior history of one or more system state variables, including transmembrane potential (V_m), ionic current gating, and ion concentrations. While prior work has represented memory either phenomenologically or with biophysical detail, in this study, we consider an intermediate approach of a minimal three-variable cardiomyocyte model, modified with fractional-order dynamics, i.e., a differential equation of order between 0 and 1, to account for history-dependence. Memory is represented via both capacitive memory, due to fractional-order V_m dynamics, that arises due to non-ideal behavior of membrane capacitance; and ionic current gating memory, due to fractional-order gating variable dynamics, that arises due to gating history-dependence. We perform simulations for varying V_m and gating variable fractional-orders and pacing cycle length and measure action potential duration (APD) and incidence of alternans, loss of capture, and spontaneous activity. In the absence of ionic current gating memory, we find that capacitive memory, i.e., decreased V_m fractional-order, typically shortens APD, suppresses alternans, and decreases the minimum cycle length (MCL) for loss of capture. However, in the presence of ionic current gating memory, capacitive memory can prolong APD, promote alternans, and increase MCL. Further, we find that reduced V_m fractional order (typically less than 0.75) can drive phase 4 depolarizations that promote spontaneous activity. Collectively, our results demonstrate that memory reproduced by a fractional-order model can play a role in alternans formation and pacemaking, and in general, can greatly increase the range of electrophysiological characteristics exhibited by a minimal model.

Analytical and empirical fluctuation functions of the EEG microstate random walk - Short-range vs. long-range correlations

We analyze temporal autocorrelations and the scaling behaviour of EEG microstate sequences during wakeful rest. We use the recently introduced random walk approach and compute its fluctuation function analytically under the null hypothesis of a short-range correlated, first-order Markov process. The empirical fluctuation function and the Hurst parameter H as a surrogate parameter of long-range correlations are computed from 32 resting state EEG recordings and for a set of first-order Markov surrogate data sets with equilibrium distribution and transition matrices identical to the empirical data. In order to distinguish short-range correlations (H ≈ 0.5) from previously reported long-range correlations (H > 0.5) statistically, confidence intervals for H and the fluctuation functions are constructed under the null hypothesis. Comparing three different estimation methods for H, we find that only one data set consistently shows H > 0.5, compatible with long-range correlations, whereas the majority of experimental data sets cannot be consistently distinguished from Markovian scaling behaviour. Our analysis suggests that the scaling behaviour of resting state EEG microstate sequences, though markedly different from uncorrelated, zero-order Markov processes, can often not be distinguished from a short-range correlated, first-order Markov process. Our results do not prove the microstate process to be Markovian, but challenge the approach to parametrize resting state EEG by single parameter models.

Revealing the activation pathway for TMEM16A chloride channels from macroscopic currents and kinetic models

TMEM16A (ANO1), the pore-forming subunit of calcium-activated chloride channels, regulates several physiological and pathophysiological processes such as smooth muscle contraction, cardiac and neuronal excitability, salivary secretion, tumour growth and cancer progression. Gating of TMEM16A is complex because it involves the interplay between increases in intracellular calcium concentration ([Ca(2+)]i), membrane depolarization, extracellular Cl(-) or permeant anions and intracellular protons. Our goal here was to understand how these variables regulate TMEM16A gating and to explain four observations. (a) TMEM16A is activated by voltage in the absence of intracellular Ca(2+). (b) The Cl(-) conductance is decreased after reducing extracellular Cl(-) concentration ([Cl(-)]o). (c) ICl is regulated by physiological concentrations of [Cl(-)]o. (d) In cells dialyzed with 0.2 μM [Ca(2+)]i, Cl(-) has a bimodal effect: at [Cl(-)]o <30 mM TMEM16A current activates with a monoexponential time course, but above 30 mM, [Cl(-)]o ICl activation displays fast and slow kinetics. To explain the contribution of Vm, Ca(2+) and Cl(-) to gating, we developed a 12-state Markov chain model. This model explains TMEM16A activation as a sequential, direct, and Vm-dependent binding of two Ca(2+) ions coupled to a Vm-dependent binding of an external Cl(-) ion, with Vm-dependent transitions between states. Our model predicts that extracellular Cl(-) does not alter the apparent Ca(2+) affinity of TMEM16A, which we corroborated experimentally. Rather, extracellular Cl(-) acts by stabilizing the open configuration induced by Ca(2+) and by contributing to the Vm dependence of activation.

Modeling magnetosensitive ion channels in the viscoelastic environment of living cells

We propose and study a model of hypothetical magnetosensitive ionic channels which are long thought to be a possible candidate to explain the influence of weak magnetic fields on living organisms ranging from magnetotactic bacteria to fishes, birds, rats, bats, and other mammals including humans. The core of the model is provided by a short chain of magnetosomes serving as a sensor, which is coupled by elastic linkers to the gating elements of ion channels forming a small cluster in the cell membrane. The magnetic sensor is fixed by one end on cytoskeleton elements attached to the membrane and is exposed to viscoelastic cytosol. Its free end can reorient stochastically and subdiffusively in viscoelastic cytosol responding to external magnetic field changes and can open the gates of coupled ion channels. The sensor dynamics is generally bistable due to bistability of the gates which can be in two states with probabilities which depend on the sensor orientation. For realistic parameters, it is shown that this model channel can operate in the magnetic field of Earth for a small number (five to seven) of single-domain magnetosomes constituting the sensor rod, each of which has a typical size found in magnetotactic bacteria and other organisms or even just one sufficiently large nanoparticle of a characteristic size also found in nature. It is shown that, due to the viscoelasticity of the medium, the bistable gating dynamics generally exhibits power law and stretched exponential distributions of the residence times of the channels in their open and closed states. This provides a generic physical mechanism for the explanation of the origin of such anomalous kinetics for other ionic channels whose sensors move in a viscoelastic environment provided by either cytosol or biological membrane, in a quite general context, beyond the fascinating hypothesis of magnetosensitive ionic channels we explore.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Principles of single-channel kinetic analysis

Single-channel recording provides high resolution information on gating mechanisms of ion channels that are generally difficult to obtain from macroscopic measurements. Analysis of the data, however, has proven to be challenging. Early approaches rely on half-amplitude threshold detection to idealize the record into dwell-times, followed by fitting duration histograms to resolve kinetics. More recent analyses exploit explicit modeling of the data to improve the idealization accuracy. The dwell-time fitting has also evolved into direct fitting of dwell-time sequences using the maximum likelihood approach while taking account of effects of missed events. Finally, hidden Markov modeling provides an ultimate approach by which both single channel amplitudes and kinetics are analyzed simultaneously without the need of idealization. The progress in theory, along with the advance in computing power as well as the development of user-friendly software, has transformed single-channel analysis, once a specialty task, now readily accessible to a broader community of scientists.

Statistical assessment of change point detectors for single molecule kinetic analysis

Change point detectors (CPDs) are used to segment recordings of single molecules for the purpose of kinetic analysis. The assessment of the accuracy of CPD algorithms has usually been based on testing them with simulated data. However, there have not been methods to assess the output of CPDs from real data independent of simulation. Here we present one method to do this based on the assumption that the elementary kinetic unit is a stationary period (SP) with a normal distribution of samples, separated from other SPs by change points (CPs). Statistical metrics of normality can then be used to assess SPs detected by a CPD algorithm (detected SPs, DSPs). Two statistics in particular were found to be useful, the z-transformed skew (S(Z)) and z-transformed kurtosis (K(Z)). K(Z)(S(Z)) plots of DSP from noise, simulated data and single ion channel recordings showed that DSPs with false negative CP could be distinguished. Also they showed that filtering had a significant effect on the normality of data and so filtering should be taken into account when calculating statistics. This method should be useful for analyzing single molecule recordings where there is no simple model for the data.

Gating of maxi channels observed from pseudo-phase portraits

Phase space has been used to visualize and analyze the dynamic behavior of stochastic and chaotic systems. We applied this concept to maxi channels recorded from excised inside-out patches of in situ interstitial cells of Cajal. Pseudo-phase portraits of channel current were fairly homogeneous from patch to patch. They showed three main peaks, α, β, and γ, in increasing conductance. These represented single or near aggregated states. The α-peak was the closed state. The β-peak was small, consisting of a single conductance state, or in some cases two (a doublet). The β-peak state(s) had a long lifetime and displayed a characteristic behavior of frequent short transitions to γ but not to α. It was always preceded by a short series of α/γ-transitions. The γ-peak was the largest and consisted of a large number of conductance states with fast state transitions, sometimes to the extent of causing a diffusive-type behavior. Phase portraits allowed us to construct a provisional gating scheme for the maxi channel and suggest that further analysis of recordings in higher dimensional phase space and with related techniques may be promising.

Test of normality for integrated change point detection and mixture modeling

Single-molecule data often show step-like changes in the quantity measured between constant levels. Analysis of this data consists of detecting the steps, i.e., change point detection (CPD), and determining the levels, i.e., clustering. We describe a novel algorithm which integrates these two analyses, based on a statistical test of a normal distribution. The test of normality (TON) algorithm integrates statistical CPD with gaussian mixture model clustering. We used TON with both simulated data and ion channel patch-clamp recordings. It performed well with simulated data except at a high signal-to-noise ratio and when the frequency of steps was high compared to the sampling frequency. TON has advantages over separate CPD and mixture modeling algorithms, especially for complex single-molecule data. This was illustrated by its application to the maxichannel, an ion channel with multiple subconductance states.

On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory

Several approaches to ion-channel gating modelling have been proposed. Although many models describe the dwell-time distributions correctly, they are incapable of predicting and explaining the long-term correlations between the lengths of adjacent openings and closings of a channel. In this paper we propose two simple random-walk models of the gating dynamics of voltage and Ca(2+)-activated potassium channels which qualitatively reproduce the dwell-time distributions, and describe the experimentally observed long-term memory quite well. Biological interpretation of both models is presented. In particular, the origin of the correlations is associated with fluctuations of channel mass density. The long-term memory effect, as measured by Hurst R/S analysis of experimental single-channel patch-clamp recordings, is close to the behaviour predicted by our models. The flexibility of the models enables their use as templates for other types of ion channel.

Quantitative analysis of DNA-looping kinetics from tethered particle motion experiments

In this chapter we show the application of a maximum-likelihood-based method to the reconstruction of DNA-looping single-molecule time traces from tethered particle motion experiments. The method does not require time filtering of the data and improves the time resolution by an order of magnitude with respect to the threshold-crossing approach. Moreover, it is not based on presumed kinetic models, overcoming the limitations of other approaches proposed previously, and allowing its applications to mechanisms with complex kinetic schemes. Numerical simulations have been used to test the performances of this analysis over a wide range of time scales. We have then applied this method to determine the looping kinetics of a well-known DNA-looping protein, the lambda-repressor.

Viscoelastic subdiffusion: from anomalous to normal

We study viscoelastic subdiffusion in bistable and periodic potentials within the generalized Langevin equation approach. Our results justify the (ultra)slow fluctuating rate view of the corresponding bistable non-Markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells. The transition kinetics is asymptotically stretched exponential when the potential barrier V0 several times exceeds thermal energy k(B)T [V(0) approximately (2-10)k(B)T] and it cannot be described by the non-Markovian rate theory (NMRT). The well-known NMRT result approximates, however, ever better with the increasing barrier height, the most probable logarithm of the residence times. Moreover, the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power-law exponent of free subdiffusion alpha . Such a potential-free subdiffusion is ergodic. Surprisingly, in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit. However, the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height. The time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude. All these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps.

Position-dependent stochastic diffusion model of ion channel gating

A position-dependent stochastic diffusion model of gating in ion channels is developed by considering the spatial variation of the diffusion coefficient between the closed and open states. It is assumed that a sensor which regulates the opening of the ion channel experiences Brownian motion in a closed region Rc and a transition region Rm, where the dynamics is described by probability densities pc(x,t) and pm(x,t) which satisfy interacting Fokker-Planck equations with diffusion coefficient Dc(x)=Dcexp(gammacx) and Dm(x)=Dmexp(-gammamx). The analytical solution of the coupled equations may be approximated by the lowest frequency relaxation, a short time after the application of a depolarizing voltage clamp, when Dm<<Dc or the diffusion parameter gammam is sufficiently large. Thus, an empirical rate equation that describes gating transitions may be derived from a stochastic diffusion model if there is a large diffusion (or potential) barrier between open and closed states.

Statistical properties of the dichotomous noise generated in biochemical processes

Dichotomous noise detected with the help of various single-molecule techniques convincingly reveals the actual occurrence of a multitude of conformational substates composing the native state of proteins. The nature of the stochastic dynamics of transitions between these substates is determined by the particular statistical properties of the noise observed. These involve nonexponential and possibly oscillatory time decay of the second order autocorrelation function, its relation to the third order autocorrelation function, and a relationship to dwell-time distribution densities and their correlations. Processes gated by specific conformational substates are distinguished from those with fluctuating barriers. This study throws light on the intriguing matter of the possibility of multiple stepping of the myosin motor along the actin filament per ATP molecule hydrolyzed.

Dissipative stochastic mechanics for capturing neuronal dynamics under the influence of ion channel noise: formalism using a special membrane

Based on the idea conveyed in the author's prior study [Fluct. Noise Lett. 6, L147 (2006)], a physical approach for the description of neuronal dynamics under the influence of ion channel noise is developed in the realm of Nelson's stochastic mechanics when open to dissipative environments. The formalism therein is scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. Led by the presence of multiple number of gates in an ion channel, a dual viewpoint of channel noise is established. Then, stochastic mechanics is adopted to model those channel fluctuations emerging from the uncertainty in accessing the permissible topological states of open gates. A mutual interaction between the above fluctuations and the noise, emerging from the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, is portrayed within a system plus reservoir strategy. Induced by the interaction, renormalizations of the membrane capacitance and of a membrane voltage dependent potential are found to arise. Consequently, the equations of motion, for the expectation values of the variables and the pair correlation functions, are obtained in the collective membrane voltage space.

Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.

Principles of single-channel kinetic analysis

Single-channel recording provides molecular insights that are nearly unattainable from macroscopic measurements. Analysis of the data, however, has proven to be a difficult challenge. Early approach relies on the half-amplitude threshold detection to idealize the data into dwell-times, followed by fitting of the duration histograms to resolve the kinetics. More recent analysis exploits explicit modeling of both the channel and noise statistics to improve the idealization accuracy. The dwell-time fitting has also evolved into direct fitting of the dwell-time sequences using the full maximum likelihood approach while taking account of the effects of missed events. Finally, hidden Markov modeling provides a new paradigm in which both the amplitudes and kinetics can be analyzed simultaneously without the need of idealization. The progress in theory, along with the advance in computing power and the development of user-friendly software, has made single-channel analysis, once a specialty task, now readily accessible to a broader community of scientists.

Equivalence of trans paths in ion channels

We explore stochastic models for the study of ion transport in biological cells. Analysis of these models explains and explores an interesting feature of ion transport observed by biophysicists. Namely, the average time it takes ions to cross certain ion channels is the same in either direction, even if there is an electric potential difference across the channels. It is shown for simple single ion models that the distribution of a path (i.e., the history of location versus time) of an ion crossing the channel in one direction has the same distribution as the time-reversed path of an ion crossing the channel in the reverse direction. Therefore, not only is the mean duration of these paths equal, but other measures, such as the variance of passage time or the mean time a path spends within a specified section of the channel, are also the same for both directions of traversal. The feature is also explored for channels with interacting ions. If a system of interacting ions is in reversible equilibrium (net flux is zero), then the equivalence of the left-to-right trans paths with the time-reversed right-to-left trans paths still holds. However, if the system is in equilibrium, but not reversible equilibrium, then such equivalence need not hold.

Fractional diffusion modeling of ion channel gating

An anomalous diffusion model for ion channel gating is put forward. This scheme is able to describe nonexponential, power-law-like distributions of residence time intervals in several types of ion channels. Our method presents a generalization of the discrete diffusion model by Millhauser, Salpeter, and Oswald [Proc. Natl. Acad. Sci. U.S.A. 85, 1503 (1988)] to the case of a continuous, anomalous slow conformational diffusion. The corresponding generalization is derived from a continuous-time random walk composed of nearest-neighbor jumps which in the scaling limit results in a fractional diffusion equation. The studied model contains three parameters only: the mean residence time, a characteristic time of conformational diffusion, and the index of subdiffusion. A tractable analytical expression for the characteristic function of the residence time distribution is obtained. In the limiting case of normal diffusion, our prior findings [Proc. Natl. Acad. Sci. U.S.A. 99, 3552 (2002)] are reproduced. Depending on the chosen parameters, the fractional diffusion model exhibits a very rich behavior of the residence time distribution with different characteristic time regimes. Moreover, the corresponding autocorrelation function of conductance fluctuations displays nontrivial power law features. Our theoretical model is in good agreement with experimental data for large conductance potassium ion channels.

Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws

We review recent evidence illustrating the fundamental difference between cytoplasmic and test tube biochemical kinetics and thermodynamics, and showing the breakdown of the law of mass action and power-law approximation in in vivo conditions. Simulations of biochemical reactions in non-homogeneous media show that as a result of anomalous diffusion and mixing of the biochemical species, reactions follow a fractal-like kinetics. Consequently, the conventional equations for biochemical pathways fail to describe the reactions in in vivo conditions. We present a modification to fractal-like kinetics following the Zipf-Mandelbrot distribution which will enable the modelling and analysis of biochemical reactions occurring in crowded intracellular environments.

Theory of non-Markovian stochastic resonance

We consider a two-state model of non-Markovian stochastic resonance (SR) within the framework of the theory of renewal processes. Residence time intervals are assumed to be mutually independent and characterized by some arbitrary nonexponential residence time distributions which are modulated in time by an externally applied signal. Making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal. These equations generalize earlier integral renewal equations by Cox and others to the case of driving-induced nonstationarity. On the basis of these equations a response theory of two-state renewal processes is formulated beyond the linear response approximation. Moreover, a general expression for the linear response function is derived. The connection of the developed approach with the phenomenological theory of linear response for manifest non-Markovian SR put forward [I. Goychuk and P. Hänggi, Phys. Rev. Lett. 91, 070601 (2003)] is clarified and its range of validity is scrutinized. The theory is then applied to SR in symmetric non-Markovian systems and to the class of single ion channels possessing a fractal kinetics.

Ligand-gated ion channel currents in a nonstationary lyotropic model

Transmembrane currents in ligand-gated ion channels are calculated in a nonstationary, chemically open whole cell system or patch of a membrane. The model is lyotropic in the sense that dynamics, and parameters such as the ligand concentration for half-maximal response (scale of response), and threshold for firing, such as in neurons, become nonlinear functions of the reactant concentrations. The derived total currents fit recorded data significantly better than those derived from mass action, Ising, and other stationary type models, in which the derived response is often displaced from the assessed response by several orders in the ligand concentration. Also, the derived slope of response is in perfect agreement with the values assessed.

Ion channel gating: a first-passage time analysis of the Kramers type

The opening rate of voltage-gated potassium ion channels exhibits a characteristic knee-like turnover where the common exponential voltage dependence changes suddenly into a linear one. An explanation of this puzzling crossover is put forward in terms of a stochastic first passage time analysis. The theory predicts that the exponential voltage dependence correlates with the exponential distribution of closed residence times. This feature occurs at large negative voltages when the channel is predominantly closed. In contrast, the linear part of voltage dependence emerges together with a nonexponential distribution of closed dwelling times with increasing voltage, yielding a large opening rate. Depending on the parameter set, the closed-time distribution displays a power law behavior that extends over several decades.

Universality of receptor channel responses

Rate parameters estimated for neurotransmitter-gated receptor channel opening and receptor desensitization are classified according to their dependence on the temporal resolution of the techniques applied in the measurements. Because allosteric proteins constituting receptor channels impose restrictions on the types of model suitable to describe the dynamic response of channels to neurotransmitters, Markovian, non-linear or fractal dynamic models and their possible extension to receptor channel response in excitable membranes are discussed.

Fractal methods to analyze ion channel kinetics

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.

Time course of reactions controlled and gated by intramolecular dynamics of proteins: predictions of the model of random walk on fractal lattices

Computer simulations of random walk on the Sierpinski gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser flash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.

A synthetic picture of intramolecular dynamics of proteins. Towards a contemporary statistical theory of biochemical processes

An increasing body of experimental evidence indicates the slow character of internal dynamics of native proteins. The important consequence of this is that theories of chemical reactions, used hitherto, appear inadequate for description of most biochemical reactions. Construction of a contemporary, truly advanced statistical theory of biochemical processes will need simple but realistic models of microscopic dynamics of biomolecules. In this review, intended to be a contribution towards this direction, three topics are considered. First, an intentionally simplified picture of dynamics of native proteins which emerges from recent investigations is presented. Fast vibrational modes of motion, of periods varying from 10(-14) to 10(-11) s, are contrasted with purely stochastic conformational transitions. Significant evidence is adduced that the relaxation time spectrum of the latter spreads in the whole range from 10(-11) to 10(5) s or longer, and up to 10(-7) s it is practically quasi-continuous. Next, the essential ideas of the theory of reaction rates based on stochastic models of intramolecular dynamics are outlined. Special attention is paid to reactions involving molecules in the initial conformational substrates confirmed to the transition state, which is realized in actual experimental situations. And finally, the two best experimentally justified classes of models of conformational transition dynamics, symbolically referred to as "protein glass" and "protein machine", are described and applied to the interpretation of a few simple biochemical processes, perhaps the most important result reported is the demonstration of the possibility of predominance of the short initial condition-dependent stage of protein involved reactions over the main stage described by the standard kinetics. This initial stage, and not the latter, is expected to be responsible for the coupling of component reactions in the complete enzymatic cycles as well as more complex processes of biological free energy transduction.

Non stochastic distribution of single channels in planar lipid bilayers

The selectivity of the planar lipid bilayers modified by two channel-forming proteins (alpha-toxin S. aureus and colicin Ia) was examined. It was established that in all cases the value of zero current potential depended on the amount of open ion channels and increased with the number of channels (from one to about 5-7). These facts point out both the interactions among ion channels and their non stochastic distribution on the membrane surface.

Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism

Markov kinetic models were used to synthesize a complete description of synaptic transmission, including opening of voltage-dependent channels in the presynaptic terminal, release of neurotransmitter, gating of postsynaptic receptors, and activation of second-messenger systems. These kinetic schemes provide a more general framework for modeling ion channels than the Hodgkin-Huxley formalism, supporting a continuous spectrum of descriptions ranging from the very simple and computationally efficient to the highly complex and biophysically precise. Examples are given of simple kinetic schemes based on fits to experimental data that capture the essential properties of voltage-gated, synaptic and neuromodulatory currents. The Markov formalism allows the dynamics of ionic currents to be considered naturally in the larger context of biochemical signal transduction. This framework can facilitate the integration of a wide range of experimental data and promote consistent theoretical analysis of neural mechanisms from molecular interactions to network computations.

Gating kinetics of the quisqualate-sensitive glutamate receptor of locust muscle studied using agonist concentration jumps and computer simulations

Outside-out patches excised from extrajunctional membrane of locust muscle were subjected to "concentration jumps" of L-glutamate, using the liquid filament switch technique, to study channel opening and closing rates, desensitization onset, and recovery from desensitization of a quisqualate-sensitive glutamate receptor (qGluR). Based on data obtained from these experimental studies, computer modeling techniques have been used in an attempt to simulate the behavior of qGluR during a concentration jump of L-glutamate. A linear model with three closed states (one unliganded, one monoliganded, and one biliganded), one open state (binding two molecules of L-glutamate), and two desensitization states (the one monoliganded, the other biliganded) leading from the unliganded closed state simulated all of the experimentally observed behavior. The results are discussed in the context of previous equilibrium studies in which desensitization was inhibited with concanavalin A and for which a ten-state model was required to simulate the behavior of qGluR.

Single ion channel models incorporating aggregation and time interval omission

We present a general theoretical framework, incorporating both aggregation of states into classes and time interval omission, for stochastic modeling of the dynamic aspects of single channel behavior. Our semi-Markov models subsume the standard continuous-time Markov models, diffusion models and fractal models. In particular our models allow for quite general distributions of state sojourn times and arbitrary correlations between successive sojourn times. Another key feature is the invariance of our framework with respect to time interval omission: that is, properties of the aggregated process incorporating time interval omission can be derived directly from corresponding properties of the process without it. Even in the special case when the underlying process is Markov, this leads to considerable clarification of the effects of time interval omission. Among the properties considered are equilibrium behavior, sojourn time distributions and their moments, and auto-correlation and cross-correlation functions. The theory is motivated by ion channel mechanisms drawn from the literature, and illustrated by numerical examples based on these.

Stochastic models for ion channels: introduction and bibliography

This paper provides an introduction to and overview of the use of stochastic models and statistical analysis in the study of ion channels in cell membranes. An extensive bibliography is included.

Cooperative gating of chloride channel subunits in endothelial cells

New methods are described to detect subconductance levels and to analyse ion channel gating. These methods are applied to simulated and experimental data. Single chloride channel records from inside-out membrane patches excised from human umbilical venous endothelial cells (HUVEC) exhibit, in addition to the full closed and full open configurations, intermediate subconductance levels which are multiple of an elementary conductance of 112.5 pS. Analysis of transitions from one state to another and the comparison of real data with simulated data leads to the proposal of a cooperative model of gating for the observed subunits of a chloride channel.