Fractal methods to analyze ion channel kinetics

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.

New Diagnostic Tool for Ion Channel Activity Hidden Behind the Dwell-Time Correlations

The patch-clamp technique is a powerful tool that allows for a long observation of transport protein activity in real time. Experimental traces of single-channel currents can be considered as a record of the channel's conformational switching related to its activation and gating. In this work, we present a mathematically simple method of patch-clamp data analysis that assesses the connectivity and occupancy of distinct conformational substates of the channel. The proposed approach appears to be a big step forward due to its possible applications in the determination of channel substates related to disease and in the analysis of drug-channel interactions on the level of repetitive sequences of channel conformations. This is especially important in cases when molecular dynamics docking is impossible and Markovian modeling requires ambiguous optimization tasks.

A Minimal PKPD Interaction Model for Evaluating Synergy Effects of Combined NSCLC Therapies

This paper introduces a mathematical compartmental formulation of dose-effect synergy modelling for multiple therapies in non small cell lung cancer (NSCLC): antiangiogenic, immuno- and radiotherapy. The model formulates the dose-effect relationship in a unified context, with tumor proliferating rates and necrotic tissue volume progression as a function of therapy management profiles. The model accounts for inter- and intra-response variability by using surface model response terms. Slow acting peripheral compartments such as fat and muscle for drug distribution are not modelled. This minimal pharmacokinetic-pharmacodynamic (PKPD) model is evaluated with reported data in mice from literature. A systematic analysis is performed by varying only radiotherapy profiles, while antiangiogenesis and immunotherapy are fixed to their initial profiles. Three radiotherapy protocols are selected from literature: (1) a single dose 5 Gy once weekly; (2) a dose of 5 Gy × 3 days followed by a 2 Gy × 3 days after two weeks and (3) a dose of 5 Gy + 2 × 0.075 Gy followed after two weeks by a 2 Gy + 2 × 0.075 Gy dose. A reduction of 28% in tumor end-volume after 30 days was observed in Protocol 2 when compared to Protocol 1. No changes in end-volume were observed between Protocol 2 and Protocol 3, this in agreement with other literature studies. Additional analysis on drug interaction suggested that higher synergy among drugs affects up to three-fold the tumor volume (increased synergy leads to significantly lower growth ratio and lower total tumor volume). Similarly, changes in patient response indicated that increased drug resistance leads to lower reduction rates of tumor volumes, with end-volume increased up to 25–30%. In conclusion, the proposed minimal PKPD model has physiological value and can be used to study therapy management protocols and is an aiding tool in the clinical decision making process. Although developed with data from mice studies, the model is scalable to NSCLC patients.

Multifractal Properties of BK Channel Currents in Human Glioblastoma Cells

Potassium channels play an important physiological role in glioma cells. In particular, voltage- and Ca2+-activated large-conductance BK channels (gBK in gliomas) are involved in the intensive growth and extensive migrating behavior of the mentioned tumor cells; thus, they may be considered as a drug target for the therapeutic treatment of glioblastoma. To enable appropriate drug design, molecular mechanisms of gBK channel activation by diverse stimuli should be unraveled as well as the way that the specific conformational states of the channel relate to its functional properties (conducting/nonconducting). There is an open debate about the actual mechanism of BK channel gating, including the question of how the channel proteins undergo a range of conformational transitions when they flicker between nonconducting (functionally closed) and conducting (open) states. The details of channel conformational diffusion ought to have its representation in the properties of the experimental signal that describes the ion-channel activity. Nonlinear methods of analysis of experimental nonstationary series can be useful for observing the changes in the number of channel substates available from geometrical and energetic points of view at given external conditions. In this work, we analyze whether the multifractal properties of the activity of glioblastoma BK channels depend on membrane potential, and which states, conducting or nonconducting, affect the total signal to a larger extent. With this aim, we carried out patch-clamp experiments at different levels of membrane hyper- and depolarization. The obtained time series of single channel currents were analyzed using the multifractal detrended fluctuation analysis (MFDFA) method in a standard form and incorporating focus-based multifractal (FMF) formalism. Thus, we show the applicability of a modified MFDFA technique in the analysis of an experimental patch-clamp time series. The obtained results suggest that membrane potential strongly affects the conformational space of the gBK channel proteins and the considered process has nonlinear multifractal characteristics. These properties are the inherent features of the analyzed signals due to the fact that the main tendencies vanish after shuffling the data.

Probing Thermotropic Phase Behavior of Dipalmitoylphosphatidylcholine Bilayers from Electrical and Topographic Data in a Horizontal Black Lipid Membrane Model

Here, a homemade device allowed preparing horizontal lipid bilayer membranes (hBLMs) for recording electrical and topographical data simultaneously and in real-time, under temperature (T)-controlled conditions along a cooling process of dipalmitoylphosphatidylcholine (DPPC) bilayers. Electrical parameters (ionic current intensity, I, and transmembrane resistance, R = ΔV/I) plotted against T exhibited discontinuities at the main transition (T_Pβ'→Lα) and pretransition (T_Lβ→Pβ') temperatures of DPPC. The T-dependent sensitivity to ΔV-induced electrostriction was revealed by capacitance measurements. The patterns of I fluctuation described long-range correlations reflected by 1/f-type noise in the ripple phase (P_β') and Brownian-type fluctuations in the liquid-crystalline (L_α) phase at voltage intensities lower than a voltage threshold (ΔV_th = ±160 mV), indicating that autocorrelations arise from an underlying structural connectivity that takes place within ordered phases. At |V| ≥ V_th, the fluctuation dynamics exhibited a 1/f behavior over the whole temperature range analyzed, suggesting that upon a certain intensity of external electrical perturbation, the membrane system evolves toward a voltage-induced percolated-pore state. At T > T_Pβ'→Lα, differential interference contrast micrographs showed droplet-like structures, probably containing solvent traces of the lipid solution, which were reverted upon cooling. However, droplets did not interfere with the thermotropic equilibrium of the bilayer phase. This suggested that the temperature-induced changes in the electrical properties of the bilayer, as well as in the complexity of the fluctuation patterns (emergency of long- and short-range correlations), were strongly associated with the characteristic thermotropic behavior of DPPC, without significant deviations induced by the presence of residual n-decane in the bilayer. Our hBLM model membrane proved useful for correlating thermotropic phase changes with electro-biophysical and topographical information.

Dual-induced multifractality in online viewing activity

Although recent studies have found that the long-term correlations relating to the fat-tailed distribution of inter-event times exist in human activity and that these correlations indicate the presence of fractality, the property of fractality and its origin have not been analyzed. We use both detrended fluctuation analysis and multifractal detrended fluctuation analysis to analyze the time series in online viewing activity separating from Movielens and Netflix. We find long-term correlations at both the individual and communal levels and that the extent of correlation at the individual level is determined by the activity level. These long-term correlations also indicate that there is fractality in the pattern of online viewing. We first find a multifractality that results from the combined effect of the fat-tailed distribution of inter-event times (i.e., the times between successive viewing actions of individuals) and the long-term correlations in online viewing activity and verify this finding using three synthesized series. Therefore, it can be concluded that the multifractality in online viewing activity is caused by both the fat-tailed distribution of inter-event times and the long-term correlations and that this enlarges the generic property of human activity to include not just physical space but also cyberspace.

ModFossa: A library for modeling ion channels using Python

The creation and simulation of ion channel models using continuous-time Markov processes is a powerful and well-used tool in the field of electrophysiology and ion channel research. While several software packages exist for the purpose of ion channel modeling, most are GUI based, and none are available as a Python library. In an attempt to provide an easy-to-use, yet powerful Markov model-based ion channel simulator, we have developed ModFossa, a Python library supporting easy model creation and stimulus definition, complete with a fast numerical solver, and attractive vector graphics plotting.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Voltage-Gated Lipid Ion Channels

Synthetic lipid membranes can display channel-like ion conduction events even in the absence of proteins. We show here that these events are voltage-gated with a quadratic voltage dependence as expected from electrostatic theory of capacitors. To this end, we recorded channel traces and current histograms in patch-experiments on lipid membranes. We derived a theoretical current-voltage relationship for pores in lipid membranes that describes the experimental data very well when assuming an asymmetric membrane. We determined the equilibrium constant between closed and open state and the open probability as a function of voltage. The voltage-dependence of the lipid pores is found comparable to that of protein channels. Lifetime distributions of open and closed events indicate that the channel open distribution does not follow exponential statistics but rather power law behavior for long open times.

The role of the circadian system in fractal neurophysiological control

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system's role in fractal regulation.

Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

The effect of nonspecific binding of lambda repressor on DNA looping dynamics

The λ repressor (CI) protein-induced DNA loop maintains stable lysogeny, yet allows efficient switching to lysis. Herein, the kinetics of loop formation and breakdown has been characterized at various concentrations of protein using tethered particle microscopy and a novel, to our knowledge, method of analysis. Our results show that a broad distribution of rate constants and complex kinetics underlie loop formation and breakdown. In addition, comparison of the kinetics of looping in wild-type DNA and DNA with mutated o3 operators showed that these sites may trigger nucleation of nonspecific binding at the closure of the loop. The average activation energy calculated from the rate constant distribution is consistent with a model in which nonspecific binding of CI between the operators shortens their effective separation, thereby lowering the energy barrier for loop formation and broadening the rate constant distribution for looping. Similarly, nonspecific binding affects the kinetics of loop breakdown by increasing the number of loop-securing protein interactions, and broadens the rate constant distribution for this reaction. Therefore, simultaneous increase of the rate constant for loop formation and reduction of that for loop breakdown stabilizes lysogeny. Given these simultaneous changes, the frequency of transitions between the looped and the unlooped state remains nearly constant. Although the loop becomes more stable thermodynamically with increasing CI concentration, it still opens periodically, conferring sensitivity to environmental changes, which may require switching to lytic conditions.

Comparing ion conductance recordings of synthetic lipid bilayers with cell membranes containing TRP channels

In this article we compare electrical conductance events from single channel recordings of three TRP channel proteins (TRPA1, TRPM2 and TRPM8) expressed in human embryonic kidney cells with channel events recorded on synthetic lipid membranes close to melting transitions. Ion channels from the TRP family are involved in a variety of sensory processes including thermo- and mechano-reception. Synthetic lipid membranes close to phase transitions display channel-like events that respond to stimuli related to changes in intensive thermodynamic variables such as pressure and temperature. TRP channel activity is characterized by typical patterns of current events dependent on the type of protein expressed. Synthetic lipid bilayers show a wide spectrum of electrical phenomena that are considered typical for the activity of protein ion channels. We find unitary currents, burst behavior, flickering, multistep-conductances, and spikes behavior in both preparations. Moreover, we report conductances and lifetimes for lipid channels as described for protein channels. Non-linear and asymmetric current-voltage relationships are seen in both systems. Without further knowledge of the recording conditions, no easy decision can be made whether short current traces originate from a channel protein or from a pure lipid membrane.

The what and where of adding channel noise to the Hodgkin-Huxley equations

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage--and the highly nonlinear impact of that voltage back on the currents themselves--into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

Bi-stability, hysteresis, and memory of voltage-gated lysenin channels

Lysenin, a 297 amino acid pore-forming protein extracted from the coelomic fluid of the earthworm E. foetida, inserts constitutively open large conductance channels in natural and artificial lipid membranes containing sphingomyelin. The inserted channels show voltage regulation and slowly close at positive applied voltages. We report on the consequences of slow voltage-induced gating of lysenin channels inserted into a planar Bilayer Lipid Membrane (BLM), and demonstrate that these pore-forming proteins constitute memory elements that manifest gating bi-stability in response to variable external voltages. The hysteresis in macroscopic currents dynamically changes when the time scale of the voltage variation is smaller or comparable to the characteristic conformational equilibration time, and unexpectedly persists for extremely slow-changing external voltage stimuli. The assay performed on a single lysenin channel reveals that hysteresis is a fundamental feature of the individual channel unit and an intrinsic component of the gating mechanism. The investigation conducted at different temperatures reveals a thermally stable reopening process, suggesting that major changes in the energy landscape and kinetics diagram accompany the conformational transitions of the channels. Our work offers new insights on the dynamics of pore-forming proteins and provides an understanding of how channel proteins may form an immediate record of the molecular history which then determines their future response to various stimuli. Such new functionalities may uncover a link between molecular events and macroscopic processing and transmission of information in cells, and may lead to applications such as high density biologically-compatible memories and learning networks.

Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons

The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. E 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can capture the stochastic dynamics of Markov chain models.We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effects on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case. The temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, but the subunit-based approaches do not correctly account for this structure. Our study confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.

Beyond the death of linear response: 1/f optimal information transport

Nonergodic renewal processes have recently been shown by several authors to be insensitive to periodic perturbations, thereby apparently sanctioning the death of linear response, a building block of nonequilibrium statistical physics. We show that it is possible to go beyond the "death of linear response" and establish a permanent correlation between an external stimulus and the response of a complex network generating nonergodic renewal processes, by taking as stimulus a similar nonergodic process. The ideal condition of 1/f noise corresponds to a singularity that is expected to be relevant in several experimental conditions.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Ion-channel-like behavior in lipid bilayer membranes at the melting transition

It is well known that at the gel-liquid phase-transition temperature a lipid bilayer membrane exhibits an increased ion permeability. We analyze the quantized currents in which the increased permeability presents itself. The open time histogram shows a "-3/2" power law which implies an open-closed transition rate that decreases like k(t)∝t(-1) as time evolves. We propose a "pore freezing" model to explain the observations. We discuss how this model also leads to the 1/fα noise that is commonly observed in currents across biological and artificial membranes.

Analysis of proton exchange kinetics with time-dependent exchange rate

Mass spectrometry is used to probe the kinetics of hydrogen-deuterium exchange in lysozyme in pH 5, 6 and 7.4. An analysis based on a Verhulst growth model is proposed and effectively applied to the kinetics of the hydrogen exchange. The data are described by a power-like function which is based on a time-dependence of the exchange rate. Experimental data ranging over many time scales is considered and accurate fits of a power-like function are obtained. Results of fittings show correlation between faster hydrogen-deuterium exchange and increase of pH. Furthermore a model is presented that discriminates between easily exchangeable hydrogens (located in close proximity to the protein surface) and those protected from the exchange (located in the protein interior). A possible interpretation of the model and its biological significance are discussed.

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.

A geometric comparison of single chain multi-state models of ion channel gating

Multi-state models of ion channel gating have been used extensively, but choosing optimally small yet sufficiently complex models to describe particular experimental data remains a difficult task. In order to provide some insight into appropriate model selection, this paper presents some basic results about the behavior of solutions of multi-state models, particularly those arranged in a chain formation. Some properties of the eigenvalues and eigenvectors of constant-rate multi-state models are presented. A geometric description of a three-state chain is given and, in particular, differences between a chain equivalent to an Hodgkin-Huxley model and a chain with identical rates are analyzed. One distinguishing feature between these two types of systems is that decay from the open state in the Hodgkin-Huxley model is dominated by the most negative eigenvalue while the identical rate chain displays a mix of modes over all eigenvalues.

On the possible methods for the mathematical description of the ball and chain model of ion channel inactivation

Ion channels are large transmembrane proteins that are able to conduct small inorganic ions. They are characterized by high selectivity and the ability to gate, i.e. to modify their conductance in response to different stimuli. One of the types of gating follows the ball and chain model, according to which a part of the channel’s protein forms a ball connected with the intracellular side of the channel by a polypeptide chain. The ball is able to modify the conductance of the channel by properly binding to and plugging the channel pore. In this study, the polypeptide ball is treated as a Brownian particle, the movements of which are limited by the length of the chain. The probability density of the ball’s position is resolved by different diffusional operators — parabolic (including the case with drift), hyperbolic, and fractional. We show how those different approaches shed light on different aspects of the movement. We also comment on some features of the survival probabilities (which are ready to be compared with electrophysiological measurements) for issues based on the above operators.

Self-organized critical gating of ion channels: on the origin of long-term memory in dwell time series

We present the model of an ion channel, operating in a regime of self-organized criticality. It is suggested that complex cooperative dynamics takes place in the protein and the overall tension of it facilitates an open or closed state of the rigid gates in the pore-making domain. For the first time multifractal spectra of ion channel dynamics are presented. Our model well reproduces the multifractal properties of ion channel dwell time series and provides an insight on the origin of the long-term correlations in these series.