Nonequilibrium voltage fluctuations in biological membranes. I. General framework of charge transport in discrete systems and related voltage noise

This paper continues our work on the theory of nonequilibrium voltage noise generated by electric transport processes in membranes. Introducing the membrane voltage as a further variable, a system of kinetic equations linearized in voltage is derived by which generally the time-dependent behaviour of charge-transport processes under varying voltage can be discussed. Using these equations, the treatment of voltage noise can be based on the usual master equation approach to steady-state fluctuations of scalar quantities. Thus, a general theoretical approach to nonequilibrium voltage noise is presented, completing our approach to current fluctuations which had been developed some years ago. It is explicitly shown that at equilibrium the approach yields agreement with the Nyquist relation, while at nonequilibrium this relation is not valid. A further general property of voltage noise is the reduction of low-frequency noise with increasing number of transport units as a consequence of the interactions via the electric field. In a second paper, the approach will be applied for a number of special transport mechanisms, such as ionic channels, carriers or electrogenic pumps.     

Current fluctuations in discrete transport systems far from equilibrium. Breakdown of the fluctuation dissipation theorem

A recently developed theoretical approach to transport fluctuations around stable steady states in discrete biological transport systems is used in order to investigate general fluctuation properties at nonequilibrium. An expression for the complex frequency dependent admittance at nonequilibrium is derived by calculation of the linear current response of the transport systems to small disturbances in the applied external voltage. It is shown that the Nyquist or fluctuation dissipation theorem, by which at equilibrium the macroscopic admittance or linear response can be expressed in terms of fluctuation properties of the system, breaks down at nonequilibrium. The spectral density of current fluctuations is decomposed into one term containing the macroscopic admittance and a second term which is bilinear in current. This second term is generated by microscopic disturbances, which cannot be excited by external macroscopic perturbations. At special examples it is demonstrated that this second term is decisive for the occurrence of excess noise e.g. the 1/f(2)-Iorentzian noise generated by the opening and closing of nerve channels in biological membranes.     

Linear noise approximation method for calculating nonequilibrium fluctuations of the occupation numbers in radiative-collisional average ion models

A novel method is proposed for calculating nonequilibrium fluctuations of the mean occupation numbers of the electron shells in the radiative-collisional average-ion models of multicharged plasma kinetics. For the class of Slater ionic models, equations are derived for the mean occupation numbers of the electron shells and their fluctuations in the Fokker-Planck approximation. To calculate the fluctuations, the Fokker-Planck equation is linearized in the vicinity of the steady-state nonequilibrium solution to the kinetic equations (linear noise approximation). The method proposed allows one to take into account both the nonequilibrium correlations of the occupation-number fluctuations and the thermodynamically equilibrium statistical correlations related to the Coulomb interaction among bound electrons. The relation among the coefficients in the Fokker-Planck equation for the occupation-number fluctuations of the electron shells is discussed based on the fluctuation-dissipative theorem.     

Gauging the strength of power frequency fields against membrane electrical noise

The possible physiological effect of power frequency fields (60 Hz in the US, 50 Hz in most other countries) is still a hotly debated issue. These relatively slow fields distribute themselves across cell membranes and a common approach has been to compare the strength of these fields to the strength of the electric noise that the membrane generates itself through Brownian motion. However, there has been disagreement among researchers on how to evaluate the membrane electric noise. In the first part of this article three major models are discussed. In the second part an ab initio modeling of membrane electric fields finds that different manifestations of Brownian noise lead to an electric noise intensity that is many times larger than what conventional estimates have yielded. Finally, the legitimacy of gauging a nonequilibrium external signal against internal equilibrium noise is questioned.     

Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating.

We describe a new electrophysiological technique called nonequilibrium response spectroscopy, which involves application of rapidly fluctuating (as high as 14 kHz) large-amplitude voltage clamp waveforms to ion channels. As a consequence of the irreversible (in the sense of Carnot) exchange of energy between the fluctuating field and the channel protein, the gating response is exquisitely sensitive to features of the kinetics that are difficult or impossible to adequately resolve by means of traditional stepped potential protocols. Here we focus on the application of dichotomous (telegraph) noise voltage fluctuations, a broadband Markovian colored noise that fluctuates between two values. Because Markov kinetic models of channel gating can be embedded within higher-dimensional Markov models that take into account the effects of the voltage fluctuations, many features of the response of the channels can be calculated algebraically. This makes dichotomous noise and its generalizations uniquely suitable for model selection and kinetic analysis. Although we describe its application to macroscopic ionic current measurements, the nonequilibrium response method can also be applied to gating and single channel current recording techniques. We show how data from the human cardiac isoform (hH1a) of the Na+ channel expressed in mammalian cells can be acquired and analyzed, and how these data reveal hidden aspects of the molecular kinetics that are not revealed by conventional methods.     

Nonequilibrium ion transport through pores

In this paper is presented an investigation of the influence of the internal structure of pores in membranes on a) the time dependent macroscopic relaxation current after a voltage jump, b) the macroscopic frequency dependent admittance and c) the microscopic current fluctuations around stationary (nonequilibrium) states. All these quantities are determined by the time dependent transport equations, which are calculated with the use of the eigenvectors and eigenvalues of the matrix of coefficients, occurring in the transport equations. Numerical calculations for channels with up to 31 barriers are presented. The treatment of the fluctuations is done with the use of a general approach to nonequilibrium transport noise recently developed by one of the authors. It is shown that the influence of the internal barrier structure as, e.g., the height of central or decentral barriers in the pores is of great complexity. Nevertheless we hope that the calculations lead to a better understanding especially of the microscopic nonequilibrium transport fluctuations in complex systems.This work has been supported by the Deutsche Forschungsgemeinschaft     

On the Stabilizing Role of Stage Structure in Piscene Consumer–Resource Interactions

A stage-structured consumer–resource model is investigated using plausible biological parameter estimates. The model, developed from patterns in energy allocation across fish life-history strategies, explicitly considers the effects of delays in maturation on juvenile growth and mortality. It is found that stage structure in the consumer fish population tends to locally stabilize consumer–resource dynamics for realistic parameters. Additionally, it is shown that stage structure bounds nonequilibrium behavior relative to the case without stage structure. Finally, it is shown that the increased stability and bounding of solutions has the seemingly paradoxical consequence of promoting nonequilibrium dynamics when even small amounts of noise are added to the system.     

Study of the iron-sulfur center N-2 from NADH-dehydrogenase in different metabolic states of the mitochondria from liver and heart

The structural--equilibrium and nonequilibrium forms of the center N-2 from NADH-dehydrogenase differ in their parametres of the spin-lattice relaxation. The curves of the temperature dependence of the ESR signal intensity become the effective method of the study of the iron-sulphur proteins. The structural nonequilibrium form of the center N-2 was observed in the "4th" metabolic (by Chance) state, but equilibrium form of the center N-2 prevailed in the "3d" state or in the uncoupled state.     

Speed-dependent cellular decision making in nonequilibrium genetic circuits

Despite being governed by the principles of nonequilibrium transitions, gene expression dynamics underlying cell fate decision is poorly understood. In particular, the effect of signaling speed on cellular decision making is still unclear. Here we show that the decision between alternative cell fates, in a structurally symmetric circuit, can be biased depending on the speed at which the system is forced to go through the decision point. The circuit consists of two mutually inhibiting and self-activating genes, forced by two external signals with identical stationary values but different transient times. Under these conditions, slow passage through the decision point leads to a consistently biased decision due to the transient signaling asymmetry, whereas fast passage reduces and eventually eliminates the switch imbalance. The effect is robust to noise and shows that dynamic bifurcations, well known in nonequilibrium physics, are important for the control of genetic circuits.     

Adaptive Responses Limited by Intrinsic Noise

Sensory systems have mechanisms to respond to the external environment and adapt to them. Such adaptive responses are effective for a wide dynamic range of sensing and perception of temporal change in stimulus. However, noise generated by the adaptation system itself as well as extrinsic noise in sensory inputs may impose a limit on the ability of adaptation systems. The relation between response and noise is well understood for equilibrium systems in the form of fluctuation response relation. However, the relation for nonequilibrium systems, including adaptive systems, are poorly understood. Here, we systematically explore such a relation between response and fluctuation in adaptation systems. We study the two network motifs, incoherent feedforward loops (iFFL) and negative feedback loops (nFBL), that can achieve perfect adaptation. We find that the response magnitude in adaption systems is limited by its intrinsic noise, implying that higher response would have higher noise component as well. Comparing the relation of response and noise in iFFL and nFBL, we show that whereas iFFL exhibits adaptation over a wider parameter range, nFBL offers higher response to noise ratio than iFFL. We also identify the condition that yields the upper limit of response for both network motifs. These results may explain the reason of why nFBL seems to be more abundant in nature for the implementation of adaption systems.     

A maximum entropy criterion of filtering and coding for stationary autoregressive signals: Its physical interpretations and suggestions for its application to neural information transmission

The operations of encoding and decoding in communication agree with filtering operations of convolution and deconvolution for Gaussian signal processing. In an analogy with power transmission in thermodynamics, an autoregressive model of information transmission is proposed for representing a continuous communication system which requires a pair of an internal noise source and a signal source to encode or decode a message. In this model transinformation (informational entropy) equals the increase in stationary nonequilibrium organization formed through the amplification of white noise by a positive feedback system. The channel capacity is finite due to the existence of inherent noise in the system. The maximum entropy criterion in information dynamics corresponds to the 2^nd law of thermodynamics. If the process is stationary, the communication system is invertible, and has the maximum efficiency of transformation. The total variation in informational entropy is zero in the cycle of the invertible system, while in the noninvertible system the entropy of decoding is less than that of encoding. A noisy autoregressive coding which maximizes transinformation is optimum, but is also ideal.     

Nonequilibrium,multiple-timescale simulations of ligand-receptor interactions in structured protein systems

Predicting the long-time, nonequilibrium dynamics of receptor-ligand interactions for structured proteins in a host fluid is a formidable task, but of great importance to predicting and analyzing cell-signaling processes and small molecule drug efficacies. Such processes take place on timescales on the order of milliseconds to seconds, so "brute-force" real-time, molecular or atomic simulations to determine absolute ligand-binding rates to receptor targets and over a statistical ensemble of systems are not currently feasible. In the current study, we implement on real protein systems a previously developed 3-5 hybrid molecular dynamics/Brownian dynamics algorithm, which takes advantage of the underlying, disparate timescales involved and overcomes the limitations of brute-force approaches. The algorithm is based on a multiple timescale analysis of the total system Hamiltonian, including all atomic and molecular structure information for the system: water, ligand, and receptor. In general, the method can account for the complex hydrodynamic, translational-orientational diffusion aspects of ligand-docking dynamics as well as predict the actual or absolute rates of ligand binding. To test some of the underlying features of the method, simulations were conducted here for an artificially constructed spherical protein "made" from the real protein insulin. Excellent comparisons of simulation calculations of the so-called grand particle friction tensor to analytical values were obtained for this system when protein charge effects were neglected. When protein charges were included, we found anomalous results caused by the alteration of the spatial, microscopic structure of water proximal to the protein surface. Protein charge effects were found to be highly significant and consistent with the recent hypothesis of Hoppert and Mayer (Am Sci 1999;87:518-525) for charged macromolecules in water, which involves the formation of a "water dense region" proximal to the charged protein surface followed by a "dilute water region." We further studied the algorithm on a D-peptide/HIV capside protein system and demonstrated the algorithms utility to study the nonequilibrium docking dynamics in this contemporary problem. In general, protein charge effects, which alter water structural properties in an anomalous fashion proximal to the protein surface, were found to be much more important than the so-called hydrodynamic interaction effects between ligand and receptor. The diminished role of hydrodynamic interactions in protein systems allows for a much simpler overall dynamic algorithm for the nonequilibrium protein-docking process. Further studies are now underway to critically examine this simpler overall algorithm in analyzing the nonequilibrium protein-docking problem.     

Nonlinear theory of large-amplitude stationary solitary waves in symmetric unmagnetized e − e + and C 60 − C 60 + plasmas

It is shown that large-amplitude stationary solitary electrostatic waves can exist in a symmetric plasma (e ^? e ⁺ or C ₆₀ ^? C ₆₀ ⁺ ), and the relevant parameter ranges (i.e., the range of the Mach numbers and the degree to which the plasma should be nonequilibrium) are determined. The basic requirement for the existence of such waves, specifically, that the symmetric plasma be in a nonequilibrium state, can easily be satisfied in low-density collisionless ideal plasmas under laboratory conditions.     

Zygotic associations and multilocus statistics in a nonequilibrium diploid population

The usual approach to characterizing and estimating multilocus associations in a diploid population assumes that the population is in Hardy-Weinberg equilibrium. The purpose of this study is to develop a set of summary statistics that can be used to characterize and estimate the multilocus associations in a nonequilibrium population. The concept of "zygotic associations" is first expanded to facilitate the development. The summary statistics are calculated using the distribution of a random variable, the number of heterozygous loci (K) found in diploid individuals in the population. In particular, the variance of K consists of single-locus and multilocus components with the latter being the sum of zygotic associations between pairs of loci. Simulation results show that the multilocus associations in the variance of K are detectable in a sample of moderate size (> or =30) when the sum of all pairwise zygotic associations is greater than zero and when gene frequency is intermediate. The method presented here is a generalization of the well-known development for the Hardy-Weinberg equilibrium population and thus may be of more general use in elucidating the multilocus organizations in nonequilibrium and equilibrium populations.     

Electronic Circular Dichroism of [16]Helicene With Simplified TD‐DFT: Beyond the Single Structure Approach

The electronic circular dichroism (ECD) spectrum of the recently synthesized [16]helicene and a derivative comprising two triisopropylsilyloxy protection groups was computed by means of the very efficient simplified time‐dependent density functional theory (sTD‐DFT) approach. Different from many previous ECD studies of helicenes, nonequilibrium structure effects were accounted for by computing ECD spectra on "snapshots" obtained from a molecular dynamics (MD) simulation including solvent molecules. The trajectories are based on a molecule specific classical potential as obtained from the recently developed quantum chemically derived force field (QMDFF) scheme. The reduced computational cost in the MD simulation due to the use of the QMDFF (compared to ab‐initio MD) as well as the sTD‐DFT approach make realistic spectral simulations feasible for these compounds that comprise more than 100 atoms. While the ECD spectra of [16]helicene and its derivative computed vertically on the respective gas phase, equilibrium geometries show noticeable differences, these are “washed” out when nonequilibrium structures are taken into account. The computed spectra with two recommended density functionals (ωB97X and BHLYP) and extended basis sets compare very well with the experimental one. In addition we provide an estimate for the missing absolute intensities of the latter. The approach presented here could also be used in future studies to capture nonequilibrium effects, but also to systematically average ECD spectra over different conformations in more flexible molecules. Chirality 28:365–369, 2016. © 2016 Wiley Periodicals, Inc.     

Production models for nonlinear stochastic age-structured fisheries

The overriding feature of stock-recruitment data for most fisheries is the amount of variability involved. Previous production models have assumed either an underlying linear stock-recruitment relationship [11] or an equilibrium condition [23]. Here a production model is derived for an age-structured fishery exhibiting nonlinear stochastic recruitment under nonequilibrium conditions. In the first section deterministic age-structured production models are reviewed, and in the next section corresponding random variable models are presented. Equations for the first and second order moments for each age class, for the stock, and for the yield are then derived using two approaches. The first approach assumes that third and order higher moments associated with the noise can be neglected (thus extending the “small noise” approach in [23]). The second approach assumes that the distributions associated with the random variables can be characterized by a particular two parameter distribution. This latter approximation can be applied to systems with “large noise,” and precision will not be lost for situations where the exact form of the distribution, associated with the stock-recruitment data, is unknown. Equations are derived for the solution under equilibrium recruitment and constant harvesting conditions. Detailed expressions are also obtained for the case where the random variables are assumed to satisfy a gamma distribution.     

Absorption and magnetic circular dichroism spectra of hemoproteins in nonequilibrium states. V. Cytochrome P450 and its substrate complex

It was shown that ferrocytochrome P450 forms a nonequilibrium state if ferrocytochrome P450 and its complexes are reduced in freezed water-glycerol solutions by thermolysed electrons, arising during gamma-radiolysis of the matrix at 77 degrees K. Unlike the equilibrium form of ferrocytochrome P450 with the heme iron at the high-spin state the reduced nonequilibrium form of the protein contains the heme iron at a low-spin state. The absorption spectrum of ferrocytochrome P450 in the nonequilibrium state is characterized by alpha and beta-bands at 562 and 534 nm, respectively, whereas the magnetic circular dichroism spectra exhibit type A effect at 562 nm. Upon temperature increasing the nonequilibrium state is relaxed to the equilibrium one. Type 1 substrates had practically no influence on the spectral characteristic of the nonequilibrium form of ferrocytochrome P450. Binding of type 2 substrates results in an essential decrease of the intensity ratio of the alpha- and beta-bands (A alpha/A beta) and is accompanied by a red-shift of the alpha-band and corresponding magnetic circular dichroism effect. It was shown that mercaptoethanol complex of hemoglobin, formed by reduction at 77 degrees K is spectrally similar to the nonequilibrium ferrocytochrome P450 complex with type 2 substrates. From analysis of experimental data one can conclude that (i) the ligand environment of heme iron in oxidased and reduced cytochrome P450 are different; (ii) the sixth axial ligand of the heme iron in the oxidised protein is probably a water molecule (OH-) attached by a hydrogen bond to the neighbouring histidine. It is assumed that a similar nonequilibrium form of cytochrome P450 can be formed in physiological conditions.     

Determination of binding parameters in the presence of coupled reactions

Components of a binding reaction may undergo nonbinding reactions: receptors may be degraded, internalized, or exchanged with cryptic sites; ligand may be degraded or compartmented. In such cases the parameters that characterize the system are not obtained from the usual equilibrium analyses. We have simulated the reactions of such systems and generated association curves, "Scatchard" plots, and "Scatchard-like" plots that permit the calculation of binding affinity and receptor number not normally calculable under nonequilibrium binding conditions. In particular, we show that certain coupled reactions produce local maxima and sigmoid shapes in association curves and that the maxima can be used to obtain affinities and receptor numbers.