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.

Nonequilibrium voltage fluctuations in biological membranes. II. Voltage and current noise generated by ion carriers, channels and electrogenic pumps.

As applications of the general theoretical framework of charge transport in biological membranes and related voltage and current noise, a number of model calculations are presented for ion carriers, rigid channels, channels with conformational substates and electrogenic pumps. The results are discussed with special reference to the problem of threshold values for sensory transduction processes and their limitations by voltage fluctuations. Furthermore, starting from the special results of model calculations, an attempt is made to determine more general aspects of electric fluctuations generated by charge-transport processes in biological membranes: different frequency dependences of voltage and current noise, and dependence of noise intensities with increasing distance from the equilibrium state.

Non-equilibrium voltage noise generated by ion transport through pores.

In this paper, we describe a systematic approach to the theoretical analysis of non-equilibrium voltage noise that arises from ions moving through pores in membranes. We assume that an ion must cross one or two barriers in the pore in order to move from one side of the membrane to the other. In our analysis, we consider the following factors: a) surface charge as a variable in the kinetic equations, b) linearization of the kinetic equations, c) master equation approach to fluctuations. To analyze the voltage noise arising from ion movement through a two barrier (i.e., one binding site) pore, we included the effects of ions in the channel's interior on the voltage noise. The current clamp is considered as a white noise generating additional noise in the system. In contrast to what is found for current noise, at low frequencies the voltage noise intensity is reduced by increasing voltage across the membrane. With this approach, we demonstrate explicitly for the examples treated that, apart from additional noise generated by the current clamp, the non-equilibrium voltage fluctuations can be related to the current fluctuations by the complex admittance.

Modelling a two-dimensional random alarm process.

This paper deals with the mathematical modelling of two-dimensional alarm processes randomly spreading, amplifying and switching off within limited distributions of particles (individuals). It has been stimulated by recent studies on the enemy alarm behavior upon disturbance in Australian bull-dog ants (Myrmecia). The alarm within a random distribution of a limited number of resting particles in a finite two-dimensional region starts with the excitation, i.e. stochastic movement of a single particle. The excitation or alarm is spread over the distribution by excitation transfer, which occurs if the distance between the moving and a resting particle is below a fixed value. The mathematical model proceeds in three steps: (a) modelling of the stochastic movement of a single excited particle; (b) quantitative description of the area scanned by a single particle; (c) simulation of the whole many-particle process, i.e. amplification and switching off of the alarm. The essential parameters characterizing the single particles' motion are the particle velocity nu, and the turning frequency beta for the statistically independent changes in the direction of movement. Further parameters of the model, which determine the spread of the alarm, are the excitation period T, the capture radius Rc, the particle density rho and the extent of the distribution. The sensitivity of the process to variations of these parameters has been studied by averaging over a great number of stochastic simulations. The results show that the parameters as realistically estimated for the case of the bull-dog ants (nu = 10 cm/s, P = 2/s, T = 5s, Rc = 10 cm, 10 particles within a circular region of radius Rp = 50 cm) represent a possible set which on the average leads to a successful spread of the alarm.

Rate theory models for ion transport through rigid pores. III. Continuum vs discrete models in single file diffusion.

In studying the single file model in its discrete as well as in its continuum form the relationship between the phenomenological continuum theory of diffusion and the rate theory approach is analyzed. The single file model in its original form is discrete and represents the most general rate theory model for ion transport through rigid pores in biological membranes. In neglecting the interionic interactions which the single file model takes into account, the Nernst-Planck equation of macroscopic free diffusion can be derived from single file by means of the procedure n leads to infinity (where n is the number of binding sites within a pore) and the classical diffusion theory can thereby be integrated into the more general concept of single filing transport. Moreover, the single file model has been transformed in the limit n leads to infinity into the corresponding continuum form involving interionic interactions. The essential differences between the two derived continuum forms are: In the macroscopic diffusion model, the interionic interactions are regarded in the form of a "mean field". Thus we only get one equation of motion (Nernst-Planck equation) for the ionic concentration c(x, t) within the membrane. In the continuum version of the single file model, however, we obtain a hierarchy of Fokker-Planck equations for the probability density functions Pm(x1, . . . , xm, t) (where m is the number of ions within a pore). The interactions of the single file system are incorporated in detail into the Fokker-Planck equation as well as into the corresponding boundary conditions. As a consequence, the boundary conditions are highly complex in comparison with periodic conditions or Dirichlet conditions often used for the Nernst-Planck equation in electrophysiology. Two types of boundary conditions have been found which are principally different: The first one is to regulate the entry and exit of the ions at the pore mouth by a negative feedback mechanism, the second one describes the collisions of the ions within multiply occupied pores. In this context the question is discussed of whether the continuum version of single file has advantages over the discrete one.

Rate theory models for ion transport through rigid pores. II. Time-dependent analysis of the single-file model with special reference to oscillatory behavior.

This article deals with the time-dependent evolution of the single-file movement of ions through channels of both biological and artificial membranes. The single-file transport process may exhibit not only the usual relaxation behaviour but also oscillatory behaviour as a steady state is approached after an initial perturbation. A necessary condition for the occurrence of oscillations is that the system acts sufficiently far from equilibrium. The occurrence of oscillations is due to the interactions within the transport system which are taken into account by the single-file model; these are the electrostatic repulsion between the ions being transported, and the competition of the ions for the free binding sites within the pore. Information about the strength of the interactions can be obtained by measuring the damping of the transport observables (e.g. the electric current): The stronger the inter-ionic repulsion, the more apparent the oscillatory behaviour will become. Furthermore, the damping is influenced by the microscopic structure of the transport system (i.e. the energy profile of the pores). With an increasing degree of microscopicity, i.e. with a decreasing number of binding sites and an increasingly irregular pore profile, the oscillations become more damped. However, a considerable oscillatory behaviour can only be predicted for pores with both a sufficiently regular structure and a sufficiently large number of binding sites. For this class of pores, however, the measurement of the damping represents an appropriate method of gaining information which could exceed that obtainable from the usual methods of measuring stationary quantities (e.g. stationary conductance). Moreover, our goal is to explain theoretically how the oscillatory behaviour can be interpreted in terms of the order inherent in the ionic movement, which is determined by both the external and internal forces and the microscopic properties of the system.

Competitive blocking of apical sodium channels in epithelia.

Apical sodium-selective channels in frog skin, when blocked by amiloride or triamterene, exhibit fluctuations in current, the spectra of which are Lorentzian. These effects have been modeled previously with two-state and three-state models by Lindemann and Van Driessche. A recent observation by Hoshiko and Van Driessche that corner frequencies are lowered by increasing the apical sodium concentration cannot be accounted for by these models. We explore the possibility that sodium (S) and amiloride (A) compete for a site at the mouth of the channel. A new three-state channel model (sodium-occupied, open/unoccupied, open/amiloride-blocked) is analyzed. Its corner frequency is of the form fc = fco [1 + (A/KA)/(1 + S/KS)], consistent with the observed sodium dependence of the corner frequency. The minimum frequency, fco, and the inhibition constants, KA and KS, are expressed in terms of the rate constants of the model. To account for sodium self-inhibition, we postulate that two sodium ions in the channel may result in clogging--a fourth state. The two corner frequencies are calculated; so are the plateau values of the noise power. The noise power shows a maximum as a function of blocker concentration, as observed previously using triamterene. The four-state model predicts the observed suppression by small amounts of blocker of the low-frequency sodium (clogging) noise.

Rate theory models for ion transport through rigid pores. I. Time-dependent analysis in the case of vanishing interactions.

In this first of a series of papers concerning the theoretical analysis of rate theory models for ion transport through rigid pores, the case of vanishing interactions is investigated. "Rigidity" means that ions crossing membranes through pores see a fixed structure of the pores, not changing in time. A single pore is considered to be a sequence of (n + 1) activation barriers separated by n energy minima. The explicit analytical treatment is restricted to pores with regular internal barrier structure, including the nonequilibrium situation of an applied electric field. In this case the connection with continuum diffusion models is demonstrated by performing in the limit n leads to infinity (n = number of binding sites within the pores) the transition to continuum. Thus, from diffusion equations describing a discrete number of jumps, the corresponding diffusion-like partial differential equations and boundary conditions are generated. For regular pores, from the time dependent solutions of the discrete equations, the corresponding solutions of the continuum equations are explicitly generated. The time-dependent relaxation behaviour of the discrete model is in good agreement with the continuum model if one assumes more than two binding sites in the pores.

Steady-state fluorescence polarization in planar lipid membranes. Experimental and theoretical analysis of the fluorophores 8-anilino-1-naphthalenesulfonate, 1,6-Diphenyl-1,3,5-hexatriene, dansyllysine-valinomycin and n-(9-anthroyloxy) fatty acids.

In continuation of earlier work, the steady-state fluorescence polarization in a globally oriented system of planar lipid membranes was analyzed experimentally and theoretically for the fluorophores 8-anilino-1-naphthalenesulfonate, 1,6-diphenyl-1,3, 5-hexatriene, dansyllysine-valinomycin and n-(9-anthroyloxy) fatty acids. The theoretical analyses of experiments were mainly done in terms of the mean orientation of transition moments with respect to the membrane normal, an angle describing the region of hindered rotational diffusion and the coefficients of rotational diffusion perpendicular to the membrane and around the membrane normal. The nonvanishing angle between the moments of absorption and emission was taken into account. In the case of n-(9-anthroyloxy) fatty acids it was found that the orientational disorder increases significantly with the depth of the fluorophore within the membrane. In order to compare with recent results from time-dependent fluorescent polarization in globally isotropic membrane suspensions and with 2H-NMR experiments, the second moment ('order parameter') of the steady-state orientational distribution of absorption dipoles was calculated. For all fluorophores the theoretical analysis indicates a preferred orientation of absorption moments within the membrane plane.

Fluctuations of barrier structure in ionic channels.

In rate-theory analysis of ion transport in channels, the energy of binding sites and the height of activation barriers are usually considered to be time-independent and not influenced by the movement of the ion. The assumption of a fixed barrier structure seems questionable, however, in view of the fact that proteins may exist in a large number of conformational states and may rapidly move from one state to the other. In this study, some of the effects of multiple conformational states of a channel on ion transport are analyzed. In the first part of the paper, the ion permeability of a channel with n binding sites is treated on the assumption that interconversion of channel states is much faster than ion transfer between binding sites. Under this condition, the form of the flux equation remains the same as for a channel with fixed barriers, provided that the rate constants for ion jumps are replaced by weighted averages over the rate constants for the individual conformational states. In the second part, a channel with two (main) barriers and a single (main) binding site is considered, with the rates of conformational transitions being arbitrary. This case, in particular, includes the situation where a jump of the ion is followed by a slow transition to a more polarized state of the binding site. Under this condition, the conductance of the channel exhibits a nonlinear dependence on ion concentration which is different from a simple saturation behavior. Under non-stationary conditions damped oscillations may occur.

Noise-current generated by carrier-mediated ion transport at non-equilibrium.

The measured spectral intensity Sj(f) of noise-current generated by carrier-mediated ion transport on bilayer membranes agrees under equilibrium and nonequilibrium conditions with the theoretically predicted behavior. It is shown that the shot noise intensity due to this ion transport mechanism yields a frequency independent level of Sj(f) at higher frequencies. The intensity of the shot noise contribution decreases with increasing voltage. Both experimentally and theoretically it could be shown that at nonequilibrium the Nyquist-theorem can no longer be applied for a description of the spectral intensity. Especially, the low frequency tail of Sj(f) is not proportional to the mean macroscopic steady state conductance. The transition of Sj(f) between the low and high frequency limit occurs in a frequency range which is related to the relaxation time constants of the transport system. In contrary to voltage jump current relaxation experiments where two relaxation times are predicted the theory predicts for the noise analysis a third relaxation time constant of a non-zero contribution to Sj(f). Besides the measurement of Sj(f) the corresponding autocorrelation function was determined. Comparison of both methods of noise analysis shows that for the carrier-mediated ion transport the determination of the autocorrelation function is the less appropriate approach due to principal methodical difficulties.

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.

Nonequilibrium ion transport through pores. The influence of barrier structures on current fluctuations, transient phenomena and admittance.

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.

Theory of single-file noise.

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consquence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.

Theory of transport noise in membrane channels with open-closed kinetics.

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels. For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.

Current noise around steady states in discrete transport systems.

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact outer reservoirs. The state of the systems is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual "jumps" of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n leads to infinity continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuation in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carriers mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance. The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g. pores in membranes, carrier mediated ion transport).

Theory of fluorescence polarization of oriented pigment molecules in spherical arrays.

Theoretical results are presented which are appropriate for the analysis of the static polarized fluorescence experiment with oriented pigment molecules in spherical arrays (vesicles). Though the global orientation mediated over the whole sphere is isotropic, the fluorescent molecules may have preferred local orientation with respect to the local plane. As in a former paper, concerning fluroescence polarization in planar arrays, three basic (local) orientation distributions of the electronic transition moments are investigated, which may be expected to describe a wide class of real cases with sufficient accuracy. Analytic expressions for the degree of polarization are derived. One important result is that the degree of polarization may be extremely dependent on the local orientation of transition moments. Hence the usual method of determination of microviscosities from experiments with vesicles with the use of the theory of fluorescence polarization for macromolecules in solutions should be regarded with great caution.

Dynamical theory of fluorescence polarization in a planar array of oriented pigment molecules.

A theory is developed appropriate for the analysis of fluorescence polarization experiments with pigment molecules in a planar array (plane membrane). Especially rotatory and oscillatory dynamics of the pigment molecules are considered. Three model calculations are performed, which describe the following different situations: (a) Rotational diffusion of molecules around the normal to the plane membrane. (b) Oscillatory diffusion of molecules with respect to this normal. (c) As a two-dimensional example the independent superposition of both types of motion. Central point of these model calculations is the determination of an intensity of emission function, from which in practical application the measured fluorescence intensities may uniquely be calculated.

Fluorescence polarization in a planar array of pigment molecules: theoretical treatment and application to flavins incorporated into artificial membranes.

A quantitative fluorescence polarization theory of molecules bound to two-dimensional plane layers has been developed when the electronic transition moments of absorption and emission are parallel within the fluorescent molecules. The transition moments are assumed to be in preferred orientation with respect to the normal to the plane and to be randomly oriented within the plane (rotational symmetry with the normal as axis of symmetry). Three basic model distributions of transition moments are investigated quantitatively. These model distributions represent a simplification but in most cases may be expected to describe reality with sufficient accuracy. For all distributions, two cases of different mobility of molecules are treated: (a) the lifetime of fluorescence is small compared with the characteristic relaxation time of the distribution, and (b) the lifetime of fluorescence is long, so that a complete reorientation of transition moments during the excited state can take place. From the quantitative calculations four characteristic quantities are derived, which are appropriate for the analysis of experimental data. Experiments are carried out with phosphatidylcholine bilayer membranes which contain three differently substituted amphiphilic flavins. All three flavins yield similar data. Their analyses predict free and fast mobility of the flavin chromophore.