A channel model with fluctuating barrier structures.

Following the theory 'Fluctuations of barrier structure in ionic channels' (Läuger, P., Stephan, W. and Frehland, E. (1980) Biochim. Biophys. Acta 602, 167-180), we constructed a model of a channels with several conformational states. The origin of these conformational states and the source for the transitions from one to the other are given explicitly for the presented model. In this work the effect of multiple conformational states on the ion transport process is analyzed. We considered a channel protein with two main barriers and one binding site. The site is surrounded by dipolar groups. The dipole moment of these groups can be reoriented by thermal activity and also by electrical interaction with the transported ions. Differently polarized states generate different activation energy barriers for the ions. The set of conformational states of the channel is constituted by all the possible polarized states of the binding site. Using the rate-theory analysis of ion transport (Glässtone, S., Laider, K.J. and Eyring, H. (1941) The theory of rate processes, McGraw-Hill, New York), the possible coupling between ion flux and the channel conformational transitions has been incorporated into the model by considering the dependence of the rate constants on the heights of the energy barriers. The resulting multistate kinetic equations have been solved numerically. It was shown that the simple saturation characteristic of the flux-concentration curve was obtained. For certain values of the model parameters, the channel shows a strongly different conductance for anions compared to cations. In fact, the model contains an interesting mechanism that exhibits selectivity with respect to the charge of the ions.

Monte-Carlo-simulations of voltage fluctuations in biological membranes in the case of small numbers of transport units.

Some years ago a theory of non-equilibrium voltage fluctuations in biological membranes was developed (Frehland and Solleder 1985, 1986) under a linearisation condition which is valid for a great number of transport units. In order to get an insight into the stochastic behaviour of such systems, consisting of small numbers of transport units, we carried out Monte-Carlo-simulations and compared the mean voltage course and the spectral density with the results of the previous theory. Under parameter conditions of biological relevance no significant differences from the behaviour of systems with large numbers, as predicted from the earlier theory, could be found in the case of rigid pores and ion carriers. However, in the case of small numbers, channels with open-closed-kinetics showed great deviation. With increasing number of transport units agreement with the previous theory was obtained.

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.