Stochastic multiresonance in the coupled relaxation oscillators.

We study the noise-dependent dynamics in a chain of four very stiff excitable oscillators of the FitzHugh-Nagumo type locally coupled by inhibitor diffusion. We could demonstrate frequency- and noise-selective signal acceptance which is based on several noise-supported stochastic attractors that arise owing to slow variable diffusion between identical excitable elements. The attractors have different average periods distinct from that of an isolated oscillator and various phase relations between the elements. We explain the correspondence between the noise-supported stochastic attractors and the observed resonance peaks in the curves for the linear response versus signal frequency.

Oscillatory amplification of stochastic resonance in excitable systems.

We study systems which combine both oscillatory and excitable properties, and hence intrinsically possess two internal frequencies, responsible for standard spiking and for small amplitude oscillatory limit cycles (Canard orbits). We show that in such a system the effect of stochastic resonance can be amplified by application of an additional high-frequency signal, which is in resonance with the oscillatory frequency. It is important that for this amplification one needs much lower noise intensities as for conventional stochastic resonance in excitable systems.

Coherence resonance and polymodality in inhibitory coupled excitable oscillators.

We have analyzed the firing activity of two and three excitable FitzHugh-Nagumo oscillators, coupled via slow variable diffusion and under the action of an external noise. We find a different form of coherence resonance in this system, which is, in contrast to previous studies, intrinsically based on the antiphase behavior of coupled elements. Additionally, we show that an exchange, performed by this form of coupling, is remarkably rhythmogenic and results in polymodal interspike distributions without any external periodic stimuli. The dependence of these distributions on the noise amplitude and the coupling strength is studied.

Frequency-dependent stochastic resonance in inhibitory coupled excitable systems.

We study frequency selectivity in noise-induced subthreshold signal processing in a system with many noise-supported stochastic attractors which are created due to slow variable diffusion between identical excitable elements. Such a coupling provides coexisting of several average periods distinct from that of an isolated oscillator and several phase relations between elements. We show that the response of the coupled elements under different noise levels can be significantly enhanced or reduced by forcing some elements in resonance with these new frequencies which correspond to appropriate phase relations.

Multirhythmicity generated by slow variable diffusion in a ring of relaxation oscillators and noise-induced abnormal interspike variability.

The deterministic and noise-dependent dynamics of a ring of three Ohmically coupled electronic relaxation oscillators are considered by means of numerical simulations. Each isolated oscillator is described by a set of two ordinary differential equations with very different characteristic times. The emergence of the limit cycle via the Hopf bifurcation results from the N-shaped current-versus-voltage characteristic of the nonlinear resistor. The phase diagram is calculated for a ring of three such oscillators in the presence of small detuning. Special attention is focused on two parameter areas, one near a transition to the homogeneous and the other near the inhomogeneous stable steady state. Along with other nontrivial limit cycles, essentially asymmetrical limit cycles termed dynamic traps may arise in these two areas. A dynamic trap is a regime in which one or two oscillators do not perform full-amplitude oscillations and, correspondingly, do not generate spikes. The interspike interval (ISI) distribution in the presence of noise is calculated as a function of the coupling strength in both areas of the parameter plane. The distributions are extremely polymodal near the homogeneous steady state even if the in-phase limit cycle is dominating. The origins of this abnormal enhancement of ISI variability are discussed in detail. A similar analysis shows that nontrivial periodic attractors are observable in the vicinity of the inhomogeneous stable steady states only if the level of noise is relatively low. In this case, the dominance of the in-phase limit cycle basin results in an almost unimodal distribution of interspike intervals.

Temporal variability in a system of coupled mitotic timers.

Cell proliferation is considered a periodic process governed by a relaxation timer. The collective behavior of a system composed of three identical relaxation oscillators in numerically studied under the condition that diffusion of the slow mode dominates. We demonstrate: (1) the existence of three periodic regimes with different periods and phase relations and an unsymmetrical, stable steady-state (USSS); (2) the coexistence of in-phase oscillations and USSS; (3) the coexistence of periodic attractors; and (4) the emergence of a two-loop limit cycle coexisting with both in-phase oscillations and a stable steady-state. The qualitative reasons for such a diversity and its possible role in the generation of cell cycle variability are discussed.

Quantized cell cycle times: interaction between a relaxation oscillator and ultradian clock pulses.

Control of the timing of cell division is considered to result from a relaxation cell cycle oscillator: this has one slow and one rapid component and obeys a system of two ordinary differential equations. Interactions of the slow component with an ultradian oscillator leads to quantization of cell cycle times when the free parameters of the cell cycle oscillator are chosen close to its bifurcation point. This model fits the experimental results previously reported.

[Interaction of mitotic oscillators as a source of variability of cell cycle duration]. / Vzaimodeistvie mitoticheskikh ostsilliatorov kak istochnik variabelnosti dlitelnosti kletochnogo tsikla.

Interaction between membrane mitotic oscillators at the expense of exchange with the molecules of lipids (slow variable) and antioxidants (fast variable) was considered. Parameters of all the oscillators are equal, excluding a small noise added to the equation for lipids. These parameters are chosen in such a way that the oscillators are not far from the transition to the stable stationary state. The numerical modeling has shown that the exchange with lipids brings about the appearance of an additional limit cycle whose period is significantly greater than that of an autonomous oscillator. The addition of noise averages the behaviour of oscillators, and distribution according to cycle duration becomes broad and bimodal. Thus the exchange of the slow variable increases the dispersion of distribution of cell generation times. This conclusion seems to be true for any oscillator with similar dynamic properties.

[Oxygen dependence of the kinetics of nonenzymatic lipid peroxidation]. / Kislorodnaia zavisimost' kinetiki nefermentativnogo perekisnogo okisleniia lipidov.

Kinetics of lipid peroxidation was considered theoretically taking into account a reversible pattern of peroxide radical formation. It has been shown that at reasonable values of the reactions rate constants discrepancy between oxygen absorption and the expenditure of polyunsaturated fatty acids can be explained. The PL equation system was written for an open system, its parametric analysis was carried out and the limit of autooscillation regime existence in relation to the rates of lipids additions and oxygen concentration were found.

The role of lipid and antioxidant exchanges in cell division synchronization (mathematical model).

Cell-cycle synchronization of two diffuse-coupled cells has been studied in the framework of the membrane model for the cell division cycle, proposed by Chernavskii et al. (1977). It has been shown semi-analytically (using the averaging principle) and by computer stimulation that a) if the duration of the G1-phase (TG1) for two identical cells is comparable with the duration of the remaining cycle (TS + G2 + M), the lipid (L)-exchange results in a synchronization with phase difference phi = 0. The antioxidant (A)-exchange leads to a phase-locking with phi = T0/2 (where T0 is the cell cycle period; b) if TG1 much greater than TS + G2 + M (or TG1 much less than TS + G2 + M) the L-exchange makes synchronization possible both with phi = 0 and phi = T0/2 while the A-exchange results in phase-locking with phi confined to the region 0 to T0/2; c) for non-identical cells differing in the values of kinetic parameters, the locking band narrows as the population density increases (when some model parameters are close to the bifurcation thresholds). We expect that the cells selected artificially at a definite phase of cycle might maintain the synchronous division for a long time if the lipid exchange between cells were stimulated.

[Membrane model of the regulation of proliferation: or predictions of the theory]. / Membrannaia model' reguliatsii proliferatsii: predskazaniia teorii.

Experimental consequences predicted by the membrane theory of cell proliferation are considered. In this theory, the conformational transitions are conditioned by the existence of the mechanical frame embracing normal cells as a whole and they induce the phase transitions in the lipid bilayer. It is interesting to record the lateral mobility of lipid molecules, because the theory predicts the principle difference between the line dependence of lipid fluidity in membranes of normal and tumour cells. Possible measurements of lateral mobility of non-lipid membrane components are described in detail, and the enhancement of diffusion is expected under mitogenic stimulation. Special attention is given to a possible method of overcoming G1-preferability and to obtaining a large population of non-proliferating cells in G2 state.

[Membrane model of the regulation of proliferation: the theory and interpretation of an experiment]. / Membrannaia model' reguliatsii proliferatsii: teoriia i interpretatsiia éksperimenta.

The role of cell surface physical organization in the cell cycle regulation is analyzed within the framework of the earlier proposed theory (Chernavskii et al., 1982). Two models of cell surface are considered: hard-frame fluid-mosaic model (latticemosaic) and the fluid-mosaic one. The former deals with normal cells. The existence of integral carcasse or "frame" which is formed by the essential part of cross-linked membrane components and may have at least two different conformational states is hypothesized. The second model describes membranes of tumour cells. With the latter theory any mitogen (excluding the restoration of nutrient depletion) reduces the mechanical tensile strength of the frame and stimulates the general structural rearrangement of the plasma membrane. There are only two conformational transitions during the cell cycle which serve as signals for the beginning of S and M phases. If the values of tensile strength are great enough and therefore the conformational transitions are impossible, the cells pass into the resting (prereplicative--G01, or premitotical--G02) state. Three types of experiments are interpreted in the proposed theory: a) on differences in the action of growth factors on normal and tumour cell cycle, b) on the necessary condition for mitogenicity of lectins, c) on the stimulation of proliferation by mechanical deformation of cells.

Cell surface and cell division.

A mathematical model of the regulation of cell division is suggested. The model is based on the hypothesis that the process giving rhythm to cell division is located in the cell membrane: i.e., the process of free-radical oxidation of membrane lipids. Much depends on the physical state of the membrane. In the membrane, phase transitions take place because of the changes in lipid composition. These transitions differ in normal and tumor cells: in normal cells they are sharp and hysteretic owing to the presence of a framework (membrane skeleton) on the surface of the membrane, while in tumor cells the integrity of the surface is violated so that the transitions are smooth. This model makes it possible to explain differences in the regulation of normal and cancer cell proliferation. Within the limits of the model, such phenomena as density dependent inhibition of growth, reverse transformation, influence of cyclic AMP and ions of Ca2+ on the cell cycle, the actions of serum and of proteases on the cycle, and so on, are explained. A rational scheme for the appearance of the selective damage found in tumor cells is proposed.

[Fluctuations of cell cycle duration]. / O mekhanizme fluktuatsii dlitel'nosti kletochnogo tsikla.

The problems whether the cell cycle is a deterministic or probabilistic process is widely discussed in current literature [1, 3, 7]. Present work deals with fluctuations of the cell cycle in terms of the mathematical model of membrane regulation of cell division. The presence of white noise in the parameters describing the influxes of lipids and antioxidants into the membrane is studied by the methods of Markovian processes, as well as by direct computer simulation. The equation for the distribution function of cell generation times is obtained and the increase of dispersion and the mean cell cycle time during the change of the system parameters which can be related to the density of cell culture is calculated. The theoretical distributions qualitatively agree with experimental data.

On the distribution of cell cycle generation times.

The problem of whether the cell cycle is a deterministic or probabilistic process is widely discussed in the current literature (P. Nurse, Nature, 286, pp. 9-10, 1980). In this report the question of fluctuations of cell cycle period is treated in the limits of the membrane model of cell division regulation. The parametric analysis of the equations set both for normal and tumour cells is carried out. We describe the bifurcation parameters in the neighbourhood of which the system can amplify the small fluctuations. The presence of white noise in parameters describing the lipids and antioxidants influxes into membrane is examined by methods of Marcovian processes and also by direct stochastic computer simulation. The equation for the distribution function of generation times is obtained and the increase of dispersion and mean cycle time during the changes of those parameters which would be connected with cell culture density is calculated. The influence of parameter fluctuations upon the cycle period for both normal and tumour cells is compared in the framework of model assumptions. The ratio of dispersion of generation time distribution to mean period value for an ensemble of tumour cells is shown to be several times greater than that for normal ones. In the discussion the problem of the presence of a premitotical (G02) resting state and of the possibility of its experimental detection is considered.

Biological consequences of the physical organization of the plasma membranes of normal and tumor cells.

The differences in the physical organization of the membranes of normal and tumor cells are explained within the framework of the hypothesis of the presence of an intact framework on the surface of normal cells and the absence of it in tumor cells. An intact framework determines the possibility of existence of metastable states of the membrane and hysteresis phenomena in the lipid bilayer. The signals for transition to the S- and M-phases of the cycle are breakoffs of the metastable states, which occur only in the membranes of normal cells. The cell cycle in tumors is constructed only on physicochemical processes in the membrane without hysteresis phenomena, and the possibilities of regulating it are greatly weakened. This hypothesis permits: a) prediction of differences in the change in the microviscosity of the lipids of normal and tumor cells during their movement along the cycle and refinement of the concept of intactness of cells in the presence of a change in the microviscosity; b) elucidation of the role of the spreading of cells for the initiation of division in vitro; c) the proposal of a method of selective destruction of tumor cells with the aid of proteolytic and lipolytic enzymes; d) the proposal of an explanation for the weak antigenicity of many spontaneous tumors and the possibility of enhancing it. Experimental data on the properties of the membranes are discussed, and they are compared with the theoretical premises.