Autowave mode of the functioning of the nitric oxide + free iron + thiols system as a basis for control of the biological action of nitric oxide and its endogenous compounds

The NO + Fe²⁺ + thiols system in an aqueous solution has been found earlier to exhibit temporal oscillatory changes in the concentration of paramagnetic dinitrosyl iron complexes with thiol-containing ligands and S-nitrosothiols, as well as in the concentration of free iron (not included in the complexes). It is proposed that autowaves can appear in this system characterized by periodic changes in the concentrations of its components in time and space. Such changes may form a basis for the control of the physiological effects of nitric oxide, dinitrosyl iron complexes, and S-nitrosothiols as agents affecting various cellular and tissue targets.     

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.     

Autowave mode of functioning of the system nitric oxide + free iron + thiols may ensure the regulation of biological action of nitric oxide and its endogenic compounds

It has been shown earlier that, in a system NO + Fe2+ + thiols in aqueous solution, an oscillatory mode of changes with time in the concentration of paramagnetic dinitrosyl iron complexes with thiol-containing legends and S-nitrosothiols formed in this system and in the concentration of free iron (not included into dinitrosyl iron complexes) can be realized. It is assumed that, in this system, autowaves can arise, which ensure periodic changes with time and space in the concentration of the system constituents. These changes may underlie the regulation of the physiologic effect of nitric oxide, dinitrosyl iron complexes, and S-nitrosothiols as agents affecting various intracellular and tissue targets.     

Entrainment of marginally stable excitation waves by spatially extended sub-threshold periodic forcing

We introduce a novel approach of stabilizing the dynamics of excitation waves by spatially extended sub-threshold periodic forcing. Entrainment of unstable primary waves has been studied numerically for different amplitudes and frequencies of additional sub-threshold stimuli. We determined entrainment regimes under which excitation blocks were transformed into consistent 1:1 responses. These responses were spatially homogeneous and synchronized in the entire excitable medium. Compared to primary pulses, pulses entrained by secondary stimulations were stable at considerably shorter periods which decreased at higher amplitudes and greater number of secondary stimuli. Our results suggest a practical methodology for stabilization of excitation in reaction-diffusion media such as nerve tissue with regions of reduced excitability.     

Slow and fast invasion waves in a model of acid-mediated tumour growth

This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.     

Periodic solutions and refractory periods in the soliton theory for nerves and the locust femoral nerve

Close to melting transitions it is possible to propagate solitary electromechanical pulses which reflect many of the experimental features of the nerve pulse including mechanical dislocations and reversible heat production. Here we show that one also obtains the possibility of periodic pulse generation when the constraint for the nerve is the conservation of the overall length of the nerve. This condition generates an undershoot beneath the baseline ('hyperpolarization') and a 'refractory period', i.e., a minimum distance between pulses. In this paper, we outline the theory for periodic solutions to the wave equation and compare these results to action potentials from the femoral nerve of the locust (Locusta migratoria). In particular, we describe the frequently occurring minimum-distance doublet pulses seen in these neurons and compare them to the periodic pulse solutions.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.     

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.     

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.     

P53脉冲的研究现状与展望

P53脉冲是指细胞内p53蛋白水平周期性或重复性的涨落.该脉冲产生的途径是调节p53的各种正负反馈环,其核心的两个负反馈环是p53-Mdm2环和Wip1-ATM-p53环.这些负反馈环能产生极限环振荡,在极限环振荡区,P53水平成脉冲型变化.随着P53脉冲的增多,不同形式的p53蛋白和促凋亡蛋白逐渐积累并到达一定阈值水平,可打开凋亡"开关",引发不可逆的细胞命运.除了P53脉冲的数目,其频率、振幅、波形等物理学参数也与细胞命运存在密切关系.这一研究进展对阐明诸多疾病发生机理和防治研究有重要意义.     

Detection of autowave distribution of the concentration of a dinitrosyl iron complex with glutathione formed in an aqueous solution of S-nitrosoglutathione after addition of a mixture of glutathione and ferrous iron

The formation of dark green concentric autowaves of the distribution of the concentration of dinitrosyl iron complex (DNIC) with glutathione in a thin (0.3 mm thick) layer of 0.5 M solution of S-nitrosoglutathione in 15 mM HEPES buffer (pH 7.7) after applying on its surface a drop of a solution of glutathione (0.5 mM) and ferrous iron (1 mM) in the same buffer of volume 10 μL was detected. At regular intervals, the picture of autowaves changed in time intervals of 0.4–0.6 s over a period of 3 s after the application of the drop onto the solution. Then the structured picture of the distribution of DNIC dissipated, followed by a uniform green coloring of the solution caused by a uniform distribution of DNIC in it. It is assumed that the formation of autowaves is a consequence of the autooscillatory mode of the existence of a chemical system formed in a mixture of NO, low-molecular-weight thiols, and ferrous iron ions. DNIC with thiolate ligands and S-nitrosothiols arising in this system have a capacity for interconversion, and it is this process that may underlie the autooscillatory, autowave mode of functioning of the system. It is not ruled out that the existence of this system in cells and tissues of living organisms may provide the spatial and temporal organization of the regulation of the biological action of NO and its different endogenous compounds and derivatives.     

Bistable waves for a class of cooperative reaction-diffusion systems

In this paper, we consider a class of coupled cooperative reaction-diffusion systems, in which one population (or subpopulation) diffuses while the other is sedentary. We use the shooting method to prove the existence of the bistable travelling wave, and then obtain its global attractivity with phase shift and uniqueness (up to translation) via the dynamical system approach. The results are applied to some specific examples of reaction-diffusion population models.     

Refinement and Pattern Formation in Neural Circuits by the Interaction of Traveling Waves with Spike-Timing Dependent Plasticity

Traveling waves in the developing brain are a prominent source of highly correlated spiking activity that may instruct the refinement of neural circuits. A candidate mechanism for mediating such refinement is spike-timing dependent plasticity (STDP), which translates correlated activity patterns into changes in synaptic strength. To assess the potential of these phenomena to build useful structure in developing neural circuits, we examined the interaction of wave activity with STDP rules in simple, biologically plausible models of spiking neurons. We derive an expression for the synaptic strength dynamics showing that, by mapping the time dependence of STDP into spatial interactions, traveling waves can build periodic synaptic connectivity patterns into feedforward circuits with a broad class of experimentally observed STDP rules. The spatial scale of the connectivity patterns increases with wave speed and STDP time constants. We verify these results with simulations and demonstrate their robustness to likely sources of noise. We show how this pattern formation ability, which is analogous to solutions of reaction-diffusion systems that have been widely applied to biological pattern formation, can be harnessed to instruct the refinement of postsynaptic receptive fields. Our results hold for rich, complex wave patterns in two dimensions and over several orders of magnitude in wave speeds and STDP time constants, and they provide predictions that can be tested under existing experimental paradigms. Our model generalizes across brain areas and STDP rules, allowing broad application to the ubiquitous occurrence of traveling waves and to wave-like activity patterns induced by moving stimuli.     

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator–prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator–prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.     

Reaction–diffusion model of atherosclerosis development

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.     

Frequency specificity in intercellular communication. Influence of patterns of periodic signaling on target cell responsiveness.

Cells often communicate by means of periodic signals, as exemplified by a large number of hormones and by the aggregation of Dictyostelium discoideum amebas in response to periodic pulses of cyclic AMP. Periodic signaling allows bypassing the phenomenon of desensitization brought about by constant stimuli. To gain further insight into the efficiency of pulsatile signaling, we analyze the effect of periodic stimulation on the dynamic behavior of a receptor system capable of desensitization toward its ligand. We first show that the receptor system adapts to square-wave stimuli, i.e., the response eventually reaches a steady, periodic pattern after a transient phase. By analyzing the dependence of the response on the characteristics of the square-wave stimulation, we show that there exist a waveform and a period of that signal that result in maximum responsiveness of the target system. Similar results are obtained when the signal takes the more realistic form of a periodically repeated stimulation followed by exponential decay of the ligand. The results are discussed with respect to the role of pulsatile secretion of gonadotropin-releasing hormone (GnRH) by the hypothalamus and of periodic signaling by cyclic AMP pulses in Dictyostelium. The analysis accounts for the existence, in both cases, of an optimal frequency and waveform of the periodic stimulus that correspond to maximum target cell responsiveness.