Min-protein oscillations in Escherichia coli with spontaneous formation of two-stranded filaments in a three-dimensional stochastic reaction-diffusion model

We introduce a three-dimensional stochastic reaction-diffusion model to describe MinD/MinE dynamical structures in Escherichia coli. This model spontaneously generates pole-to-pole oscillations of the membrane-associated MinD proteins, MinE ring, as well as filaments of the membrane-associated MinD proteins. Experimental data suggest MinD filaments are two-stranded. In order to model them we assume that each membrane-associated MinD protein can form up to three bonds with adjacent membrane associated MinD molecules and that MinE induced hydrolysis strongly depends on the number of bonds MinD has established.Comment: Corrected typos, changed conten

Overlapped KAM patterns for linearly coupled asymmetric oscillators

The pattern of energy dependence for the onset of chaos is investigated for conservative system of two linearly coupled asymmetric oscillators, harmonic oscillator and a two-well nonlinear oscillator. With increase of energy, the amount of chaoticity first grows up to a certain critical energy, according to the KAM scenario, and above this point, with a further increase of energy, the amount of chaoticity decreases according to the inverse KAM scenario. At the point of transition, there is an overlap of the two scenarios. The position of the critical energy increases with increasing value of the coupling strength between oscillators

Missing preimages for chaotic logistic map with a hole

Chaotic transients and preimages are investigated for a new map proposed recently, having a hole in the unit interval of the r = 4 logistic map. This map is characterized by deviations from Frobenius-Perron equation for average lifetimes in dependence on hole position in the form of bursts of average lifetime. We present classification of these bursts on the basis of average lifetimes. Using time maps it is investigated how these bursts are caused by missing preimages of the hole interval I ^( 0 ). We derive approximate expression for the ratio of lifetimes deduced from Frobenius-Perron and from Kantz-Grassberger equations

Overlapping of two truncated crisis scenarios: Generator of peaks in mean lifetimes of chaotic transients

Maxima of mean transient time versus driving amplitude were found for weakly dissipated Duffing oscillator. In the neighborhood of peak of mean transient time an approximate power-law dependence was found. This behavior was compared with scaling in the vicinity of crisis point and interpreted as crossing of two neighboring crisis points which appears with decrease of driving amplitude. At this point chaotic attractor was destroyed and chaotic transient, exhibiting a maximum in the lifetime was borned. It was shown that the peak of mean lifetime has a regular behavior described by quadratic function

conservative systems, linearly coupled harmonic and nonlinear oscillators, KAM scenario, inverse KAM scenario Preklapajući KAM-scenariji za linearno vezane asimetrične oscilatore

The pattern of energy dependence for the onset of chaos is investigated for conservative system of two linearly coupled asymmetric oscillators, harmonic oscillator and a two-well nonlinear oscillator. With increase of energy, the amount of chaoticity first grows up to a certain critical energy, according to the KAM scenario, and above this point, with a further increase of energy, the amount of chaoticity decreases according to the inverse KAM scenario. At the point of transition, there is an overlap of the two scenarios. The position of the critical energy increases with increasing value of the coupling strength between oscillators.Struktura energijske ovisnosti evolucije kaosa istražuje se za konzervativni sustav dvaju linearno vezanih asimetričnih oscilatora, harmonijskog oscilatora i nelinearnog oscilatora s dvije jame. S porastom energije stupanj kaotičnosti najprije raste do određene kritične vrijednosti energije, sukladno KAM scenariju, a iznad te energije, s daljnjim porastom energije, udio kaotičnosti pada sukladno inverznom KAM scenariju. Pri prijelazu iz jednog scenarija u drugi dolazi do njihovog preklopa. Vrijednost kritične energije raste s porastom jakosti vezanja među oscilatorima

Nedostajuće predslike za kaotičnu logističku mapu s rupom

Chaotic transients and preimages are investigated for a new map proposed recently, having a hole in the unitinterval of the r 4 logistic map. This map is characterized by deviations from Frobenius-Perron equation for average lifetimes in dependence on hole position in the form of bursts of average lifetime. We present classification of these bursts on the basis of average lifetimes. Using time maps it is investigated how these bursts are caused by missing preimages of the hole interval I 0 . We derive approximate expression for the ratio of lifetimes deduced from Frobenius-Perron and from Kantz-Grassberger equationsKaotični tranzijenti i predslike istražuju se za novu mapu koja ima rupu unutar jediničnog intervala logističke mape za r 4. Ovu mapu karakteriziraju odstupanja od Frobenius-Perronove jednadžbe za vrijeme poluraspada u ovisnosti o položaju rupe, u obliku skokova u vremenu poluraspada. Primjenom vremenske mape istražuje se kako ti skokovi nastaju kao posljedica nedostajućih predslika rupnog intervala. Izvodi se približan izraz za omjer vremena poluraspada dobivenog pomoću Frobenius-Perronove i Kantz-Grassbergerove jednadžbe

conservative systems, linearly coupled harmonic and nonlinear oscillators, KAM scenario, inverse KAM scenario Preklapajući KAM-scenariji za linearno vezane asimetrične oscilatore

The pattern of energy dependence for the onset of chaos is investigated for conservative system of two linearly coupled asymmetric oscillators, harmonic oscillator and a two-well nonlinear oscillator. With increase of energy, the amount of chaoticity first grows up to a certain critical energy, according to the KAM scenario, and above this point, with a further increase of energy, the amount of chaoticity decreases according to the inverse KAM scenario. At the point of transition, there is an overlap of the two scenarios. The position of the critical energy increases with increasing value of the coupling strength between oscillators.Struktura energijske ovisnosti evolucije kaosa istražuje se za konzervativni sustav dvaju linearno vezanih asimetričnih oscilatora, harmonijskog oscilatora i nelinearnog oscilatora s dvije jame. S porastom energije stupanj kaotičnosti najprije raste do određene kritične vrijednosti energije, sukladno KAM scenariju, a iznad te energije, s daljnjim porastom energije, udio kaotičnosti pada sukladno inverznom KAM scenariju. Pri prijelazu iz jednog scenarija u drugi dolazi do njihovog preklopa. Vrijednost kritične energije raste s porastom jakosti vezanja među oscilatorima

Bursts in average lifetime of transients for chaotic logistic map with a hole

Chaotic transients are studied for a logistic map at r=4, with an inserted narrow hole. We find that average lifetime τ of chaotic transients that are dependent on the hole position roughly follows the Frobenius-Perron semicircle pattern in most of the unit interval, but at the positions that correspond to the low period (1,2,3,ldots), unstable periodic orbits of the logistic map at r=4 there are bursts of τ. An asymptotic relation between the Frobenius-Perron and Kantz-Grassberger average lifetimes, at these positions, is obtained and explained in terms of missing preimages determined from a transient time map. The addition of noise leads to the destruction of bursts of average lifetime

Nedostajuće predslike za kaotičnu logističku mapu s rupom

Chaotic transients and preimages are investigated for a new map proposed recently, having a hole in the unitinterval of the r 4 logistic map. This map is characterized by deviations from Frobenius-Perron equation for average lifetimes in dependence on hole position in the form of bursts of average lifetime. We present classification of these bursts on the basis of average lifetimes. Using time maps it is investigated how these bursts are caused by missing preimages of the hole interval I 0 . We derive approximate expression for the ratio of lifetimes deduced from Frobenius-Perron and from Kantz-Grassberger equationsKaotični tranzijenti i predslike istražuju se za novu mapu koja ima rupu unutar jediničnog intervala logističke mape za r 4. Ovu mapu karakteriziraju odstupanja od Frobenius-Perronove jednadžbe za vrijeme poluraspada u ovisnosti o položaju rupe, u obliku skokova u vremenu poluraspada. Primjenom vremenske mape istražuje se kako ti skokovi nastaju kao posljedica nedostajućih predslika rupnog intervala. Izvodi se približan izraz za omjer vremena poluraspada dobivenog pomoću Frobenius-Perronove i Kantz-Grassbergerove jednadžbe

Energy dependence of selfsimilarity truncation in a system of weakly coupled dissipative oscillators relevant for biological systems

Biological fractals are truncated, i.e., their selfsimilarity extends at most over a few orders of magnitude of separation. In J. Theor. Biology 212 (2001) p. 47, we have shown that the nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. We show here that the degree of truncation decreases with increasing energy. We point out that at the level of a sufficiently fine precision technique, the truncated fractality acts as a smooth (nonfractal) structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long term behaviour of biological systems. Consequently, a possible erratic nature of the system's behaviour due to truncated fractality may disappear once the experimental errors in the measurement and/or treatment of biological system reaches a certain level of precision. We point out a possible significance of this result for the biological control of processes