Stability and Dissipative Structures in Open Systems far from Equilibrium
14 Mar 2007-pp 209-324
About: The article was published on 2007-03-14. It has received 94 citations till now. The article focuses on the topics: Dissipative system & Stability theory.
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TL;DR: In this paper, the authors generalized the chemical mechanism of Field, Koros, and Noyes for the oscillatory Belousov reaction by a model composed of five steps involving three independent chemical intermediates.
Abstract: The chemical mechanism of Field, Koros, and Noyes for the oscillatory Belousov reaction has been generalized by a model composed of five steps involving three independent chemical intermediates. The behavior of the resulting differential equations has been examined numerically, and it has been shown that the system traces a stable closed trajectory in three dimensional phase space. The same trajectory is attained from other phase points and even from the point corresponding to steady state solution of the differential equations. The model appears to exhibit limit cycle behavior. By stiffly coupling the concentrations of two of the intermediates, the limit cycle model can be simplified to a system described by two independent variables; this coupled system is amenable to analysis by theoretical techniques already developed for such systems.
1,172 citations
TL;DR: It is suggested that competition solves a sensitivity problem that confronts all cellular systems: the noise-saturation dilemma.
Abstract: This article is the first of a series to globally analyse competitive dynamical systems. The article suggests that competition solves a sensitivity problem that confronts all cellular systems: the noise-saturation dilemma. Low energy input patterns can be registered poorly by cells due to their internal noise. High energy input patterns can be registered poorly by cells because their sensitivity approaches zero when all their sites are turned on. How do cells balance between the two equally deadly, but complementary, extremes of noise and saturation? How do cells achieve a Golden Mean?
648 citations
TL;DR: This study suggests that diffusive instabilities should also be sought in ecological interactions by presenting an example of their occurrence when cooperating prey interact with predators.
622 citations
332 citations
TL;DR: In this paper, the behavior of a single chemical oscillator coupled in series illustrates the phenomena of synchronization, multiply periodic and almost-periodic oscillations, and subharmonic resonance.
Abstract: That chemical reactions can exhibit all of the interesting and well known behavior of nonlinear oscillators is shown by collecting the scattered results in the literature for a simple reaction mechanism, supplementing them with some new results, and analyzing the behavior of the single chemical oscillator using the method of isoclines. Furthermore, two oscillators coupled in series illustrate the phenomena of synchronization, multiply periodic and almost‐periodic oscillations, and subharmonic resonance. Finally the reaction mechanism is modified to a form which is biochemically realistic and still behaves similarly to the original scheme.
204 citations
References
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TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Abstract: It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
9,015 citations
TL;DR: The synthesis of enzymes in bacteria follows a double genetic control, which appears to operate directly at the level of the synthesis by the gene of a shortlived intermediate, or messenger, which becomes associated with the ribosomes where protein synthesis takes place.
5,588 citations
TL;DR: In this article, a general reciprocal relation applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility, and certain average products of fluctuations are considered.
Abstract: A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the derivation, certain average products of fluctuations are considered. As a consequence of the general relation $S=k logW$ between entropy and probability, different (coupled) irreversible processes must be compared in terms of entropy changes. If the displacement from thermodynamic equilibrium is described by a set of variables ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$, and the relations between the rates ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ and the "forces" $\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{1}},\ensuremath{\cdots},\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{n}}$ are linear, there exists a quadratic dissipation-function, $2\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}{\ensuremath{\rho}}_{j}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{\mathrm{ij}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{i}=\frac{\mathrm{dS}}{\mathrm{dt}}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}(\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{j}}){\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{j}$ (denoting definition by $\ensuremath{\equiv}$). The symmetry conditions demanded by microscopic reversibility are equivalent to the variation-principle $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{-}\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})=\mathrm{maximum},$ which determines ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ for prescribed ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$. The dissipation-function has a statistical significance similar to that of the entropy. External magnetic fields, and also Coriolis forces, destroy the symmetry in past and future; reciprocal relations involving reversal of the field are formulated.
5,505 citations
TL;DR: This work deals with patterns in a thin layer of solution, in which cerium ions catalyse the oxidation of analogues of malonic acid by bromate by oscillations in the solution colour.
Abstract: OSCILLATING chemical reactions are interesting, not only in themselves but as models of a number of important biological processes1–5. Oscillating reactions have been described in which cerium (or manganese) ions catalyse the oxidation of analogues of malonic acid by bromate6–8. Oscillations in the concentrations of oxidized and reduced forms of the catalyst cause oscillations in the solution colour, while stirring leads to forced synchronization of oscillations throughout the whole volume. In the absence of stirring, periodic propagation of concentration waves occurs in certain conditions, and such a phenomenon in a one-dimensional system (a long tube) has been described9,10. Our work deals with patterns in a thin layer of solution (two-dimensional system).
1,563 citations
TL;DR: In this paper, the probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.
Abstract: The probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.This probability can be expressed in terms of the dissipation function; the resulting relation, which is an extension of Boltzmann's principle, shows the statistical significance of the dissipation function. From the form of the relation, the principle of least dissipation of energy becomes evident by inspection.
1,544 citations