Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the average connectivity.
Deep Dive into Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics.
Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the averag
Many complex systems in general -and living systems and cells in particular -display remarkable stability, i.e. a capacity to sustain their spatial and temporal molecular organization. Yet, their stability is dynamic, i.e. these systems -to a certain degree -are capable of adapting to changes in their physical and chemical environment. This has led several authors [1,2,3,4] to interpret such systems as existing at the edge of chaos. Mathematically the edge of chaos refers to regions in parameter space, where the system dynamics is characterized by a maximal Lyapunov exponent (MLE), λ 1 , equal to zero. In this case small changes in parameters may cause the dynamics to switch between regular and chaotic behavior, thereby being able to explore large portions of the system's phase-space. This possibility is most relevant for living systems existing in fluctuating environments. In many dynamical systems the edge of chaos exists only for a tiny portion of parameter space, typically in sets of singular points, i.e. sets of measure zero. The dynamics of systems at the edge of chaos can become highly nontrivial, even for simple maps like the logistic map [5]. It has been argued that living systems have evolved towards the edge of chaos by natural selection [2], however it is not clear which mechanisms allow self-organization around these exceptional regions in parameter space.
Living systems have exist in the state of quasi-stationary nonequilibrium and therefore can not be closed systems. They require a flow of substrate and energy to and from the system. Since long [6], rate equations for molecular dynamics have been considered. For systems to be self-sustaining, such rate equations need to be autocatalytic, i.e. some molecular species directly or indirectly catalyze their own production. For living systems, cells in particular, to be in a stationary state, production, decay and flow rates of intercellular components effectively have to balance each other, [7,8]. Replicating, living systems therefore in general balance between stationary states (nonreplicating modes) and growth (replicating mode), limited by constraints posed by the environment. This balance provides a natural selection criterion.
Autocatalytic systems are frequently governed by nonlinear equations for enzyme-kinetics, e.g. Michaelis-Menten differential equations [9], or more general replicator equations, see e.g. [10]. For various reasons linearized autocatalytic networks have been considered, for the case of abundant substrate, see e.g. [11,12], or for reverse engineering [13,14]. Systems with linearized dynamics can be easily depicted in terms of directed reaction networks, where nodes represent molecular species. Two nodes, where one node directly influences (production or inhibition) the other, are connected by a directed link. Weights of such links quantify associated reaction rates; negative rates indicate inhibitory links. Weights of self-loops in the reaction network, i.e. links of a node onto itself, quantify decay rates. Recent progress in genomic and proteomic technology begins to reveal facts about regulatory networks of organisms. There is some evidence that these directed networks show scale-free topological organization [15,16,17,18]. More recent evidence suggests topological differences between in-and out-degree distributions [19,20]. Basically two main approaches for modeling catalytic networks have been pursued: Discrete state approaches, e.g. Boolean networks [21], and continuous approaches, relying on ordinary or stochastic differential equations [22,23,24,25,26]. The relevance of noise has been experimentally demonstrated [27,28,29,30,31].
Interestingly, various models of disordered recurrent networks [21,32] seem to share three distinguished modes of operation: (i) stable, (ii) critical, and (iii) chaotic super-critical. These properties could be generic or even universal. The importance of determining the minimum complexity of models exhibiting these properties has been pointed out [21] and the question has been raised whether these properties can already be found in linear systems. Following this philosophy we have recently introduced a model for genetic regulatory dynamics [33]. This model is governed by sets of linear equations
where A ij is the weighted adjacency matrix of the full autocatalytic reaction network, whose entries may be zero, positive and negative -indicating that i either has no influence on j or the production of molecular species i is stimulated or suppressed by j, respectively. This means that if substrate j exists, i gets produced (or reduced) at rate A ij . x i is the concentration of the molecular species i (e.g. proteins or mRNA). J i corresponds to a flow-vector. The molecular species i flows into the system if J i > 0 and out of the system if J i < 0. ν i is a suitable noise term. Negative molecular concentration values x i do not make any sense, hence we impose the positivity condition
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