Abstract
Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an 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 us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding 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 a system depend on the average connectivity.
Highlights
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
The edge of chaos refers to regions in parameter space, where the system dynamics is characterized by a maximal Lyapunov exponent (MLE), λ1, equal to zero
We analyzed stability properties by computing the probabilities for finding exponentially growing, decaying and non-exponentially growing dynamics and found that stable dynamics plays a dominant role in the plateau interval, [k−, k+]
Summary
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. J where Aij 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 Aij. xi is the concentration of the molecular species i (e.g. proteins or mRNA). This effect gives random strategies of evolutionary phase-space sampling a finite chance of locating this particular region in parameter space This may offer an explanation for why and how complex chemical reaction systems may have found the vicinity of the edge of chaos at all, before evolutionary self-organization could take over for an eventual fine tuning.
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