Dynamically Stable Systems

> The question of dynamical stability and stochastic behavior of large biochemical networks is discussed. It is argued that stringent conditions of asymptotic stability have very little chance to materialize in a multidimensional system described by the differential equations of chemical kinetics. The reason is that the criteria of asymptotic stability (Routh-Hurwitz, Lyapunov criteria, Feinberg’s Deficiency Zero theorem) would impose the limitations of very high algebraic order on the kinetic rates and stoichiometric coefficients, and there are no natural laws that would guarantee their unconditional validity. Highly nonlinear, dynamically unstable systems, however, are not necessarily doomed to collapse, as a simple Jacobian analysis would suggest. It is possible that their dynamics may assume the form of pseudo-random fluctuations quite similar to a shot noise, and, therefore, their behavior may be described in terms of Langevin and Fokker-Plank equations. We have shown by simulation that the resulting pseudo-stochastic processes obey the heavy-tailed Generalized Pareto Distribution with temporal sequence of pulses forming the set of constituent-specific Poisson processes. Being applied to intracellular dynamics, these properties are naturally associated with burstiness, a well documented phenomenon in the biology of gene expression.

The goal of this paper is to demonstrate that extremely stringent conditions of dynamical stability have very little chance to materialize in the realm of large biochemical networks. *We claim that any large biochemical network almost for sure is dynamically unstable.* It does not mean, however, that such a system is doomed to collapse due to implosion or explosion, as a simple linear analysis would suggest. It is possible that the system maintains a mode of existence in which the events of instability occur in a more or less random order, but generally, over a period of time, compensate each other.

The question of stability is of primary importance in studying genetic regulatory networks (GRN). The analogy with traffic in a big city discussed above is fully relevant to GRN. Dense interconnectedness of GRN is the reason why smooth behavior of GRN as a whole may be strongly dependent on seamless functioning of each gene. Proteins translated from mRNAs of some genes serve as transcription factors for many other genes; therefore, large sections of gene expression machinery may be halted by mere shortage, even temporary, of a few proteins that resulted from the transcription of other genes. Since the processes of protein production and delivery to appropriate regulatory sites are essentially random and involve many fluctuations and uncertainties, the traffic jams in such a system are rather mundane events. Since we assume (at least in the mainstream science, see) that there is no supervisory intelligent system in the cell which knows where the bottleneck has occurred and which has independent resources to eliminate it, each such event may cause development of an avalanche of secondary events threatening to bring the entire system to collapse. […]