Nonlinear Dyanmics in Biological Systems: Actin Networks and Gene Networks
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Two problems in biological systems are studied: (i) experiments in microscale deformations of actin networks and (ii) a theoretical treatment of the stability of discrete state network models of genetic control. In the experiments on actin networks, we use laser tweezers to locally deform actin networks at the micron scale as a model of the action of molecular motors and other cellular components, and we image the network during deformation using confocal microscopy. Using these tools, we observe two nonlinear effects. The first observation is that there are two time scales of relaxation in the network: the stress induced by deformation relaxes rather quickly, however, the strain relaxes at a different rate. Additionally, upon removing the deforming force, the initial rate at which the strain relaxes seems to be independent of the amount of stress still in the network. The second observation is that large deformations are irreversible, and imaging the network implies that a large-scale snapping event seems to accompany this irreversibility. In the theoretical treatment of gene networks, we focus on the stability of their dynamics in response to small perturbations. Previous approaches to stability have assumed uncorrelated random network structure. Real gene networks typically have nontrivial topology significantly different from the random network paradigm. In order to address such situations, we present a general method for determining the stability of large Boolean networks of any specified network topology and predicting their steady-state behavior in response to small perturbations. Additionally, we generalize to the case where individual genes have a distribution of `expression biases,' and we consider non-synchronous update, as well as extension of our method to non-Boolean models in which there are more than two possible gene states. We find that stability is governed by the maximum eigenvalue of a modified adjacency matrix (&lambda<sub>Q<\sub>), and we test this result by comparison with numerical simulations. We also discuss the possible application of our work to experimentally inferred gene networks and present approximations to &lambda<sub>Q<\sub> in several cases.