Dynamical Systems Problems Inspired by Biology

After very briefly surveying studies in complex systems biology, I discuss studies the other way around, i.e., studies of dynamical systems inspired by biology instead of biology from dynamical systems. Five topics are discussed.

The first issue concerns with reluctance to relaxation to equilibrium. Biological systems, in general, are kept out from falling to equilibrium. Put differently, is there some mechanism so that relaxation to equilibrium is hindered even in a closed physico-chemical system? We show that “transient dissipative structure” at macroscopic catalytic reaction systems show hindrance to relaxation to equilibrium, and then discuss the mechanism for it.

The second issue concerns with a Hamiltonian system with large degrees of freedom coupled globally with each other. In contrast to the naive expectation on equilibrium systems, we provide an example in which collective macroscopic oscillation continues over a large time span before relaxing to equilibrium, whose duration increases with the system size. This collective oscillation is explained by a self-consistent 'swing' mechanism.

The third issue is with regards to many degrees of freedom, or concerns with the number of dimension beyond which the system is regarded as many. We show that there is a critical number at 5~10, beyond which attractors often touch with the basin boundary. The number is discussed as magnitude relation between exponential and factorial.

The fourth problem is motivated by cell differentiation. We give examples of such differentiation of states, which are represented as internal bifurcation in coupled dynamical systems.

Last, if I have time, design of robust dynamical systems is discussed in relationship with evolution of gene regulatory networks, where linking between robustness to noise and to structural change is shown which, implies evolutionary congruence between developmental and mutational robustness.