Differential equations biological network dynamics

We repeat these simple examples from earlier work since they form a major justification for the elaborations we report below, and if they are understood well, then much of what follows should be clear. The dynamics in the differential equations are in qualitative agreement with the dynamics in the synthetic genetic networks in E. coU. One important aspect of biology is to understand the ways in which the organization and structure of the control networks can be used to predict the dynamics. Thus, we would like to develop methods that could be used to predict the dynamic behaviors just demonstrated without integrating or carrying out the stability analysis of the differential equations. 

Our initial studies of dynamics in biochemical networks included spatially localized components [32]. As a consequence, there will be delays involved in the transport between the nuclear and cytoplasmic compartments. Depending on the spatial structure, different dynamical behaviors could be faciliated, but the theoretical methods are useful to help understand the qualitative features. In other (unpublished) work, computations were carried out in feedback loops with cyclic attractors in which a delay was introduced in one of the interactions. Although the delay led to an increase of the period, the patterns of oscillation remained the same. However, delays in differential equations that model neural networks and biological control systems can introduce novel dynamics that are not present without a delay (for example, see Refs. 57 and 58). 

---