Symbolic dynamics and computation in model gene networks

We analyze a class of ordinary differential equations representing a simpli fied model of a genetic network. In this network, the model genes control t he production rates of other genes by a logical function. The dynamics in t hese equations are represented by a directed graph on an n-dimensional hype rcube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the bou ndary between neighboring orthants, the equation is integrated until the bo undary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time th at same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequen ce of orthants that arise from integration of the equation starting and end ing on the same boundary. The union of the words defines the language. Lett ers and words correspond to analytically computable Poincare maps of the eq uation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of d ynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation. (C) 2001 American Institute of Physics.