Structural principles for dynamics of glass networks




Lu, Linghong

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Gene networks can be modeled by piecewise-linear (PL) switching systems of differential equations, called Glass networks after their originator. Networks of interacting genes that regulate each other may have complicated interactions. From a `systems biology' point of view, it would be useful to know what types of dynamical behavior are possible for certain classes of network interaction structure. A useful way to describe the activity of this network symbolically is to represent it as a directed graph on a hypercube of dimension $n$ where $n$ is the number of elements in the network. Our work here is considering this problem backwards, i.e. we consider different types of cycles on the $n$-cube and show that there exist parameters, consistent with the directed graph on the hypercube, such that a periodic orbit exists. For any simple cycle on the $n$-cube with a non-branching vertex, we prove by construction that it is possible to have a stable periodic orbit passing through the corresponding orthants for some sets of focal points $F$ in Glass networks. When the simple cycle on the $n$-cube doesn't have a non-branching vertex, a structural principle is given to determine whether it is possible to have a periodic orbit for some focal points. Using a similar construction idea, we prove that for self-intersecting cycles where the vertices revisited on the cycle are not adjacent, there exist Glass networks which have a periodic orbit passing through the corresponding orthants of the cycle. For figure-8 patterns with more than one common vertex, we obtain results on the form of the return map (Poincar{\'e} map) with respect to how the images of the returning cones of the 2 component cycle intersect the returning cone themselves. Some of these allow complex behaviors.



Switching network, Structural principles, Periodic orbit