Negative probabilities in Gene Regulatory Networks
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We introduce a natural framework to identify sign-indefinite co-expressions between genes based on the known expressions and given the sign of their respective correlations. Specifically, given information concerning the affinity among genes (i.e., connectivity in the gene regulatory network) and knowledge whether they promote/inhibit co-expression of the respective protein production, we seek rates that may explain the observed stationary distributions at the level of proteins. We propose to encapsulate their “promotion vs. inhibition” functionality in a sign-indefinite probability transition matrix–a matrix whose row-sums equal to one, but is otherwise sign indefinite. The purpose of constructing such a representation for the interaction network with sign-indefinite contributions in protein regulation, is to quantify the structure and significance of various links, and to explain how these may affect the geometry of the network, highlighting the significance of the regulatory functions of certain genes. We cast the problem of finding the interaction (sign-indefinite) transition matrix as a solution to a convex optimization problem from which all the relevant geometric properties may be easily derived.
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