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We present a novel combinatorial problem that arises from mathematical biology. In order to understand the dynamics of models of gene regulatory networks over a parameter space, a problem of constructing linear extensions of a partial order with algebraic constraints arises naturally. We formulate the problem for a class of algebraic constraints related to the form of nonlinearities in the gene regulation model. We provide an algorithm that partially solves the problem. We formulate adoi:10.1137/17s015872 fatcat:37ucxodlhzc45bbf6nmulc25ye