Linear extensions of a partial order subject to algebraic constraints

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 a conjecture on the special role of additive constraints in the class of all considered constraints. We present several examples where we show that the number of solutions is much smaller than the number of unconstrained linear extensions.