A Sparse Update Method for Solving Underdetermined Systems of Nonlinear Equations Applied to the Manipulation of Biological Signaling Pathways

Abstract

We are interested in sparse solutions of underdetermined systems of nonlinear equations and present an iterative method that updates the current iterate by a sparse vector defined as the solution of a constrained $\ell_1$-minimization problem. Local quadratic convergence of the iterates toward a solution of the underdetermined system of nonlinear equations is guaranteed under standard differentiability assumptions. Our work is motivated by the pharmaceutical modulation of defective biological signaling pathways as linked to various pathological conditions. Based on differential equation models of signal transduction networks the inverse problem of correcting qualitative biological behavior such as bistability or oscillation typically leads to an underdetermined system of nonlinear equations. In this context sparse solutions point to a low number of network intervention sites that may serve as manageable drug target candidates. The practicability of our method is demonstrated by means of a defective biological switch associated to the intrinsic apoptotic signaling pathway.