Augmented Sparse Reconstruction of Protein Signaling Networks

The problem of reconstructing and identifying intracellular protein signaling and biochemical networks is of critical importance in biology today. We sought to develop a mathematical approach to this problem using, as a test case, one of the most well-studied and clinically important signaling networks in biology today, the epidermal growth factor receptor (EGFR) driven signaling cascade. More specifically, we suggest a method, augmented sparse reconstruction, for the identification of links among nodes of ordinary differential equation (ODE) networks from a small set of trajectories with different initial conditions. Our method builds a system of representation by using a collection of integrals of all given trajectories and by attenuating block of terms in the representation itself. The system of representation is then augmented with random vectors, and minimization of the 1-norm is used to find sparse representations for the dynamical interactions of each node. Augmentation by random vectors is crucial, since sparsity alone is not able to handle the large error-in-variables in the representation. Augmented sparse reconstruction allows to consider potentially very large spaces of models and it is able to detect with high accuracy the few relevant links among nodes, even when moderate noise is added to the measured trajectories. After showing the performance of our method on a model of the EGFR protein network, we sketch briefly the potential future therapeutic applications of this approach.Comment: 24 pages, 6 figure

A Lagrangian relaxation approach to the edge-weighted clique problem

The $b$-clique polytope $CP^n_b$ is the convex hull of the node and edge incidence vectors of all subcliques of size at most $b$ of a complete graph on $n$ nodes. Including the Boolean quadric polytope $QP^n$ as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over $QP^n_n$. The problem of optimizing linear functions over $CP^n_b$ has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of $CP^n_b$ in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented. \u