  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.
