Reverse Engineering Gene Networks with Microarray Data

We consider the question of how to solve inverse problems of the form ex(0) = x(t) for an unkown matrix A, given measurements of x(t) at different time points. Problems of this form have applications in reverse engineering gene networks. In particular, we examine the cases where A is circulant and Toeplitz. We are also able to extend our findings of the circulant case to some generalizations of circulant matrices. Introduction DNA Microarrays can be used to determine measurements of cellular gene products at a given point in time. These concentrations of gene product provide clues to the overall interaction of the genes in the gene network being studied. We can measure the perturbations x1, . . . , xn of the gene mRNA expression concentrations from the steady state, which are governed by the equation dx(t) dt = Ax(t), (1) where x(t) = (x1(t), . . . , xn(t)) T . In other words, the rates of change of the gene product concentrations are determined by the deviations from the steady state of all the gene products present. If A is unstructured, then A contains n degrees of freedom, so we would expect that n concentration measurements would suffice to determine A uniquely. However, measuring the given gene product concentrations at a particular time is both time consuming and expensive. Gene networks can be on the order of 10,000 genes, so taking 10 measurements is impossible. Hence, we would like to impose some structure on A that will allow us to take fewer gene concentration measurements, enabling us to determine A more easily. This will provide a solution to our gene network which is easy to find and can provide a starting point for determining the exact structure of the gene network. In particular, we will first study the case where A is circulant and the case where A is Toeplitz. It is expected that if A is circulant, then A can be uniquely determined from a measurement of the n gene products at a single point in time. This seems intuitive since a circulant A contains only n degrees of freedom. If A is Toeplitz, we expect A may be determined by the measurement of the n gene products 1 at two points in time since Toeplitz A contains 2n− 1 degrees of freedom. Once A is determined, the gene product concentrations for any time t are given by x(t) = e x0, (2) where x0 = x(0) is the vector of gene product deviations from the steady state caused by a perturbation to the system at t = 0. Circulant Matrices Circulant matrices are those square matrices C of the form C = circ(c1, c2, . . . , cn) = 