  The Chemical Master Equation (CME) is used to stochastically model
biochemical reaction networks, under the Markovian assumption. The low-order
statistical moments induced by the CME are often the key quantities that one is
interested in. However, in most cases, the moments equation is not closed; in
the sense that the first $n$ moments depend on the higher order moments, for
any positive integer $n$. In this paper, we develop a moment closure technique
in which the higher order moments are approximated by an affine function of the
lower order moments. We refer to such functions as the affine Moment Closure
Functions (MCF) and prove that they are optimal in the worst-case context, in
which no a priori information on the probability distribution is available.
Furthermore, we cast the problem of finding the optimal affine MCF as a linear
program, which is tractable. We utilize the affine MCFs to derive a finite
dimensional linear system that approximates the low-order moments. We quantify
the approximation error in terms of the $% l_{\infty }$ induced norm of some
linear system. Our results can be effectively used to approximate the low-order
moments and characterize the noise properties of the biochemical network under
study.
