We use center manifold theory to analyze a model of gene transcription and protein synthesis which consists of an ordinary differential equation (ODE) coupled to a delay differential equation (DDE). The analysis involves reformulating the problem as an operator differential equation which acts on function space, with the result that an infinite dimensional system is reduced to one of two dimensions. This work extends a previous CNSNS paper in which this problem was treated by Lindstedt's method. The present work is shown to provide approximations of general motions, including the approach to a periodic motion, in contrast to Lindstedt's method, which approximates only the periodic motion itself. In particular we show that the origin is asymptotically stable for the critical (bifurcation) value of the delay parameter.