In this paper, we investigate the treatment of constraints in rate equations describing the temporal evolution of biological populations or chemical reactions. We present a formulation for arbitrary holonomic and linear nonholonomic constraints which ensures the positivity of the dynamical variables and which is an analog of Gauss' principle of least constraint in classical mechanics. The approach is illustrated for the replication of molecular species in the Schuster-Eigen hypercycle model, imposing the conservation of the total number of molecules and the entropy production as constraints. The latter is used to model the behavior of an isolated system tending toward equilibrium and, for comparison, a stationary nonequilibrium state of an open system, which is characterized by undamped oscillations.