We consider a class of matrices with a specific structure that arises, among other examples, in dynamic models for biological regulation of enzyme synthesis (Tyson and Othmer, 1978). We first show that a stability condition given in (Tyson and Othmer, 1978) is in fact a necessary and sufficient condition for diagonal stability of this class of matrices. We then revisit a recent generalization of (Tyson and Othmer, 1978) to nonlinear systems given in (Sontag, 2005), and recover the same stability condition using our diagonal stability result. Unlike the input-output based arguments employed in (Sontag, 2005), our proof gives a procedure to construct a Lyapunov function. Finally we study static nonlinearities that appear in the feedback path, and give a stability condition that mimics the Popov criterion.