We present a numerical framework for computing the evolution of curves subject to a class of generalized singular interface conditions from nonlocal elliptic problems. A classic example is Mullins-Sekerka dynamics, which comes as the sharp interface limit of the Cahn-Hilliard equations from materials science. The motivating example for the current work is a particular, singular, asymptotic limit of a system of reaction-diffusion equations: the saturated Gierer-Meinhardt system. Using fully implicit time-stepping and a finite difference approximation of the curve points, which are constrained to remain parameterized by scaled arc length, we solve a suitably discretized singular boundary integral formulation of the elliptic problem. The resulting approach can be used on a wide class of problems with little additional effort. This is joint work with Iain Moyles.