We compute changes in the writhe of a polygonal space curve when one of the vertices is displaced. The resulting expressions can be used in simulations of supercoiled DNA. For Brownian dynamics simulations, the expressions can be used to eliminate the explicit twisting degree of freedom. For Monte Carlo simulations, they can be used in fast local moves. Preliminary Monte Carlo simulations using only such fast local moves show that these can be used to efficiently simulate small DNA supercoils.