Within a recently developed framework of dynamical Monte Carlo algorithms, we compute the roughness exponent $ζ$ of driven elastic strings at the depinning threshold in 1+1 dimensions for different functional forms of the (short-range) elastic energy. A purely harmonic elastic energy leads to an unphysical value for $ζ$. We include supplementary terms in the elastic energy of at least quartic order in the local extension. We then find a roughness exponent of $ζ ≃ 0.63$, which coincides with the one obtained for different cellular automaton models of directed percolation depinning. The quartic term translates into a nonlinear piece which changes the roughness exponent in the corresponding continuum equation of motion. We discuss the implications of our analysis for higher-dimensional elastic manifolds in disordered media. ; Comment: 4 pages, 2 figures