Calibration of the Timestep for the Dynamics of Supercoiled DNA modeled by B-Splines

We address here the correspondence between the physical timestep and the simulation timestep in our molecular dynamics simulations of supercoiled DNA modeled by cubic B-splines. The correspondence is not straightforward since the independent variables in our dynamics are the B-spline control points rather than the DNA chain points. Since each DNA curve point depends on four neighboring control points, the effective B-spline mass matrix is nondiagonal. We derive this matrix rigorously and show that each row corresponding to a control point has seven nonzero elements. These elements are configuration dependent and must be computed numerically. We obtain the rough estimate of  = 100-150 ps for our timestep by equating the expected kinetic energy for the spline system of DNA - from the classical equipartition theorem - with the computed kinetic energy using the velocity vectors of the dynamics trajectory and the spline mass matrix. This estimate lies between our rigorously calculated lower and upper bounds for. A lower bound is obtained by assuming full energy contributions for all the system's degrees of freedom. An upper bound for  is obtained by estimating the effects of solvent damping and quantum effects, which together reduce the expected kinetic energy of DNA in its natural environment. Our resulting timestep estimate corresponds to a difference of more than three orders of magnitude between our program timestep t (0.1 ps) and the physical value .

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