Beyond Optimization: Simulating the Dynamics of Supercoiled DNA by a Macroscopic Model

The development and application of macroscopic dynamic models for large biomolecules is as much of an art as it is a science. Here we discuss scaling and parameterization issues for a macroscopic B-spline model of supercoiled DNA in the context of the Langevin framework. We show how scaling the masses and the damping constant, ,  is computationally advantageous (for enhanced sampling) and how, by calibration, the corresponding physical timestep and damping constant are obtained. Our timestep estimate for a mass scaled model, roughly 20 ps, is consistent with a rigorous  calibration for the dynamic B-spline model performed earlier. An interesting finding here is that the B-spline model, though different in details from a bead model, is essentially equivalent in the Langevin framework; thus, its computational  simplicity can be advantageous.  Another finding is that the optimal  for sampling is smaller (by two orders of magnitude) than the value needed to reproduce the experimental translational diffusion constant, Dt. This has been noted previously for proteins and appears to be an important aspect of modeling macromolecules in view of the sampling problem, as well as numerical integration. Using the smaller  has the effect, in the diffusive regime, of scaling the times computed in a Langevin trajectory by a corresponding factor. This leads to an effective timestep in our simulation around 10 ns. Performance of the implicit integrator  used here is also discussed, demonstrating that numerical damping effects are negligible for this macroscopic model at the  used for maximal sampling. Furthermore, computational gain can be achieved over explicit integration. Finally, an estimate on the effects of hydrodynamics on the translational friction constant, fT, is presented, on the basis of the  Kirkwood-Riseman equation. We find that fT is reduced by approximately a factor of two for DNA of 1000 base pairs, making the free-draining limit (i.e., no hydrodynamics) quite reasonable. The results presented here - scaling, parameterization, and integration in the context of a dynamic macroscopic model - have general applicability to simplified models of biopolymers. They also add the needed qualitative information so that slow dynamic processes involving supercoiled DNA, such as slithering, can be interpreted in terms of physical timescales from such macroscopic dynamic simulations.

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