Brownian dynamics of double-stranded DNA in periodic systems with discrete salt

Steven P. Mielke, Niels Grønbech-Jensen, and Craig J. Benham

Phys. Rev. E 77, 031924 – Published 31 March 2008

Abstract

Numerical models of mesoscale DNA dynamics relevant to in vivo scenarios require methods that incorporate important features of the intracellular environment, while maintaining computational tractability. Because the explicit inclusion of ions leads to electrostatic calculations that scale as the square of the number of charged particles, such models typically handle these calculations using low-potential, mean-field approaches, rather than by considering the discrete interactions of ions. This allows approximation of the long-range, screened self-repulsion of DNA, but is unable to capture detailed electrostatic phenomena, such as short-range attractions mediated by ion-ion correlations. Here, we develop a dynamical model of explicitly double-stranded, sequence-specific DNA in a bulk environment consisting of other polyions and explicitly represented counterions and coions. DNA is represented as two interwound chains of charged Stokes spheres, and ions as free, monovalently charged Stokes spheres. Brownian dynamics simulations performed at salt concentrations of 0.1, 1, 10, and 100 mM demonstrate this model captures anticipated behaviors of the system, including increasing compaction of the polyion by the ionic atmosphere with increasing ionic strength. The decay of the distance dependence of the ion concentrations as one moves away from the polyion approaches their equilibrium values in quantitative agreement with predictions of Poisson-Boltzmann theory. The simulation results also demonstrate quantitative agreement with experimental measurements of the persistence length of B-DNA, which increases significantly at low ionic strengths. The model also captures behaviors intimating the importance of explicitly representing ionic and polyionic structure. These include penetration of the polyion interior by both coions and counterions, and counterion-mediated accumulation of coions near the surface of the polyion. Such phenomena are likely to play an important role in the formation of alternative DNA secondary structures, suggesting the present methods will prove valuable to dynamic models of superhelical stress-induced DNA structural transitions.

Received 2 May 2007

DOI:https://doi.org/10.1103/PhysRevE.77.031924

Authors & Affiliations

Steven P. Mielke^1,*, Niels Grønbech-Jensen², and Craig J. Benham¹

¹UC Davis Genome Center, University of California, Davis, California 95616, USA
²Department of Applied Science, University of California, Davis, California 95616, USA

^*Author to whom correspondence should be addressed.

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Vol. 77, Iss. 3 — March 2008

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Figure 1
Double-chain model. (a) Three-dimensional representation of the interwound bead chains. One DNA polynucleotide is black and the other gray. The particles comprising the $i th$ duplex unit are labeled. (b) Corresponding two-dimensional schematic. Complementary beads (e.g., $i$ and $i + N$ ) touch—i.e., are separated by 2 times the bead radius, $R$ —at mechanical equilibrium. The intrachain separation of particles $i$ and $i + 1$ is labeled as $l_{i}$ . In the three-dimensional structure, noncomplementary interchain beads (e.g., $i$ and $i + N + 1$ ) approximately touch at equilibrium.Reuse & Permissions
Figure 2
Volume elements. Volume elements used to obtain radial counterion distributions are represented by gray surfaces. These were defined with respect to local coordinate frames ascribed to the duplex axis (black line) vertices and end points (see Sec. 2). To vertex $i$ are associated radial $(ρ_{k, i})$ , axial $(z_{i})$ , and azimuthal $(ϕ_{i})$ cylindrical coordinates, where $k$ indexes the volume elements, and an ion is associated to the $k th$ element if its position, $r$ , with respect to the local frame satisfies [ $0 < r_{z_{i}} < 10.2 Å$ ; $ρ_{k} < r_{ρ_{i}} < ρ_{k} + 5 Å$ ; $0 < r_{ϕ_{i}} < 2 π$ ]. Similarly, to the end points (duplex midpoints 1 and $N$ ) are associated radial $(ρ_{k, end})$ , azimuthal $(ϕ_{end})$ , and polar $(θ)$ spherical coordinates, where $k$ again indexes the volume elements, and an ion is associated to the $k th$ element if its position with respect to the local frame satisfies [ $- π / 2 < r_{θ} < π / 2$ ; $ρ_{k} < r_{ρ_{end}} < ρ_{k} + 5 Å$ ; $0 < r_{ϕ_{end}} < 2 π$ ].Reuse & Permissions
Figure 3
Configuration snapshots and system geometries. The pictured unit cells (dimensions indicated) correspond to simulations at 0.1 mM (a), 1 mM (b), 10 mM (c), and 100 mM (d) 1:1 salt concentrations. Counterions and coions are represented as gray and white spheres, respectively. In each case, a free, 147 base pair, $G + C$ copolymeric DNA molecule is represented as interwound, black and white bead chains. Relative sizes of the particles are chosen for clarity. All four snapshots were taken after at least $2.0 \times 10^{6}$ configurations with salt had been generated.Reuse & Permissions
Figure 4
Salt concentrations versus radial distance. Averaged concentrations of counterions (solid error bars) and coions (dashed error bars) in volume shells of $5 − Å$ radial width surrounding the central macroion are plotted (on logarithmic scale) at radial distances out to the half-width of the simulation boxes. The bulk concentration of 1:1 salt is indicated in each figure by a solid horizontal line: (a), (b), (c), and (d) correspond to the simulations at 0.1 mM, 1 mM, 10 mM, and 100 mM, respectively. The solid curves represent concentrations predicted by numerically solving the Poisson-Boltzmann equation for a cylinder with uniform surface charge density and a radius of $12 Å$ . This radius, and the distances at which the predicted counterion concentrations reach their bulk values, are indicated by dashed vertical lines. The latter decrease with increasing ionic strength, suggesting that the ionic atmosphere becomes increasingly compact, as anticipated. The simulations overestimate the counterion concentrations in the bulk region owing to the 294 additional counterions required to satisfy electroneutrality.Reuse & Permissions
Figure 5
Persistence lengths. Mean values of the persistence length and their standard deviations calculated from simulations at four salt concentrations, 0.1 mM, 1 mM, 10 mM, and 100 mM (plotted on logarithmic scale). At 1, 10, and 100 mM, $P \approx 500 Å$ , in agreement with the familiar value associated to B-DNA. At 0.1 mM, the persistence length is seen to be significantly larger $(P \approx 4700 Å)$ , consistent with the experimental results described in Ref. [24].Reuse & Permissions

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