Intrachain reactions of supercoiled DNA simulated by Brownian dynamics.

We considered an irreversible biochemical intrachain reaction of supercoiled DNA as a random event that occurs, with certain probability, at the instant of collision between two reactive groups bound to distant DNA sites. Using the Brownian dynamics technique, we modeled this process for a supercoiled DNA molecule of 2.5 kb length in dilute aqueous solution at an NaCl concentration of 0.1 M. We calculated the mean reaction time tau(Sigma) as a function of the intrinsic second-order rate constant k(I), the reaction radius R, and the contour separation S of the reactive groups. At the diffusion-controlled limit (k(I) --> infinity), the kinetics of reaction are determined by the mean time tau(F) of the first collision. The dependence of tau(F) on R is close to inversely proportional, implying that the main contribution to the productive collisions is made by bending of the superhelix axis. At sufficiently small k(I), the mean reaction time can be satisfactory approximated by tau(Sigma) = tau(F)(app) + 1/(k(I)c(L)), where c(L) is the local concentration of one reactive group around the other, and tau is an adjustable parameter, which we called the apparent time of the first collision. The value of tau depends on R very weakly and is approximately equal to the mean time of the first collision caused by mutual reptation of two DNA strands forming the superhelix. The quasi-one-dimensional reptation process provides the majority of productive collisions at small k(I) values.

Diffusion-controlled intrachain reactions of supercoiled DNA: Brownian Dynamics simulations.

The Brownian Dynamics technique was used to model a diffusion-controlled intramolecular reaction of supercoiled DNA (2500 basepairs) in 0.1 M sodium chloride solution. The distance between the reactive groups along the DNA contour was 470 basepairs. The reaction radius was varied from 6 to 20 nm. The results are presented in terms of the probability distribution P(F)(t) of the first collision time. The general form of the function P(F)(t) could be correctly predicted by a simple analytical model of one-dimensional diffusion of the superhelix ends along the DNA contour. The distribution P(F)(t) is essentially non-exponential: within a large initial time interval, it scales as P(F)(t) approximately t(-1/2), which is typical for one-dimensional diffusion. However, the mean time of the first collision is inversely proportional to the reaction radius, as in three dimensions. A visual inspection of the simulated conformations showed that a considerable part of the collisions is caused by the bending of the superhelix axis in the regions of the end loops, where the axis is most flexible. This fact explains why the distribution P(F)(t) combines the features of one- and three-dimensional diffusion. The simulations were repeated for a DNA chain with a permanent bend of 100 degrees in the middle position between the reactive groups along the DNA contour. The permanent bend changes dramatically the form of the distribution P(F)(t) and reduces the mean time of the first collision by approximately one order of magnitude.

Salt-dependent DNA superhelix diameter studied by small angle neutron scattering measurements and Monte Carlo simulations.

Using small angle neutron scattering we have measured the static form factor of two different superhelical DNAs, p1868 (1868 bp) and pUC18 (2686 bp), in dilute aqueous solution at salt concentrations between 0 and 1.5 M Na+ in 10 mM Tris at 0% and 100% D2O. For both DNA molecules, the theoretical static form factor was also calculated from an ensemble of Monte Carlo configurations generated by a previously described model. Simulated and measured form factors of both DNAs showed the same behavior between 10 and 100 mM salt concentration: An undulation in the scattering curve at a momentum transfer q = 0.5 nm-1 present at lower concentration disappears above 100 mM. The position of the undulation corresponds to a distance of approximately 10-20 nm. This indicated a change in the DNA superhelix diameter, as the undulation is not present in the scattering curve of the relaxed DNA. From the measured scattering curves of superhelical DNA we estimated the superhelix diameter as a function of Na+ concentration by a quantitative comparison with the scattering curve of relaxed DNA. The ratio of the scattering curves of superhelical and relaxed DNA is very similar to the form factor of a pair of point scatterers. We concluded that the distance of this pair corresponds to the interstrand separation in the superhelix. The computed superhelix diameter of 16.0 +/- 0.9 nm at 10 mM decreased to 9.0 +/- 0.7 nm at 100 mM salt concentration. Measured and simulated scattering curves agreed almost quantitatively, therefore we also calculated the superhelix diameter from the simulated conformations. It decreased from 18.0 +/- 1.5 nm at 10 mM to 9.4 +/- 1.5 nm at 100 mM salt concentration. This value did not significantly change to lower values at higher Na+ concentration, in agreement with results obtained by electron microscopy, scanning force microscopy imaging in aqueous solution, and recent MC simulations, but in contrast to the observation of a lateral collapse of the DNA superhelix as indicated by cryo-electron microscopy studies.

Looping dynamics of linear DNA molecules and the effect of DNA curvature: a study by Brownian dynamics simulation.

A Brownian dynamics (BD) model described in the accompanying paper (Klenin, K., H. Merlitz, and J. Langowski. 1998. A Brownian dynamics program for the simulation of linear and circular DNA, and other wormlike chain polyelectrolytes. Biophys. J. 74:000-000) has been used for computing the end-to-end distance distribution function, the cyclization probability, and the cyclization kinetics of linear DNA fragments between 120 and 470 basepairs with optional insertion of DNA bends. Protein-mediated DNA loop formation was modeled by varying the reaction distance for cyclization between 0 and 10 nm. The low cyclization probability of DNA fragments shorter than the Kuhn length (300 bp) is enhanced by several orders of magnitude when the cyclization is mediated by a protein bridge of 10 nm diameter, and/or when the DNA is bent. From the BD trajectories, end-to-end collision frequencies were computed. Typical rates for loop formation of linear DNAs are 1.3 x 10(3) s(-1) (235 bp) and 4.8 x 10(2) s(-1) (470 bp), while the insertion of a 120 degree bend in the center increases this rate to 3.0 x 10(4) s(-1) (235 bp) and 5.5 x 10(3) s(-1) (470 bp), respectively. The duration of each encounter is between 0.05 and 0.5 micros for these DNAs. The results are discussed in the context of the interaction of transcription activator proteins.

Modulation of intramolecular interactions in superhelical DNA by curved sequences: a Monte Carlo simulation study.

A Monte Carlo model for the generation of superhelical DNA structures at thermodynamic equilibrium (Klenin et al., 1991; Vologodskii et al., 1992) was modified to account for the presence of local curvature. Equilibrium ensembles of a 2700-bp DNA chain at linking number difference delta Lk = -15 were generated, with one or two permanent bends up to 120 degrees inserted at different positions. The computed structures were then analyzed with respect to the number and positions of the end loops of the interwound superhelix, and the intramolecular interaction probability of different segments of the DNA. We find that the superhelix structure is strongly organized by permanent bends. A DNA segment with a 30 degrees bend already has a significantly higher probability of being at the apex of a superhelix than the control, and for a 120 degrees bend the majority of DNAs have one end loop at the position of the bend. The entropy change due to the localization of a 120 permanent bend in the end loop is estimated to be -17 kJ mol-1 K-1. When two bends are inserted, the conformation of the superhelix is found to be strongly dependent on their relative positions: the straight interwound form dominates when the two bends are separated by 50% of the total DNA length, whereas the majority of the superhelices are in a branched conformation when the bends are separated by 33%. DNA segments in the vicinity of the permanent bend are strongly oriented with respect to each other.

Conformational and thermodynamic properties of supercoiled DNA.

We used Monte Carlo simulations to investigate the conformational and thermodynamic properties of DNA molecules with physiological levels of supercoiling. Three parameters determine the properties of DNA in this model: Kuhn statistical length, torsional rigidity and effective double-helix diameter. The chains in the simulation resemble strongly those observed by electron microscopy and have the conformation of an interwound superhelix whose axis is often branched. We compared the geometry of simulated chains with that determined experimentally by electron microscopy and by topological methods. We found a very close agreement between the Monte Carlo and experimental values for writhe, superhelix axis length and the number of superhelical turns. The computed number of superhelix branches was found to be dependent on superhelix density, DNA chain length and double-helix diameter. We investigated the thermodynamics of supercoiling and found that at low superhelix density the entropic contribution to superhelix free energy is negligible, whereas at high superhelix density, the entropic and enthalpic contributions are nearly equal. We calculated the effect of supercoiling on the spatial distribution of DNA segments. The probability that a pair of DNA sites separated along the chain contour by at least 50 nm are juxtaposed is about two orders of magnitude greater in supercoiled DNA than in relaxed DNA. This increase in the effective local concentration of DNA is not strongly dependent on the contour separation between the sites. We discuss the implications of this enhancement of site juxtaposition by supercoiling in the context of protein-DNA interactions involving multiple DNA-binding sites.

Computer simulation of DNA supercoiling.

We treat supercoiled DNA within a wormlike model with excluded volume. A modified Monte Carlo approach has been used, which allowed computer statistical-mechanical simulations of moderately and highly supercoiled DNA molecules. Even highly supercoiled molecules do not have a regular shape, though with an increase in writhing the chains look more and more like branched interwound helixes. The averaged writhing (Wr) approximately 0.7 delta Lk. The superhelical free energy F is calculated as a function of the linking number. Lk. The calculations have shown that the generally accepted quadratic dependence of F on Lk is valid for a variety of conditions, though it is by no means universal. Significant deviations from the quadratic dependence are expected at high superhelical density under ionic conditions where the effective diameter of DNA is small. The results are compared with the available experimental data.

Variance of writhe for wormlike DNA rings with excluded volume.

We have calculated the variance of the equilibrium distribution of a circular wormlike polymer chain over the writhing number, less than (Wr)2 greater than, with allowance for the excluded volume effects. Within this model the less than (Wr)2 greater than value is a function of the number of Kuhn statistical segments, n, and the chain diameter, d measured in Kuhn statistical lengths, b. Simulated DNA chains varied from 200 to 10,000 base pairs and the d value varied from 0.02 to 0.2. Theory predicts a considerable ionic strength dependence of the DNA superhelix energy as a consequence of the change in the DNA diameter. A comparison with the available experimental data has yielded an estimate of the DNA torsional rigidity, the Kuhn statistical length, and the effective diameter of the double helix under conditions of the complete screening of the DNA electrostatic potential.

Effect of excluded volume on topological properties of circular DNA.

We have performed computer simulations of closed polymer chains with allowance for the excluded volume effects within the framework of the free-joint model. The probability of knot formation, the linking probability of a pair of chains and the variance in the writhing number proved to be significantly affected by the excluded volume effects. This is true even for DNA with completely screened charges for which the b/d ratio (where b is the Kuhn statistical length and d is the diameter of the double helix) is as large as 50. Allowance for the electrostatic repulsion (change of the DNA effective diameter d) further increases the effects. The most dramatic dependence on d is found for the probability of knot formation. The data on the dependence of the variance of writhing, mean value of (WR)2, on d indicate that the DNA superhelix energy should be significantly ionic strength-dependent. Special calculations have shown that the free-joint model underestimates the mean value of (Wr)2 value by about 20% as compared with the wormlike model.