Computer simulation of DNA supercoiling

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.

Mathematical relationships among DNA supercoiling, cation concentration, and temperature for prokaryotic transcription

DNA twist has been proposed to affect transcription from some promoters of Escherichia coli, but involvement of twist has been difficult to test because it cannot be measured in transcription reaction mixtures. However, changes in other factors affect both DNA twist and transcription. These parameters are expected to be related when maximum transcription initiation is considered. In the present work, mathematical relationships among supercoiling, cation concentration, and temperature are derived for prokaryotic transcription initiation. The relationships indicate that as DNA becomes more negatively supercoiled, maximal initiation occurs at a higher cation concentration and at a lower temperature. For example, when superhelical density becomes more negative by 0.0025, a 1.6-fold increase in potassium concentration is predicted to be required to maintain transcription initiation at its maximum rate. Experimental verification of the relationships should provide a useful test of the idea that transcription initiation is sensitive to DNA twist.

A Brownian dynamics program for the simulation of linear and circular DNA and other wormlike chain polyelectrolytes

For the interpretation of solution structural and dynamic data of linear and circular DNA molecules in the kb range, and for the prediction of the effect of local structural changes on the global conformation of such DNAs, we have developed an efficient and easy way to set up a program based on a second-order explicit Brownian dynamics algorithm. The DNA is modeled by a chain of rigid segments interacting through harmonic spring potentials for bending, torsion, and stretching. The electrostatics are handled using precalculated energy tables for the interactions between DNA segments as a function of relative orientation and distance. Hydrodynamic interactions are treated using the Rotne-Prager tensor. While maintaining acceptable precision, the simulation can be accelerated by recalculating this tensor only once in a certain number of steps.

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.

On the origin of the temperature dependence of the supercoiling free energy

Monte Carlo simulations using temperature-invariant torsional and bending rigidities fail to predict the rather steep decline of the experimental supercoiling free energy with increasing temperature, and consequently fail to predict the correct sign and magnitude of the supercoiling entropy. To illustrate this problem, values of the twist energy parameter (E(T)), which governs the supercoiling free energy, were simulated using temperature-invariant torsion and bending potentials and compared to experimental data on pBR322 over a range of temperatures. The slope, -dE(T)/dT, of the simulated values is also compared to the slope derived from previous calorimetric data. The possibility that the discrepancies arise from some hitherto undetected temperature dependence of the torsional rigidity was investigated. The torsion elastic constant of an 1876-bp restriction fragment of pBR322 was measured by time-resolved fluorescence polarization anisotropy of intercalated ethidium over the range 278-323 K, and found to decline substantially over that interval. Simulations of a 4349-bp model DNA were performed using these measured temperature-dependent torsional rigidities. The slope, -dE(T)/dT, of the simulated data agrees satisfactorily with the slope derived from previous calorimetric measurements, but still lies substantially below that of Duguet's data. Models that involve an equilibrium between different secondary structure states with different intrinsic twists and torsion constants provide the most likely explanation for the variation of the torsion constant with T and other pertinent observations.

Winding of the DNA helix by divalent metal ions

When supercoiled pBR322 DNA was relaxed at 0 or 22 degrees C by topoisomerase I in the presence of the divalent cations Ca2+, Mn2+ or Co2+, the resulting distributions of topoisomers observed at 22 degrees C had positive supercoils, up to an average delta Lk value of +8.6 (for Ca2+at 0 degrees C), corresponding to an overwinding of the helix by 0.7 degrees/bp. An increase of the divalent cation concentration in the reaction mixture above 50 mM completely reversed the effect. When such ions were present in agarose electrophoresis gels, they caused a relaxation of positively supercoiled DNA molecules, and thus allowed a separation of strongly positively supercoiled topoisomers. The effect of divalent cations on DNA adds a useful tool for the study of DNA topoisomers, for the generation as well as separation of positively supercoiled DNA molecules.

Configurational transitions in Fourier series-represented DNA supercoils

A new Fourier series representation of supercoiled DNA is employed in Langevin dynamics simulations to study large-scale configurational motions of intermediate-length chains. The polymer is modeled as an ideal elastic rod subject to long-range van der Waals' interactions. The van der Waals' term prevents the self-contact of distant chain segments and also mimics attractive forces thought to stabilize the association of closely spaced charged rods. The finite Fourier series-derived polymer formulation is an alternative to the piecewise B-spline curves used in past work to describe the motion of smoothly deformed supercoiled DNA in terms of a limited number of independent variables. This study focuses on two large-scale configurational events: the interconversion between circular and figure-8 forms at a relatively low level of supercoiling, and the transformation between branched and interwound structures at a higher superhelical density.

Extension of torsionally stressed DNA by external force

Metropolis Monte Carlo simulation was used to study the elasticity of torsionally stressed double-helical DNA. Equilibrium distributions of DNA conformations for different values of linking deficit, external force, and ionic conditions were simulated using the discrete wormlike chain model. Ionic conditions were specified in terms of DNA effective diameter, i.e., hard-core radius of the model chain. The simulations show that entropic elasticity of the double helix depends on how much it is twisted. For low amounts of twisting (less than about one turn per twist persistence length) the force versus extension is nearly the same as in the completely torsionally relaxed case. For more twisting than this, the molecule starts to supercoil, and there is an increase in the force needed to realize a given extension. For sufficiently large amounts of twist, the entire chain is plectonemically supercoiled at low extensions; a finite force must be applied to obtain any extension at all in this regime. The simulation results agree well with the results of recent micromanipulation experiments.

The effect of ionic conditions on DNA helical repeat, effective diameter and free energy of supercoiling

We determined the free energy of DNA supercoiling as a function of the concentration of magnesium and sodium chloride in solution by measuring the variance of the equilibrium distribution of DNA linking number,<(DeltaLk)2>. We found that the free energy of supercoiling changed >1.5-fold over the range of ionic conditions studied. Comparison of the experimental results with those of computer simulations showed that the ionic condition dependence of<(DeltaLk)2>is due mostly to the change in DNA effective diameter, d, a parameter characterizing the electrostatic interaction of DNA segments. To make this comparison we determined values of d under all ionic conditions studied by measuring the probability of knot formation during random cyclization of linear DNA molecules. From the topoisomer distributions we could also determine the changes in DNA helical repeat, gamma, in mixed NaCl/MgCl2 solutions. Both gamma and d exhibited a complex pattern of changes with changing ionic conditions, which can be described in terms of competition between magnesium and sodium ions for binding to DNA.

The effect of ionic conditions on the conformations of supercoiled DNA. I. Sedimentation analysis

We studied the conformations of supercoiled DNA as a function of superhelicity and ionic conditions by determining its sedimentation coefficient both experimentally and by calculation. To cancel out unknown parameters from both calculations and experiments, we determined the ratio of the sedimentation coefficient, s, to that of open circular DNA, s(oc). Calculations of the sedimentation coefficient were based on direct solution of the Burgers-Oseen problem for an equilibrium set of DNA conformations generated for each condition by the Metropolis Monte Carlo procedure. There were no adjustable parameters in the Monte Carlo simulations because all three parameters of the DNA model used, bending and torsional elasticity of DNA and DNA effective diameter specifying electrostatic interactions, were known from independent data. The good agreement between measured and calculated values of s/s(oc) allowed us to interpret the sedimentation results in terms of DNA conformations, with particular emphasis on the marked effect of ionic conditions. As NaCl concentration decreases, s/s(oc) increases because the superhelix becomes less regular and more compact. In the presence of just 10 mM MgCl(2), supercoiled DNA adopts essentially the same set of conformations as in moderate to high concentrations of NaCl. Our simulations showed that s is a strong function of the superhelix branching frequency. At near physiological ionic conditions, there are about four branches in the 7 kb DNA molecule used in this work. We found no indication of superhelix collapse in any ionic conditions even remotely approaching physiological ones. For all ionic conditions studied, we conclude that the electrostatic interaction of DNA segments specified by the DNA effective diameter is the primary determinant of supercoiled DNA conformations.

Visualization of supercoiled DNA with atomic force microscopy in situ

Tertiary structure of supercoiled DNA is a significant factor in a number of genetic functions and is apparently affected by environmental conditions. We applied atomic force microscopy (AFM) for imaging the supercoiled DNA deposited at different ionic conditions. We have employed a technique for the sample preparation that permits high-resolution AFM imaging of DNA bound to the surface in buffer solutions without drying the sample (AFM in situ). The AFM data show that at low ionic strength, DNA molecules are loosely interwound supercoils with an irregular shape. Plectonemic superhelices are formed in high-concentration, near-physiological salt solutions. At such ionic conditions, superhelical loops are typically separated by regions of close helix-helix contacts. The data obtained show directly and unambiguously that overall geometry of supercoiled DNA depends dramatically on ionic conditions. This fact and the formation of close contacts between DNA helices are important features of supercoiled DNA related to its biological functions.

Molecular modeling of closed circular DNA thermodynamic ensembles

Many modeling studies of supercoiled DNA are based on equilibrium structures from theoretical calculations or energy minimization. Since closed circular DNAs are flexible, it is possible that errors are introduced by calculating properties from a single minimum energy structure, rather than from a complete thermodynamic ensemble. We have investigated this question using molecular dynamics simulations on a low resolution molecular mechanics model in which each base pair is represented by three points (a plane). This allows the inclusion of sequence-dependent variations of tip, inclination, and twist. Three kinds of sequences were tested: (1) homogeneous DNA, in which all base pairs have the helicoidal parameters of an ideal, average B-DNA; (2) random sequence DNA; and (3) curved DNA. We examined the rate of convergence of various structural parameters. Convergence for most of these is slowest for homogeneous sequences, more rapid for random sequences, and most rapid for curved sequences. The most slowly converging parameter is the antipodes profile. In a plasmid with N base pairs (bp), the antipodes distance is the distance dij from base pair i to base pair j halfway around the plasmid, j = i + N/2. The antipodes profile at time tau is a plot of dij over the range i = 1, N/2. In a homogeneous plasmid, convergence requires that the antipodes profile averaged over time must be flat. Even in the small plasmids examined here, the average properties of the ensembles were found to differ from those of static equilibrium structures. These effects will be even more dramatic for larger plasmids. Further, average and dynamic properties are affected by both plasmid size and sequence.

Molecular dynamics simulations of small DNA plasmids: effects of sequence and supercoiling on intramolecular motions

Small (600 base pair) DNA plasmids were modeled with a simplified representation (3DNA) and the intramolecular motions were studied using molecular mechanics and molecular dynamics techniques. The model is detailed enough to incorporate sequence effects. At the same time, it is simple enough to allow long molecular dynamics simulations. The simulations revealed that large-scale slithering occurs in a homogeneous sequence. In a heterogeneous sequence, containing numerous small intrinsic curves, the centers of the curves are preferentially positioned at the tips of loops. With more curves than loop tips (two in unbranched supercoiled DNA), the heterogeneous sequence plasmid slithers short distances to reposition other curves into the loop tips. However, the DNA is immobilized most of the time, with the loop tips positioned over a few favored curve centers. Branching or looping also appears in the heterogeneous sequence as a new method of repositioning the loop tips. Instead of a smooth progression of increasing writhing with increasing linking difference, theoretical studies have predicted that there is a threshold between unwrithed and writhed DNA at a linking difference between one and two. This has previously been observed in simulations of static structures and is demonstrated here for dynamic homogeneous closed DNA. Such an abrupt transition is not found in the heterogeneous sequence in both the static and dynamic cases.

Effect of supercoiling on the juxtaposition and relative orientation of DNA sites

There are many proteins that interact simultaneously with two or more DNA sites that are separated along the DNA contour. These sites must be brought close together to form productive complexes with the proteins. We used Monte Carlo simulation of supercoiled DNA conformations to study the effect of supercoiling and DNA length on the juxtaposition of DNA sites, the angle between them, and the branching of the interwound superhelix. Branching decreases the probability of juxtaposition of two DNA sites but increases the probability of juxtaposition of three sites at branch points. We found that the number of superhelix branches increases linearly with the length of DNA from 3 to 20 kb. The simulations showed that for all contour distances between two sites, the juxtaposition probability in supercoiled DNA is two orders of magnitude higher than in relaxed DNA. Supercoiling also results in a strong asymmetry of the angular distribution of juxtaposed sites. The effect of supercoiling on site-specific recombination and the introduction of supercoils by DNA gyrase is discussed in the context of the simulation results.

Thermodynamics of the first transition in writhe of a small circular DNA by Monte Carlo simulation

Monte Carlo simulations are employed to investigate the thermodynamics of the first transition in writhe of a circular model filament corresponding to a 468 base-pair DNA. Parameters employed in these simulations are the torsional rigidity, C = 2.0 x 10(-19) dyne cm2, and persistence length, P = 500 A. Intersubunit interactions are modeled by a screened Coulomb potential. For a straight line of subunits this accurately approximates the nonlinear Poisson-Boltzmann potential of a cylinder with the linear charge density of DNA. Curves of relative free energy vs writhe at fixed linking difference (delta l) exhibit two minima, one corresponding to slightly writhed circles and one to slightly underwrithed figure-8's, whenever delta l lies in the transition region. The free energies of the two minima are equal when delta lc = 1.35, which defines the midpoint of the transition. At this midpoint, the free energy barrier between the two minima is found to be delta Gbar = (0.20) kBT at 298 K. Curves of mean potential energy vs writhe at fixed linking difference similarly exhibit two minima for delta l values in the transition region, and the two minimum mean potential energies are equal when delta l = 1.50. At the midpoint writhe, delta lc = 1.35, the difference in mean potential energy between the minimum free energy figure-8 and circle states is (1.3) kBT, and the difference in their entropies is 1.3 kB. Thus, the entropy of the minimum free energy figure-8 state significantly exceeds that of the circle at the midpoint of the transition. The first transition in writhe is found to occur over a rather broad range of delta l values from 0.85 to 1.85. The twist energy parameter (ET), which governs the overall free energy of supercoiling, undergoes a sigmoidal decrease, while the translational diffusion coefficient undergoes a sigmoidal increase, over this same range. The static structure factor exhibits an increase, which reflects a decrease in radius of gyration associated with the circle to figure-8 transition.

Action at a distance in supercoiled DNA: effects of sequence on slither, branching, and intramolecular concentration

We report a computer modeling study of DNA supercoiling in model plasmids over the size range of 140-1260 bp. We used a computer model with basepair resolution. Molecular dynamics was used to produce ensembles at 300 K and to investigate intramolecular motions. The plasmid models varied by their sequence. The sequence types employed for comparison included a curve-bearing plasmid, a heterogenous sequence plasmid, and a homogenous sequence. Within the three sequence types tested at the 1260-bp plasmid size, we observed several sequence-dependent phenomena. Writhe, radius of gyration, slither motion, and branching probability were seen to be sequence dependent. Branching probability was the least in the homogenous plasmid and the greatest in the curve-bearing plasmid. The curve imposed a symmetry on the plasmid that was absent in the heterogenous sequence. Significant localizations and enhancements of intramolecular concentration were seen to a persistence length. Molecular dynamics allowed us to observe the mechanism of branch formation and reabsorption. We observed a size-dependent change in the types of motion observed in plasmids. Slither motion predominated in plasmids up to 600 bp in size, whereas global rearrangements were more important in the 1260 mer.

Monte Carlo simulations of supercoiling free energies for unknotted and trefoil knotted DNAs

A new Monte Carlo (MC) algorithm is proposed for simulating inextensible circular chains with finite twisting and bending rigidity. This new algorithm samples the relevant Riemann volume elements in a uniform manner, when the constraining potential vanishes. Simulations are performed for filaments comprising 170 subunits, each containing approximately 28 bp, which corresponds to a DNA length of 4770 bp. The bending rigidity is chosen to yield a persistence length, P = 500 A, and the intersubunit potential is taken to be a hard-cylinder potential with diameter d = 50 A. This value of d yields the same second virial coefficient as the electrostatic potential obtained by numerical solution of the Poisson-Boltzmann equation for 150 mM salt. Simulations are performed for unknotted circles and also for trefoil knotted circles using two different values of the torsional rigidity, C = (2.0 and 3.0) x 10(-19) dyne cm2. In the case of unknotted circles, the simulated supercoiling free energy varies practically quadratically with linking difference delta l. The simulated twist energy parameter ET, its slope dET/dT, and the mean reduced writhe <w>/delta l for C = 3 x 10(-19) dyne cm2 all agree well with recent simulations for unknotted circles using the polygon-folding algorithm with identical P, d, and C. The simulated ET vs. delta l data for C = 2.0 x 10(-19) dyne cm2 agree rather well with recent experimental data for p30 delta DNA (4752 bp), for which the torsional rigidity, C = 2.07 x 10(-19) dyne cm2, was independently measured. The experimental data for p30 delta are enormously more likely to have arisen from C = 2.0 x 10(-19) than from C = 3.0 x 10(-19) dyne cm2. Serious problems with the reported experimental assessments of ET for pBR322 and their comparison with simulated data are noted. In the case of a trefoil knotted DNA, the simulated value, (ET)tre, exceeds that of the unknotted DNA, (ET)unk, by approximately equal to 1.40-fold at magnitude of delta l = 1.0, but declines to a plateau about 1.09-fold larger than (ET)unk when magnitude of delta l > or = 15. Although the predicted ratio, (ET)tre/(ET)unk approximately equal to 1.40, agrees fairly well with recent experimental measurements on a 5600-bp DNA, the individual measured ET values, like some of those reported for pBR322, are so large that they cannot be simulated using P = 500 A, d = 50 A, and any previous experimental estimate of C.

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.

The influence of salt on the structure and energetics of supercoiled DNA

We present a detailed computational study of the influence of salt on the configurations, energies, and dynamics of supercoiled DNA. A potential function that includes both elastic and electrostatic energy components is employed. Specifically, the electrostatic term, with salt-dependent coefficients, is modeled after Stigter's pioneering work on the effective diameter of DNA as a function of salt concentration. Because an effective charge per unit length is used, the electrostatic formulation does not require explicit modeling of phosphates and can be used to study long DNAs at any desired resolution of charge. With explicit consideration of the electrostatic energy, an elastic bending constant corresponding to the nonelectrostatic part of the bending contribution to the persistence length is used. We show, for a series of salt concentrations ranging from 0.005 to 1.0 M sodium, how configurations and energies of supercoiled DNA (1000 and 3000 base pairs) change dramatically with the simulated salt environment. At high salt, the DNA adopts highly compact and bent interwound states, with the bending energy dominating over the other components, and the electrostatic energy playing a minor role in comparison to the bending and twisting terms. At low salt, the DNA supercoils are much more open and loosely interwound, and the electrostatic components are dominant. Over the range of three decades of salt examined, the electrostatic energy changes by a factor of 10. The buckling transition between the circle and figure-8 is highly sensitive to salt concentration: this transition is delayed as salt concentration decreases, with a particularly sharp increase below 0.1 M. For example, for a bending-to-twisting force constant ratio of A/C = 1.5, the linking number difference (delta LK) corresponding to equal energies for the circle and figure-8 increases from 2.1 to 3.25 as salt decreases from 1.0 to 0.005 M. We also present in detail a family of three-lobed supercoiled DNA configurations that are predicted by elasticity theory to be stable at low delta Lk. To our knowledge, such three-dimensional structures have not been previously presented in connection with DNA supercoiling. These branched forms have a higher bending energy than the corresponding interwound configurations at the same delta Lk but, especially at low salt, this bending energy difference is relatively small in comparison with the total energy, which is dominated by the electrostatic contributions. Significantly, the electrostatic energies of the three-lobed and (straight) interwound forms are comparable at each salt environment. We show how the three-lobed configurations change slowly with ALk, resulting in branched interwound forms at higher salt. In longer chains, the branched forms are highly interwound, with bent arms. At low salt, the branched supercoils are asymmetric, with a longer interwound stem and two shorter arms. From molecular dynamics simulations we observe differences in the motions of the DNA as a function of salt. At high salt, the supercoiled chain is quite compact but fairly rigid, whereas at low salt the DNA is loosely coiled but more dynamic. Especially notable at low salt are the large-scale opening and closing of the chain as a whole and the rapid "slithering"of individual residues past one another. Toroidal forms are not detected under these conditions. However, the overall features of the open, loose supercoils found at low salt are more similar to those of toroidal than interwound configurations. Indeed,simulated x-ray scattering profiles reveal the same trends observed experimentally and are consistent with a change from closed to open forms as salt is decreased. Like the minimization studies, the dynamics reveal a critical point near 0.1 M associated with the collapse of loose to tight supercoils. Near this physiological concentration, enhanced flexibility of the DNA is noted. The collective observations suggest a potential regulatory role for salt on supercoiled DNA function, not only for closed circular DNA,but also for linear DNA with small looped regions.

Effect of ethidium binding and superhelix density on the supercoiling free energy and torsion and bending constants of p30 delta DNA

Topoisomer distributions created by the action of topoisomerase I on p30 delta DNA in the presence of various concentrations of ethidium are measured and analyzed using recently developed theory to obtain the twist energy parameter (ET) that governs the free energy of supercoiling in each case. Competitive dialysis experiments to investigate the relative affinity of ethidium for linear and supercoiled DNAs at different binding ratios are assayed fluorometrically and the results are analyzed using related theory. The topoisomer distributions and fluorescence intensity ratios agree well with the theory, which is based on the assumption that the supercoiling free energy varies quadratically with the effective linking difference, regardless of ethidium binding or superhelix density. The topoisomer distribution experiments alone yield an average best-fit value, ET = 950 +/- 80, independent of ethidium binding ratio from r = 0 to 0.082, while the combined topoisomer distribution and ethidium binding experiments yield an average best-fit value, ET = 1030 +/- 90, which is essentially independent of ethidium binding ratio from r = 0 to 0.082 and superhelix density from sigma = 0 to (-)0.053. One may conclude that the supercoiling free-energy-varies quadratically with effective linking difference over the entire range of observed ethidium binding ratios and superhelix densities. The independently measured torsion constant (alpha) of p30 delta DNA is likewise essentially independent of superhelix density and ethidium binding ratio. The observed invariance of ET and alpha implies that the bending constant kappa beta is similarly invariant to superhelix density and ethidium binding ratio. The apparently ideal behavior displayed by p30 delta DNA is not exhibited by pBR322 DNA, which is discussed in the following companion paper.

Circularization of small DNAs in the presence of ethidium: a theoretical analysis

A rigorous theory is developed for ethidium binding to linear and circular DNAs and for the ratios of topoisomers produced upon ligation of an equilibrium population of noncovalently closed circles in the presence of ethidium. Assuming an unwinding angle theta E = 26 degrees for intercalated ethidium, optimum values of the intrinsic binding constant, KE = 7.16 x 10(4) M-1, the intrinsic twist, l0 = 23.746 turns, and twist energy parameter, Et = 5250, are obtained by fitting the present theory to the data of Shore and Baldwin [(1993) Journal of Molecular Biology, Vol. 170, pp. 983-1007] for a 247 base pair DNA. A very good fit is achieved with these optimum values, but a poor fit results when the parameters estimated by Shore and Baldwin are employed in the same theory. Three assumptions employed in the analysis of Shore and Baldwin are found to be not strictly valid. Adoption of the present substantially larger Et value as representative of their short DNAs would allow the Et vs N data of Shore and Baldwin to conform to the shape predicted by Shimada and Yamakawa [(1985) Journal of Molecular Biology, Vol. 184, pp. 319-329] and Frank-Kamenetskii et al. [(1985) Journal of Biomolecular Structure and Dynamics, Vol. 2, pp. 1005-1012], and would imply that all of their DNAs exist in a substantially stiffer than normal state.

Electrostatic effects in short superhelical DNA

We present Monte Carlo simulations of the equilibrium configurations of short closed circular DNA that obeys a combined elastic, hard-sphere, and electrostatic energy potential. We employ a B-spline representation to model chain configuration and simulate the effects of salt on chain folding by varying the Debye screening parameter. We obtain global equilibrium configurations of closed circular DNA, with several imposed linking number differences, at two salt concentrations (specifically at the extremes of no added salt and the high salt regime), and for different chain lengths. Minimization of the composite elastic/long-range potential energy under the constraints of ring closure and fixed chain length is found to produce structures that are consistent with the configurations of short supercoiled DNA observed experimentally. The structures generated under the constraints of an electrostatic potential are less compact than those subjected only to an elastic term and a hard-sphere constraint. For a fixed linking number difference greater than a critical value, the interwound structures obtained under the condition of high salt are more compact than those obtained under the condition of no added salt. In the case of no added salt, the electrostatic energy plays a dominant role over the elastic energy in dictating the shape of the closed circular DNA. The DNA supercoil opens up with increasing chain length at low salt concentration. A branched three-leaf rose structure with a fixed linking number difference is higher in energy than the interwound form at both salt concentrations employed here.

On higher buckling transitions in supercoiled DNA

A combination of detailed energy minimization and molecular dynamics studies of closed circular DNA offers here new information that may be relevant to the dynamics of short DNA chains and/or low superhelical densities. We find a complex dependence of supercoiled DNA energies and geometries on the linking number difference delta Lk as physiological superhelical densities (magnitude of sigma approximately 0.06) are approached. The energy minimization results confirm and extend predictions of classical elasticity theory for the equilibria of elastic rods. The molecular dynamics results suggest how these findings may affect the dynamics of supercoiled DNA. The minimization reveals sudden higher order configurational transitions in addition to the well-known catastrophic buckling from the circle to the figure-8. The competition among the bending, twisting, and self-contact forces leads to different families of supercoiled forms. Some of those families begin with configurations of near-zero twist. This offers the intriguing possibility that nicked DNA may relax to low-twist forms other than the circle, as generally assumed. Furthermore, for certain values of delta Lk, more than one interwound DNA minimum exists. The writhing number as a function of delta Lk is discontinuous in some ranges; it exhibits pronounced jumps as delta Lk is increased from zero, and it appears to level off to a characteristic slope only at higher values of delta Lk. These findings suggest that supercoiled DNA may undergo systematic rapid interconversions between different minima that are both close in energy and geometry. Our molecular dynamics simulations reveal such transitional behavior. We observe the macroscopic bending and twisting fluctuations of interwound forms about the global helix axis as well as the end-over-end tumbling of the DNA as a rigid body. The overall mobility can be related to magnitude of sigma and to the bending, twisting, and van der Waals energy fluctuations. The general character of molecular motions is thus determined by the types of energy minima found at a given delta Lk. Different time scales may be attributed to each type of motion: The overall chain folding occurs on a time scale almost an order of magnitude faster than the end-over-end tumbling. The local bending and twisting of individual chain residues occur at an even faster rate, which in turn correspond to several cycles of local variations for each large-scale bending and straightening motion of the DNA.

Modulation of DNA supercoiling by interaction with netropsin and other minor groove binders

The assay of DNA unwinding by ethidium, followed by sedimentation velocity techniques, was applied to complexes of supercoiled plasmid DNA with different non-intercalating drugs which strongly and sequence-specifically bind to DNA. Compared with the behaviour of naked DNA, most of the complexes exhibit an increase in the critical EB/nucleotide binding ratio associated with the principal minimum in the sedimentation profile. Using netropsin (Nt) as the paradigm of the minor groove binders investigated, the drug-induced alterations in various structural parameters of both the relaxed and supercoiled form of DNA are described. Whereas winding number, helical repeat (both being defined with reference to a surface normal), and linking number of the superhelical DNA remain constant in our experiments, its twist number, surface twist, number of superhelical turns as well as the absolute values of linking number difference, superhelix density, and writhing number increase on binding of Nt. Correspondingly, compared with the naked relaxed DNA a higher linking number (or twist number, or winding number), a higher average duplex winding angle and a lower helical repeat have to be assigned to the relaxed Nt-DNA complex. The various minor groove binders investigated were found to differ considerably in their efficiency to alter the structure of supercoiled DNA.

Lagrangian molecular dynamics using selected conformational degrees of freedom, with application to the pseudorotation dynamics of furanose rings

Using internal conformational degrees of freedom for biopolymers as natural variables, and introducing a Lagrangian dynamics approach, one can simulate time-dependent processes over a much longer time scale than in classical Newtonian molecular dynamics (MD) techniques. Two factors contribute to this: a substantial reduction in the number of degrees of freedom and a very large increase in the size of the time step. We present the Lagrangian equations of motion for repuckering transitions in model furanose (F), ribose (R), and 2'-deoxyribose (dR) ring systems using the pseudorotation phase angle as the single dynamic variable. As in most Lagrangian analyses, the effective masses for the R and dR models are dependent on conformation, and we test the behavior of this variable mass (VM) model. Since the variation in effective mass is small, the VM model is compared with a simplified constant mass (CM) model, which is shown to be an excellent approximation. The equations of motion for the CM and VM models are integrated with the leapfrog and the iterative leapfrog algorithms, respectively. The Lagrangian dynamics approach reduces the number of degrees of freedom from about 40 to 1, and allows the use of time steps on the order of 20 fs, about an order of magnitude greater than is used in conventional MD simulations.

Probability of DNA knotting and the effective diameter of the DNA double helix

During the random cyclization of long polymer chains, knots of different types are formed. We investigated experimentally the distribution of knot types produced by random cyclization of phage P4 DNA via its long cohesive ends. The simplest knots (trefoils) predominated, but more complex knots were also detected. The fraction of knots greatly diminished with decreasing solution Na+ concentration. By comparing these experimental results with computer simulations of knotting probability, we calculated the effective diameter of the DNA double helix. This important excluded-volume parameter is a measure of the electrostatic repulsion between segments of DNA molecules. The calculated effective DNA diameter is a sensitive function of electrolyte concentration and is several times larger than the geometric diameter in solutions of low monovalent cation concentration.

Chirality of DNA supercoiling assigned by scanning force microscopy

Reproducible images of pBR322 plasmid molecules have been recorded by scanning force microscopy under 1-propanol. Most of the plasmids were found in a coiled state. The supercoiled molecules of our samples look like branched or unbranched interwound superhelixes. This is consistent with available electron microscopy data on circular DNA molecules. By applying a stratigraphic analysis which takes advantage of the height information contained in the scanning force microscopy images, it is possible to assign the chirality of the local supercoiling of the individual molecules.

DNA torsional dynamics by multifrequency phase fluorometry

The time decay of the fluorescence polarization anisotropy of calf thymus DNA-ethidium complexes is obtained from measurements with sine-modulated excitation employing the so-called multifrequency phase fluorometry. A torsional dynamics model developed by J. M. Schurr [(1984) Chemical Physics, Vol. 84, pp. 71-96] and translated into the frequency domain is found here to describe accurately DNA-ethidium fluorescence data collected under modulated excitation. At a low dye/DNA ratio (1:400) the value of the DNA torsional constant (alpha = 4.63 +/- 0.2 10(-12) dyne cm) fitting the data agrees very well with the known values of alpha. When the measurements are extended to a higher ethidium/DNA ratio, energy transfer effects between intercalated dyes are observed. A theoretical prediction of the donor and acceptor dye contributions to the fluorescence polarization anisotropy is made here, taking into account also dye-dye distance distributions.

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.

Trefoil knotting revealed by molecular dynamics simulations of supercoiled DNA

Computer simulations of the supercoiling of DNA, largely limited to stochastic search techniques, can offer important information to complement analytical models and experimental data. Through association of an energy function, minimum-energy supercoiled conformations, fluctuations about these states, and interconversions among forms may be sought. In theory, the observation of such large-scale conformational changes is possible, but modeling and numerical considerations limit the picture obtained in practice. A new computational approach is reported that combines an idealized elastic energy model, a compact B-spline representation of circular duplex DNA, and deterministic minimization and molecular dynamics algorithms. A trefoil knotting result, made possible by a large time-step dynamics scheme, is described. The simulated strand passage supports and details a supercoiled-directed knotting mechanism. This process may be associated with collective bending and twisting motions involved in supercoiling propagation and interwound branching. The results also demonstrate the potential effectiveness of the Langevin/implicit-Euler dynamics scheme for studying biomolecular folding and reactions over biologically interesting time scales.

DNA twist as a transcriptional sensor for environmental changes

A variety of reports describe shifts in the environment which cause a corresponding change in the measured linking number of plasmid DNA isolated from bacterial cells. This change in linking number is often attributed to a change in superhelical density. This, coupled with the observation that transcription is often dependent upon the superhelical density of the DNA template seen in vitro, has led to the suggestion that superhelical density may control expression of certain genes. However, since many environmental changes could, in principle, influence DNA twist itself, then the measured differences in linking number, delta Lk, may simply be a consequence of variation in twist according to the relationship delta Lk = delta Tw + delta Wr, where delta Tw and delta Wr are changes in twist and writhe, respectively. In fact, we show that when an environmental change causes a change in the helical pitch of the DNA, and if the superhelical density of DNA is regulated to remain constant according to the homeostatic model of Menzel and Gellert, then delta Lk approximately delta Tw. We have found that there are a number of published reports describing variation in promoter activity as a function of linking number that can be explained by considering twist. We suggest that there are classes of sigma 70 promoters whose ability to be recognized by RNA polymerase is exquisitely sensitive to the relative orientation of the -35 and -10 regions, and environmental conditions can control this relative orientation by changing DNA twist.(ABSTRACT TRUNCATED AT 250 WORDS)

Supercoiled DNA energetics and dynamics by computer simulation

A new formulation is presented for investigating supercoiled DNA configurations by deterministic techniques. Thus far, the computational difficulties involved in applying deterministic methods to supercoiled DNA studies have generally limited computer simulations to stochastic approaches. While stochastic methods, such as simulated annealing and Metropolis-Monte Carlo sampling, are successful at generating a large number of configurations and estimating thermodynamic properties of topoisomer ensembles, deterministic methods offer an accurate characterization of the minima and a systematic following of their dynamics. To make this feasible, we model circular duplex DNA compactly by a B-spline ribbon-like model in terms of a small number of control vertices. We associate an elastic deformation energy composed of bending and twisting integrals and represent intrachain contact by a 6-12 Lennard Jones potential. The latter is parameterized to yield an energy minimum at the observed DNA-helix diameter inclusive of a hydration shell. A penalty term to ensure fixed contour length is also included. First and second partial derivatives of the energy function have been derived by using various mathematical simplifications. First derivatives are essential for Newton-type minimization as well as molecular dynamics, and partial second-derivative information can significantly accelerate minimization convergence through preconditioning. Here we apply a new large-scale truncated-Newton algorithm for minimization and a Langevin/implicit-Euler scheme for molecular dynamics. Our truncated-Newton method exploits the separability of potential energy functions into terms of differing complexity. It relies on a preconditioned conjugate gradient method that is efficient for large-scale problems to solve approximately for the search direction at every step. Our dynamics algorithm is numerically stable over large time steps. It also introduces a frequency-discriminating mechanism so that vibrational modes with frequencies greater than a chosen cutoff frequency are essentially frozen by the method. With these tools, we rapidly identify corresponding circular and interwound energy minima for small DNA rings for a series of imposed linking-number differences. These structures are consistent with available electron microscopy data. The energetic exchange of stability between the circle and the figure-8, in very good agreement with analytical results, is also detailed. Molecular dynamics trajectories at 100 femtosecond time steps then reveal the rapid folding of the unstable circular state into supercoiled forms. Significant bending and twisting motions of the interwound structures are also observed. Such information may be useful for understanding transition states along the folding pathway and the role of enzymes that regulate supercoiling.(ABSTRACT TRUNCATED AT 400 WORDS)

Torsional rigidity of positively and negatively supercoiled DNA

Time-correlated single-photon counting of intercalated ethidium bromide was used to measure the torsion constants of positively supercoiled, relaxed, and negatively supercoiled pBR322 DNA, which range in superhelix density from +0.042 to -0.123. DNA behaves as coupled, nonlinear torsional pendulums under superhelical stress, and the anharmonic term in the Hamiltonian is approximately 15 percent for root-mean-square fluctuations in twist at room temperature. At the level of secondary structure, positively supercoiled DNA is significantly more flexible than negatively supercoiled DNA. These results exclude certain models that account for differential binding affinity of proteins to positively and negatively supercoiled DNA.