A theory of thermal fluctuations in DNA miniplasmids

Normal-Mode Analysis of Circular DNA at the Base-Pair Level. 2. Large-Scale Configurational Transformation of a Naturally Curved Molecule

Fine structural and energetic details embedded in the DNA base sequence, such as intrinsic curvature, are important to the packaging and processing of the genetic material. Here we investigate the internal dynamics of a 200 bp closed circular molecule with natural curvature using a newly developed normal-mode treatment of DNA in terms of neighboring base-pair "step" parameters. The intrinsic curvature of the DNA is described by a 10 bp repeating pattern of bending distortions at successive base-pair steps. We vary the degree of intrinsic curvature and the superhelical stress on the molecule and consider the normal-mode fluctuations of both the circle and the stable figure-8 configuration under conditions where the energies of the two states are similar. To extract the properties due solely to curvature, we ignore other important features of the double helix, such as the extensibility of the chain, the anisotropy of local bending, and the coupling of step parameters. We compare the computed normal modes of the curved DNA model with the corresponding dynamical features of a covalently closed duplex of the same chain length constructed from naturally straight DNA and with the theoretically predicted dynamical properties of a naturally circular, inextensible elastic rod, i.e., an O-ring. The cyclic molecules with intrinsic curvature are found to be more deformable under superhelical stress than rings formed from naturally straight DNA. As superhelical stress is accumulated in the DNA, the frequency, i.e., energy, of the dominant bending mode decreases in value, and if the imposed stress is sufficiently large, a global configurational rearrangement of the circle to the figure-8 form takes place. We combine energy minimization with normal-mode calculations of the two states to decipher the configurational pathway between the two states. We also describe and make use of a general analytical treatment of the thermal fluctuations of an elastic rod to characterize the motions of the minicircle as a whole from knowledge of the full set of normal modes. The remarkable agreement between computed and theoretically predicted values of the average deviation and dispersion of the writhe of the circular configuration adds to the reliability in the computational approach. Application of the new formalism to the computed modes of the figure-8 provides insights into macromolecular motions which are beyond the scope of current theoretical treatments.

Normal-Mode Analysis of Circular DNA at the Base-Pair Level. 1. Comparison of Computed Motions with the Predicted Behavior of an Ideal Elastic Rod

We have extended a newly developed approach to study the low-frequency normal modes of mesoscopic fragments of linear DNA in order to investigate the dynamics of closed circular molecules of comparable size, i.e., a few hundred base pairs. We have added restraint energy terms and a global minimization step to treat the more complicated, spatially constrained duplex in terms of the intrinsic conformation and flexibility of the constituent base-pair "step" parameters. Initial application of the methodology to the normal modes of an ideal closed circular DNA molecule [Formula: see text] which is naturally straight in its relaxed open linear state, inextensible, and capable of isotropic bending and independent twisting at the base-pair level [Formula: see text] matches theoretical predictions of elastic rod dynamics. The energy-optimized closed circular states and the types of low frequency motions follow expected behavior, with (1) uniform twist density and uniform energy density in the minimum energy state; (2) a near-zero frequency torsional mode with "free" rotation about the global helical axis; (3) higher-order torsional modes accompanied by global rocking motions and pure in-plane and out-of-plane bending motions in the torsionally relaxed circle; and (4) mixed modes of bending when the chain is supercoiled (over- or undertwisted). Furthermore, the computed changes in normal-mode frequencies with imposed supercoiling or with variation of chain length are virtually identical to theoretically predicted values.

Effect of the boundary conditions and influence of the rotational inertia on the vibrational modes of an elastic ring

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.

Total positive curvature of circular DNA

The properties of double-stranded DNA and other chiral molecules depend on the local geometry, i.e., on curvature and torsion, yet the paths of closed chain molecules are globally restricted by topology. When both of these characteristics are to be incorporated in the description of circular chain molecules, e.g., plasmids, it is shown to have implications for the total positive curvature integral. For small circular micro-DNAs it follows as a consequence of Fenchel's inequality that there must exist a minimum length for the circular plasmids to be double stranded. It also follows that all circular micro-DNAs longer than the minimum length must be concave, a result that is consistent with typical atomic force microscopy images of plasmids. Predictions for the total positive curvature of circular micro-DNAs are given as a function of length, and comparisons with circular DNAs from the literature are presented.

Multiscale modeling of nucleic acids: insights into DNA flexibility

The elastic rod theory is used together with all-atom normal mode analysis in implicit solvent to characterize the mechanical flexibility of duplex DNA. The bending, twisting, stretching rigidities extracted from all-atom simulations (on linear duplexes from 60 to 150 base pairs in length and from 94-bp minicircles) are in reasonable agreement with experimental results. We focus on salt concentration and sequence effects on the overall flexibility. Bending persistence lengths are about 20% higher than most experimental estimates, but the transition from low-salt to high-salt behavior is reproduced well, as is the dependence of the stretching modulus on salt (which is opposite to that of bending). CTG and CGG trinucleotide repeats, responsible for several degenerative disorders, are found to be more flexible than random DNA, in agreement with several recent studies, whereas poly(dA).poly(dT) is the stiffest sequence we have encountered. The results suggest that current all-atom potentials, which were parameterized on small molecules and short oligonucleotides, also provide a useful description of duplex DNA at much longer length scales.

EcoR124I: from plasmid-encoded restriction-modification system to nanodevice

SUMMARY

Plasmid R124 was first described in 1972 as being a new member of incompatibility group IncFIV, yet early physical investigations of plasmid DNA showed that this type of classification was more complex than first imagined. Throughout the history of the study of this plasmid, there have been many unexpected observations. Therefore, in this review, we describe the history of our understanding of this plasmid and the type I restriction-modification (R-M) system that it encodes, which will allow an opportunity to correct errors, or misunderstandings, that have arisen in the literature. We also describe the characterization of the R-M enzyme EcoR124I and describe the unusual properties of both type I R-M enzymes and EcoR124I in particular. As we approached the 21st century, we began to see the potential of the EcoR124I R-M enzyme as a useful molecular motor, and this leads to a description of recent work that has shown that the R-M enzyme can be used as a nanoactuator. Therefore, this is a history that takes us from a plasmid isolated from (presumably) an infected source to the potential use of the plasmid-encoded R-M enzyme in bionanotechnology.

Mapping the phase diagram of the writhe of DNA nanocircles using atomistic molecular dynamics simulations

We have investigated the effects of duplex length, sequence, salt concentration and superhelical density on the conformation of DNA nanocircles containing up to 178 base pairs using atomistic molecular dynamics simulation. These calculations reveal that the partitioning of twist and writhe is governed by a delicate balance of competing energetic terms. We have identified conditions which favour circular, positively or negatively writhed and denatured DNA conformations. Our simulations show that AT-rich DNA is more prone to denaturation when subjected to torsional stress than the corresponding GC containing circles. In contrast to the behaviour expected for a simple elastic rod, there is a distinct asymmetry in the behaviour of over and under-wound DNA nanocircles. The most biologically relevant negatively writhed state is more elusive than the corresponding positively writhed conformation, and is only observed for larger circles under conditions of high electrostatic screening. The simulation results have been summarised by plotting a phase diagram describing the various conformational states of nanocircles over the range of circle sizes and experimental conditions explored during the study. The changes in DNA structure that accompany supercoiling suggest a number of mechanisms whereby changes in DNA topology in vivo might be used to influence gene expression.

Conformational Analysis of Stiff Chiral Polymers with End-Constraints

We present a Lie-group-theoretic method for the kinematic and dynamic analysis of chiral semi-flexible polymers with end constraints. The first is to determine the minimum energy conformations of semi-flexible polymers with end constraints, and the second is to perform normal mode analysis based on the determined minimum energy conformations. In this paper, we use concepts from the theory of Lie groups and principles of variational calculus to model such polymers as inextensible or extensible chiral elastic rods with coupling between twisting and bending stiffnesses, and/or between twisting and extension stiffnesses. This method is general enough to include any stiffness and chirality parameters in the context of elastic filament models with the quadratic elastic potential energy function. As an application of this formulation, the analysis of DNA conformations is discussed. We demonstrate our method with examples of DNA conformations in which topological properties such as writhe, twist, and linking number are calculated from the results of the proposed method. Given these minimum energy conformations, we describe how to perform the normal mode analysis. The results presented here build both on recent experimental work in which DNA mechanical properties have been measured, and theoretical work in which the mechanics of non-chiral elastic rods has been studied.

Thermal fluctuations of small rings of intrinsically helical DNA treated like an elastic rod

It is shown that the writhe distribution in a collection of thermally fluctuating rings formed from intrinsically helical DNA treated like an elastic rod can be determined completely from a knowledge of the dependence of the configuration integral on the intrinsic geometric torsion of the rods. When the length of the rods is of the order of the persistence length or less, the configuration integral can be expressed in terms of the frequencies of the normal modes of the rings, and so these torsion-dependent frequencies then become the quantities of interest. The nature of the normal modes of rings formed from intrinsically helical elastic rods, as deduced from the equations of motion in the Kirchhoff-Clebsch theory of the dynamics of elastic rods, is described. Examples of computed writhe distributions are given. The observed broadening of the writhe distribution as the intrinsic curvature of the rods is increased is shown to be related to the stability properties of the circular equilibrium configuration of the elastic rods. These results thus imply that thermal fluctuations in small rings of intrinsically curved DNA will be more pronounced than in rings of the same size formed from intrinsically straight DNA.

Sequence-dependent nucleosome structural and dynamic polymorphism. Potential involvement of histone H2B N-terminal tail proximal domain

Relaxation of nucleosomes on an homologous series (pBR) of ca 350-370 bp DNA minicircles originating from plasmid pBR322 was recently used as a tool to study their structure and dynamics. These nucleosomes thermally fluctuated between three distinct DNA conformations within a histone N-terminal tail-modulated equilibrium: one conformation was canonical, with 1.75 turn wrapping and negatively crossed entering and exiting DNAs; another was also "closed", but with these DNAs positively crossed; and the third was "open", with a lower than 1.5 turn wrapping and uncrossed DNAs. In this work, a new minicircle series (5S) of similar size was used, which contained the 5S nucleosome positioning sequence. Results showed that DNA in pBR nucleosomes was untwisted by approximately 0.2 turn relative to 5S nucleosomes, which DNase I footprinting confirmed in revealing a approximately 1 bp untwisting at each of the two dyad-distal sites where H2B N-terminal tails pass between the two gyres. In contrast, both nucleosomes showed untwistings at the dyad-proximal sites, i.e. on the other gyre, which were also observed in the high-resolution crystal structure. 5S nucleosomes also differ with respect to their dynamics: they hardly accessed the positively crossed conformation, but had an easier access to the negatively crossed conformation. Simulation showed that such reverse effects on the conformational free energies could be simply achieved by slightly altering the trajectories of entering and exiting DNAs. We propose that this is accomplished by H2B tail untwisting at the distal sites through action at a distance ( approximately 20 bp) on H3-tail interactions with the small groove at the nucleosome entry-exit. These results may help to gain a first glimpse into the two perhaps most intriguing features of the high-resolution structure: the alignment of the grooves on the two gyres and the passage of H2B and H3 N-terminal tails between them.

Fluctuating elastic rings: statics and dynamics

We study the effects of thermal fluctuations on elastic rings. Analytical expressions are derived for correlation functions of Euler angles, mean-square distance between points on the ring contour, radius of gyration, and probability distribution of writhe fluctuations. Since fluctuation amplitudes diverge in the limit of vanishing twist rigidity, twist elasticity is essential for the description of fluctuating rings. We discover a crossover from a small scale regime in which the filament behaves as a straight rod, to a large scale regime in which spontaneous curvature is important and twist rigidity affects the spatial configurations of the ring. The fluctuation-dissipation relation between correlation functions of Euler angles and response functions, is used to study the deformation of the ring by external forces. The effects of inertia and dissipation on the relaxation of temporal correlations of writhe fluctuations, are analyzed using Langevin dynamics.

DNA rings with multiple energy minima

Within the context of DNA rings, we analyze the relationship between intrinsic shape and the existence of multiple stable equilibria, either nicked or cyclized with the same link. A simple test, based on a perturbation expansion of symmetry breaking within a continuum elastic rod model, provides good predictions of the occurrence of such multiple equilibria. The reliability of these predictions is verified by direct computation of nicked and cyclized equilibria for several thousand DNA minicircles with lengths of 200 and 900 bp. Furthermore, our computations of equilibria for nicked rings predict properties of the equilibrium distribution of link, as calculated by much more computationally intensive Monte Carlo simulations.

Topological effects of the TATA box binding protein on minicircle DNA and a possible thermodynamic linkage to chromatin remodeling

DNA ring closure experiments on short restriction fragments ( approximately 160 bp) bound by the TATA box binding protein (TBP) have demonstrated the formation of negative topoisomers, consistent with crystallographically observed TBP-induced DNA untwisting but in contrast to most previous results on topological effects in plasmid DNA. The difference may be due to the high free energy cost of substantial writhe in minicircles. A speculative mechanism for the loss of TBP-induced writhe suggests that TBP is capable of inducing DeltaTw between 0 and -0.3 in minicircles, via loss of out-of-plane bending upon retraction of intercalating Phe stirrups, and that TBP can thus act as a "supercoil shock absorber". The proposed biological relevance of these observations is that they may model the behavior of DNA in constrained chromatin environments. Irrespective of the detailed mechanism of TBP-induced supercoiling, its existence suggests that chromatin remodeling and enhanced TBP binding are thermodynamically linked. Remodeling ATPases or histone acetylases release some of the negative supercoiling previously restrained by the nucleosome. When TBP takes up the supercoiling, its binding should be enhanced transiently until the unrestrained supercoiling is removed by diffusion or topoisomerases. The effect is predicted to be independent of local remodeling-induced changes in TATA box accessibility.

Nucleosome dynamics. VI. Histone tail regulation of tetrasome chiral transition. A relaxation study of tetrasomes on DNA minicircles

We have recently described the relaxation of mononucleosomes on an homologous series of 351-366 bp DNA minicircles, as a tool to study nucleosome structure and dynamics in vitro. Nucleosomes were found to have a tail-regulated access to three distinct DNA conformations, depending on the crossing between the entering and exiting DNAs, and its polarity. This approach was now used to explore tetrasome chiral transition, and the influence of the histone tails. The data confirmed the existence of two states, with linking number differences DeltaLk(t)=-0.74(+/-0.01) and +0.51(+/-0.06). As expected, the particle free energy is higher in the right-handed state (DeltaG(t)=1.9(+/-0.I) kT), but it decreased (to 1.3(+/-0.1) kT) upon histone acetylation and the addition of phosphate, a potent tail destabilizer. Removal of the tails with trypsin further decreased DeltaG(t) (to 0.6 kT), and also induced a loss of supercoiling in both states, to DeltaLk(t)=-0.64(+/-0.03) and +0. 35(+/-0.05). The loop end-conditions, and hence the parameters of the DNA superhelix, were then calculated for both states using the explicit solutions to the equations of the mechanical equilibrium in the theory of elastic rod model for DNA. Whereas the pitch of the DNA superhelix may be approximately equal and opposite in the two conformations, its radius (r) was 20% larger in the right-handed conformation, confirming previous observations by electron microscopy of a tetrasome lateral opening in that conformation. The above supercoiling losses were found to reflect a further 3 % increase in r (to 23 %) upon removal of the tails in the right-handed conformation, and a 14 % increase in the left-handed conformation. The use of composite tetramers with one histone tail intact and the other removed showed these effects to be essentially due to the H3 tails. Altogether, these results show that the H3 tails oppose the tetrasome opening which is expected to be required to relieve the clash between the entering and exiting DNAs in the course of the transition, but which also appears to be intrinsic to the protein reorientation mechanism. We propose that the block against opening results from the H3 tails intercalating into the small groove of the double helix at +/-10 bp from the dyad, and acting as wedges against local DNA straightening. The tails (especially H3) may therefore regulate tetrasome chiral transition in vivo.

Elastic stability of DNA configurations. I. General theory

Results are presented in the theory of the elastic rod model for DNA, among which are criteria enabling one to determine whether a calculated equilibrium configuration of a DNA segment is stable in the sense that it gives a local minimum to the sum of the segment's elastic energy and the potential of forces acting on it. The derived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring end conditions. Their utility is illustrated with an example from the theory of configurations of the extranucleosomal loop of a DNA miniplasmid in a mononucleosome, with emphasis placed on the influence that nicking and ligation on one hand, and changes in the ratio of elastic coefficients on the other, have on the stability of equilibrium configurations. In that example, the configurations studied are calculated using an extension of the method of explicit solutions to cases in which the elastic rod modeling a DNA segment is considered impenetrable, and hence excluded volume effects and forces arising from self-contact are taken into account.

Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact

Configurations of protein-free DNA miniplasmids are calculated with the effects of impenetrability and self-contact forces taken into account by using exact solutions of Kirchhoff's equations of equilibrium for elastic rods of circular cross section. Bifurcation diagrams are presented as graphs of excess link, DeltaL, versus writhe, W, and the stability criteria derived in paper I of this series are employed in a search for regions of such diagrams that correspond to configurations that are stable, in the sense that they give local minima to elastic energy. Primary bifurcation branches that originate at circular configurations are composed of configurations with D(m) symmetry (m=2,3,...). Among the results obtained are the following. (i) There are configurations with C2 symmetry forming secondary bifurcation branches which emerge from the primary branch with m=3, and bifurcation of such secondary branches gives rise to tertiary branches of configurations without symmetry. (ii) Whether or not self-contact occurs, a noncircular configuration in the primary branch with m=2, called branch alpha, is stable when for it the derivative dDeltaL/dW, computed along that branch, is strictly positive. (iii) For configurations not in alpha, the condition dDeltaL/dW>0 is not sufficient for stability; in fact, each nonplanar contact-free configuration that is in a branch other than alpha is unstable. A rule relating the number of points of self-contact and the occurrence of intervals of such contact to the magnitude of DeltaL, which in paper I was found to hold for segments of DNA subject to strong anchoring end conditions, is here observed to hold for computed configurations of protein-free miniplasmids.

The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes

Explicit solutions to the equations of equilibrium in the theory of the elastic rod model for DNA are employed to develop a procedure for finding the configuration that minimizes the elastic energy of a minicircle in a mononucleosome with specified values of the minicircle size N in base pairs, the extent w of wrapping of DNA about the histone core particle, the helical repeat h(0)b of the bound DNA, and the linking number Lk of the minicircle. The procedure permits a determination of the set Y(N, w, h(0)b) of integral values of Lk for which the minimum energy configuration does not involve self-contact, and graphs of writhe versus w are presented for such values of Lk. For the range of N of interest here, 330 < N < 370, the set Y(N, w, h(0)b) is of primary importance: when Lk is not in Y(N, w, h(0)b), the configurations compatible with Lk have elastic energies high enough to preclude the occurrence of an observable concentration of topoisomer Lk in an equilibrium distribution of topoisomers. Equilibrium distributions of Lk, calculated by setting differences in the free energy of the extranucleosomal loop equal to differences in equilibrium elastic energy, are found to be very close to Gaussian when computed under the assumption that w is fixed, but far from Gaussian when it is assumed that w fluctuates between two values. The theoretical results given suggest a method by which one may calculate DNA-histone binding energies from measured equilibrium distributions of Lk.