DIMACS/PMMB/MBBC Workshop on DNA Topology II

April 3-4, 1997
Rutgers University

Organizers:

Wilma Olson (Chair), Rutgers University, olson@rutchem.rutgers.edu

Bernard Coleman, Rutgers University, bcoleman@stokes.rutgers.edu

Tamar Schlick, NYU, schlick@nyu.ed

Irwin Tobias, Rutgers University, tobias@rutchem.rutgers.edu

Abstracts

Superhelical stresses imposed on DNA can destabilize the B-form duplex, causing local strand separations to occur at sites where its thermodynamic stability is lowest. This effect is important because strand separation is an obligatory step in many essential biological processes, including the initiation of transcription and of replication. So the locations and conditions under which strand separations occur must be stringently controlled in vivo.

Theoretical methods will be described for predicting the locations and extents of destabilization in superhelical DNA sequences. The predictions of these analyses are in quantitatively precise agreement with experimental results. This enables their use to predict the destabilization experienced by other sequences on which experiments have not been performed.

The sites of predicted destabilizations in genomic sequences are found to be closely associated with specific classes of regulatory regions. The promoters of all genes in the E. coli SOS system are destabilized at their repressor binding sites. This is not an attribute of the sequence alone, highly homologous sequences present elsewhere on the E. coli genome are not destabilized. The sequences encoding yeast genes exhibit a tripartite pattern, in which the 3'- and 5'-regions flanking the coding sequence are destabilized, but the coding region itself is not. Putative replication origins exhibit characteristic destabilization patterns, as do two classes of DNA matrix binding sites. In two cases where the experiments have been performed, sites in transcriptionally active molecules where duplex destabilization was predicted have been shown to be actually separated in vivo. A range of theoretical predictions will be described, and their implications regarding the possible involvement of DNA strand separation in specific mechanisms of activity will be discussed.

Force-extension (F-x) relationships have been measured for single molecules of DNA under a variety of buffer conditions, using an optical trapping interferometer modified to incorporate feedback control. One end of a single DNA molecule is fixed to a cover glass surface by means of a stalled RNA polymerase complex. The other end is linked to a microscopic bead, which is captured and held in an optical trap. The DNA is stretched by moving the cover glass with respect to the trap using a piezo-driven stage, while the position of the bead is recorded at nanometer-scale resolution. An electronic feedback circuit is activated to prevent bead movement beyond a preset clamping point by modulating the light intensity, altering the strap stiffness dynamically. This arrangement permits rapid determination of the F-x relationship for individual DNA molecules as short as ca. 1 mm with unprecedented accuracy, subjected to both low and high loads: complete data sets are acquired in under a minute. Experimental F-x relationships can be fit over much of their range by entropic elasticity theories based on worm-like chain models. Fits yield a persistence length of ca. 47 nm in a buffer containing 10 mM sodium ions. Multivalent cations, such as divalent magnesium or spermidine, reduce the persistence length to ca. 40 nm. Although multivalent ions shield most of the negative charges on the DNA backbone, they do not further reduce the persistence length significantly, suggesting that the intrinsic persistence length remains close to 40 nm. An elasticity theory incorporating both enthalpic and entropic contributions to stiffness fit the experimental results extremely well throughout the full range of extensions and return an elastic modulus of ca. 1100 pN.

Many polycylic aromatic hydrocarbons (PAH), environmental pollutants that are mutagenic and tumorigenic, are activated to mirror image pairs of diol epoxides. Each member of the pair can react with DNA to form a covalent reaction product known as an adduct. Such adducts can cause mutations when the DNA replicates, which may finally lead to tumors. However, a fascinating observation has puzzled researchers for decades: even though both members of the pair can react with DNA to the same site, the tumorigenicity of one member is always much greater than that of the other. A structural distinction has long been sought as the underlying origin to the biological difference. Benzo[a]pyrene, present in automobile exhaust and tobacco smoke, is the prototype PAH which manifests this intriguing phenomenon. We have employed potential energy searches with the torsion angle space molecular mechanics program DUPLEX to compute structural distinctions between DNA adducts of tumorigenic and non-tumorigenic activated benzo[a]pyrene. We have also carried out a molecular dynamics simulation with AMBER for the major adduct of the tumorigenic member of the pair in a base sequence context that models an arm of a DNA replication fork; this simulation includes aqueous solvent and a mammalian replication enzyme. These studies have correctly predicted long-sought structural underpinnings behind the strikingly different biological effects of a mirror image pair of activated benzo[a]pyrenes, and have suggested structural distortions that could produce mutations when the DNA replicates, which ultimately could cause cancer.

The "writhing number" of a curve in 3-space, introduced by Calugareanu (1959-61) and named by Fuller (1971), is the standard measure of the extent to which the curve wraps and coils around itself. It is important in molecular biology because, for example, the writhing of DNA is closely connected with the interlinking of the two strands of the double helix...an interlinking which must be overcome by the action of cut-and-paste enzymes during replication and reproduction.

The "helicity" of a vector field defined on some region in 3-space, introduced by Woltjer (1958) and named by Moffatt (1969), is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics.

In a series of papers, we provide upper bounds for the writhing number of a curve in terms of its length and thickness, upper bounds for the helicity of a vector field in terms of its energy and the geometry of its domain, and descriptions of the patterns of maximum writhing and helicity.

In the first paper (available at the conference...just ask), we introduce the subject, describe computer experiments which search for and discover patterns of maximum writhing and helicity, derive rough upper bounds for both these quantities in the form of a 4/3 power growth law, develop analytical tools for obtaining sharp upper bounds for helicity and for revealing the vector fields which attain them, and apply these methods in one specific case, the flat solid torus, which will then serve as a model for the following paper.

Eukaryotic genes are assembled into nucleosomes even when active in transcription. Nucleosomes are highly compact, containing nearly two turns of DNA (about 146 bp) tightly wrapped around a central core histone octamer. Moreover, nucleosomes are very stable structures. How does RNA polymerase II transcribe through such a structure? The polymerase is larger than the nucleosome, it must rotate once approximately every 10 bp with respect to the DNA helix as it transcribes and, as it does so, it must drag the nascent transcript around with it, probably with other proteins attached. Clearly, major disruption of the nucleosome is required if transcription of nucleosomal DNA is to occur.

Two types of models have been proposed: (1) displacement/loss of the histones to allow polymerase free passage; (2) unfolding of the nucleosome to ease steric problems for the polymerase. We designed an experiment to distinguish between these two models: a single nucleosome was placed at a unique site on a plasmid and then transcribed using a phage polymerase 2E. The nucleosome was displaced and re-formed elsewhere on the plasmid as a result of transcription.

In the next series of experiments, we used a very short DNA fragment with a precisely positioned nucleosome as a template for transcription. Transcription resulted in the transfer of the nucleosomal histones from one end of the template to the other without leaving the DNA. This was a surprising result because the polymerase and the nucleosome are traveling in opposite directions and neither of them dissociate from DNA! However, simple model building suggested a "spooling" mechanism for transfer of the nucleosome: The polymerase approaches the nucleosome, causing displacement of proximal DNA from the surface of the histone octamer. The polymerase continues into the nucleosome and then DNA behind it binds to the exposed DNA-binding residues on the histone surface. Thus, a DNA loop is formed within the nucleosome to which the transcribing polymerase is bound. As polymerase continues, the DNA ahead uncoils from the octamer and the DNA behind coils around it, eventually resulting in transfer of the octamer behind the polymerase and allowing it to complete the transcript. Some preliminary evidence for this loop intermediate will be presented.

A brief survey will be given of some recent results in the theory of the elastic rod model of DNA, with the emphasis on the use of exact and explicit solutions to the equations of mechanical equilibrium of Kirchhoff's theory of elastic rods to calculate the dependence of the tertiary structure of an otherwise free segment of DNA on conditions imposed at its end points.

The theory of the elastic rod model, when applicable, implies that in appropriate circumstances small changes in end conditions can cause a nearly planar loop of DNA to undergo a continuous and reversible transition that can be described as a 180 degree rotation of the loop about its axis of symmetry, with the effect of transforming the loop from an uncrossed to a singly crossed structure [1]. Explicit solutions in the theory give, in addition to the configurations of the duplex axis during such a "loop flip" [1], expressions relating the sequential location of the points of crossing and the angles of crossing to the end conditions that gave rise to the crossed structure [2].

Among the protein aggregates that appear to impose end conditions of the type that have been treated in the theory are (i) the type II topoisomerase, DNA gyrase, and (ii) the histone octamer. Applications to these two cases will be discussed.

(i) In the discussion of the gyrase it will be shown that, by examining sequences of configurations that result from small changes in tangent angles at points where a DNA molecule departs from its region of contact with a bending protein [1], one can visualize hypothetical mechanisms for the activity of topoisomerases that require neither the controlled passage of a segment of DNA through a gap resulting from the severing of two strands, nor the rotation of a nicked DNA molecule at fixed writhe about its intact strand.

(ii) In the theory of mononucleosomes on DNA minicircles, one faces the problem of calculating the writhe of the duplex axis of a minicircle and both the configuration and elastic energy of the extranucleosomal loop (i.e., the part of the minicircle not bound to the histone) for specified values of three quantities: the linking difference of the minicircle, the twist density of nucleosomal DNA, and the amount of wrapping (in turns) of the bound DNA around the histone core particle. Recent work [3] on the use of explicit solutions to solve the problem will be presented.

[1] I. Tobias, B. D. Coleman, and W. Olson, J. Chem. Phys., 101, 10990-10996 (1994).

[2] B. D. Coleman, I. Tobias, and D. Swigon, J. Chem. Phys., 103, 9101-9109 (1995).

[3] D. Swigon, B. D. Coleman, and I. Tobias, manuscript in preparation.

Biological enzymes such as topoisomerases and recombinases perform topological operations on circular DNA resulting in the production of knots and/or links. Topoisomerases are enzymes that can change crossings of a knot or link by passing one strand of DNA through another. This enzyme action can be studied using a generalization of the unknotting number: a strand passage metric on knots (links) [M] in which one computes the minimum number of strand passages necessary to inter-convert a pair of knots (links). Another distance of biological interest is the positive distance between knots (links) where only changes from a positive to a negative crossing are allowed in order to study chirality issues. Recombination metrics can be used to study recombinases which are enzymes that perform smoothings by strand exchange resulting in an exchange of genetic information. The main tools used in calculating these metrics include signature, linking number, and Dehn surgery on the double branched cover of the knot (link).

We have carried out Monte Carlo/simulated annealing and energy minimization studies in order to examine the minimum energy configurations of simplified models of closed circular DNA at various levels of supercoiling as a function of chain length and salt concentration. Both elastic and electrostatic (i.e., modified Debye-Hueckel) energy contributions are considered and a repulsive potential is employed to avoid distant points along the chain from coming into close contact. The net phosphate charges (reduced to account for counterion condensation) are placed at evenly spaced increments along the double-helical axis of the DNA and the solvent is treated as a dielectric continuum. For long DNA chains we employ a fast adaptive multiple method for the computation of the modified Debye-Hueckel energy and its derivatives in both Monte Carlo/simulated annealing and energy minimization codes.

In vivo and in vitro experiments show that regulation of a single gene exhibits highly nonlinear phenomena, such as synergy, threshold and saturation. The initiation of transcription involves activators binding to specific sites on the DNA upstream of the gene and binding of the transcription complex to both the TATA box and to the activators via DNA looping. In this study we construct a mathematical model which predicts the level of transcription, given the concentrations of the transcription factors and activator, and the number, affinity and location on the DNA of the activator binding sites. Protein-protein interactions and DNA-protein interactions are modeled using chemical kinetics. DNA looping plays a critical role in the model. We consider three approaches for incorporating it into the model: Elastic rod theory (static model), Monte-Carlo simulations (semi-dynamic model) and Langevin dynamics (dynamic model).

As has been demonstrated by multiple sequence alignment (Ioshikhes et al., 1996), the main feature of nucleosome DNA sequence patterns is periodic distribution of AA (TT) dinucleotides along the sequence. The periodic distribution of CC (GG) dinucleotides along nucleosome sites has been recently demonstrated as well (Bolshoy, A., 1995). Since the periods of these distributions are the same, one can assume that periodicity would be the main feature of some dinucleotide patterns. For evaluation of the contributions of other dinucleotides to the nucleosome pattern, an original match resonance analysis was applied. In this procedure the sequences were aligned to harmonic distributions of the dinucleotides of interest with a period of 10.3 bp, and the summed output patterns were compared to similar outputs generated for random sequences. By varying period and phase shifts between the periodic distributions we maximized the sum of the amplitudes of the oscillating components in the output dinucleotide distributions, thus, calculating the resonance parameters. In this multidimensional optimization one could use many different ascent strategies to reach the global maximum.

In one procedure the dinucleotides were introduced one after another, according to their effect on the output expressed as standard deviates of the output amplitudes of the dinucleotides from the corresponding reshuffled controls. The AA (TT) dinucleotides were confirmed as substantial contributors. Their relative phase shift and period obtained by this technique are, within the error bars, identical to ones obtained by multiple alignment. Four other dinucleotide complementary pairs were detected as substantial contributors to the nucleosome signal: AC/GT, AG/CT, CC/GG, and GA/TC.

In the second procedure the dinucleotides are introduced one by one according to the difference in the overall outputs caused by a given dinucleotide. The AG/CT, AA/TT, AC/GT, CC/GG, and GA/TC dimers have been confirmed as the main contributors to the nucleosome signal. GC appears as a contributor as well. The set of contributing dinucleotides with their respective phases and amplitudes will be presented as the nucleosome dinucleotide matrix suitable for prediction of nucleosome positions in the sequences.

References:

1. Bolshoy, A., "CC dinucleotides contribute to the bending of DNA in chromatin". Nature Struct. Biol. 2, 446-448 (1995).

2. Ioshikhes, I., Bolshoy, A., Derenshteyn, K., Borodovsky, M., and Trifonov, E.N., "Nucleosome DNA sequence pattern revealed by multiple alignment of experimentally mapped sequences". J. Mol. Biol. 262, 129-139 (1996).

Living cells possess a language of their own called "cell language" [1]. The molecule-based call language has been found to be isomorphic with the sound- and visual signal-based human language with respect to 9 out of 13 design features [2]. Since the linguistic competence of the cell is obviously heritable, it must be encoded in DNA.

F. de Saussure recognized two fundamental aspects of human language--diachronic (i.e., historical) and synchronic (i.e., ahistorical) [3]. The former refers to a set of properties that has evolved over a long history such as lexicon and grammar, while the latter is related to the ability of individuals to generate an infinite number of novel sentences utilizing a finite set of words and grammatical rules.

Cell language also possesses diachronic and synchronic properties encoded in DNA. It is here postulated (1) the diachronic structures of DNA consist of both structural genes encoded in coding DNA and hypothetical "spatiotemporal genes" [4] regulating the evolution of gene expressions in space and time that are thought to be encoded either in noncoding DNA or over the whole DNA molecule, and (2) that the synchronic structures are identifiable with sequence-dependent and conformon-driven folding patterns of the DNA double helix that determine the timing of gene expression. (Conformons are conformational strains of biopolymers that carry both information and energy, such as negative supercoils of DNA.) Thermodyamically speaking, diachronic structures are equilibrium structures, while synchronic structures are dissipative structures extensively studied by Prigogine and his colleagues [5]. It is further postulated that the synchronic structures of DNA have a 1:1 correspondence with IDS's (intracellular dissipative structures), namely intracellular or transmembrane gradients of ions, metabolites, structural proteins, and enzymic activities. Experimental evidence indicated that IDS's are in turn related to cell functions in a 1:1 manner. Therefore, the sequence of events linking DNA to cell functions can be schematized as follows:

Diachronic DNA - Conformons - Synchronic DNA - 1:1 - IDS's - 1:1 - Cell Functions

According to the above scheme, the final form of expression of genetic information is not polypeptides but IDS's, in agreement with the Bhopalator model of the living cell [6]. In addition, the coupling between diachronic and synchronic structures of DNA mediated by conformons not only provides rational mechanisms for linking evolution (which determines diachronic DNA) and function (which is determined by synchronic DNA) on the molecular level but also accounts for individual creativity of living cells within the constraint of their common genome.

The recent attempt by J. Collado-Vides [7,8] to apply the Chomskyan grammatical theory to the analysis of prokaryotic genomic data has dealt primarily with diachronic structures of DNA and did not distinguish between diachronic and synchronic structures on the one hand, nor between equilibrium and dissipative structures on the other, both of which being essential for explaining cell functions in molecular terms.

References:

1. Ji, S., "A cell linguistic analysis of apoptosis". Comments on Toxicology. (1997) in press.

2. Ji, S., "Isomorphism between cell and human languages". (1997) submitted.

3. Culler, J., "Ferdinand de Saussure". Revised Edition. Cornell University Press, Ithaca, NY. 45-57 (1986).

4. Ji, S., "Biocybernetics: a machine theory of biology". Molecular Theories of Cell Life and Death. Rutgers University Press, New Brunswick NJ 177 (1991)

5. Prigogine, I., "From Being to Becoming". W.H. Freeman and Co., San Francisco, CA (1980).

6. Ji, S., "The bhopalator--a molecular model of the living cell based on the concepts of conformons and dissipative structures." J. Theoret. Biol. 116, 399-426 (1985).

7. Collado-Vides, J., "A linguistic representation of the regulation of transcription initiation". BioSystems 29, 87-104 (1993).

8. Collado-Vides, J., "A linguistic representation of the regulation of transcription initiation." BioSystems 29, 105-128 (1993).

A variety of DNA models use some sort of 'elastic' curve as a basis. We will discuss mathematical and computational issues associated with the kinematics and dynamics of such curves in the context of DNA models, such as the representation of twisted curves, the energetics of twisted curves, and computation of the dynamics of twisted curves.

The enzymatically "activated" unwound and stretched forms of DNA involved in transcription and recombination are high energy states in the absence of proteins. Here the distortions of the double helix are thought to be much more severe than those observed in regulatory protein-DNA crystal complexes; the Watson-Crick base pairs are expected to be either completely or partially disrupted during biochemical events. As a first step towards visualizing these high energy conformations, we have undertaken systematic all-atom potential energy studies of stretching and compression of the A- and B-DNA double helices. The simple model that we use not only reproduces the compressed double helical form with highly inclined base pairs observed in protein-DNA complexes but also suggests the existence of different DNA families that become energetically favorable in highly stretched forms including an "activated" highly stretched unwound form.

Single- and double-stranded DNAs exhibit a rich variety of knotted and linked structures that are classic examples of topologically significant objects. Topological chirality and achirality of DNA knots and links to some extent determine molecular conformation types that in turn have important effects on the molecular biology of DNA.

Novel symmetry properties of topological achiral knots with up to 12 crossings are presented. By use of an algorithm that involves the development of appropriate vertex-bicolored knot graphs, rigidly achiral presentations have been found for all topologically achiral invertible prime knots with up to 10 crossings and for a selected number of such knots with 12 crossings. The 1994th member in the 12-crossing knot family provides the first example of a topological achiral prime knot whose rigidly achiral presentation is also a minimal diagram.

Empirical and analytical methods employed in the detection of topological chirality and achirality in oriented and non-oriented links are examined. U-polynomials of non-oriented links are modified for use in the detection of topological chirality. By use of this method, all but eight non-oriented links with up to four components and nine crossings are proven to be topologically chiral. The topological chirality of certain Borromean links is similarly proven. The above methods have been applied to known DNA knots and links. The recently synthesized single-stranded DNA Borromean link (this quotation is permitted by Professor Nadrian Seeman) is shown to be topologically chiral.

Traditionally several obstacles are associated with locating minimum elastic energy configurations of partially geometrically-constrained circular DNA chains, such as minichromosomes, using a stochastic search procedure. Presently, these difficulties are eliminated by constructing chains of alternating linker and nucleosomal segments in which short lengths of spacer DNA are modeled as having linear trajectories, while regions of DNA bound into nucleosomes are defined by geometrically fixed supercoils. Using trigonometric equations which relate internal molecular parameters to helical parameters for polymers, derived by Miyazawa, circular DNA chains (i.e., single turn helices of zero pitch) are located directly. Sample structures presented here suggest a variety of situations in which large chromatin domains undergo sudden configurational shifts through small changes in linker length and in the shape of the histone octamer, suggesting possible mechanisms for activation of genetic domains. Also, the amount of DNA bound into the nucleosome complex appears to define the folding motif for the minichromosome, relatively independent of the number of nucleosome repeats in the chain. Finally, low writhe structures have been located which support particular resolutions to the linking number problem of minichromosomes and account for the reduced change in linking number associated with nucleosome formation using hyperacetlyated histone cores.

The symmetries of the DNA double helix require a new term in its linear response to stress: the coupling between twist and stretch. Recent experiments with torsionally-constrained single molecules give the first direct measurement of this important material parameter. We extract its value from a recent experiment of Strick et al. and find rough agreement with an independent experimental estimate recently given by Marko. We also present a very simple microscopic theory predicting a value comparable to the one observed.

Our laboratory has recently reported (1) the ability of the histone (H3-H4)*(H3-H4) tetramer of the nucleosome to form monomeric particles (MT) on both negatively and positively supercoiled DNA minicircles, while relaxed minicircles were a poor substrate. This is in contrast to the histone (H2A-H2B-H3-H4)*(H2A-H2B-H3-H4) octamer which forms mononucleosomes only on negatively supercoiled and, more poorly, on relaxed minicircles (2). Consistently, relaxation with topoisomerase I led to equilibria containing only negatively supercoiled topoisomers in the case of mononucleosomes (3), and both negatively and positively supercoiled topoisomers in the case of MT particles. This data, together with the known left-handed superhelical conformation of the tetramer within the octamer (4), pointed to the possibility for the tetramer to flip to a right-handed superhelical form under modest positive DNA supercoiling. Moreover, similar data were obtained after addition of a large amount of naked DNA prior to relaxation, and cross-linking of the two H3's through a disulfide bridge. This further suggested that the transition occurred without dissociation, or even significant reshuffling of the histones. A model was proposed in which tetramer transition from a left- to a right-handed conformation involved a rotation of the two H3-H4 dimers, or a localized deformation about their H3-H3 interface (1).

In more recent work, we wished to test the possibility for the transition to be induced by the positive supercoiling generated by ethidium bromide (EtBr) intercalation in the loop. Because the number of EtBr molecules intercalated can be increased progressively, and at the same time accurately measured by fluorimetry, we hoped not only to reinforce the above model but also to gain some understanding of the dynamics of the transition. Results showed 1) that the transition can indeed be efficiently induced by EtBr intercalation; and 2) that the transition occurs through four successive steps, each characterized by a specific tetramer conformation and apparent flexibility of loop DNA. These steps allowed in turn to model the large dependence of the linking number reduction on the minicircle size which was observed for MT particles (article in preparation).

1) Hamiche A., Carot V., Alilat M., De Lucia F., O'Donohue M.-F., Revet B2E & Prunell A. (1996) Proc. Nat. Acad. Sci. USA, 93, 7588-7593.

2) Goulet I., Zivanovic Y., Prunell A. & Revet B. (1988) J. Mol. Biol. 20, 253-266.

3) Zivanovic Y., Goulet I., Revet B., Le Bret M. & Prunell A. (1988) J. Ml. Biol. 200, 267-290.

4) Arents G., Burlingame R. W., Wang B. C., Love W. E. & Moudrianakis E. N. (1991) Proc. Nat. Acad. Sci. USA, 88, 10148-10152.

Randomly cyclized DNA molecules can be found in one of the numerous topological states which cannot be changed without breaking a backbone of one or two DNA strands. Several levels of parameters describe a topological state of circular DNA molecules. If the cyclization reaction is slow enough, the topological parameters will accept their equilibrium values. These equilibrium values reflect DNA conformations in solution and depend on DNA length and concentration as well as solution conditions. Quantitative analysis of the topological equilibrium, based on parallel use of computer simulations and experimental measures, has proved to be a fruitful approach to study DNA properties in solution and DNA interactions with the proteins that modify topology of the molecule.

Yeast Telomere Repeat Sequence (TRS) DNA is highly conserved [(C1-3A)n] and known to be packaged into non-nucleosomal structures in vivo. We have used total intensity light scattering and electron microscopy (EM) to monitor the effects of yeast TRS on in vitro DNA condensation by cobalt (III) hexaammine. Insertion of 72 bp of TRS into a 3.3 kb plasmid depresses condensation as seen by light scattering and results in a 35-50 percent decrease in condensate size as measured by EM. the depression in total light scattering intensity is greater when the plasmid is linearized with the TRS at an end (39-49 percent) than when linearized with the TRS in the interior (18-22 percent). The condensation kinetics of TRS-containing plasmids show an unusually abrupt plateau around 5 minutes after addition of cobalt (III) hexaammine. These results, along with analysis of toroidal condensate dimensions suggest that the growth stages of condensation are inhibited by the presence of a TRS insert, perhaps as a result of incompatible packing geometries. This hypotheses is being explored further through both structural studies and co-condensation experiments in which a kb-sized plasmid is condensed in the presence of 100 bp fragments containing the TRS.

Knots may be viewed as idealized one-dimensional filaments in three-dimensional space. In the last few years, however, we have begun (or, historically, resumed) the study of knots as "real" physical objects, capable of having physical-like properties such as "thickness" (i.e., "rope-length" if you think of making knots out of rope with a fixed diameter) or "self-repelling energy". We are developing a theoretical foundation, e.g., studying various notions of size or energy, and also developing computer environments in which to visualize, manipulate, and energy-minimize knots. The provocative correlations between computer estimated minima for these rather naive energies and observed physical behavior of DNA knots in gel electrophoresis suggest that the naive models are capturing a significant amount of the variation in the "real" systems. We expect that further elaboration of the models will lead to more subtle ability to predict actual behavior, along with greater understanding of the dynamics of electrophoresis of DNA loops.

Superhelical parameters (N, the number of base pairs in one superhelical turn; H, the vertical displacement of residues along the superhelix axis; R, the radius of the superhelix) of double-helical DNA structures are investigated by modeling regular repeating sequences of the type XnY10-n. The XX and the XY steps are assumed to be B-DNA like (twist = 36 deg; tilt = roll = 0 deg; shift = slide = 0 Å; rise = 3.4 Å) and identical base sequence dependent perturbations are introduced in the YY and YX steps. Superhelical structures are generated as a function of twist, roll, and slide. Each superhelical repeating unit is identified by a virtual bond linking the origins of the first and the eleventh base pairs. Perturbations are confined, initially, within the limits observed from the analysis of single crystal structures of DNA oligomers. Two sets of structures emerge with dimensions close to the ideal nucleosome structure. These two groups with opposite superhelical sense differ only by small changes in the independent parameters. These models are being used to study folding and nucleosome formation of naturally occurring sequences. (Supported by USPHS Grant GM20861.)

During the process of genetic recombination DNA structure is dramatically changed by interaction with proteins mediating DNA strand exchange. The central player in the process of DNA recombination in bacteria is the RecA protein. This protein binds to DNA forming right-handed helical filaments in which double-stranded DNA is stretched and unwound, resulting in a structure with 18.6 base pairs per 95 Angstrom pitch. Up to now the molecular structure of RecA-DNA complexes has not been established and it is not know by what mechanism the average axial spacing between base pairs is increased from 3.4 Angstrom to 5.1 Angstrom. Recent studies of DNA stretching by external force suggest the existence of a specifically stretched DNA structure with ca. 5.1 Angstrom axial spacing between base pairs. Another biologically interesting form of DNA is created during the process of homologous recognition between single and double-stranded DNA molecules. During this process the RecA protein seems to coaxially wind single and double-stranded DNA molecules within RecA synaptic filaments. Recent experiments characterizing DNA structure within synaptic RecA-DNA filaments will be presented.

In joint research with B. D. Coleman and I. Tobias [1], recent results [2] in the theory of the elastic rod model of DNA were employed to calculate the elastic energy of the extranucleosomal loop of a DNA minicircle in a mononucleosome as functions of the linking difference (also called "delta link") of the minicircle, the amount w of wrapping of DNA around the histone core, and the twist density of nucleosomal (wrapped) DNA. By comparing the results with published experimental observations [3] of distributions of delta link obtained by relaxation of mononucleosomes with topoisomerase I, one can estimate the DNA-histone binding energy for the two additional DNA segments (each with a length equal to the cleavage periodicity) that bind to the core particle when w increases from ~1.5 to ~1.75 turns. Our calculations indicate that for this pair of extremal segments the binding energy is -1.5±0.3 kcal (or -0.75±0.15 kcal per segment). In particular, for a persistence length A/kT of 500 Å and a ratio A/C of elastic constants equal to 1.32, we find that if the twist related helical repeat of nucleosomal DNA is taken to be 10.31 bp/turn (which corresponds to a cleavage periodicity of 10.18 bp), when the binding energy G is assigned the value -0.75 kcal per segment, our calculations yield the following conclusions: (i) For 341 base pair plasmids, relaxation of mononucleosomes with topoisomerase I should give topoisomers with delta links -0.3 and -1.3 in the ratio 1:3. (ii) For 354 base pair plasmids, relaxation should give topoisomers -0.5 and -1.5 in the ratio 1:1.5. Both ratios are in accord with experimental observations of Zivanovic et al. [3]. Stein [4] estimated that the total DNA-histone binding energy for 14 segments of nucleosomal DNA is -15 kcal, which yields an average of -1.1 kcal per segment. In view of the expectation that less energy should be required to separate an extremal segment from the core particle, an average of -1.1 kcal per segment and our estimate of -0.75 kcal per extremal segment are compatible.

1. D. Swigon, B. D. Coleman, and I. Tobias, manuscript in preparation.

2. B. D. Coleman, I. Tobias, and D. Swigon, J. Chem. Phys., 103, 9101-9109 (1995).

3. Y. Zivanovic, I. Goulet, B. Revet, M. LeBret, and A. Prunell, J. Mol. Biol., 200, 267-290 (1988).

4. A. Stein, J. Mol. Biol., 130, 103-134 (1979).

This talk is based on recent research done in collaboration with B.D. Coleman of Rutgers and M. Lembo of Rome III. We consider a ring formed >from a naturally straight elastic rod, the ends of which may have been twisted with respect to each other before closure to generate a linking difference. The partial differential equations governing the small-amplitude displacements of the axis of the rod from its circular configuration are used to characterize all of the normal modes of vibration of the ring. The analysis also yields an explicit expression, the dispersion relation, connecting the wave length and frequency of each mode, and a formula for the dependence of the energy of the mode on its amplitude and frequency. It is observed that when the linking number difference is not zero, the modal oscillations acquire a chiral aspect not present when the linking number difference is zero. In addition, when the linking number difference is zero, there are some modes in which the excess twist density undergoes a spatial and temporal oscillation, and others in which the twist density is constant. When the linking number difference does not equal zero, however, in all of the modes the twist density varies with time and position along the axial curve. These results, and the apparatus of classical statistical mechanics, are used to study the thermal fluctions in the configuration of DNA minicircles of about 200 bp. A calculation of the average of the writhe, , over a canonical ensemble of vibrating elastic rings, all having the same value of the linking number difference, shows that this quantity vanishes only when the linking number difference is zero. That is, is not zero even when the stable equilibrium state of lowest energy, for the value of the linking number difference considered, is circular, and hence of zero writhe. An expression is derived for the free energy of the supercoiling of minicircles. Its dependence on the linking number difference and the number of base pairs N, when these quantities are small, is found to be consistent with experiment.

The equilibrium formation of catenanes can be used to study DNA conformations. We investigated, by this approach, the properties of supercoiled DNA as a function of ionic conditions and supercoiling density. The fraction of cyclizing molecules that becomes topologically linked with supercoiled DNA was measured experimentally. This fraction is the product of the concentration of the supercoiled DNA and a proportionality constant, B, that depends on conformations of supercoiled molecules. In parallel with these experimental studies, we calculated the values of B using Monte Carlo simulations of the equilibrium distribution of DNA conformations. We found very good agreement between measured and simulated values of B for all the ionic conditions and DNA superhelix densities studied. The value of B decreases nearly exponentially with increasing superhelicity, this dependence being especially strong at low salt concentration. The dependence of B on the concentration of NaCl, magnesium chloride, and spermidine can be described with good accuracy in terms of changes of the DNA effective diameter.

The main objective of this research is to understand and visualize the dynamics of supercoiled DNA. Since supercoiled DNA is hundreds, thousands, tens of thousands of base pairs or longer, traditional molecular dynamics cannot be used to simulate such DNA because it is not fast enough to treat such large DNA for long time scales. Thus, developing methods which can treat long DNA for long time scales is important. Classical continuum elasticity theory provides a good, yet simple model of DNA. Elastic rod theory is used to develop equations for DNA dynamics using arguments based upon the balance of forces and the balance of momentum at each cross section of the rod. The dynamical equations can include external forces such as electrostatic interactions, energy dissipation terms to account for viscous drag, and a random force to account for thermal fluctuations. It is also possible to study chains containing regions of intrinsic curvature, altered twist, differences in intrinsic bending and twisting stiffness, or bound drugs and/or proteins. (Supported by Sloan-DOE Joint Postdoctoral Fellowship in Computational Molecular Biology).

The double helix is significantly distorted in nucleo-protein complexes the most noticeable are DNA bending, twisting, and variations in the groove dimensions. These deformations are important for achieving the structural complementarity between the protein and the cognate DNA site. We are interested in characterizing these structural rearrangements of DNA at two levels: the local level of several base pairs, and the mesoscopic level of several helical turns of DNA (20-30 bp). To this aim, we have analyzed ca. 40 available X-ray complexes between DNA and proteins and modeled the p53-DNA complex.

LOCAL DNA ANISOTROPY: DNA bending flexibility is anisotropic and sequence-dependent. The duplex bends preferentially toward the major groove when complexed with proteins. The strongest bends (or kinks) occur predominantly in the pyrimidine-purine YR dimers (CA:TG, TA, CG) and in AG:CT steps. Bending toward the major groove is accompanied by DNA unwinding. This sequence-dependence of DNA bending and the Bend-Twist correlation are entirely consistent with the rules of DNA "conformational mechanics" found earlier for "pure" unbound B-DNA.

OVERALL TERTIARY FOLDS (WRITHING): In most of the complexes with transcription factors, DNA is bent into non-planar loops, closely resembling the path of the double helix in the nucleosome. This non-planarity correlates with, and appears to be functionally related to the periodic distribution of the YR and AG dimers in the binding sites, separated normally by 8-10 bp spacers. This feature is expected to facilitate binding of regulatory proteins to negatively supercoiled DNA.

In general, our data are consistent with the common idea that the proteins select those DNA sites which have intrinsic tendency to adopt the structures observed in the complexes.

PUTATIVE ARCHITECTURAL ROLE OF P53 IN THE SPATIAL ORGANIZATION OF THE DNA RESPONSE ELEMENT: The ubiquitous tumor suppresser protein, cooperatively binds to DNA as a tetramer. The solution data indicate that binding of the p53 tetramer to the cognate 20 bp long response element results in the DNA bending by 50-60 deg. On the other hand, according to the X-ray structure of the DNA site bound specifically to a single p53 monomer, the duplex remains straight. To resolve this discrepancy, we built the stereochemically feasible model of the complex between DNA and the p53 tetramer.

Our modeling of the complex using the observed peptide-DNA contacts shows that when four p53 monomers bind the response element in the B-DNA configuration, the steric clashes occur among different p53 subunits. To relieve these clashes, DNA has to be bent and unwound. In our model, the DNA bending occurs toward the major groove in the highly conserved tetramers C(A/T)(T/A)G separated by 10 bp, thus causing the left-handed DNA superstructure. This bending is consistent with the anisotropy of the CA:TG and AG:CT dimers described above and with recent DNA cyclization studies. The overall lateral arrangement of the four p53 subunits with respect to the DNA loop comprises a novel nucleoprotein assembly which has not been reported previously in other complexes.

We suggest this kind of the nucleoprotein superstructure may be important for the p53 binding to the response elements packed in chromatin and subsequent trans-activation.

For years it was believed that at low ionic strength the chromatin fiber is organized as an extended beads-on-a string filament. The evidence came mainly from elecron microscopy (EM) observations and could not be reconciled with physical data that suggested a three-dimensional loose arrangement of nucleosomes. We have combined biochemical approaches with the new imaging capabilities of scanning force microscopy (SFM) and mathematical modeling to approach these structural issues. The data showed unambiguously that even at low ionic strength, chromatin is organized as loosely coiled, irregular fibers with an average diameter of 30 nm. The linker histones play a major role in maintaining this organization, since their removal leads to flattening of the fiber into a beads-on-a-string configuration. Parallel mathematical modeling of intact and linker-histone depleted fibers makes it possible to define, to a first approximation, the major structural determinant of extended fibers as being angle between DNA strands entering and exiting the nucleosome and linker length.

We have also considered the structure of the condensed fiber obtained by raising the salt concentration of the solution. Biochemical data from experiments employing protease and nuclease digestion suggest that the linker histones are not internalized within the fiber, but the accessibility of linker DNA to nucleases gradually decreases upon compaction. SFM imaging of fibers at increasing salt concentrations reveal a gradual loss in individual nucleosome resolution and a highly irregular folding into structures resembling the "superbeads" previously reported by EM.

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