On higher buckling transitions in supercoiled DNA

On Higher Buckling Transitions in Supercoiled DNA

TAMAR SCHLICK,' WILMA K. OLSON,' TIMOTHY WESTCOTT,' and JERRY P. GREENBERG'

'Chemistry Department and Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012; 'Department of Chemistry, Rutgers, the State University of New Jersey,, Wright-Rieman Laboratories, New Brunswick, New Jersey 08903; and 3San Diego Supercomputer Center, P. 0. Box 85608, San Diego, California 921 38

SYNOPSIS

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 U k as physiological superhelical densities ( I u I - 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 U k , more than one interwound DNA minimum exists. The writhing number as a function of U k is discontinuous in some ranges; it exhibits pronounced jumps as U k is increased from zero, and it appears to level off to a characteristic slope only a t higher values of U k . 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 I u I 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 U k . 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. 0 1994 John Wiley & Sons, Inc.

I NTRO D UCTlO N available free energy associated with supercoiling are crucial for key biological functions, such as rep- lication, transcription, and r e c ~ m b i n a t i o n . ~ ~ ~ - ' ~ While both the torsional stress leading to supercoiling and the resulting folded products are fairly well understood, the dynamical details involved are largely unknown.

The onset of supercoiling and supercoiling transitions are intriguing phenomena from both a theoretical and a biological viewpoint. It is well known that closed circular DNA adopts highly coiled superhelical forms in solution 1-7; the compaction and

The catastrophic transition from the circular form to the figure-8 interwound has been studied

565

Biopolymers, Vol. 34, 565-597 (1994) 0 1994 John Wiley & Sons, Inc. ccc 0006-3525/94/050565-33

566 SCHLICK ET AL.

analyti~ally.’~-’~ An elastic energy model composed of bending and twisting components provides a good approximation in this case. Self-contacts can essentially be ignored in this regime, and the catastrophic “buckling,” or collapse, of the open circle can be predicted to occur a t a critical linking number difference A L k , which depends on the ratio of the bending to twisting elastic energy constants, p

For more highly coiled structures, the self-contacts among pairs of points along the DNA chain must be considered. Such treatments have been studied by various approaches, both analytical and via simulation. Recent analytical work has focused on highly writhed A variety of computer simulation techniques using stochastic methods have employed energy minimization by simulated annealing or ensemble averages from Metropolis Monte Carlo calculations.20~31 Finite element technique^^',^^ as well as deterministic schemes for minimization ( e.g., truncated Newton ) have also been used to identify favorable supercoiled config- u r a t i o n ~ . ~ ~ More recently, Brownian dynamics has been used to explore DNA motion35 and molecular dynamics has been applied to gain insight into folding and knotting p r o c e ~ s e s . ~ ~ , ~ ~

In most of these investigations, however, the superhelical density u ( a = ALk/Lk,,‘where ALk is the difference between the topological invariant Lk and the associated value for the relaxed state, Lk,) has been set to only a small number of values that roughly sample the physiologically observed range for DNA. The corresponding equilibrium geometries as a function of ALk have been presented as single minimized states, or Boltzmann distributions of forms about isolated points in configuration space. Thus, subtle details of supercoiling transitions, particularly as physiological superhelical densities are approached, may have been missed in these studies. A more detailed picture is now possible with pow- erful methods such as finite element analysis and molecular dynamics.

In this paper, we analyze results from detailed energy minimization and molecular dynamics studies that focus on the transition behavior in a series of configurations-circular and interwound-as physiological superhelical densities ( I u 1 - 0.06) are approached. The minimization results confirm and extend predictions of classical elasticity theory for equilibria of elastic rods, and the molecular dynamics results suggest how these findings may affect the dynamics of supercoiled DNA. In addition, the computations show striking qualitative similarities to recent time-resolved fluorescence polarization and

= A/C.15-17

dynamic light scattering measurements of DNA plasmids at increasing superhelical density.37 The experimental data point to two rather abrupt structural transitions that may involve some coupling of secondary and tertiary structure in supercoiled DNA. In other words, the net transition from relaxed circle to native supercoil appears to proceed via one or more distinct intermediate structures.

From minimization we find other sudden configurational transitions in analogy to the first catastrophic buckling from the circle to the figure-8 form. Similar trends have also been obtained very recently in the finite element analyses of Yang, Tobias, and Olson,32 although details are different. While we find that the transitions beyond the circle to figure-8 are not as dramatic as the first, the interplay among the bending, twisting, and self-contact forces leads to different families of supercoiled forms. These families overlap. That is, we observe more than one interwound supercoiled minimum may correspond to the same value of ALk. This suggests systematic interconversions among forms that are close in both energy and geometry. Furthermore, we find the variation of the writhing number as a function of ALk to be discontinuous: It exhibits pronounced jumps as ALk is increased from zero and appears to level off to a characteristic slope at higher values of ALk. This picture suggests a somewhat more complex dependence of the writhing number on ALk than the linear correlation ( Wr - a A L k ) , proposed on the basis of a few isolated pointsZ6s3’ or assumed with idealized helical model^.^,^^ Significantly, some of these families begin a t states with near-zero twist energy. This suggests that for small DNA circles and/or low superhelical densities, nicked DNA may relax to forms other than the circle as generally assumed.

Our emphasis in this work is on the analyses of the energy minimization findings and the molecular dynamics trajectories. A detailed minimization study is essential for describing systematically the favorable three-dimensional forms corresponding to our DNA chain model and associated energy. The equilibrium forms serve as a framework for analysis of the dynamical fluctuations and range of motions observed in the dynamics. Such equilibria information is typically absent in alternative statistical- mechanical treatments of DNA supercoiling. The dynamics trajectories serve to generate an ensemble of configurations according to the force field, revealing the energetic and geometric fluctuations, and thereby further elucidating the significance of the minimization findings. Moreover, since additional parameters are involved in the dynamics protocol,

ON HIGHER BUCKLING TRANSITIONS 567

the minimization serves as a reference against which behavior during the dynamics can be tested for ar- tifacts. Our analysis of the dynamics identifies different relative time scales associated with global translational, bending and twisting, and rotational (tumbling) motions, as well as the local bending variations. While direct comparison with experimental measurements of supercoiled DNA dynamics is difficult a t this time, we hope that the analyses offered here will aid in the interpretation of existing data as well as stimulate further experiments. There is no information on the relative frequencies of different molecular motions in the analytical and statistical mechanical models currently used in the analysis of various measures of large-scale chain dynamics ( e.g., fluorescence polarization a n i ~ o t r o p y ~ ~ or photon correlation s p e c t r o s ~ o p y ~ ~ ) .

The next section summarizes our B-spline chain model and algorithms for energy minimization and molecular dynamics. The third section presents the energy minimization results, with each family discussed in turn. The fourth section describes the molecular dynamics results for five simulations performed at ALlZ = 1.5,2.5,3.0,4.0, and 5.5. It discusses the geometric and energetic differences in the overall dynamics as well as local motions, in view of the minimization findings. It also describes four different types of observed motions, each with a characteristic time scale. The fifth section summarizes the minimization and dynamics results, and the final section discusses current model limitations and out- lines future directions for our work.

CURVE REPRESENTATION, NUMERICAL METHODS, AND ENERGY MODEL

Our recent simulation study of supercoiled DNA34 combines an idealized elastic energy m ~ d e l ~ ' , ~ ' that has been used in supercoiled DNA studies22,41-44 with a compact B-spline representation of closed-circular duplex DNA curves, 2425 truncated-Newton mini- m i ~ a t i o n , " , ~ ~ and molecular dynamics (MD) . 4 7 3 4 8

The basic methodology is outlined below. The reader interested in further technical details is referred to our first paper.34

B-Spline Construction

The spline construction represents complex smooth curves in terms of a small number ( N ) of control points { x, } in three dimensions (see Figure 1). The ribbon-like double-helical curve r (u) = { rl (u), rp(u) , . . . , rN(u)} , 0 I u I 1, is constructed by N

0 0

Figure 1. A sample construction of the B-spline ribbon- like curve for closed circular DNA. A stereo pair for the curve is shown, with the 14 control points denoted by solid circles. Note that the control points do not necessarily fall on the curve itself. Sixteen points per spline segment are shown.

control points, each of which defines a curve segment locally by the linear combination

where coordinate indices are defined modulo N. The four basis functions are given by the following relations4':

The parametric form of the Fj( u) makes it possible to generate a smoothly bending curve at any level of refinement, including individual base pairs. The local features of our supercoiled DNA model are represented by the mesh points in Figure 1. A smoothly bending model is critical to the application of elasticity theory to DNA. The adaptation of such a model to treat chains with localized sites of high bending27 is incorrect.

The cubic polynomials in Eq. ( 2 ) ensure parametric continuity of position, slope, and curvature a t the junction points. Thus, this construction ensures chain closure and second-order continuity (important for minimization). Through local dependency of the curve on the {xi } , it also reduces considerably the number of nonzero derivative ele- m e n t ~ . ~ ~

568 SCHLICK ET AL.

Truncated Newton Minimization

The truncated Newton method for potential energy r n i n i m i z a t i ~ n ~ ' , ~ ~ combines the rapid convergence properties of Newton methods with the modest computational and storage requirements more typical of gradient methods." Application of this approach to supercoiled DNA permits very rapid energy minimization and precise identification of minima (e.g., by characterizing local convexity of the energy surface) in comparison to stochastic approaches. Detailed low-energy profiles, as described here, can be obtained, as well as descriptions of higher energy forms, such as lobed structures and knotted i n t e r ~ o u n d s . ~ ~ The latter states are difficult to locate with stochastic methods. The partial-derivative expressions, complex due to the indirect and highly nonlinear dependency of the energy on the independent variables { xi } , are obtained using various mathematical simplifications 34 and are required for both minimization and MD calculations. Full algorithmic details are given elsewhere.34

Langevin/ Implicit-Euler MD

For MD, we use the Langevin/implicit-Euler ~ c h e m e ~ ~ , ~ ' to permit the use of large time steps At typically prohibited in macromolecular simula- t i o n ~ . ~ ~ , ~ ~ This is accomplished by discretizing the Newtonian equations of motion by the implicit-Eu- ler scheme53 and using the Langevin formulation, which includes, in addition to the systematic force, a random force r and a frictional term proportional to the velocity. These additional terms effectively mimic molecular collisions and establish thermal equilibrium. The resulting scheme is described by the following pair of differential equations:

where the random force r is a stationary Gaussian process with mean and covariance matrix given by

(r") = 0, (rn(rm)') = 2ykBTkf(6 , , /At ) (312)

The superscript n represents vector values a t time ( n A t ) ; x and v are the collective position and velocity vectors, respectively; A4 is the diagonal mass matrix; y the damping constant, gE the gradient

vector of the potential energy E ; kB is Boltzmann's constant; T is the temperature (set to room temperature in these simulations) ; and 6 the Kronecker delta function. T o calculate xn+' from x n and v", we formulate a minimization subproblem for the "dynamics function" 'P( x ) , where

' P ( x ) = $ ( l + y A t ) ( x - x G ) ~ M ( x - x ; )

+ ( A t ) ' E ( x ) (4a)

x t = x " + [At/(l+yAt)][v"+AtM-'r""] (4b)

This minimization is efficiently performed using the truncated Newton variant for potential energy function^.^' After a minimum xn+l for 'P is calculated, the new velocity vector vn+' is calculated directly by Eq. (3b) .

T o damp out high-frequency modes, we choose a "cutoff frequency" w, and set the frictional damping constant to

y = (w,)'At ( 5 )

This choice establishes a regime where modes with vibrational frequencies w w, are effectively fro- en.^^ Time steps of At = 100 fs (see below) are used here in combination with a cutoff frequency of w, = 10" s- l , appropriate for observing the global bending and twisting motions of DNA with characteristic times that are three or more orders of magnitude longer than typical chemical bond vibra- t i o n ~ . ~ ~ Since standard MD techniques and all-atom representations cannot currently capture large-scale folding patterns and fluctuations of macromolecules, the combination of macroscopic representation, elastic energy model, and large time-step dynamics protocol can be used to reveal information on these low-frequency vibrational modes. The overall motions revealed during the course of these simulations, while very large in comparison to those observed in all-atom studies, represent only a small fraction of the phase space potentially available to the supercoiled DNA. As noted below, individual base pairs do not span the full range of bending and twisting angles expected on the basis of the chosen force constants.

In our present formulation, the masses for the control points of the B-spline curve are set to unity. However, with the simulation time step of s, the corresponding physical time step can be estimated from the following analysis. A rigorous analytical treatment (described separately") can be


used to formulate the nondiagonal mass matrix corresponding to the kinetic energy expressed in the frame of our spline model of DNA. Specifically, expressions for the time derivatives of the control points are derived using the definition of the curve- point vectors as given in Eq. ( 1 ). The entries of the resulting mass matrix must be determined numer- ically a t every different configuration. The kinetic energy can then be computed by using this mass matrix and the velocity vectors [ Eq. (3b) ] as computed in our program. By equating this kinetic energy to the expected energy, assuming f ( k B T ) energy per degree of freedom, we can estimate the physical time unit corresponding to our simulation time step At. Our recent analysis provides an estimate of 100 ps for our time step. A separate work describes this calibration computation in detail." This corresponds to total simulation times ap- proaching the microsecond time frame. In the present work we refer to our iteration time unit as T.

It should be emphasized that our MD simulations also serve to generate an ensemble of configurations and thereby provide insight into energetic and geometric fluctuations. Indeed, a comparison between the molecular dynamics and Monte Carlo ensembles generated for nicked DNA suggests an agreement with one another and with experiment (B. Li, T. Schlick, & A. V. Vologodskii, unpublished work). The dynamics simulations have the added advantage of revealing pathways of molecular flexibility but require a greater number of iterations to sample the conformation space that can be probed by Monte Carlo techniques with larger structural moves. Our approach omits details of the dynamics of individual atoms, but with a sufficient number of mesh points it allows us to monitor the bending and twisting of the chain a t the base-pair level. In the calculations reported here we introduce 16 mesh points per B- spline control point. With 14 control points per curve, each mesh point thus corresponds to 4.5 base pairs in a 1000 base-pair DNA. Furthermore, the assignment of uniform local twist density-while theoretically exact for the equilibrium states identified in the energy minimization-is yet to be proven correct for the nonequilibrium states sampled in the course of the dynamics simulations or the states collected in a Monte Carlo s i m u l a t i ~ n . ~ ~ * * ~ The assumption is only reasonable in the context of dynamics if the relaxation time for the local twist fluctuations is fast in relation to the time step. Current analyses with Metropolis Monte Carlo simulations will provide further information on the validity of this approximation (S. Pedersen, I. Tobias, & W. K. Olson, work in progress).

Energy Function

The energy E includes components for bending (EB) , twisting (ET) , contour length ( E L ) , and attractive/repulsive van der Waals interactions ( Ev):

In these formulas, s denotes the arc length, K the curvature, and Lo the contour length of the chain. The parameters A and C are the bending and twisting stiffness constants, respectively; a and b are the attractive and repulsive nonbonded parameters; and the symbols d, denote pairwise distances along the chain. The twist energy as written above follows from the assumption of an isotropic elastic rod model a t e q ~ i l i b r i u m ~ ~ ; the twist density w (not to be con- fused with the frequency introduced earlier) is uniform throughout the chain, and the integral term representing a quadratic deviation of the local helical twist, (C/2 ) f ( w - wo) * ds, can be expressed in terms of the writhing number W r and the linking number L k . W r is a geometric quantity that measures the shape of the curve, and L k is a topological invariant describing the number of times the DNA strands intertwine around one another.55 This formulation takes advantage of the fundamental relationship proven by White that L k is the sum of W r and T w (twist), 56 where T w is a measure of the total helical twist. By using ET in the form of Eq. ( 6b) , we con- veniently introduce the critical variable A L k , representing the imposed linking number difference as measured from a relaxed circular state L k o . Thus, only the curvature and the writhing number must be calculated for evaluation of the elastic terms. The computed energies are directly related to the local (i.e., base-pair level) fluctuations of the chain about the assumed rodlike equilibrium rest state. As noted above, the elastic treatment is appropriate only if the chain undergoes small bending and twisting de- formations.

The contour length term E L is a computational device, and its contribution has negligible value throughout the calculation [ e.g., O( 1Op4-1OpS) vs O ( l 0 ) energy units] with our choice for K = 0.1

570 SCHLICK ET AL.

kcal/( mol A’). The following arc length dependent quantities

L = ds i and the writhing number, given by the Gauss double (7a) integral,55,56

are computed by analogous discrete summations over curve points as a function of our curve parameter u. For L and EB these expressions are given by

and

Since the Gauss integral is independent of the curve parameter ( s , u , etc.), the Wr expressions simply involve a discrete analogue of Eq. (7c) in which s1 and sp are replaced by u1 and u2. In all formulas above, r ’ and rn represent the vectors dr /du and d2r/du2, respectively; the symbols - , X, and 11 - 1) denote dot products, cross products, and Euclidean vector norms, respectively. The symbol C represents

summation over all spline curve segments: i = 1, 2, . . . , N and over a uniform mesh for 0 I u < 1: u = 0, A u , 2Au, . . . , ( M - 1) A u , where M Au = 1. Full computational details are given elsewhere.34

The elastic parameters A and C have been estimated from various experimental measurements of bending and torsional A u c t ~ a t i o n s . ~ ~ ~ ~ ~ Inherently, the measured values reflect average sequence and ionic-medium effects, but they are appropriate for our first-order macroscopic approximation of supercoiled DNA. Our choice of A = 2.3 X lo-’’ erg cm (or A = 144 (kcal/mol)A) corresponds to a DNA persistence length 1 = 559 A and rms per res- idue bending angle ( l3;)’l2 = 6.3” at 298 K.34 In the results described below we set p = A / C to 1.0 and 1.5 to sample the physiologically observed range of these parameters. This range of force constants is also in agreement with the rms fluctuations of local base-pair bending and twisting angles found in 31 B-DNA crystal structures in the Nucleic Acid Data- base.59 There have been recent suggestions3’ that larger values of C may be possible with p < 1. As noted from the trends in our p = 1.0 and 1.5 series, the overall picture obtained here should also apply for other parameter ranges.

i,u

Our form of Ev accounts for both repulsive and attractive forces along the DNA. Repulsion stems from excluded volume, electrostatic, and ionic con- siderations. Attraction may be biologically relevant for duplex cross-packing schemes facilitated by the binding of magnesium ions, 6o cross-linking by excess topoisomerases,61 and compression of the effective interwound superhelix diameter at high ionic ~ t r e n g t h . ~ For modeling attractive regimes in detail, the persistence length (and hence A and p ) would have to be modified to match the observed ionic strength d e p e n d e n ~ e . ~ ~

At present, however, Ev is mainly a computational device to describe smoothly repulsive interactions along the chain. It also ensures an approximately homogeneous distribution of curve points along the chain, so that each spline segment has roughly the same arc length. For this purpose, we parameterize Ev to reproduce a shallow energy minimum of -0.05 kcal/mol at a distance of 30 A, slightly larger than the estimated helix diameter of a hydrated B-DNA in s o l ~ t i o n . 6 ~ - ~ ~ This effective helix diameter corresponds to high salt conditions. The resulting attractive/repulsive coefficients are a = 7.3 X lo7 (kcal/mol) A6 and b = 2.7 X 10l6 ( kcal /mol ) A 1 2 .

ENERGY MINIMIZATION RESULTS

Our earlier minimization study for a 1000 base-pair (bp) model described energy minima corresponding to small AL,k increments in the range 0-2 and only


integral increments in the range 2-6.34 Here we extend that study by performing a large series of minimization runs to refine the 2 I ALk I 6 range as well. The same modeling procedure is used for consistency, with 14 controlling points per curve (Figure 1 ) .

Computations were performed on an IBM RISC / 6000 model 340 computer and DEC VAX/VMS cluster system at New York University and on DEC MicroVax 3100 and Silicon Graphics 4D /210GTXB workstations a t Rutgers University. A typical MD simulation of 5000 iterations, as detailed in the next section, requires approximately 3 CPU hours on the RISC machine. The energy minimization procedure takes a small fraction of that time, such as one tenth on average, depending on the starting point.

For minimization, a wide range of starting points, consisting of the circle and various interwounds, was used. Results are presented as a series of plots show- ing the variation of Wr, the total energy E, and the energy components ( E B , E T , Ev) as a function of ALk for p = 1.0 and 1.5 (Figures 2 and 3 ) . Each point on those curves corresponds to a structure calculated by the truncated Newton minimizer to be an energy minimum. Many starting points were tested to ensure the validity of each minimum. The Hessian ( second-derivative matrix of the potential energy) for each such point was also subjected to analysis by the eigenvalue package EISPACK65,66 to verify that all the eigenvalues are non-negative. In Figures 2b, c and 3b, c points corresponding to the circle are also shown for reference. The writhing numbers for the circle are zero (not shown).

Overall Picture-Families of Supercoiled DNA

The picture that emerges is an intriguing one. Cer- tainly the two families of structures corresponding to the circle and the figure-8 forms are well understood. The bending energy is lowest a t the circle and remains constant as ALk is increased from 0.75 to a value less than 2, a t which point the torsional strain must be alleviated through nonplanar bending (writhing). At this point, ET drops sharply but then rises slowly, while EB continues to increase steadily. We find that the figure-8 form is energetically favorable beyond the critical values ALk = 1.425 and 1.875 for p = 1.0 and 1.5, respectively, where the energies of both forms are These values are in excellent agreement with the analytical results of Le BretL6 and those of Tsuru and Wadati.23 They are also, as expected, lower than the theoretically derived values of ALk = 6 p (1.732 and 2.598, respectively) where exchange of stability occurs between the circle and figure-8 forms.15-17 At these

points the circle is no longer a stable equilibrium structure.

Note in our results identification of figure-8 forms a t values of ALk as low as 0.75, with Wr - 0.9. While these forms are not predicted by elasticity theory to occur a t such low ALk, they are possible here due to the attractive nonbonded interactions in our van der Waals energy term. Now, beyond these two well-understood configurational regimes, the Wr vs ALk plots reveal distinct energy-minimum families. These families are also obtained in corresponding minimizations of a potential without the attractive van der Waals interactions (data not shown). While the energetic and geometric discontinuities among the higher order families are not as dramatic as in the case of the circle and figure-8, they nonetheless suggest extensions of the theories used to interpret the first catastrophe in the context of elasticity theory. Let us follow the behavior shown in Figures 3a-d in the case of p = 1.5. For this analysis, representative interwound minima are pictured in Figure 4. The trends with p = 1 are very similar (Figure 2a-d) , with transitions occurring a t slightly lower ALk, and higher energies and writhing numbers, as expected.

The Figure-8 Family (F1 )

The writhing numbers for the figure-8 family rise very slowly from 0.9 to 1.2 as ALk is increased from 0.75 to 2.5. The figure-8 minimum corresponding to ALk = 0.75 emerges from a bent and twisted circle forming a roughly planar form with a single point of close contact (Figure 4a) . The ALk = 2.0 minimum reveals a nonplanar structure with further interwinding of the two end loops (Figure 4b ). Thus, the catastrophic transition from the circle to figure- 8 occurs as a configuration with no self-contact forces is transformed abruptly to one with a point of contact. This behavior can be appreciated by bending a guitar string with corks a t the ends into a circle ( a model suggested by C. Benham) : as torsional strain is introduced by turning one end with respect to the other, the string “jumps” to a figure- 8 form around the two-turn threshold. This figure- 8 shape is apparently stabilized by equilibrated forces in the top and bottom loops of the structure. As slightly more torsional strain is introduced, the guitar string only bends a little. This stability of the figure-8 is evident in our plots from the plateau in bending energy corresponding to the family. The contribution from E B rises very slowly while that from E T rises more rapidly. The contact force measured by E v remains nearly constant as the figure- 8 family is traversed (Figure 3d).

572 SCHLICK ET AL.

I 0 1 2 3 4 5 6

ALk 15

E n e r g y f o r t h e Circle a n d I n t e r w o u n d s

10

5

0

(A/C = 1.0) 1 2- 4

0 1 2 3 4 5 6

ALk Figure 2. ( a ) The writhing number for energy-minimized supercoiled DNA forms as a function of A L k , for A / C = 1.0. Each point here and in the following plots corresponds to an energy minimum, and different symbols are used to denote the different families that emerge. The stable minimum energy states of the circle ( Wr = 0 ) are not shown. ( b ) The total energy (bending, twisting, and van der Waals components) for the energy-minimized supercoiled DNA forms as a function of ALk, for A / C = 1.0. The energy of the circle is shown for reference. An inset magnifies a region of family crossing. ( c ) The corresponding bending, twisting, and van der Waals components, along with the total energy, for the energy-minimized supercoiled DNA forms as a function of ALk, for A / C = 1.0. The contributions associated with each family are denoted by the same symbols used in ( a ) . The van der Waals energy (bottom portion), twisting energy (middle portion), and bending energy (upper portion) are shown. ( d ) An enlarged view of the van der Waals portion of ( c ) .

Family F2: 2.5 I ALk 5 4.0, 2 5 Wr I 3

At a critical value near ALk = 2.5, a second catastrophic transition, consistent with the predictions

of Le Bret4* occurs. As in the first catastrophe, EB can no longer remain constant, and ET drops sharply while EB increases significantly. Correspondingly, Ev drops as favorable attractive forces emerge. This


n

0 3

z L- G

2

a

v

h M k

w

15

10

5

0

E n e r $ y C o m p o n e n t s f o r I A t e r w o u n d and Circle /'

(A/C = 1.0) ,/' I'

c

- d -6

t

00 0

-7.5 t ' l t l l l l l ' I I " I " t I l l l l ' ' l l l 0 1 2 3 4 5 6

ALk

Figure 2. (Continued from the previous page.)

second sudden configurational jump corresponds to a change from a point of contact to a line of contact (termed the contact zone in our analyses of supercoiled structural families in the Appendix). The large gap in the Wr vs ALk plot (between points corresponding to { U k = 2.5, W r = 1.2) and { U k = 2.5, W r = 1.9 ) ) represents energetically unfavored states. The change in Ev a t this point (Figure 3d) is more than double the change involved in the circle to figure-8 transformation (for the circle Ev = -5.6 kcal/mol). The two minima a t ALk = 2.5 are pictured in Figure 4c, d. Note the asymmetric figure-8 of Figure 4c, exhibiting the beginning of another fold, and the interwound arrangement of Figure 4d achieving approximately one superhelical turn of

close contact. The latter structure, with near-zero twist energy-see below-is much flatter than the deformed figure-8 minimum energy structure a t ALk = 2.5 (Figure 4c).

Family F2, in the range 2.5 I ALk 5 4.0, is quite complex and may, in fact, be a collection of subtle families of its own. Indeed, contact-zone analyses reveal small differences between the forms in the lower and upper portions of F2 ( see Appendix). The long-range attractive interactions in the contact zone as well as the bending and twisting energy show several pronounced dips with increasing A L k . The overall sharp slope in the W r vs ALk curve of the F2 family reflects these energetic contributions. The W r - 2 state, an interwound form that traps one

574 SCHLICK ET AL.

F4 ,-’

W r of E n e r g y - M i n i m i z e d I n t e r w o u n d s

(A/C = 1.5)

- F1 -

- I I I I 0

1 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I l l

0 1 2 3 4 5 6

1 E n e r g y f o r the’ Circle an‘d I n t e r w o u n d s ri 10

5

0

0 1 2 3 4 5 6 ALk

Figure 3. ( a ) The writhing number for energy-minimized supercoiled DNA forms as a function of U k , for A / C = 1.5. Each point here and in the following plots corresponds to an energy minimum, and different symbols are used to denote the different families that emerge. The stable minimum energy states of the circle ( Wr = 0) are not shown. ( b ) The total energy (bending, twisting, and van der Waals components) for the energy-minimized supercoiled DNA forms as a function of U k , for A / C = 1.5. The energy of the circle is shown for reference. An inset magnifies a region of family crossing. ( c ) The corresponding bending, twisting, and van der Waals components, along with the total energy, for the energy-minimized supercoiled DNA forms as a function of U k , for A / C = 1.5. The contributions associated with each family are denoted by the same symbols used in ( a ) . The van der Waals energy (bottom portion), twisting energy (middle portion), and bending energy (upper portion) are shown. ( d ) An enlarged view of the van der Waals portion of ( c ) .

superhelical loop, is apparently much less stable (as obtained from analysis of regular helices and isolated measured by changes in Wr with increasing ALk) points.6326 compared to the figure-8 form. The slope of the F2 Note that the F2 family, like the figure-8 family, family roughly follows the approximation Wr begins with near-zero twist energy. Both the bending - 0.72 ( ALk) -shown as reference in Figure 3a- and twisting contributions to the energy increase


h M L : r d

n 4

0 x h- a 0

G k a, d

W

15

10

5

0

Ea c i rc le ,,.

I

-7 L I steadily in this regime, although the variation of EB with U k is concave and that of ET is convex in form. As noted above, the van der Waals contributions decrease in this range as favorable attractive interactions in the interwound contact zone increase. Overall, the total energy increases nearly quadrat- ically. Figure 4e pictures the minimum corresponding to ALk = 3, Wr = 2.3. This interwound appears somewhat asymmetric, with nonuniformly distributed contact forces in the top and bottom portions of the structure.

Family F3: 3.5 5 ALk I 4.7, 2.7 I Wr 5 3.2

At ALk = 3.5, an overlapping family of stable configurations (over the range 3.5 s aLk I 4.0) emerges.

This family corresponds to higher Wr values compared to the F2 states with the same value of ALk. The new family becomes energetically favorable over the lower Wr family at aLk - 3.6. Notably, this higher Wr family, extending up to a L k = 4.7, also begins with near-zero twist energy. As U k is increased, ET rises steadily, while EB increases somewhat more slowly and Ev shows an irregular (although much less pronounced) pattern of decreases and increases similar to that of family F2. The attractive forces appear to have reached a threshold.

Pictured in Figure 4f, g are the F2 and F3 minima corresponding to a L k = 3.5: Wr = 2.5 and Wr = 2.7, respectively. A greater number of turns and a rough similarity to the minimum a t ALk = 3 in the F2 family (Figure 4e) are evident in the first. The

576 SCHLICK ET AL.

(a) ALk = 0.75, W r = 0.91

( e ) ALk = 3.0, Wr = 2.32

(b) ALk = 2.0, Wr = 1.06 ( c ) ALk = 2.5, Wr = 1.22 (d) ALk = 2.5, Wr = 1.92

(j) ALk = 4.1, W r = 3.32 (h) ALk = 4.0, Wr = 2.81 (i) ALk = 4.1, Wr = 3.05

(k) ALk = 4.7, Wr = 3.23 (1) ALk = 4.7, Wr = 3.64 (m) ALk = 5.5, Wr = 3.84


structure of higher Wr (and lower ET) for ALk = 3.5, at the start of the F3 family, is clearly flatter than the corresponding minimum in the F2 family. The energies of the F2 and F3 families intersect in the same manner as both the circle and figure-8 and the figure-8 and F2 families; however, the differences in energy are now much smaller, as seen from the enlarged view in the inset of Figure 3b. Thus, the total energy corresponding to the combination of the lowest energy forms appears to vary nearly gua- dratically with A L k . This follows directly from the approximate overall linear increase in Wr with ALk.31

The existence of two overlapping families of stable interwound structures a t certain regions of ALk has also been found in the recent finite element analysis of supercoiled DNA of Yang, Tobias, and Olson.32 Although only one form is the global energy minimum, there are clearly very different ways to partition superhelicity between bending and twisting in stable states at some values of ALk. The dynamical behavior of supercoiled DNA may be affected by the existence of such similar forms, since interconversions of interwound configurations within and among different families are likely. The MD simulations, described in the next section, explore these possibilities and provide further support of this idea.

Family F4: 4.1 I ALk I 6.0, 3.3 I Wr I 4

As we continue to follow the F3 family in the range 3.5 I ALk 5 4, a new family of minimum energy

states (F4) of still higher Wr emerges at ALk = 4.1. This new family of structures appears just as the F2 family of configurational minima can no longer be detected. As with the other families of minimum energy states, the F4 family begins with near-zero twist energy and a sudden drop in Ev. This starting point is only slightly higher in overall energy than the F3 minimum energy state of the same ALk. (The energies of the two families become equal a t ALk = 4.2.) The attractive forces associated with an additional superhelical turn in the contact zone of the interwound structure lower Ev as Wr now increases more sharply. The repulsive forces become more important in this range and limit the amount of interwinding. As ALk increases to 6 , Ev shows some drop with ALk but increases very slowly overall.

The sharp increase in the twisting energy with ALk along the F4 family reflects close nonbonded contacts as well as resistance of the terminal loops to further bending. The three minima shown in Fig- ure 4h, i, j represent the different families of structures that appear near ALk = 4.0: { ALk = 4.0, Wr = 2.8) in family F2, { ALk = 4.1, Wr = 3.l} in family F3, and { ALk = 4.1, Wr = 3.3) in family F4. The energetically favorable middle state (Figure 4i) , although clearly bent overall, appears more symmetric than the other two. The terminal loops of the F2 minimum are of different sizes, while the loops of the low-twist F4 minimum lie in different planes. Thus, the low-twist F4 structure is not flat as the low twist states of other structural families. Family F3 extends to ALk = 4.7, where the energy difference

Figure 4. A series of energy-minimized configurations of supercoiled DNA for A / C = 1.5 as a function of ALk in the range 0.75-5.5. Two views are shown for each structure. All corresponding energies below are in kcal /mol.

( a ) ALk = 0.75, Wr = 0.91, EB = 5.54, ET = 0.03, Ev = -5.75, E = -0.18. ( b ) ALk = 2.0, Wr = 1.06, E B = 5.69, E T = 1.14, Ev = -5.72, E = 1.11. (c ) ALk = 2.5, Wr = 1.22,'EB = 6.15, ET = 2.11, Ev = -5.67, E = 2.59. ( d ) ALk = 2.5, Wr = 1.92, E B = 7.81, ET = 0.43, Ev = -6.26, E = 1.99. ( e ) ALk = 3.0, Wr = 2.32, EB = 8.58, ET = 0.60, Ev = -6.50, E = 2.68. ( f ) ALk = 3.5, Wr = 2.55, EB = 9.18, ET = 1.15, Ev = -6.60, E = 3.74. (g) ALk = 3.5, Wr = 2.74, EB = 9.73, ET = 0.75, Ev = -6.69, E = 3.78. ( h ) ALk = 4.0, Wr = 2.81, EB = 9.85, ET = 1.83, Ev = -6.63, E = 5.06. ( i ) ALk = 4.1, Wr = 3.05, EB = 10.44, ET = 1.42, Ev = -6.77, E = 5.09. ( j ) ALk = 4.1, Wr = 3.32, EB = 11.53, ET = 0.78, Ev = -7.13, E = 5.18. ( k ) ALk = 4.7, W r = 3.23, EB = 11.0, ET = 2.78, Ev = -6.77, E = 7.00. (1) ALk = 4.7, Wr = 3.64, EB = 12.29, ET = 1.43, EV = -7.19, E = 6.53. (m) ALk = 5.5, Wr = 3.84, EB = 12.85, ET = 3.52, Ev = -7.09, E = 9.29.

Note that overall each unit increase in ALk produces approximately one more inner loop in the interwound, but that the changes are nonuniform. These forms correspond to key points on the curves shown in Figure 3a. Images ( a ) - ( c ) lie on the figure-8 family, while images ( c ) and ( d ) correspond to the point where the second buckling transition occurs ( ALk = 2.5). Forms (f) and (g) ( ALk = 3.5) correspond to the point where another family is detected; forms (h ) - ( j ) are near the region where three families exist ( ALk - 4 ) ; forms ( k ) and (1 ) ( ALk = 4.7) correspond to the end of the lower family in this region.

578 SCHLICK ET AL.

between the corresponding F3 and F4 forms is quite large, about 0.5 kcal/mol in favor of the upper family. The greater interwinding of the F4 structure, corresponding to the { ALk = 4.7, Wr = 3.6) minimum in comparison to the F3 form a t { ALk = 4.7, Wr = 3.2 } , is evident in Figure 4k, 1. The Wr for family F4 appears to increase more slowly in the vicinity of the configuration where about three superhelical loops are entrapped in the central core. This apparently reflects the resistance of the terminal loops to further bending as noted above. A representative form of the F4 family corresponding to { ALk = 5.5, Wr = 3.8) is shown in Figure 4m. As ALk and Wr increase, an overall quadratic variation of the total energy of the F4 family approximately follows the quadratic increase in ET.

MD INSIGHTS

The purpose of our MD simulations is to provide further insight into the mobility and fluctuation behavior of supercoiled DNA. The studies address the following questions: (1) Will the dynamics of supercoiled DNA be affected by the different families of energy minima? ( 2 ) Will interconversions among forms belonging to neighboring families occur frequently? ( 3 ) Will there be specific patterns in the energetic and geometric fluctuations? (4 ) Can different types of motion be characterized? ( 5 ) And how will dynamical details vary with ALk?

Five runs of MD were performed for ALk = 1.5, 2.5,3.0,4.0, and 5.5, all with p = 1.5. Each simulation started from the circle, and the folding dynamics were followed with 5000 iterations of the Langevin/ implicit-Euler scheme, as described in Section 2 and in our previous paper.34 As mentioned, the unit of iteration 7 is used in our analyses below, and only the notion of relative, rather than absolute, time scales is presented. Based on our recent calibration,m this total trajectory length corresponds to about 0.5 x 10-6s.

Figure 5 shows the variation in Wr as a function of iteration for the five MD runs. Note for the four higher ALk runs that the torsionally stressed circle folds quickly into an interwound configuration. More regular fluctuations are noted after iteration 2000, and these fluctuations are quite different in character for the different values of ALk. Figures 6-13 provide detailed views of these simulations. Figure 6 illustrates the fluctuations in E B , in ET, and in the total energy during the five MD trajectories. Figures 7-11 show images at selected frames of the five simulations, with frame numbers chosen to cor-

, , I I I , , I T

0 1000 2000 3000 4000 5000 1teratmn

Figure 5. The writhing number as a function of iteration for five MD trajectories at ALk = 1.5, 2.5, 3.0, 4.0, and 5.5 for A / C = 1.5. The circle is the starting point in each simulation. Each iteration here and in subsequent figures corresponds to the simulation time step of At = 100 fs. The physical time frame is about three orders of magnitude higher, or 100 ps80 (see note in text ) .

respond to key states in geometries and energies. For the ALk = 2.5,3.0, and 4.0 series we show stereo views of these selected frames. For ALk = 1.5, where bent circular forms are involved, we show two images-the right view obtained by a 90" rotation about the vertical axis of the left image. For a common reference, the center of mass in each frame above is translated to the origin. For ALk = 5.5, we use an alternative representation to provide a sense of the translational, as well as rotational, motion: single frames are drawn as they are generated by the simulation, with 4-6 structures for each 1000 7 interval.

Figure 12 shows the variation of Ev for all runs. Figure 13 presents for each run the principal moments of the radius of gyration, Zl , I 2 , 13. These moments are useful for describing the overall shape of the DNA, as well as the local motions, since they characterize the axes of the best-fitting ellipsoid. For example, when the DNA is flat and circular in form, I I = Z2 and I , = 0. When the chain is folded in an extended interwound shape, Zl is much greater than both Z2 and Z3.

Below we analyze the five trajectories in turn by grouping them into three sets of ALk.

The ALk = 1.5 Trajectory

For ALk = 1.5, the circle is the global energy minimum, and the figure-8 is somewhat higher in energy, by - 1 kcal/mol (see Figure 3b). However, the value


_ - - - _ _ _ - - - _ _ _ - - ' _ _ _ _ _ - . _ - - " " ~ " " " " " " " " " '

'. E, , _ I * _

I , . - _

A U - 6.6 305

0 " ~ " " ' ~ " " ' ~ " " " " 0 1000 2000 3000 4000 V)I I I ,

Iteration

Figure 6. 2.5, 3.0, 4.0, and 5.5, A / C = 1.5, as a function of iteration.

The bending, twisting, and total energies for the MD trajectories at aLk = 1.5,

1.5 is near the ALk threshold of - 1.9 for conversion to the figure-8. Thus, the circle is expected to bend and twist considerably. Indeed, we note from Figures 5-7 fluctuations in Wr, EB, and ET that correspond to movements from the circle (e.g., frames 50, 1380, 1470, 1540, 2800, and 3800) to a bent figure-8-like form ( frames 800,2040,2280,3200, and 5000). The fluctuation range of EB is slightly greater in mag-

nitude than that of ET (Figure 6 ) , while the fluctuations in Ev (Figure 12) are very small in comparison. The range of local bending and twisting associated with these energy terms is much less than that expected from the assumed force constants. The local twist deformation, ( 27r/L0) ( ALk - W r ) , for example, is only 0.54" per base pair in the high energy Wr = 0 state. The rms twist angle of 5.5", cor-

580 SCHLICK ET AL.

1470

Figure 7 . Frames from the MD trajectory at ALk = 1.5, A /C = 1.5, at selected iterations. The right image in each frame is obtained by rotating the left image by 90' along the vertical axis.

responding to the range of B-DNA structures found in the crystallographic data base,59 translates to a conformation of the ALk = k1.5 DNA elastica with writhing number -13.8 or 16.8. Much larger changes

of overall structure are thus expected if the local bending and twisting of individual base pairs span the full range of values observed in short linear chains. Although the fluctuations of E are not reg-



ular, the variation in overall shape, as measured by the principal axis lengths of the supercoil, follows a regular pattern with time. As evident from Figure 13, the chain alternates with a period of approximately 15007 between an open circle-with two principal axes Il and I2 of comparable magnitude and a smaller third axis 13-t0 a more compact form, with one long axis and two shorter axes of comparable magnitude. The components Il , 12, and I3 are obtained, as described previously, 34 by diagonalizing the tensor of average second moments of the displacement vectors from the center of gravity of each configuration. These sinusoidal fluctuations are expected from our current choice of the damping constant y [Eq. (5)], which highlights the elastic modes. A larger value can be used to mimic solvent damping (G. Ramachandran and T . Schlick, work in progress).

The changes in Wr in Figure 5 are directly cor- related with the variation of I2 from high to low values in the circular and more compact forms, respectively. The large energy barrier between the circle and figure-8, however, prevents interconversion

between the two configurational classes. Interest- ingly, the states of higher Wr of - 0.5 are not necessarily those of the highest total energy, since EB

is high. A larger kinetic than potential energy is also noted throughout the run, with the kinetic energy appearing to drive the motion strongly.

The Alk = 2.5 and 3.0 Trajectories

In the ALk = 2.5 and 3.0 trajectories we find a shorter period in the sinusoidal fluctuations of Wr with time. Note from Figure 3a that the two minima at ALk = 2.5 correspond to the well-separated forms of the figure-8 (Fl) and the low-twist F2 interwound structure. The minimum at ALk = 3 falls along the rather “steep” F2 family of the Wr vs ALk curve. As suggested by our writhe-density analyses (Ap- pendix), subfamilies of F2 may be involved, and thus large fluctuations in geometry may be expected in the ALk = 3.0 simulation. The variation in Wr in the ALk = 2.5 simulation suggests a partial transition from the lower energy F2 interwound toward the higher energy figure-8-like configuration.

582 SCHLICK ET AL.

Figure 8. iterations.

Stereo frames from the MD trajectory at ALk = 2.5, A / C = 1.5, at selected

The states of lower Wr in the ALk = 2.5 simulation correspond to points of higher twist energy and lower bending energy (see Figures 5 and 6) . There is no apparent tendency for the ALk = 2.5

states to fluctuate in the direction of higher Wr along the F2 family of configurations, a trend consistent with the partial transformation from the F2 interwound to the lower Wr figure-8 configuration. The


I 1400

I 1600

I 2600


ALk = 3 supercoil, in contrast, shows both increases and decreases in Wr with respect to the U k = 3 minimum energy state.

The folding dynamics of the ALk = 2.5 and 3 runs show a common pattern in the evolution of Wr. A critical Wr maximum and minimum appear around iterations 300 and 400, respectively, after which steady increases in Wr occur. These critical points correspond to a maximum and minimum in EB, respectively. Note in Figures 8 and 9 that a kink of - 90O-spread over several residues is introduced around iteration 300, quickly directing the folding into a highly curved interwound form. For ALk

= 4.0 (Figure l o ) , the kinks appear even more rapidly as the circular state is more unstable. A second fold rapidly follows a t U k = 2.5 and 3, and after a region of high bending energy, EB decreases and then fluctuates around lower values. The variation of ET is roughly anticorrelated with that of EB in both runs. The fluctuations in Ev are much smaller in magnitude than both EB and ET. After larger fluctuations in the beginning of each run, where rapid folding occurs, Ev tends to level off. This tendency is more pronounced for the ALk = 2.5 run, where the fluctuations of Ev are rapidly damped. Thus, overall the elastic and potential energies evolve

584 SCHLICK ET AL.

2600

750


Stereo frames from the MD trajectory at hLk = 3.0, A / C = 1.5, at selected

2800 lf 3200 c c

similarly in both simulations. The kinetic energy is also larger in magnitude than the potential energy in each case.

Since Ev does not play an important role for these relatively low superhelical states, more mobility is

possible, and the fluctuations of Eg and ET appear to drive the motion. The sinusoidal variations of EB and ET involve interconversions between states of higher Wr, where ET has a local minimum and EB has a local maximum (e.g., frames 2200 and 4600

4200


I


for ALk = 2.5, Figure 8; frames 3200 and 4600 for U k = 3.0, Figure 9 ) , to states of lower Wr, where EB has a local minimum and ET has a local maximum (e.g., frames 2600 and 3930 for ALk = 2.5, Figure 8; frames 2600 and 3900 for ALk = 3.0, Figure 9 ) . These trends become more regular as Ev levels off. For the ALk = 3.0 run, over the states where Ev shows a narrower but still significant range of fluctuations, the higher and lower Wr states correspond, to lower and higher Ev contributions, respectively (e.g., lower Ev for frames 3200 and 4600 and higher Ev for frames 2600 and 3900 in Figure 9) . The slightly larger fluctuations in EB compared to ET in the ALk = 2.5 run dominate the total energy variation. Both terms appear to contribute equally in the U k = 3.0 case with minima in E occurring a t points of inflection on the EB and ET curves. Since attraction is important in this range, tighter interwound shapes do not penalize the overall energy significantly. The states between the high and low Wr forms (e.g., frames 2800, 3600, and 4200 for ALk = 3.0, Figure 9 ) show a maximum in the kinetic energy. Overall bending and twisting motions about the global helical axis are evident, with the end loops changing directions and appearing to drive the motion.

A common feature of the supercoiled forms in these runs with Wr - 2 is a bent and asymmetric interwound shape (see, for example, frames 2200, 3000, and 3930 for U k = 2.5, Figure 8; frames 1500, 2800, and 5000 for ALk = 3.0, Figure 9 ) . This configuration is more pronounced in the U k = 3.0 run

and more apparent in the MD simulations than in the related energy-minimized forms (Figure 4d, e ) . Indeed, none of the states in the ALk = 3.0 MD simulation are as extended in terms of the radius of gyration as the energy minimum. On average, the chain is shorter and thicker than the optimized configuration. The principal axis ( 11) is consistently lower, and the other axes ( I 2 and 13) consistently higher, than the corresponding values (44.8,8.4,5.8 nm) a t the energy minimum (Figure 13 ) . The bending, as measured by the axis lengths, is much less pronounced when a L k = 2.5. The average length of II over the course of the ALk = 2.5 simulation (41.4 nm) is only slightly less and the average length of I2 (13.3 nm) only slightly greater than the corresponding axes of the F1 (42.8 and 12.9 nm) and F2 (44.4 and 11.2 nm) minimum energy states a t this A L k . The mean length of I3 (5.5 nm) is roughly comparable to that of the F1 state (5.9 nm) but greater than that of the F2 state (1.9 nm) .

The time scale of overall chain bending (approximately 7 5 0 ~ ) ~ as measured by the sinusoidal variation of principal axis lengths, is clearly faster when ALk = 2.5 or 3.0 than when ALk = 1.5 (Figure 13) . This period of fluctuations is even shorter than that noted above for the writhing number in these simulations. The structural asymmetry in the U k = 2.5 and 3.0 runs appears to stem from a nonuniform distribution of contact forces on the upper and lower segments of these interwounds. One superhelical loop is entrapped in the contact zone of these configurations, so that each segment winds around the

586 SCHLICK ET AL.

1 100

200

300

I 400

600

70(

eoa

1020

1240

1500


Stereo frames from the MD trajectory at ALk = 4.0, A / C = 1.5, at selected

other for only one half a turn. Thus, a nonzero twist moment leading to a bent geometry can relax the structure more easily.32

Note that the variations in the ALk = 3.0 run in ET are larger in comparison to the ALk = 2.5 run

(Figure 6). This is expected since ALk = 2.5 corresponds to the beginning of a family with near-zero twist. Thus, the ALk = 2.5 simulation reveals small fluctuations about the energy minimum form: Wr - 1.9 * 0.2, with lower Wr values occupied more


1720

88

2700

2880

3870

4050

4670

5000 B Figure 10. (Continued from the previous page.)

readily than higher values. The figure-8 form computed by energy minimization at AL,k = 2.5, while comparable in energy to the states of the MD simulations, is not detected. As noted above, however, the fluctuations in Wr are in the direction of the transition to the figure-8. For ALk = 3.0, larger fluc-

tuations in Wr of size f0.5 are noted with respect to the energy minimum form ( Wr = 2.3), with both lower and higher values occurring. These motions can be interpreted as movements toward the upper and lower portions of the F2 family of structures. The nearby upper F3 family (see Figure 3a) begins

588 SCHLICK ET AL.

Figure 11. Composite images from the MD trajectory at ALk = 5.5, A / C = 1.5. Unlike the series above, the images here reveal the translational motion involved in the trajectory. They are obtained by plotting the coordi- nates of the curve as they are generated in the simulation (without translation of the center of mass to the origin). The frame numbers corresponding to each image in the five groups are as follows: group 1-0, 100, 250,520, 600, 1000; group 2-1000, 1320, 1500, 2000; group 3-2000, 2200, 2400, 2600, 2800, 3000; group 4-3000, 3200, 3400, 3600, 3800, 4000; group 5-4000, 4200, 4400,4600, 5000.

-5 , , , , / , , , , , , , , / , / , , , , , , ! -

- 5 5 ALk = 55

-5 , , , , 1 , , , , 1 , , , , , , , , / / , , , , / / , , , 1 , , , , -5 5 ALk = 4 0

-5 5

-6

-6 5

bLk = 3 0 4 3

-7 , , , , I , , , , 1 , , , , 1 , , , , / , , , , 1 , , - L d

ALk = 2 5

-6 5

-6 5

-7 1500 2000 2500 3000 3500 4000 4500 5000

Iteration

Figure 12. The van der Waals energy as a function of iteration for the five MD trajectories at ALk = 1.5, 2.5, 3.0, 4.0, and 5.5 for A / C = 1.5. The emphasis here is on the later portion of the trajectory, following the larger van der Waals fluctuations associated with the rapid folding of the circle.

with Wr = 2.7, although only at ALk = 3.5. As our Wr-contribution analyses reveal, the lower portion of the F2 family may be a closely related family of its own. The attainment of the Wr - 1.8 form around iteration 3900 in the two MD runs (Figures 5,8, and 9 ) suggests that structures within the same family may have been identified. Indeed, the overall dimensions of the supercoil, as measured by the principal axes of the radius of gyration, are comparable at the ends of the ALk = 2.5 and 3.0 simulations, where 11, Z,, and I3 are (42.9, 11.1, 6.5 nm) and (43.4, 9.9, 6.8 nm), respectively.

Thus, the highly dynamic motion in these trajectories can be understood by the low stability of the interwound form entrapping one superhelical loop, the small Ev contributions for these low superhelical densities, and the close relationship between the lower and higher Wr families in this range of U k .

The Afk = 4.0 and 5.5 Trajectories

In contrast to the ALk = 1.5, 2.5, and 3.0 runs, smaller overall fluctuations in Wr are noted in the a L k = 4.0 and 5.5 runs. The geometric and energetic fluctuations also tend to decay in time. This is more evident for the ALk = 5.5 run, with a rms deviation

50 ' ' ' ALk- 1.5 1

30

0 1000 2000 3000 4000 5000

. . : i : . . : ' . . . . -: . . : :: iJ . . . . . . . . ' ': Q'

(b) 50 ' ALk- 2.5 ' 40P-----7 30

20 c ,? -I

0 1000 2000 3000 4000 SOOO

(e) ' ALk= 5.5 ' 50

1000 2000 3000 4000 SO00 0

I t e ra t ion

i

20 1

0 1000 2000 3000 4000 5000

It era t ion

Figure 13. The principal moments of the radius of gyration at hLk = 1.5, 2.5, 3.0, 4.0, and 5.5, A / C = 1.5, as a function of iteration. The principal moments Z,, 12, Z3, are computed from our parametric curve of the DNA r ( u ) , as follows. The tensor R , containing the elements of the average second moments of the displacement vectors with respect to the center of gravity, is computed through summations of outer products:

where the center of gravity vector ( r ( u ) ) = (1 /L) C r; (u) du. The quantity L denotes

the total contour length of the chain, and the summation symbols { i, u } extend over all piecewise segments of the curve r and all mesh points considered in the unit interval over u. The nine elements of the matrix R describe the best fitting ellipsoid to the B-spline curve. To determine the principal axes of this ellipsoid, we perform the similarity transformation Q-'RQ = D, where Q is the diagonalizing eigenvector matrix relating the principal axis system to the original coordinate frame of the chain. The eigenvalues di of D ( i = 1, 2, 3 ) describe rough ellipsoidal bound areas for the closed DNA chain. The len hs of the principal ellipsoid axes are represented by the rms displacement lengths Zi = 8.

w

590 SCHLICK ET AL.

of 0.6 about a mean writhing number of 3.3. In contrast to the lower ALk runs, the greater importance of EV -tending to restrict the local fluctuations, the more plateau-shaped families of the W r vs ALk plot in this region, and the larger potential than kinetic energy contributions-may account for this overall behavior. The relatively small fluctuations in the writhing number in the ALk = 4.0 simulation (rms deviation of 0.7 about a mean writhing number of 2.6) are not surprising given the dependence of W r on ALk in the minimum energy structures over this range. The fluctuations in W r in the ALk = 4.0 MD simulation span values characteristic of the F2, F3, and F4 families of minimized structures (where W r = 2.8, 3.0, and 3.3, respectively). The limited ranges of both the writhing number and the total energy follow from the very similar energies of the minimized structures in the vicinity of ALk = 4.0. The values of ET are typical of those in the F2 and F3 configurational families (see below).

The ALk = 4.0 simulation reveals interconversions between at least two families of structures: the F3 family containing the global energy minimum (e.g., frame 3600, Figure 10) and the F2 family (e.g., frame 4050, Figure 10). The global minimum found on the F3 family for ALk = 4.0 is only 0.2 kcal/mol lower in energy than the corresponding form in the F2 family. Transitions between the intenvound forms of those families occur with sinusoidal variations of the elastic energy. Bending and twisting motions about the global DNA helix axis are most notable, with {high E B , low E T } combinations corresponding to the higher W r forms, and {low E g ,

high ET} combinations corresponding to the low W r forms. More pronounced bending tendencies- leading to overall curved forms-are evident here in contrast to the ALk = 3.0 trajectory. The amplitude of fluctuations in the principal axis lengths (Figure 13) a t ALk = 4.0 is roughly double that found a t ALk = 3.0. The frequency of bending ( - 8007), however, is comparable to that identified a t the lower ALk.

For ALk = 5.5, the fluctuations in W r observed in the dynamics are only about k0.15 with respect to the energy minimum ( W r = 3.8). According to Figure 3a, there is a single family in this region with little variation of W r with ALk. The more irregular variations of the total energy, in comparison to other runs a t ALk values near multiple families of the minimized energy structures, suggest that more random fluctuations result. The images in Figure 11, produced by the MDMOVIE program,67 reveal the very rapid folding of the circular state and significant internal bending and twisting motions, yielding both straight and highly curved forms. The

rate of interconversion between these straight and bent configurations, as measured by the sinusoidal fluctuations of the principal axes of the radius of gyration is - 8007 (Figure 13). Overall molecular tumbling is also apparent from these figures. The long axis of the ALk = 5.5 supercoil rotates - 270" during the course of the simulation, suggesting a complete end-over-end tumbling time of - 67007.

SUMMARY AND FURTHER DISCUSSION

Our minimization study reveals a more complex dependence of the writhing number on ALk for supercoiled DNA than previously assumed. A simpler linear correlation between W r and ALk has often been suggested based on sampling DNA supercoils with isolated values of ALk or analyzing idealized helical structures. While this may be a good approximation for highly supercoiled DNA, the picture presented here-distinct intenvound families and buckling transitions-may be important for inter- preting the highly dynamical biological processes of DNA for small chains or low superhelical densities. Previous approximations of the geometric variations in supercoiled DNA may have oversimplified the picture and overlooked certain favorable regions in configuration space.

New insights are also suggested by the minimization findings for buckling catastrophes in the framework of elasticity theory. We identify one catastrophic configurational transition beyond the circle to figure-8, as well as two more subtle ones at higher ALk. These results confirm and extend predictions of classical elasticity theory. These trends are also revealed in the recent finite element analysis of Yang, Tobias, and Olson. 32 Details for the writhe variation within the families are somewhat different, but are probably accounted for by the different treatment of the nonbonded contacts ( a hard-sphere term along the contact zone in that study). To our knowledge, detailed analytical treatments that con- sider self-contact contributions are yet to be devel- oped for this problem.

Our key minimization observations can be sum- marized as follows.

1. The variation of Wr as a function of ALk for energy-minimized supercoiled configurations is not a simple linear one. A more complex relationship reveals regions in configuration space that are unfavored, with favorable regions falling along distinct families of equilibrium structures.


2. Discontinuities in configuration space can be accounted for by the interplay of bending, twisting, and van der Waals forces. The circle ( W r = 0 ) to figure-8 ( W r - 1) transition occurs a t a critical bending-energy state that can no longer be sustained without writhing. Along with an abrupt collapse from a configuration with no self-contact to one with a point of contact are a sharp drop in ET, sharp rise in E g ,

and small drop in Ev. A second abrupt transition occurs between the figure-8 and the interwound family entrapping one superhelical turn of interstrand contact ( W r - 2 ) . The transition from apoint to line of contact again brings ET down and E B up sharply; the van der Waals contribution drops significantly as favorable attractive forces along the chain increase. This second buckling transition appears here to be less dependent than the first on the ratio of bending to twisting elastic constants; it occurs for both p = 1.0 and 1.5 around ALk = 2.5. In the finite element treatment of Yang, Tobias, and Olson3* with p = 1.32, the transition to the W r - 2 state occurs a t ALk = 3.0 with a smaller drop in the twist energy. Interestingly, one of the transitions of supercoiled DNA observed by Song et al.37 occurs a t ALk = -2.5, a state corresponding to the mirror image of the configurational jump found here.

Other less sharp transitions between minima are observed at higher ALk where certain W r values are unfavored. A sudden drop in ET is a noted characteristic of these jumps, as well as a decrease in Ev . However, as W r increases and the DNA become more tightly interwound, repulsive nonbonded forces become important. E v no longer exhibits occasional dips but in- stead increases slowly. This regime is characterized by more steady rises in E B and ET as well. The combined effect is to “smooth out” supercoiling behavior, i.e., merge different families and produce a more continuous structural dependence on ALk. This behavior has been recently confirmed with a Fourier-series representation of the chain (G.-H. Liu, W. K. Olson, & T. Schlick, work in progress) and will be described separately. In contrast, discontinuous higher order transitions are still found up to ALk = 6.0 in the finite element analysis of Yang, Tobias, and Olson.32

3. Certain supercoiling families overlap. That is, there are equilibrium forms that correspond to the same ALk. These forms occur, for ex-

ample, in the range of 3.5 < ALk I 4.7 for the p = 1.5 series (Figure 3a). The existence of different local minima that are close in energy but geometrically distinct suggest rapid interconversions, as revealed by our MD simulations. Since the forms of DNA supercoils within one family are quite similar, interconversions between different family ensembles a t a given A L K may complicate current inter- pretations of gel electrophoresis experiments. It is implicitly assumed that there is a unique three-dimensional supercoiled form associated with a given ALk.4 The present calculations show that the ensemble of states with given ALk does not necessarily arise from small de- formations of a single equilibrium state. In- deed, Song et al.37 suggest that their observed transitions in supercoiled DNA as physiological densities are approached indicate that a variable such as u traverses several structural transitions. As noted earlier, a coupling between changes in secondary and tertiary structure is suggested. The two abrupt structural transitions noted by Song et al.37 may be related to the two collapses noted here (around the figure-8 form and the next interwound).

4. The noted families begin with near-zero twist. This is a consequence of sudden increases of writhing numbers that are possible in the superhelical regime where repulsion is not dom- inant. In this range, twisting the DNA further costs little energy. It is possible that the near zero-twist forms for odd integral writhing numbers (e.g., 1, 3, 5 ) may occur as relaxed forms for nicked DNA. That is, the circle may not be the only low-twist form as often assumed. It is thus possible that a nicked DNA with a covalent break in one chain may relax under certain conditions (e.g., high salt in the absence of an electric field) to a relatively flat supercoiled form with zero twist. As illustrated by these calculations, there is a significant energy barrier separating equilibrium states of comparable energy (e.g., the circle and figure- 8 at ALk = 1.5) and even larger barriers separating torsionally strained states from the equilibrium state a t a given ALk (see the graphs of W r , E , and ET vs 7 in Figures 5 and 6 ) .

5. The overall approximately quadratic supercoiling-energy variance as a function of ALk is observed despite the different families. Beyond the circle to figure-8 buckling, the intersection of energy curves belonging to different families

592 SCHLICK ET AL.

merges similar curves with increasing ALk. The composite roughly resembles a continuous quadratic curve overall when only the lowest energies from each family are included. How- ever, a t low superhelical densities (c I 0.02- 0.03, or below ALk I 2-3 in our simulations of 1000 base-pair DNA), a rigid interpretation of this dependency oversimplifies the families and the ensembles of supercoils that may result. The quadratic fit is quite good in the range of naturally occurring values of u, although the 1000 base-pair chains studied here are smaller than those of typical biological interest. In contrast, the quadratic approximation of E vs ALk does not hold over a broad range of ALk in the finite element analysis of Yang, Tobias, and Olson,32 where the variation of Wr with ALk is discontinuous up to values of ALk = 6. In general, the writhing number a t a given a L k is related to the slope of the E vs ALk curve a t that value of ALk, 31 and so a quadratic variation of the elastic energy with ALk follows if dWr/d( A L k ) is constant. Thus, in the case of the finite element calculations, there are several distinct quadratic variations of E vs ALk over the range 0 I ALk I 6.0.

The MD simulations provide further insights into these variations of energies and geometries. The existence of the families appears to influence the dynamics of supercoiled DNA: The noted motions reflect the characteristics of the families a t a given ALk, and interconversions among forms in nearby families can be related to anticorrelated changes of ET and E B . In some cases, the DNA supercoils tend to twist about common axes (e.g., for ALk = 1.5), while in others the chains show a large-scale overall bending and straightening motion (see below). The internal mobility of the chain also appears to depend on the superhelical density. For lower ALk values, where repulsive nonbonded contributions are not critical and multiple families are found, significant sinusoidal-like fluctuations in EB, ET, and Wr are noted. In contrast, Ev tends to level off a t these ALk. Fluctuations are particularly great in the low-stability form corresponding to the interwound just beyond the figure-8, with Wr - 2. Indeed, both wire models and finite element analyses3' suggest the lower stability of this form in comparison to the interwound configuration with one (figure-8) and three superhelical nodes (the latter form entrapping two superhelical turns). In our simulations, such asymmetric and bent forms are revealed and have nonzero twist moments.

The kinetic energy is notably higher than the potential energy for the higher ALk simulations. At these superhelical densities, EV does not decay, and the energy fluctuations-particularly in ET-tend to decay in time. Smaller fluctuations in these simulations are consistent with the smaller variations in geometries associated with neighboring families.

The simulations reveal several different scales of molecular motion consistent with the experimental dynamics of DNA supercoils (e.g., Ref. 38). In the cited photocorrelation spectroscopy studies, only translational diffusion can be reasonably quantified. Faster internal motions corresponding to bending and twisting motions of the superhelix are assigned collectively to a separate exponentially decaying component of the scattered light vector. Here, translational motion occurs over a longer time frame than can be resolved accurately. The relative time scales of the more rapid motions, however, can be examined in our work and introduced in new models used to account for the experimental observations. The end-over-end tumbling of the chain can be derived from the time correlations of unit vectors on the principal axes of the chain, while the rate of overall chain bending is readily extracted from the time correlations of the principal axis lengths. The local bending and twisting time scales are derived from the correlations of the curvature and twist a t individual base pairs along the chain.

Over the time scale of the ALk = 5.5 simulation, the DNA chains exhibit an overall tumbling motion, with the long axis of the DNA rotating over a complete period within roughly 67007. The large-scale bending of the duplex as a whole is even faster, corresponding to periodic changes between straight and bent forms of 200-8007 depending on ALk. The local bending of the chain is a more rapid process with small (3.5") sinusoidal fluctuations in local curvature occurring with a period of 100-200~ or less depending upon overall supercoiling geometry and individual chain location (data to be presented elsewhere).

For more detailed information, the local dynamics of DNA must be considered as well. A t present, the macroscopic elastic representation essentially freezes any high-frequency motions that may be involved. The current choice of the damping constant clearly focuses on characteristic elastic (sinusoidal) motions dictated by the internal potential energy. While the hierarchy of time scales for the global motions described above (translation, rotation, collective, and local bending) will be maintained, the magnitudes of the differences among them may be


changed. A full treatment of DNA, including the solvent and salt environment, will further damp the characteristic frequencies of the internal bending motions. Indeed, an analysis of the effects of solvent on the collective motions of a protein reveal a dramatic change from sinusoidal to damped behavior in vacuum versus solvent simulations (Figures 9 and 10 in Ref. 81). The “noise” introduced by the solvent remains to be investigated in the future.

FUTURE DIRECTIONS

Our analysis has only dealt with the interwound supercoil regime for small DNA chains involving no branch points. For larger chains, branching transitions are important and likely to introduce other interesting catastrophic phenomena in both geometry and energy.21 While positive superhelical densities are used here for comparison with our earlier work, the minima obtained with negative ALk are simply mirror images of the forms detailed here, and all other analyses apply.

Our current representation of the repulsive interactions by a van der Waals potential with a minimum at 30 A (around the geometric diameter of DNA) is a reasonable approximation for high salt conditions (around 1 M ) . Clearly, the “effective” diameter of the DNA helix is not simply its geometric diameter 62-64,68; the value depends strongly on the ionic environment. The recent findings of Rybenkov, Cozzarelli, and Vologodskii6’ and of Shaw and Wang7’ suggest a rapidly decaying curve for the effective DNA diameter as a function of increasing sodium-ion concentration. At high ionic strength, cryo-electron microscopic images reveal the virtual collapse of the DNA strands against one another in the central contact zone of the interwound superhelix, yielding structures qualitatively similar to those found in these calculations. None- theless, as the experimentally determined bending and twisting constants already incorporate sequence-averaged and ionic-medium factors, the right choice of nonbonded parameters for the simulations is more difficult. Moreover, the bending and twisting force constants as well as the intrinsic helical twist are sensitive to the ionic ~ t r e n g t h . ~ ~ , ~ ’ , ~ ~

A better formulation of the nonbonded energy that reflects the long-range interactions of the DNA polyelectrolyte backbone is possible with an explicit Debye-Huckel term.73 Monte Carlo simulations in- corporating Debye-Huckel interactions of the neg- atively charged phosphate groups along short DNA

supercoils reveal the importance of these effects.74 Equilibrium forms are found to open up considerably and the critical collapse of the circle to figure-8 is shifted to higher ALk as salt concentration decreases. Our recent work reports detailed results over a wide range of salt concentrations with a Debye- Huckel potential parameterized after the pioneering work of Stigter73 (T. Schlick, B. Li, and W. K. Olsen, in preparation).

More direct treatments of the electrostatic interactions, as mentioned above, as well as of base sequence-dependent bending and twisting of the chain backbone (e.g., Refs. 75-78) are undoubtedly needed for finer resolutions of DNA structure and dynamics. These local details will be possible in the future through implicit modeling of the second DNA strand and assignment of spatial-dependent bending and twisting constants. Such models will also provide an important check of the treatment of DNA as a homogeneous isotropic thin rod (with uniform twist density) in nonequilibrium states. This approximation is important not only to the relatively limited motions of long DNAs in these and otherz8,*’ MD trajectories but also to the large-scale flexibility of supercoiled DNA probed in Metropolis Monte Carlo s i r n ~ l a t i o n s . ~ ’ ~ ~ ~ , ~ ~ The low-order out-of-plane vi- brations of a closed isotropic elastic rod are driven by inhomogeneities in the local twist (M. Lembo, I. Tobias, and B. Coleman, private communication). Thus, it is possible that some of the abrupt configurational transitions noted here may be even more intriguing when local factors are incorporated in the calculations. For example, the folding behavior in closed circular DNA containing highly curved segments may exhibit abrupt transitions in geometries and energies that are highly sensitive to the spatial and size distributions of intrinsically curved seg- m e n t ~ . ~ ~ Transitions between the circle and figure-

Table I Interwound Families, p = 1.5”

Range of Contact Zones for

Family Contact Zone (nm)

F1 F2 F3 F4

5 29-48 50-61 69-11

a See text and Figure A for discussion ofcon- tact zone. The error in the measured length of the contact zone (which depends on the number of segments used for the chain analysis) is tl nm.

594 SCHLICK ET AL.

_ _

0.06

0.05

. f

I . I .

' Family F3 - ALk - 4.1

-

t

0.06

(b) 0.07 . , . , . , . , .

' F m l y F4 -

Luk- 5.1

F m l y FZ

ALk- 2.7

0.05 t

Famly FZ

ALk- 3.5

0.05

0.04 ' 0.03

0.02

0.01 1 t \

100 0.00

20 40 60 80

60 80 100 0 20 40

Contact (nm)

Figure A. An analysis of the contributions to the writhing number for different pairs of points along the DNA chain. The painvise contributions to the Gauss double integral are illustrated here as intensity peaks within a chain contact zone. The contact zone and intensity peaks vary for the different families detected for the energy-minimized interwounds. Intensity peaks are shown for the following representative minima in the families discussed in the text for the A / C = 1.5 series.

( a ) Family F1, figure-8 (0.75 I ALk I 2.5): ALk = 2.4; Wr = 1.1. ( b ) Family F2, lower portion (2.5 5 ALk I 3.1): ALk = 2.7; Wr = 2.1. ( c ) Family F2, upper portion (3.2 5 ALk 5 4.0): ALk = 3.5; Wr = 2.6. ( d ) Family F3 (3.5 I ALk I 4.7): hLk = 4.1; Wr = 3.1. ( e ) Family F4 (4.1 I ALk I 6.0): ALk = 5.1: Wr = 3.8.


8 may also be more abrupt if the electrostatic re- pulsions of the sugar-phosphate backbone are treated more explicitly. Such effects, and the char- acterization of the different types of motions of supercoiled DNA at ambient temperatures and ionic atmospheres, will be the focus of future work.

APPENDIX: ANALYSIS O F THE Wr DENSITY FOR THE DIFFERENT FAMILIES

To help classify the different interwound configurations associated with the various families of minima, we performed the following analysis of the writhing-number contributions along the chain. For a given energy minimum, the DNA curve is divided into 128 evenly spaced points. By choosing a single point in each segment, we can determine the different pairwise contributions to the Gauss double integral [ Eq. (7c) ]. While most pairs contribute nearly zero to the integral, pairs of points in close contact as a result of interwinding contribute significantly. Significant contributions occur within the central contact zone of the interwound. This zone is defined in this analysis as the region within which the pairwise contributions to Wr are greater than 0.015. The contact zone has different lengths for the various families of interwounds (see Table I ) , and within it, the contributions to the writhing number are nonuniform: they vary in intensity from one end of the zone to the other. These intensities can be used to classify the intenvound minima into families. The differences in intensities further show that the superhelical core of the interwound structures is not uniform, as is frequently assumed in analytical

Figure A shows the intensity distributions within the contact zone for a representative minimum in each family of structures. Part ( a ) of the figure, corresponding to the figure-8 family ( F l ) , displays a single very high peak in the center of a short contact zone ( 5 nm) . Parts (b ) and ( c ) show the intensities associated with the lower (2.5 I ALk I 3.1) and upper (3.2 I ALk I 4.0) portions of F2. The first has two weak intensity peaks a t about one third and two thirds the length of the total contact zone ( - 32 nm). The second has two weak intensity peaks but they are shifted toward the end of the contact zone; the length of the contact zone varies from 43 to 48 nm as ALk increases. The existence of the subfamilies is further supported by increased attractive van der Waals interactions within the F2 family, around ALk = 3.1 (Figure 3d).

The intensity distributions in the writhing number for family F3 (part d ) resemble the latter portion

work. 18.19.23

of F2 except that the two end loops of the intenvound have significantly different lengths and the contact zone varies from 50 to 61 nm with increase of ALk. For example, the minimum corresponding to ALk = 4.1, Wr = 3.1 has a contact zone of 58 nm, with one looped region of size 128 nm and another of about 96 nm (see Figure 4i). The characteristics of family F4 (part e ) are two strong peaks-one at each end of the contact zone-and three weak peaks evenly distributed between the two strong peaks. The contact zone has length of 69 to 77 nm. Points of close contact and orientation of superhelical strands are clearly nonuniform in this optimum structure with ALk = 5.1 and Wr = 3.8.

We are grateful to Drs. William Bauer, Nicholas Cozza- relli, Jorg Langowski, Irwin Tobias, and James Wang for sharing the results of recent work. We thank Bin Li for assisting with the computations. Our research was gen- erously supported by the National Science Foundation ( CHE-9002146 and Presidential Young Investigator Award ASC-9157582), U.S. Public Health Service (GM 34809), Searle Scholars Program/The Chicago Commu- nity Trust, Whitaker Foundation, and the Faculty of Arts and Sciences at New York University. Computational support was provided by the Academic Computing Facility at New York University and the Rutgers University Center for Computational Chemistry.

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