The Elastic Resilience of DNA Can Induce All-or-None Structural Transitions in the Nucleosome Core Particle NANCY 1. MARKY and GERALD S. MANNING

Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, New Jersey 08903

SYNOPSIS

DNA on the surface of the histone octamer in the native nucleosome core particle is modeled as a circumferentially wound elastic line on the surface of a cylinder. In a model for the radial transition, the line is allowed to straighten, and thus lose energy, by swinging off the surface, but it is impeded in such an excursion by a radial force field representing the attractive interaction between DNA and histone octamer. In a model for the axial transition, the line may straighten by becoming more parallel to a generator of the cylinder while remaining on the surface. In this mode of straightening, dimer-tetramer or tetramer-tetramer interfaces are disrupted, and the resulting energy gain impedes the transition. Both radial and axial transitions are predicted to occur in all-or-none fashion. We propose that these models are nelated to the abrupt transitions actually observed in the nucleosome core particle.

INTRODUCTION At full extension each of the 46 DNA molecules in a human cell would be about 2.2 cm long. If one of these polymers is placed free in a dilute solution environment, however, thermal fluctuations prevent full extension. A standard formula from polymer theory gives a value of about 47 pm for its rms endto-end distance, a length on the cellular scale. Comparison of the size of the genome at full extension with cell size thus overstates the biological “packaging problem,” which, in iOs crude sense of compatibilization of scale, is largely solved by the existence of temperature. It would appear that the intricate multilevel folding of DNA in chromatin is more likely t o reflect the organization required for the precise temporal a n d spatial functioning of the mechanisms of transcription, replication, and cell differentiation. We devote attention in this paper to the primary level of organization exhibited by the nucleosome core particle. T h e core partiicle exists of 146 base pairs (bp) of DNA wound in 1.8 left-handed, roughly superhelical, turns on the outside of a protein kernel. The overall shape resembles a disk with diameter Biopolymers. Vol. 31, 1543-1557 (1991) 0 1991 John Wiley & Sons, Inc. CCC OoOS-3525/91/131S43-15$04.00

106 A and height 57 A.’-4 T h e protein is a n octameric complex of histones, two each of H2A, H2B, H3, and H4. A rough picture has the DNA wound on the surface of a protein cylinder consisting of a stack of four discs (checkers). Each checker is a histone dimer. “Red” H2A/ H2B checkers are a t each end of the stack, and a black “king” ( H 3 / H4)2 tetramer is sandwiched between them. Actually, this model does not wreak gross violence on reality. Perhaps the major discrepancy is the implied coincidence of the DNA superhelical axis with the cylindrical axis of the histone octamer. In Fig. 4 of Uberbacher and Bunick’s report on the x-ray crystallographic structure,2 we see a n acute angle between these axes; otherwise, the picture of DNA wound on the outside of a stack of checkers does not seem so bad, depending, of course, on one’s purpose^.^ T h e nucleosome core particle is not strongly stable. An incisive overview3 has been formulated by van Holde: The core particle is a fragile object. Outside of a narrow environmental range, it is unstable, unfolding in a variety of ways. This should not be surprising, for the particle represents a precarious marriage of basically incompatible partners. The histone octamer is not itself stable at low or even moderate ionic strengths; in the absence of DNA, it will dissociate into dimers and tetramers. The DNA, on 1543

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MARKY AND MANNING

the other hand, has been bent to a degree inconsistent with its normal propensity. Only the strong interaction between the DNA and the histones compensates for these tendencies to unfold or dissociate. In the physiological range of pH, ionic strength, and temperature the core particle is marginally stable. . . . It seems reasonable that this incipient instability . . . is an essential feature of nucleosome function. An irreversibly condensed. . .structure would be physiologically useless; what is seemingly required and provided is a structure that can exist in several levels of stability. Van Holde, and also Camerini-Otero and Felsenfeld,6 assign an important role in marginalizing the stability of the core particle to the energy required to bend the DNA to a radius of curvature much smaller than “normal.” Indeed, the persistence length of DNA free in solution (160 bp at 0.1M salt) is remarkably similar to the 146 bp length of nucleosome core particle DNA, which may therefore be likened, from a coarse-grained point of view, to an elastic rod with resiliency against bending. From standard formulas of polymer theory, we learn that thermal energy can bend a persistence length segment to about a fifth of a complete circular turn. Core particle DNA is wound in nearly two turns, and thus is under considerable elastic stress above that automatically supplied by thermal forces. (On the contrary, if core particle DNA were, say, ten times longer than a persistence length, each of its two turns would contain five persistence lengths, and each persistence length would therefore complete only about a fifth of a turn. The DNA would then not be strained beyond its typical thermal fluctuations.) The marginal stability of the core particle as it exists in nature may be partly related to the congruence of the length of its DNA with a persistence length of free DNA. The thesis of this paper is that the incipient instability of the nucleosome core particle may be described as an elastic instability, driven by the tendency of core particle DNA to straighten and thus lose its stored (unfavorable) energy of elastic bending. A feature of elastic instability is its all-or-none character. A tall column stands upright until it is built to a critical height, whereupon its rectilinear shape abruptly becomes unstable, and it buckles. A narrow rod may be twisted yet remain straight, but when the twist reaches a critical amount, the rod abruptly assumes the shape of a helix. For the nucleosome core particle we will discuss the possibility of two elastic instabilities. In one of them, the ends of the DNA abruptly straighten out and away from the histone surface when the forces attracting the DNA to the surface weaken to a critical level (or

when the stiffness of the DNA is augmented to a critical level). In the other, the DNA abruptly straightens in the direction of the superhelical axis, that is, the axis of the “stack-of-checkers” cylindrical model for the histone octamer. In this latter mode the DNA is assumed to remain rigidly attached to the protein cylinder. The result of straightening, then, is that the checker stack is stretched and torn apart. The onset of this instability is marked by an increase of the stiffness of DNA to a critical level, or by a decrease of histone-histone association forces to a critical level. Histone-histone forces, histone-DNA forces, and DNA stiffness can be modified by environmental changes, chemical modification, or protein binding, thus triggering the elastic instabilities discussed. ( DNA-DNA interaction presumably arising from ionic repulsion between DNA segments on different superhelical turns may also be of importance. It can be accommodated in the “catch-all’’ energy parameter of the theory and is discussed in the Discussion section). There is experimental evidence for both types of instability in the core particle, but we again turn to van Holde for general motivation and defer discussion of specific examples to the final section of our paper: There has been a great deal of interest in studying the ways in which [core particle] structure responds to changes in the solvent environment. It would seem that this interest is generated by two expectations.On the one hand, it is hoped that such responses can tell us something about the forces stabilizing the particles. Further, it is believed by some that such conformational changes can yield clues as to how the nucleosome might be rearranged in uiuo to allow such processes as transcription or replication to proceed. We would add only that abrupt all-or-none conformational transitions triggered by small perturbations perhaps make it easier to think about the precision with which in vivo rearrangements must be regulated.

THE RADIAL TRANSITION Description of the Model

When the core particle is viewed end on, that is, from the top or bottom of the stack of checkers, the appearance is remarkably disc-like’ (see also van H ~ l d e Fig. , ~ 6-15). Although there are irregularities-after all, the crystallographic resolution is approaching atomic scale-the path of the DNA is ob-

THE ELASTIC RESILIENCE OF DNA

served essentially as a circle wrapped around a circular disc filled with electron density from the histones. In this perspective, of course, one does not see the pitch of the DNA superhelix, which, however, is fairly shallow (mean value about 28 A, perhaps a bit more than required to avoid collision of two hydrated gyres ) . Motivated by the end-on image of the core particle from x-ray diffraction, we adopt the model shown in Figure 1 for the movement of the DNA tails (end segments) on and off the histone octamer, represented by the solid circle. We call the movement “radial” because it occurs in the plane of the circle, perpendicular to the axis of the cylindrical stack of checkers. Three possible positions for the DNA are drawn, all with the same anchoring point on the circle, perhaps a particularly tight binding site (the central segment of DNA is known to adhere more tightly than the tails, so that this “clamped” boundary condition, which fixes the initial point and direction of the elastic line, is perhaps appropriate for our problem). In one trajeatory, the DNA is fully associated and wraps around the circle. In another, representing complete dissociation of the tail, the DNA extends straight out as a tangent from the point of anchor. A third shows an intermediate path

Figure 1. Geometry of the radial transition. The solid circle is the cylinder-like histone octamer, viewed by looking down on the cylindrical axis. Curves 1, 2, and 3 represent an end segment (tail) of DNA of some chosen fixed length. In our calculations, we used 10, 20, and 40 bp as tail lengths. Curve 1 represents a completely associated tail. Curve 3 is the same DNA tail when completely dissociated from the histone surface. Curve 2 shows the tail in a possible intermediate position (not necessarily an equilibrium position). The thin circles are isopotential lines representing attractive histone-DNA forces. At arc length s along the path of the DNA, measured from the common starting point of the three curves, the angle of inclination of the path to the horizontal, is designated as I 3 ( s ) .

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(and is meant to represent generically any path intermediate between the two extremes of complete association and complete dissociation). Which of the three is realized as the position of stable mechanical equilibrium depends on the energy balance between the forces pulling the DNA toward the circle (histone-DNA attraction) and the stresses in the DNA that resist bending. Note that DNA melting is not part of the problem as formulated; the DNA is considered to move on and off the histone surface as an intact duplex (see the discussion section). We give the mathematical formulation of the problem. Let s be length along the trajectory (path) of the double-helical axis of the DNA tail, measured from the point of anchor a t s = 0. A segment of DNA of length ds, centered a t position s along the path, has curvature K ( S ) , with units of reciprocal length, and positive, or unfavorable, elastic bending energy ( 1/ 2 ) b ~ * d swhere , b is the Hooke’s law constant (its units are energy times length). The entire tail segment, of total length L, has bending energy obtained by integration along its path, Elastic bending energy

= ( 1/ 2 ) b

[

K2ds

( 1)

On the one extreme of complete dissociation (curve 3 in Figure 1) ,there is zero curvature, and the bending energy is zero. On the other extreme of complete association (curve 1, which lies entirely along the circle), the curvature a t each point of the DNA is constant at ri’, where ro is the radius of the circle, and the bending energy equals ( 1/ 2 ) bLro2. In Eq. ( 1 ) , we have taken b as a constant, an average value along the length of the DNA. We get its numerical value from multiplication of the persistence length of DNA (160 bp a t 0.1M salt) by IzT, according to a standard prescription of polymer theory. With this simplification, we are unable to handle the effects of nonuniform local bendability. The DNA tail also has negative, or favorable, potential energy from the forces attracting it toward the histone surface. We represent it, per unit length, by the expression - y w ( s ), where the constant y, with units energy per unit length, measures the maximum amount of favorable energy available from attractive histone-DNA forces, attained when the DNA lies on the surface. Accordingly, the function w ( s )is dimensionless and has maximum value unity on the surface. The attractive energy of a small segment of length ds centered at path length s from the starting point of the DNA tail is - y w ( s ) d s , and the total attractive energy is obtained by integrating along the trajectory of the tail,

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MARKY AND MANNING

Histone-DNA energy

= -7

wds

(2

For the case of curve 3 in Figure 1,the histone-DNA energy does not simply vanish, since the portion of the tail close to the anchor point continues to lie close to the circle; the completely dissociated tail has a small amount of negative histone-DNA energy. The completely associatedtail lies entirely along the circle, so its histone-DNA energy equals - yL. For reference in what follows, it is useful to define a dimensionless ratio g.

When the DNA lies entirely on the histone surface (curve 1 in Figure l ) ,the numerator is its energy per unit length due to histone-DNA attractive forces (taken without sign). The denominator is the work per unit length required to bend it onto the surface. When g = 1, the net energy of DNA on the histone surface is zero. (It is not the case, however, that DNA lies on the histone surface when g = 1, for, as we have indicated, the fully straightened tail, curve 3 in Figure 1, has a small negative energy, and an intermediate path, such as curve 2 in Figure 1,could possess still more negative net energy under these conditions and thus be stable relative to curve 1with zero energy. Thus, the histone-DNA attraction must be strengthened into the range g > 1 if curve 1 on the surface is to be the lowest energy path.) When g = 0, the histone-DNA attraction has been “turned off,’’ and the lowest energy path is the completely dissociated curve 3 in Figure 1;under these conditions, the energy is entirely from bending and proceeds from zero toward increasingly unfavorable positive values as we go from curve 3 through curve 2 to curve 1. Implicit in the description given for the histoneDNA energy is an idealized model for the protein residues that attract the DNA. They are envisaged as uniformly smeared along a circle of radius ro representing the histone surface (see the comments below in the next section). The attractive energy - ywds of a small DNA segment of length ds located at distance r from the center of the circle then depends only on r . We adopt an idealized expression for w ,

where p is the radial distance 1 r - ro1 / r o of the DNA

segment from the circle in units of ro, and 6 is a dimensionless parameter inversely correlated with the width of the potential energy ring. For indeed, what we have just described is a “ring” of potential energy -yw, which parallels the circle of radius ro and has its minimum (maximum of negative energy) at r = r o . The histone-DNA contacts that hold the DNA on the protein surface in opposition to its tendency to straighten has been replaced by a uniform ring of negative potential energy. We have drawn the ring in Figure 1as a set of isopotential concentric circles of radii r > ro. Equation ( 4 ) , however, indicates symmetry of the ring about the radius ro and allows the DNA segment to be located at a position r < ro. We will see that this aspect of the idealized model is almost without effect on our results. In numerical work we use the round value ro = 50 A for the radius of the potential energy minimum and the value 200 for the dimensionless parameter 6. Then the depth of the potential energy ring decays within 5 A to the fraction e-’ of its maximum value y a t ro,when, in terms of the actual nucleosome core particle, perhaps a water molecule or two intervenes between the DNA and the histone. Short Tails Dissociate Gradually

If the length L of the dissociating tail is very short, in practice, less than 10 bp, the model is not particularly realistic. Histone-DNA attractive forces may be produced by as little as one or two amino acid residues per double helical turn.’v3 Under these circumstances, the modeling approximation of forces smeared uniformly along a DNA segment of length less than 10 bp may be especially severe. Besides, if we cannot average over at least duplex turn, the whole idea of using the continuum elasticity theory of isotropically bending rods as embodied in Eq. ( 1) may not be too convincing. We present the results for short tails primarily because they exhibit what might be deemed the “expected” behavior-as the strength of the attractive forces gradually decreases, the tail passes progressively through all intermediate positions from complete winding on the histone surface (curve 1in Figure 1) to complete straightening (curve 3 ) . For a very short tail, all intermediate positions may be represented approximately by circular a r m 7 The uniform curvature of the completely associated tail is r ; ’ , since it lies entirely along the histone surface with the same curvature. The curvature vanishes for the completely dissociated and straightened tail. A general formula for the equilibrium curvature R-’ of partially dissociated tails is

THE ELASTIC RESILIENCE OF DNA

where g is the dimensionless ratio defined by Eq. ( 3 ) , and may be used as a measure of strength of the histone-DNA attractive forces relative to the resistance of the DNA to bending. The formula is valid for short tails, L / r o 6 1, and for curvatures R-' Iro', which means that the DNA tail is on or outside the histone surface. The derivation7is based on the direct analytical integration of Eqs. (1) and ( 2 ) , made possible by restriction to small L . The value g required for the tail to wrap on the histone surface can be obtained from Eq. ( 5 )by setting its left-hand side equal to unity. [ This value of g is then seen to be proportional to ( r o / L ) 4 and , with the requirement of short lengths, L 6 ro, the value of g for complete association can be considerably in excess of unity]. Equation ( 5 ) now tells us that when the histone-DNA attraction gradually weakens from the range of values required for complete wrapping, the DNA tail progressively dissociates from the surface, with gradually decreasing curvature R - l , until it entirely straightens as g vanishes. To summarize, if the tail is short, then small environmental changes affecting the histone-DNA energy (or the stiffness of the DNA) produce small changes in the position of the tail. We have exemplified this type of behavior with numerical calculations for a 10-bp tail. The possible paths of a short DNA tail were represented by the first term of a series expansion in powers of the length variable s, 8(s)

=

xIs/L

0 I s IL

(6)

where 8(s) is the angle of inclination toward the horizontal made by the tangent line of the DNA path drawn at length s along the path (see Figure 1) . Equation ( 6 ) , with constant coefficient xl, represents an arc of a circle of constant curvature x l / L , that is, with radius L / x l . (We use this form for subsequent reference when more complicated shapes are considered for longer DNA tails.) For example, when the DNA tail of length L lies entirely on the circle of radius ro representing the histone surface, the value of x1 is L / ro. [ Substitution of this value into Eq. ( 6 ) shows that L 6 ro is the condition for validity of the first-order expansion (uniformly small angles 8 ) , a requirement shared with Eq. ( 5 ) J . The completely dissociated tail lies straight along the horizontal axis and is therefore characterized by xl = 0. An intermediate position is represented by a value of xi between 0 and L / ro.

1547

The computer was given a set of environmental conditions in the form of a fixed numerical value for g and asked to evaluate the total energy of the DNA tail [the elastic bending energy plus the histoneDNA energy from Eqs. ( 1) and ( 2 ) ] for various paths (various values of x1) ,and then to report back the path of minimum energy. The resulting series of minimum-energy paths for different values of g is qualitatively similar to the series drawn in Figure 2 ( a ) ,which gives the results of a more accurate calculation for the 10-bp tail, described in the next section. As the histone-DNA attraction weakens from values resulting in complete winding of the DNA on the surface, the DNA tail assumes positions of gradually decreasing curvature until it has entirely straightened out and away from the surface. Figure 2 ( b ) gives a graphical representation of the dissociation process. The radial distance pL between the end point s = L of the DNA tail and the histone surface is taken from Figure 2 ( a ) and plotted against g. As the histone-DNA attraction weakens (from right to left in the figure), the distance, a measure of dissociation, increases progressively. The results of the analytical theory leading to Eq. ( 5 ) have thus been confirmed for short tails by direct numerical means. Small environmental changes produce small changes of DNA path. Tails of longer length Dissociate in an All-or-None Transition Numerical Analysis. A detailed mathematical development led to the conjecture that elastic instability could trigger an all-or-none dissociation of DNA tails from the histone core.* This approach will be summarized below, but the results of direct numerical analysis, which confirm the conjecture, make the issue physically and visually transparent, and are presented first. As in the previous section, the computer was asked to evaluate numerically the sum of the two energy contributions specified by Eqs. ( 1) and ( 2 ) , and to report the minimum-energy path of the DNA tail. The selection, however, was from a broader class of paths:

For very short paths we used only the first term of the series, thus restricting the class of paths to circular arcs. Here we allow more complicated shapes, while leaving the computer free to choose xi = 0 for all but the first term if the circular arc indeed turns

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MARKY AND MANNING

1

g = 99-00

Negative values of x2 allow the path to straighten from the circular arc as it proceeds from small s, thus decreasing its positive bending energy at, perhaps, relatively low cost of negative histone-DNA energy. The results are displayed as Figures 2, 3, and 4 for DNA tail lengths 10,20,and 40 bp, respectively.

V

"

0

2

6

4

8

10

9.90 > g > 0.99

U

0.2

I

PL

0.1

lo

0

20 U

0.0 0

2

6

4

8

lo

g

Figure 2. (a) Minimum-energy paths for a 10-bp DNA tail. The view is from above, as in Figure 1. The quantity g is a measure of the strength of the histone-DNA forces relative to the stiffness of the DNA. The path for g = 99 is, like curve 1 in Figure 1, on the histone surface. As g decreases, the 10-bp tail is progressively released. The coordinates u and u are ordinary planar Cartesian, measured in units of base pairs of DNA, 3.4 A/ bp. ( b ) The radial distance p L of the end point of the DNA tail to the histone surface as a function of g. The progressive release of the tail is portrayed from right to left on the graph, that is, as g decreases.

out to have a minimum energy (it does not). We have worked with the sum of the first four terms, and verified that no numerically significant contribution to the problem is made by inclusion of terms for i = 3 and 4. The computer's task may therefore be lightened to selection from paths of the form

0.8

0.9

1.0

1.1

1.2

1.3

B

Figure 3. ( a ) Behavior of a 20-bp DNA tail (labeling as in Figure 2). The g = 9.9 curve is on the histone surface. As g decreases to a value near unity, the DNA is progressively released, but remains close. It then jumps to a position far from the surface, and continues gradually toward the horizontal position of complete dissociation as g continues to decrease. ( b ) Radial distance pL of the end point of the 20-bp tail to the histone surface as a function of g. The abrupt release of the tail near a g value of unity is clearly seen (right to left on the graph).

THE ELASTIC RESILIENCE OF DNA

40 V

1.26 >

g

1

> 1.04

30

a0

10

a

P

30

40 U

1649

surface to a trajectory far removed from it as g decreases by a small increment, from 0.99 to 0.95. In other words, to within the numerical precision considered, when the histone-DNA attraction decreases past the critical value g = 1.0, the 20-bp DNA tail undergoes all-or-none dissociation. Figure 3 ( b ) shows the dependence on g of the radial distance p L between the end of the DNA tail and the histone surface. The abrupt dissociation jump is evident. The 40-bp tail dissociates in two stages, as demonstrated in Figure 4. We see in Figure 4 ( a ) that about half of the tail nearest the end jumps off the histone surface a t g = 1.3. Little change occurs as g continues to decrease from 1.3 to 1.0. Then, at g = 1.0, the rest of the 40-bp tail suddenly dissociates. The two-tier dissociation is also clearly represented by the pL vs g plot in Figure 4 ( b ) . See the discussion section for comment on the possible significance of this result for nucleosome core particle structure. Euler-Lagrange Analysis. We mentioned that our

PL

“ I

0.9

‘

I

1.0

‘~

I

.

,

1.2

1.1

’

,

1.3

.

,

1.4

g

Figure 4. ( a ) Behavior of a 40-bp DNA tail (labeling as in Figure 2 ) . As g decreases from large values to a value

near 1.31, the DNA stays essentially associated with the histone surface. At this value of g, a segment of about 20 bp jumps off the surface. Release continues progressively until g falls to a value near 1.04, whereupon the rest of the tail is abruptly released. (b) The double transition of Figure 4 ( a ) is clearly seen in this pL vs g plot. In Figures 2 ( a ) , 3 ( a ) , and 4 ( a ) , each path is an equilibrium path, having minimum energy for a given value of the ratio g, which characterizes the relative influence of histone-DNA attraction and DNA elastic bending. In each case the curves computed for the largest value of g are almost coincident with circular arcs of radius ro and thus represent the fully associated tails. Figure 2 shows the gradual dissociation of a 10-bp tail described in the previous section. In Figure 3 ( a ) ,by contrast, we see an abrupt jump of a 20-bp tail from a path close to the histone

numerical research was motivated by-and confirms-a conjecture drawn from an earlier analytical theory? A descriptive review is warranted, since the analysis is evidently capable of yielding some insights. The “result-oriented” reader may, however, skip over at this point to the section on the axial transition. The net energy of the DNA tail is given as an integral by summing Eqs. ( 1)and ( 2). The EulerLagrange equation corresponding to the energy may be derived and turns out to be a third-order, highly nonlinear, differential equation. The solutions of the equation give the function O( s) for equilibrium paths, that is, paths representing local energy minima or maxima. Solution of an already difficult equation is complicated still further by the appearance in it of the histone-DNA potential energy a t the position of the end of the tail. Since the path of the tail is not known beforehand, so that the location of the end of the tail in relation to the histone surface is unknown, this parameter can only be determined in an exact procedure by a self-consistent step after the solution (containing the unknown parameter) is found. Before describing the approximate analysis that was devised, we can gain some useful knowledge from the boundary conditions on the Euler-Lagrange equation. Because this equation is third order, there are three of them. The first, O(0) = 0, is one that we impose ourselves and requires all paths from which the equilibrium path is sought to be initially directed along the horizontal [see Figure 1, for example, where all three paths are drawn this

1550

MARKY AND MANNING

way, and the representation of O( s) in Eq. (7), which satisfies this condition]. The other two boundary conditions arise from the theoretical analysis. They are B’( L ) = O”( L ) = 0, that is, the first two derivatives of B with respect to s must vanish at the end of the DNA tail. Now 0 specifies the direction of the path. If its first derivative, i.e., the rate of change of direction, vanishes at some point s along the path, the path is “straight” at the location. If the second derivative also vanishes, the path is even straighter. So the theory tells us there is a very strong tendency for the elastic line (the DNA tail) to straighten near its end in its equilibrium trajectory. The computed minimum-energy paths in Figures 3 and 4 certainly have the appearance of being straight near their ends. To solve the Euler-Lagrange equation approximately, 0 ( s ) was represented by Eq. ( 7 ) with n = 5, which for technical reasons was the smallest possible number of terms that could be used. For the equilibrium trajectory, that is, for the solution of the equation, the coefficients x i ,i = 2, . . . , 5, could be found as functions of the first coefficient xl. In turn, the value of x1 for the equilibrium path could be determined as the physically relevant root of a ninthorder polynomial. For small values of g, which we recall is a measure of the relative strength of the histone-DNA attraction, the resulting equilibrium trajectories for the DNA tail were very much like those computed by direct numerical means in Figure 3 ( a ) (the Euler-Lagrange method, with the approximation used, did not lend itself to tail lengths greater than 20 bp ) ;as g was increased progressively from zero through small values, the tail was gradually pulled in toward the histone circle. The interesting event occurred when g continued to increase-the equilibrium trajectory stopped a t an intermediate position, like curve 2 in Figure 1. More precisely, the equilibrium trajectories continued to be pulled in further toward the histone circle as g increased, but at a smaller and smaller rate relative to the uniform rate of increase of g, such that a limiting “barrier” trajectory, well outside the histone circle (like curve 2 in Figure 1) , was approac hed. Now it is obvious that as g is allowed to increase without limitation, the trajectory lying entirely o n the surface (curve 1 in Figure 1) must ultimately attain lower energy than any intermediate trajectory (the bending energy “saturated’ on approach to the surface if the Hooke’s law constant b is fixed, but the potential energy trough representing the histone-DNA attraction may, a t least in the mathematical model, be taken as deep as we wish at the

surface). The barrier trajectory, which is well off the histone surface and is the only equilibrium path that could be found for very large g, must therefore mark a position of unstable equilibrium, that is, a local energy maximum rather than the required minimum, which is on the surface. How this situation can lead to an abrupt all-ornone transition is illustrated schematically in Figure 5. The vertical axis of the figure measures the net energy of the DNA tail, positive bending energy plus negative histone-DNA interaction energy. The horizontal axis is a pseudo-coordinate measuring distance of the tail from the histone surface. The lefthand end represents complete wrapping on the surface, like curve 1of Figure 1. The right-hand end is the completely dissociated tail (curve 3 of Figure 1 ) . Intermediate points on the horizontal axis represent intermediate trajectories, like curve 2 of Figure 1. The different curves in Figure 5 are distinguished by different values of g. When g is zero, the lowest energy position of the DNA tail is at the extreme right; it is completely dissociated. For a small but nonzero value of g, there is an energy minimum near, but not at, the right-hand end. This situation corresponds to a stable path for the DNA near curve 3 of Figure 1. As g increases toward a critical valuewhich we take (schematically) as exactly equal to unity in Figure 5-the energy minimum becomes shallow and shifts to the left, indicating a decreasingly stable path further in toward, but still far from, the histone surface (curve labeled 1 - in Figure 5). At the critical g value the energy curve is flat all the way from complete association to complete dissociation. This situation is one of “neutral equilibrium,” in which a broad range of paths all have the same energy. As g is increased past the critical value,

g=o

g < ‘ g = lE=O

g= I g

=

9’

I+

’

d i s t a n c e from s u r f a c e

-

Figure 5. Evolution from stable to unstable equilibrium (see text for explanation).

THE ELASTIC RESILIENCE OF DNA

an energy maximum emerges in a position that remains far from the histone surface. The position of lowest energy is now on the histone surface (lefthand end of the horizontal axis in Figure 5 ) . Therefore as g is increased from a value just less than critical (curve 1- in Figure 5 ) to a value just greater (curve 1+) , the stable path of the DNA tail undergoes an abrupt transition from nearly complete dissociation (right side of horizontal axis) to complete wrapping on the surface (left end of axis). Of course, this description is purely mechanical; in the real system thermal fluctuations will act to blur the transition to some extent.

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1

THE AXIAL TRANSITION Description of the Model

Nucleosomal DNA is about equal to a persistencelength segment, and hence retains elastic resiliency toward bending. Its two turns around the histone octamer core are highly strained, and there is a strong tendency to straighteh. When straightening occurs in the plane of the disklike core, out and away from the surface of the histone, we speak of a “radial” transition. Visualization has been provided in Figures 1-4. The radial transition is the basis for our ideas on partial dissociation of DNA from the histone core. The DNA can also straighten in an orthogonal direction, perpendicular to the page in Figures 1-4, that is, along the cylindrical axis of the histone core. The “axial” transition is illustrated in Figure 6. It will be the basis of a model of disruption of the histone core triggered by DNA straightening. The plane of the page in Figure 6 should be visualized as circumferentially wrapped around the surface of a cylinder of radius ro-the same ro as in the radial transition problem-with the vertical u axis parallel to the axis of the cylinder, while the horizontal u axis curves circumferentially along a base circle of the cylinder. Thus, all vertical lines u = constant (parallel to the u axis) are straight-line generators of the surface of the cylinder, and all horizontal curves u = constant (parallel to the u axis) are base circles of the cylinder. A segment of the DNA is represented by a thick curve in two possible positions on the surface of the cylinder. Curve 1 winds circumferentially. It has zero pitch. It corresponds exactly to curve 1in Figure 1, which simply shows it in another perspective. Curve 2 in Figure 6 begins at the same point and in the same direction as curve 1. Like curve 1,it winds around the cylinder, but with a pitch that rises from zero and increases

U

Figure 6. Geometry of the axial transition. The plane of the page is to be visualized as smoothly wrapped around the cylindrical surface of the histone octamer, so that the ( u,u) “plane” actually represents the curved surface of the cylinder. The u axis, and all lines parallel to it, are circumferences of the cylinder. The u axis, and all lines parallel to it, are generators of the cylinder, parallel to its axis. The DNA segment in position 1 is circumferentially wrapped on the histone surface, approximately (if the pitch is ignored) as in the native structure of the nucleosome core particle. In trajectory 2 the DNA has shifted “axially”; it winds on the cylinder, but with a pitch that increases with winding. For the DNA to assume this path, disruption of histone-histone contacts is required, since the cylinder must be stretched. The angle O( s ) is between two surface tangents; one is tangent to the circumferential direction, and the other is tangent to the path at arc length s along the path.

as winding proceeds, becoming progressively more parallel to a straight-line generator. Curve 2 does not have a representation in Figures 1-4. Indeed, it is important to emphasize that both curves in Figure 6 lie on the cylindrical surface. An “axial transition,” for example, from curve 1 to curve 2, does not involve dissociation of the DNA from the histones. To understand the structural implications of axial straightening of nucleosomal DNA, we return to the device of representing the histone octamer as a stack of four checkers, each representing a dimer of histones. A pair of black checkers, the king, is the (H3/ H4)2 tetramer and is sandwiched between red checkers, the H2A/H2B dimers, at each end. The cylinder of Figure 6 is the curved surface of the stack. In the nucleosome core particle the central portion of the DNA loops once around the black king, and each end of the DNA completes somewhat less than half a turn around each red checker. We assume

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MARKY AND MANNING

environmental conditions such that the ends of the DNA cannot dissociate from the histone stack; not only is there no radial transition, but, further, the DNA is not allowed to run vertically off the top or bottom of the stack. In this case, the axial transition such as curve 1 + curve 2 in Figure 6 must involve separation of the histone checkers along the cylindrical axis; the original intact cylinder “telescopes” to accommodate the vertically straightened DNA. The emerging gap at a checker-checker interface then becomes bounded by a “virtual” cylindrical surface. If the size of the gaps between checkers is small relative to the persistence length of the DNA, which is the case, then it is reasonable to imagine the DNA as wrapped on the virtual parts of the telescoped cylinder just as on the real parts. Thus, axial straightening of the DNA involves disruption of histone-histone contacts, or, in our rough model, the formation of gaps between checkers. If histone-histone association is relatively strong, the axial transition is inhibited, and the native structure of the nucleosome core particle (intact stack of checkers) is maintained. If the histone-histone stabilizing forces are weakened sufficiently, the axial transition can sunder the core particle, separating dimer from hexamer, two dimers (red checkers) from the central tetramer (black king), or heterotypic tetramer (mixed red/ black king) from heterotypic tetramer. To proceed in quantitative fashion, the energetics of the system must be formulated. The various possible trajectories of the DNA, all of which are wrapped with varying degrees of pitch on the curved surface of a cylinder, are stressed by bending energy. This part of the total energy is given by Hooke’s law in Eq. ( 1) ,just as for the problem of the radial transition. The trajectory with greatest bending stress corresponds to the native conformation, which is taken as curve 1 in Figure 6. By neglecting the shallow pitch of native nucleosomal DNA, just as we did for the radial transition, we therefore have the native DNA trajectory winding circumferentially, that is, along the u axis in Figure 6. The origin of coordinates in Figure 6 could be near a red/ black checker interface, in which case the portion of the DNA shown is about 30 bp and corresponds to the segment wrapped primarily on the red checker (H2A/H2B dimer). Or it could be near the black/ black checker interface, with segment length of about 73 bp. (An axial transition in the first case would separate a dimer of histones from a hexamer, while two symmetrical transitions in the second case would cause rupture of heterotypic tetramers.) The minimum of potential energy of histone-his-

tone association is also taken as the native structure of the core particle, with the DNA positioned along a base circle ( u axis in Figure 6 ) bounding the interface between two checkers. As the DNA straightens axially, from curve 1toward something like curve 2 in Figure 6, the checkers separate. The histonehistone restoring forces are transmitted to the DNA, which is then impeded in its straightening. The bending energy of the DNA decreases, but it must work against histone-histone association forces, and its potential energy from the latter source increases. ( A computer simulation on the atomic level would, of course, not describe the energetics this way.) The right-hand side of Eq. ( 2 ) can be adopted for this contribution to the total energy of the DNA. Analogy with Eq. (4)allows a tractable model for the net histone-histone potential energy function,

A trough, or ramp, of potential energy is constructed with base circles of the cylinder as isopotential curves. A DNA segment of length ds centered at distance s along the DNA trajectory has potential energy - ywds. Each such segment along trajectory 1 in Figure 6 has the minimum energy, or maximum negative energy, - y d s ; the entire trajectory runs along the bottom of the energy trough. The segments in curve 2 have increasing potential energy with increasing distance along the curve, since the height u above the u axis increases with distance along the curve. Equation ( 9 ) is not meant to be on the same level of accuracy as the potentials that would go into a high-resolution computer simulation. The remarks following Eq. (4)may be consulted here. Note that the parameter 6 in Eq. ( 9 ) has the dimensions of inverse square length, whereas in Eq. (4),it is dimensionless. As in Eq. ( 4 ) ,however, it is a measure of the width of the energy trough, and we have elected to use the same symbol in both formulas. With Eqs. ( l ) ,( 2 ) , and (9), the axial transition problem may be formulated within the context of the Euler-Langrange procedure? We present the principal result, then outline the analysis. The Axial Transition Has All-or-None Character

Define a dimensionless ratio h,

a measure of the resistance raised by the histone-

THE ELASTIC RESILIENCE OF DNA

histone association forces to axial straightening of the DNA. The Euler-Lagrange method yields an exact r e ~ u l tWhen .~ h > 5.174, the elastically stressed native configuration of the DNA (along the circumference of the cylinder in our model) is a stable equilibrium path regardless of its length (for example, we mentioned 30 bp and 70-75 bp as possibly relevant lengths of portions of the DNA). When h drops below this critical value, DNA segments greater than 18bp abruptly become unstable in the native configuration. If this event occurred at the interface between red and black checkers, then the 30 bp wrapped on an H2A/ H2B dimer would undergo an axial transition, severing the dimer from the remaining hexamer of histones. If it occurred at the plane of dyad symmetry, then both symmetrically disposed 73-bp halves of the DNA would undergo axial transitions, disrupting heterotypic tetramers. There is no claim that this result of elasticity theory is obvious. It may be illustrated dramatically, however, with the aid of a flexible knitting needle deformed to fit circumferentislly inside a cylindrical cardboard carton. The curious reader must, however, hold this “apparatus” well away from anyone’s eyes, including his or her own. This is not a joke! Perhaps depending on the material from which the needle is made, it can be placed under enormous stress when thus deformed. In our hands the axial transition (in which the sharp end of the needle pokes out straight) is faster than the eye can follow. It is safer to be a pure theoretician. Is the nucleosome core pafticle near the critical state h = 5.174 under physiological conditions? We try a rough calculation. For the Hooke’s law bending constant b that appears in h, we use the value corresponding to the DNA persistence length, 160 bp in 0.1M salt. For 6, we take a value that makes the potential energy function w in Eq. ( 9 ) drop to the fraction e-l of its maximum value unity at distance u = 5.6 A, with the rationalization that the association energy between histone dimers (checkers) largely decays when each checker is hydrated with a single layer of water molecules. The radius ro of the cylinder is set at 43 A, the approximate distance from its center to the duplex axis of the DNA. When we use these estimates in the critical condition h = 5.174, together with the defining Eq. (10) for h, we arrive at a numerical value for y, the depth of the potential energy trough running around the circumference of the checker-checker interface. Finally, we multiply y by the circumference 27rro of the interface and obtain the prediction of 2.0 kcal mol-’ for the critical (minimum) bond strength re-

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quired to maintain integrity of the interface against an axial transition. If environmental perturbations, histone modification, or invasion of another protein (high-mobility group proteins, RNA polymerase), can weaken the histone-histone attraction below the critical strength, then there will occur an abrupt allor-none disruption of the histone contacts, driven by axial DNA straightening. That this value for the critical histone-histone attraction is in the range of physiological (noncovalent) energies is reassuring. A question of interpretation arises, however. The passage from van Holde’s book3 quoted in the Introduction notes that the histone octamer, in the absence of DNA, is not stable at physiological salt concentrations. Yet we are attributing suppression of the axial transition to histone-histone attractive forces. The resolution of this apparent inconsistency is almost surely to be found in the discussion by Camerini-Otero and Felsenfeld‘ of histone-histone association thermodynamics. Dissociation of histones in the absence of bound DNA is driven by a very large gain of translational entropy, as the separate histones can each sample the entire volume space of the solution. This effect is almost entirely suppressed if the dissociating histones remain bound to the same short segment of DNA, as is the case in our model. Another consideration, probably with smaller energetic consequences,‘ is the mitigation by phosphate groups of repulsive ionic forces among histone-positive charges in the DNA-bound complex. We pass now to an outline of the derivation of these results. The reader who does not wish to inspect this level of detail may skip to the Discussion, which attempts to interpret specific experimental data in light of the theory. Outline of Derivation of the Critical Condition

The Euler-Lagrange equation for this problem may be d e r i ~ e d . ~It- l is ~ yet more complicated than the equation for the radial problem, because now additional terms are generated by the curvature of the cylindrical surface. Still, the equation does describe the behavior of the angle of inclination tl( s) of equilibrium trajectories of the DNA (see Figure 6 ) , and the problem can be simplified once it has been recognized that identically vanishing 0 is always a solution, regardless of the length of the DNA, or of the depth and width of the potential energy trough. To be more precise, if the starting direction is circumferential, then continued winding along the circumference of the cylinder for the entire length of the trajectory [curve 1 in Figure 6, i.e., identically

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MARKY AND MANNING

vanishing O( s ) ] always gives an equilibrium path for the DNA. The question then becomes one of stability. Is the circumferential path a local energy minimum or a maximum? Are we dealing with stable or unstable equilibrium? If we restrict attention to the question posed, we may search only among trajectories O(s) lying near the circumferential one, so that O( s ) is always very small’. Great simplification of the Euler-Lagrange equation results, and it turns out to reduce to a linear fourth-order differential equation with constant coefficients,

+

+

(d4y/ds4) 2r02(d2y/ds2) ysb-’y = 0

(11)

where y ( s ) is an auxilliary function such that dy/ ds = L-’8( s), and L is the length of the trajectory (i.e., of the segment of DNA in question). The solutions of this equation have the form

+

+

+

y ( s ) = Alea1’ A2eazs A3ea3S A4ea4S (12)

where { Ai } is a set of four constants of integration, and { ui ] is the set of four roots of the characteristic equation for Eq. ( 11) . For the result of the previous section, these roots consist of two pairs of complex conjugates. They contain what we know of the structure of the core particle; that is, they are explicitly known functions of ro and y6/ b. There are four boundary conditions, two at s = 0 and two at the end of the trajectory s = L , which we do not list here. Substitution of Eq. (12) into any of the boundary conditions yields a linear algebraic equation with unknowns {Ai).The four boundary conditions then generate a set of four simultaneous linear equations in the four unknowns { Ai} 9

MA=O

(13)

where A is a 4-vector with elements { Ai}, 0 is a 4vector with zero as each element, and M is a 4 X 4 matrix with elements that are functions of the characteristic roots { ui}and the length L of the trajectory. The first thing to notice is that A = 0 is always a solution of Eq. (13). Thus, from Eq. (12), identically vanishing y (s), hence identically vanishing O( s), that is, the circumferential trajectory, is always a solution of the Euler-Lagrange equation for equilibrium paths. The problem now settles into an extended investigation’ of the properties of det ( M ), the de-

terminant of M . D e t ( M ) can be regarded as a function of the trajectory length L , which contains as a parameter the dimensionless ratio h defined in Eq. ( 10). We recall that h measures the strength of the potential energy ramp along the circumference of the cylinder, which, if “sturdy” enough, holds the DNA to a circumferential path against its tendency to straighten in the axial direction. Indeed, analysis shows that for sufficiently large h , det ( M ) does not vanish for any length L ; no value of L is a root of the equation det ( M ) = 0 if h is large. Therefore, if h is large, the solution A = 0 of Eq. (13) is the only solution. The circumferential trajectory, as the only equilibrium path, must be stable. A sufficiently strong ramp holds the DNA, no matter how long, to a circumferential path against its tendency (for lengths less than a persistence length) to straighten. Numerical analysis reveals a critical value of the ramp strength, h = 5.174 (the value indicated in the previous section). For ramp strengths greater than critical, there is no root L of the equation det ( M ) = 0. For h equal to the critical value, there is such a root, L = 1.418r0, or about 18 bp if the radius of the cylinder is 43 A. In this critical condition the solution { Ai } = { O,O,O,O } of Eq. ( 13) is not the only solution. There is a continuum of nonzero solutions, corresponding to trajectories that lie near, but not on, the circumference of the cylinder. All of them have net energies equal to that of the circumferential trajectory. When h drops to a value just less than critical, these noncircumferential paths have energies just less than the circumferential path. The circumferential path has become unstable, and there is a transition from it to a stable path with less curvature. In Figure 6 there is an abrupt shift of stability from curve 1 to curve 2 as the ramp strength decreases past the critical value. In the case, curve 2 in Figure 6 is to be visualized as lying near curve 1, and both curves as representing 18-bp segments of DNA. What happens if the DNA segment is 30 bp in length, or even a complete turn, about 80 bp, lengths that, as discussed above, would appear to be relevant to core particle transitions? If the ramp strength is just above critical, either length, or, for that matter, any length, is stable when circumferentially wound. But when the ramp strength drops slightly, to a value just less than critical, these segments abruptly become very unstable, since they are substantially longer than 18 bp, which now has abruptly become the maximum length for circumferential stability. There is an abrupt transition from curve 1to curve 2 in Figure 6, where these curves both represent either 30-bp, or 80-bp, segments, and where curve 2 is very far from being circumferential (except at its

THE ELASTIC RESILIENCE OF DNA

starting point). A large-scale axial transition has occurred .

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to correspond to Simpson’s experiments, in that 20 bp are also involved in the experiment. In Figure 3, a 20-bp segment dissociates because we assumed a point of anchor a t its base, attributing the reason DISCUSSION for the existence of the anchoring point to a site of particularly tight binding. But the two-stage disWithin the context of a model based on the classical sociation observed in Figure 4 for a 40-bp tail implies elastic bending behavior of a thin rod, we have prean alternate possibility. There is no point of anchor sented theoretical evidence for all-or-none radial and at 20 bp in Figure 4; the anchoring point is at the axial conformational transitions of the nucleosome base of the 40-bp tail segment. The initial dissociacore particle triggered by the emergence of elastic tion of 20 bp is entirely the result of the complex, instability at critical values of parameters that conelastically unstable behavior of our model. Simpson trol structure (the core particle radius r o ) , DNA may have observed dissociation of 20 bp, not because stiffness (the Hooke’s Law constant b ) ,and histonea tight binding site exists a t 20 bp and prevents furhistone and histone-DNA interactions ( y and 6 ) . ther dissociation, but rather because of the propMore briefly, the tendency of DNA to straighten erties of elastic instability disclosed by our calcucan create instabilities leading to sudden disruption lations. of the core particle. Van Holde3 has indicated (see Unequivocal examples of an axial transition are the Introduction) why there is interest in core parharder to find. The most likely case may be nucleoticle conformational transitions. We take up here some unfolding induced by ethidium, as studied by the experimental situation as it currently exists. Wu et al.15These authors found that a low level of The work of Simpson, l 3 reviewed by van H ~ l d e , ~ ethidium binding drives a conformational change, provides an example of a well-defined structural which they indeed were able to model, in present transition. As a first step in the thermal disruption terminology, as an axial transition. The DNA asof core particles, about 20 bp at each end of the sumes an extended superhelical form with large DNA are released from the intact histone octamer pitch, presumably accompanied by disruption of the over a narrow range of temper’atures (see Fig. 4 of histone octamer. Unlike what happens at higher Ref. 13). The released DNA tails are in the native ethidium binding levels, l6 the DNA remains assoDNA duplex form; optical melting to single strands ciated to the core histones. The authors argue from occurs only a t higher temperatures. a detailed topological and thermodynamic analysis We propose that our radial transition underlies that the axial transition requires untwisting of the Simpson’s observations. With reference to the defDNA duplex, which is energetically unfavorable. inition of the ratio g in Eq. ( 3 ) , we can go further. Since intercalation of ethidium untwists DNA at Since increased temperature makes the DNA more the binding site, it compensates for the unfavorable flexible,l4 that is, lowers its persistence length and energy from this source, and the axial transition is stiffness constant b, the transition cannot be driven then spontaneously driven by the release of superby the effect of temperature on the elastic resilience helical stress as the DNA straightens toward the of DNA; this effect would increase g, whereas what extended superhelix. Wu et al. also point to a posis needed is a decrease of g from values greater than sible complication-the observation that detectable its critical value near unity to values just less than structural change does not occur below a threshold critical. In physical terms, the radial transition is level of ethidium binding. driven by the straightening of DNA, but an increased Although the effects of twisting energetics are not temperature decreases the tendency of the DNA to explicit in our elastic theory, we can accommodate straighten. Therefore, the triggering mechanism the interpretation of Wu et al. in our description of must be a decrease in strength of the histone-DNA the axial transition. The energetically unfavorable binding forces with increased temperature. The pauntwisting is a structural consequence of the axial rameter y in g must decrease as the temperature transition. In other words, it is a consequence of increases, and there is abrupt release of the DNA extension in the axial direction. Therefore, it genends at the temperature that causes g to drop below erates an effective force opposing axial extension, unity. Of course, in the experiment, the transition just like the histone-histone attractive contacts that would not be as sharp as theotetically predicted; it are disrupted by the transition. Its energy may be is blurred somewhat by thermal fluctuations, i.e., taken as incorporated in the product y6 of the paBoltzmann statistics. rameter h in Eq. ( l o ) , along with the unfavorable In our Figure 3 we have given the results of calenergy of disrupted histone-histone contacts. We culations for a 20-bp tail. This figure would appear arrive thus a t the interpretation of Wu et al., but

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MARKY AND MANNING

with the additional feature that, as observed, the transition occurs abruptly when yS is decreased below a threshold value (corresponding to h = 5.174), presumably as a consequence of sufficient ethidium binding. There is in fact a growing body of literature on the possible role of torsion, or twisting stress, in activating genes for transcription, and we would therefore like to pursue this point. As noted, we have not yet attempted an explicit treatment of twisting energy in our model, but have confined ourselves, insofar as the elastic component of energy is concerned, to bending. We found, however, that elastic instability and consequent disruption of the core particle can be triggered by slight weakening of histone-DNA and/ or histone-histone contacts. One obvious way to weaken a contact between, say, a DNA phosphodiester group and a positively charged residue on the octamer surface, would be to break it by twisting (postively or negatively, i.e., untwisting) the DNA. Alternately, the ion pair may be effectively rigid. A twist of the DNA would then be transmitted through this rigid contact to the histone octamer, where it might weaken favorable histonehistone interactions. The regime of elastic instability of the native structure could be entered and abrupt all-or-none disruption ensue. The interpretation by Wu et al. of the energetics of their ethidium-driven axial transition, and our admendment to it, can be framed thus. We site two further experiments, which, taken together, are consistent with this suggestion. A low level of temperature-induced twisting stress (or more precisely, superhelical stress) has been found to have no effect on the structure of the nucleo80me.l~But the effect of a large twisting stress caused by DNA gyrase has been interpreted1’ as release of long DNA end segments from the histone octamer surface or disruption of the octamer. These observations are consistent with a threshold, or critical, value of the twisting stress at which an allor-none radial or axial transition occurs. Other conformational transitions of the core particle are observed to be induced by high salt and by low salt? We have attempted to sort through the various factors but have reluctantly concluded that the experimental literature currently lacks the internal consistency needed for systematic theoretical analysis. For the low-salt transition this situation is a pity, since “all workers agree that there is an abrupt conformation change a t about 1 mM salt.”3 Depsite intensive study, however, there is no agreement on the nature of the transition. In this situation we mention the low-salt electric dichroism measurements performed in Crothers’

laboratory 1920 simply as an example of how our concepts from elasticity theory can be applied. Both 146-bp and 175-bp nucleosomes were proposed to open out to expanded disks (about the same height as the native form, but a larger diameter), with about one superhelical turn of DNA instead of nearly two for the native structure. The driving force could be stiffening of the DNA at low salt?l Energetically unfavorable distortion of the histone proteins is thought to hinder the transition. The conformational changes proposed can be interpreted in present terms as a radial transition; the radially straightening DNA, instead of dissociating from the histone surface, drags the histones with it against the forces of histone-histone attraction. The 175bp nucleosome undergoes a further transition at salt less than 1 mM, apparently to an axially extended superhelical form (like the structure stabilized by ethidium) . A shift from expanded disk to extended superhelix would be a good example of what we have in mind as an axial transition. The model of Uberbacher et al?’ for the low-salt transition provides another convenient example. These authors argue that end release of the DNA from an essentially intact octamer, in a manner similar to Simpson’s thermal end release, is the event most consistent with their SANS measurements. The picture is thus of a radial transition, with rupture of DNA from the histone surface. Why should a decrease of salt below a threshold value induce release of DNA from favorable ionic contacts with the histone surface, when it is known that low salt strengthens ionic interactions (less screening) ? To answer this question, it is essential to emphasize the decisive role of g , the ratio of histone-DNA interaction strength to the stiffness of DNA [ Eq. ( 3 ) ], in the radial transition. The numerator can increase, but if the denominator increases more, then g can still decrease below its critical value, triggering the radial instability. Indeed, all current data on, and theories for, the persistence length of DNA recognize a steep increase in stiffness at very low ionic strength (see the Appendix of Manning23for references). There is another component of the structural energetics of the core particle that we have not emphasized despite its possible importance. In the native structure the two superhelical turns of DNA are close enough to interact, presumably through ionic repulsion. If, for ease of discussion, we represent the two turns as congruent with axially displaced circles on a stack of checkers, the repulsive force between DNA turns is purely along the axial direction and acts to counter the net histone-histone attractive forces between checkers. The energy parameter y in Eq. (10) must be regarded generally as a com-

THE ELASTIC RESILIENCE OF DNA

posite quantity that measure a balance among the variety of histone-histone interactions, and possibly reflects DNA torsion effects as well, Here, we see that the balance must be extended to include DNADNA repulsion. The overall net effect must be attraction along the axis under native conditions (positive net value of y ) , for otherwise the native structure would not exist. Repulsion between superhelical turns could also be important in the radial transition. When a DNA segment on one turn is radially displaced from the surface of the checker stack, a radially acting repulsive force component arises on it from the DNA on the other turn, countering the histone-DNA attraction. The energy parameter y in Eqs. ( 2 ) and ( 3 ) is therefore also a net value. To see how ionic repulsion between DNA turns can affect interpretive discussion, let us return to Simpson’sthermal end re1ea~e.l~ We concluded that increased temperature can trigger a radial transition by weakening direct histone-DNA attractive contacts. Now we must entertain the possibility that temperature increases repulsion between superhelical turns. Since purely ionic interactions in water are only minimally dependent on temperature, this consideration seems unlikely. On the other hand, both radial and axial transitions induced by lowering the ionic strength can be driven by enhanced ionic repulsion between turns as well as by decreased DNA bendability . With the exception of end release of DNA, much of what has been discussed in this section must involve disruption of the histone octamer. Current thought focuses on the possibility, and even ease, of distortion and separation of the interfaces between H2A/H2B dimers and the central (H3/H4), tetramer. In concluding, we therefore find it appropriate to recall some remarks from the review of nucleosome structure by Uberbacher and B ~ n i c k . ~ Noting the biochemical evidence [ which] has long indicated that partial or complete dissociation of 1 H2A/ H2B] dimers plays a key role in the “activation” of nucleosomes, and that this is perhaps affected by high mobility group protein interactions or other modifications of nuclesosomes, the authors attach significance to the differing degrees of interaction of dimers and (H3/ H4)2 tetTamers in three separate x-ray diffraction studies of the histone octamer structure. “For the first time, there is detailed structural information documenting the potentially dynamic nature of this interaction.”

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We have tried in our presentation to suggest how this dynamism can be controlled by the incipient elastic instability of the core particle. We are grateful to D. M. Crothers, G. Felsenfeld, and K. E. van Holde for valuable comments on the manuscript. Our research has been supported in part by NIH Grant GM36284.

REFERENCES 1. Richmond, T. J., Finch, J. T., Rushton, B., Rhodes, D. & Klug, A. (1974) Nature 311,532-537. 2. Uberbacher, E. C. & Bunick, G. J. (1989) J . Biomol. Struct. Dynam. 7, 1-18. 3. van Holde, K. E. (1989) Chromatin, Springer-Verlag, New York. 4. Widom, J. ( 1989) Ann. Rev. Biophys. Biophys. Chem. 18,365-395. 5. Uberbacher, E. C. & Bunick, G. J. (1988) Comments Mol. Cell. Biophys. 4, 339-348. 6. Camerini-Otero,R. D. & Felsenfeld, G. (1977) Nucleic Acids Res. 4 , 1159-1181. 7. Manning, G. S. (1985) Cell Biophys. 7, 177-184. 8. Manning, G. S. ( 1988) Phys. Rev. A 38, 3073-3081. 9. Manning, G. S. (1990) Quart. Appl. Math. 4 8 , 517525. 10. Manning, G. S. (1987a) Quart. Appl. Math. 4 5 , 5 1 5 527. 11. Manning, G. S. (198713) Quart. Appl. Math. 45,808815. 12. Nickerson, H. K. & Manning, G. S. (1988) Geometriae Dedicata 2 7 , 127-136. 13. Simpson, R. T. (1979) J . Biol. Chem. 254, 1012310127. 14. Gray, H. B., Jr. & Hearst, J. E. (1968) J . Mol. Biol. 35,111-129. 15. Wu, H. M., Dattagupta, N., Hogan, M. & Crothers, D. M. (1980) Biochemistry 19,626-634. 16. McMurray, C. T. & van Holde, K. E. (1986) Proc. Natl. Acad. Sci. U S A 83,8472-8476. 17. Morse, R. H. & Cantor, C. R. (1985) Proc. Natl. Acad. Sci. U S A 82,4653-4657. 18. Garner, M. M., Felsenfeld,G., O’Dea, M. H & Gellert, M. (1987) Proc. Natl. Acad. Sci. USA 84,2620-2623. 19. Wu, H. M., Dattagupta, N., Hogan, M. & Crothers, D. M. (1979) Biochemistry 18, 3960-3965. 20. Schlessinger,F. B., Dattagupta, N. & Crothers, D. M. ( 1982 ) Biochemistry 2 1,664-669. 21. Manning, G. S. (1981) Biopolymers 20, 1751-1755. 22. Uberbacher, E. C., Ramakrishnan, V., Olins, D. E. & Bunick, G. J. ( 1983) Biochemistry 22, 4916-4923. 23. Manning, G. S. (1983) Biopolymers 22,689-729.

Received June 18, 1991 Accepted July 30, 1991