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REVIEWS OF MODERN PHYSICS, VOLUME 79, JULY­SEPTEMBER 2007

Structure and interactions of biological helices
Alexei A. Kornyshev*
Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, United Kingdom

Dominic J. Lee
Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, United Kingdom

Sergey Leikin
Section of Physical Biochemistry, National Institute of Child Health and Human Development, National Institutes of Health, DHHS, Bethesda, Maryland 20892, USA

Aaron Wynveen§
Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, United Kingdom

Published 6 August 2007
Helices are essential building blocks of living organisms, be they molecular fragments of proteins -helices , macromolecules DNA and collagen , or multimolecular assemblies microtubules and viruses . Their interactions are involved in packing of meters of genetic material within cells and phage heads, recognition of homologous genes in recombination and DNA repair, stability of tissues, and many other processes. Helical molecules form a variety of mesophases in vivo and in vitro. Recent structural studies, direct measurements of intermolecular forces, single-molecule manipulations, and other experiments have accumulated a wealth of information and revealed many puzzling physical phenomena. It is becoming increasingly clear that in many cases the physics of biological helices cannot be described by theories that treat them as simple, unstructured polyelectrolytes. The present article focuses on the most important and interesting aspects of the physics of structured macromolecules, highlighting various manifestations of the helical motif in their structure, elasticity, interactions with counterions, aggregation, and poly- and mesomorphic transitions. DOI: 10.1103/RevModPhys.79.943 PACS number s : 87.15. v

CONTENTS
I. Introduction II. Structure A. Biological helices B. Helical structure factors C. Ideal helices D. Discovery of DNA structure E. Nonideal helices: Helical coherence length F. Effect of assembly on DNA structure G. Bent and supercoiled helices H. Summary and comments III. Elasticity A. Elastic rod theory B. Thermal motions C. DNA elasticity D. Summary and comments IV. Electrostatics 944 945 945 945 947 947 947 948 949 950 950 950 951 951 952 952

*Electronic address: a.kornyshev@imperial.ac.uk


Electronic address: dj.lee@imperial.ac.uk Electronic address: leikins@mail.nih.gov § Electronic address: a.wynveen@imperial.ac.uk
0034-6861/2007/79 3 /943 54 943

A. Zwitterionic helix in a nonpolar dielectric medium B. Charged cylinders in an electrolyte solution C. Counterion condensation and adsorption on DNA D. Charged helix in an electrolyte solution E. Summary and comments V. Pair Interactions A. Parallel molecules with arbitrary charge patterns B. Parallel zwitterionic helices in a nonpolar environment C. Parallel charged cylinders in an electrolyte solution 1. Mean-field results 2. Wigner crystal model 3. Standing charge-density waves 4. Counterion fluctuations D. Parallel, rigid, ideal helices in an electrolyte solution 1. Mean-field results 2. Counterion correlations and fluctuations E. Parallel nonideal helices F. Homology recognition in genetic recombination G. Skewed helices: Chiral interactions H. Supercoiling I. Nonelectrostatic forces J. Summary and comments VI. Columnar, Nematic, and Cholesteric Assemblies A. Donnan equilibrium

952 953 955 956 957 958 959 960 960 960 961 962 963 963 963 965 966 968 969 971 971 972 972 972

©2007 The American Physical Society


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973 974 975 976 978 981 982 983 985 985 985 987 988 989 989 990 991

B. Cell model for charged, cylindrical rods C. Helices: Azimuthally dependent interactions and correlations D. A new look at old pictures: X-ray evidence of strong azimuthal correlations E. Cholesteric aggregates F. Predicted and measured forces: DNA, guanosine, and collagen G. Integral screw puzzle H. Electrostatics of the B-A transition in DNA I. XY models of mesomorphic transitions in DNA aggregates J. Summary and comments VII. Counterion-Induced DNA Condensation A. Experimental observations B. Counterion-correlation models C. Electrostatic zipper model D. Summary and comments VIII. Conclusions and Outlook Acknowledgments References

I. INTRODUCTION

Genetic recombination, packaging of DNA in cells and viruses, folding of proteins, and assembly of the organic matrix of bone are just a few of the many fundamental biological reactions that involve interactions between helical macromolecules. Some of these reactions, such as self-assembly of collagen fibers, can be reproduced in a test tube without the complex biological machinery. Other reactions, like condensation and decondensation of chromosome material, are controlled by multiple factors in cells. But even the latter reactions ultimately depend on the underlying physics of helixhelix interactions. The physics of interactions between biological helices is surprisingly rich and often counterintuitive. Collagen self-assembles into highly ordered fibers at elevated temperature instead of denaturing. DNA forms a variety of liquid crystalline phases. In the presence of Mn2+ but not Ca2+ or Mg2+ , DNA liquid crystals self-assemble from solution upon heating, just like collagen fibers. A variety of polyamines and basic polypeptides also induce "condensation" of DNA from solution into liquid crystals, but the temperature does not play an important role in this case. In low-density, highly hydrated liquid crystals, DNA packing is generally cholesteric. The average orientation of the molecules is perpendicular to the cholesteric axis and rotates around this axis with a period equal to the cholesteric pitch usually several microns . The cholesteric packing is caused by chiral interactions between the helices. One would expect such interactions to strengthen with increasing density of the molecules, resulting in a monotonically decreasing cholesteric pitch. Instead, the cholesteric pitch goes through a minimum and begins to increase again. A further increase in the density causes a transition from the cholesteric to the columnar hexagonal or line-hexatic phase even though
Rev. Mod. Phys., Vol. 79, No. 3, July­September 2007

the molecules are still separated by more than a nanometer of water. When almost all water is squeezed out, a conformational transition from the B into the A structure of DNA takes place. Electrostatic interactions between DNA at such densities become very strong, but the charge density of DNA upon the transition into the A conformation does not decrease. Instead, it increases by almost 30%. Studies of liquid crystalline phases of DNA, collagen, -helices, helical viruses, and so on, as well as measurements of the chemical potential vs separation between molecules within them, revealed many more surprising phenomena and produced a wealth of experimental data. But our understanding of the underlying physics is clearly lagging behind these experiments. For instance, the possible physical mechanisms of DNA condensation into liquid crystals by counterions and the nature of the short-range exponential forces between DNA molecules observed in these liquid crystals are still debated in the literature. At least in these debates, the problem is one of choice between different models. The lack of understanding is more severe, for example, for the cholesteric to hexagonal phase transition, for which a few speculative ideas but no molecular models have been proposed. Recent advances in theory and simulations suggest that the key to the physics of these phenomena might be in understanding the relationships between intermolecular interactions and the structure of the molecules and/or counterions adsorbed onto them . In retrospect, this might seem obvious; for instance, the chirality of DNA liquid crystals is a direct consequence of the chiral structure of the double helix. Nevertheless, even now some of the most popular models represent DNA by a homogeneously charged cylinder. This review is an attempt to analyze these recent advances within a general theoretical framework that incorporates models proposed by different authors as special cases including the cylinder-based models . Our goal is to elucidate the most productive ideas through a systematic comparison of the corresponding predictions with all relevant empirical knowledge rather than hand picking a few measurements that fit the best . We focus on electrostatic interactions since much more work has been done in this area, and at least some consensus appears to be emerging. We only briefly discuss other interactions hard core, van der Waals and hydration , but we point out their potential contributions whenever necessary. DNA is the central object of the review, because its analysis is more amenable to rigorous theory and because most of the empirical information was accumulated for it. Nevertheless, we do discuss models and experimental observations for other biological helices -helices, collagen, and guanosine as well. The general theoretical framework employed here is based on the formalism of helical structure factors developed by Crick over 50 years ago Cochran, Crick, and Vand, 1952; Crick, 1953a, 1953b; Klug et al., 1958 . The Crick theory converted the art and magic of model building into an exact science for rigorous analysis of x-ray diffraction from noncrystalline aggregates of heli-


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ces. It allowed Watson and Crick 1953 to decipher the structure of DNA from the x-ray data reported by Franklin and Gosling 1953 and Wilkins et al. 1953 . This theory laid the foundation for the revolution in modern biology and medicine, but it did not find its way into the physics of helices until the last decade. The review is structured as follows. In Sec. II, we introduce the Crick structure factors and generalize them for nonideal helices. We briefly discuss the discovery of the DNA structure and point out some features of the Franklin and Gosling diffraction pattern that were not noticed at the time, although they contain important information about intermolecular interactions. In Sec. IV, we focus on the electrostatics of an isolated helix, relate modern theories of counterion condensation to counterion structure factors, and discuss available experimental data. We address different models of pair interaction potentials and phenomena related to the interaction between just two helices in Sec. V. In Sec. VI, we analyze models of multimolecular, liquid crystalline assemblies for which most of the experimental observations were reported. We reconsider the interpretation of the classical DNA diffraction patterns and provide a detailed comparison of the predictions of different models with measured intermolecular forces. We also briefly describe advances in the statistical mechanics of such assemblies, which might present some additional interest due to the unusual, frustrated form of the underlying interaction potentials. In Sec. VII, we concentrate on a critical comparison of different proposed mechanisms of counterion-induced DNA condensation. We briefly depart from the main theme in Sec. III to introduce the elasticity theory of helical macromolecules. This area of research has been rapidly pushed forward by recent advances in single-molecule manipulation techniques. For interested readers we provide several references to some of the latter studies, but we describe only those aspects of helix elasticity that are essential for understanding the physics discussed in subsequent sections. In an effort to keep the analysis self-contained, whenever practical we provide the derivations in the main text of the review. However, to simplify the task for readers more interested in the physics than mathematical details of the theory, the derivations and formulas that are too cumbersome or elaborate are described in supplementary material refer to the EPAPS Document at the end of the Reference section .
II. STRUCTURE A. Biological helices

FIG. 1. Atomic structure of a a sample polypeptide -helix, b a double-stranded helix of B-DNA, and c triple-helix collagen. The gray ribbons are guides to the eye showing the protein or DNA helical backbones. The two grooves separating the sugar-phosphate backbone strands ribbons in DNA are often referred to as minor and major grooves based on their size in the B form of DNA.

Many biological macromolecules and their structural domains consist of one or several interwoven helical chains of atoms helical strands . Examples of some of the most common molecular helices are shown in Fig. 1. The connected chain of polypeptide backbone atoms forms a single, right-handed helical line in a polypeptide -helix Fig. 1 a . Polyaminoacids and polypeptides may have a pure -helical conformation. More comRev. Mod. Phys., Vol. 79, No. 3, July­September 2007

monly, however, shorter -helices form the building blocks of proteins. Two sugar-phosphate chains form the backbone of DNA, the macromolecular carrier of the genetic information Fig. 1 b . The two strands are connected together via hydrogen bonds between the nucleotide "side chains." The hydrogen-bonded pairs of nucleotides base pairs are stacked, forming the inner core of the molecule Fig. 1 b . Depending on its environment and mechanical strain, the DNA double helix may have several different helical forms, of which the most common are right-handed A- and B-DNA, and the most peculiar is the left-handed Z-DNA. Several different helical conformations of DNA may even coexist as domains of the same long molecule Ha et al., 2005 . Collagen Fig. 1 c is a triple helix formed by three interwoven, left-handed polypeptide chains connected together through hydrogen bonds between backbone amide and carbonyl groups. Collagen triple helices selfassemble into fibers which form tendons, ligaments, and the organic scaffold of bone, skin matrix, and other structures of connective tissues. More complex supramolecular helices are formed upon self-assembly of small molecules. For instance, self-assembly of guanosine-phosphate nucleotides produces a four-stranded guanosine helix, which mimics the structure of chromosome telomeres. Some proteins selfassemble into multimolecular helices such as microtubules and actin filaments in cytoskeleton. Some viral particles e.g., tobacco mosaic virus TMV are also multimolecular helices formed by self-assembly of several different proteins, encapsulating a nucleic acid in the viral particle core.
B. Helical structure factors

A rigorous description of helical macromolecules in terms of their structure factors in reciprocal space was


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Kornyshev et al.: Structure and interactions of biological helices

FIG. 2. X-ray-diffraction pattern from calf thymus B-DNA Franklin and Gosling, 1953 showing the location of layer lines, Eq. 9 , on the scattering wave-vector axis. The lines of constant kz thin solid lines are slightly curved due to the specific geometry of x-ray film mounting. From Franklin and Gosling, 1953.

proposed by Cochran, Crick, and Vand 1952 to describe x-ray scattering from -helical polypeptides. It was crucial for the discovery of the DNA structure by Watson and Crick 1953 and formed the basis of modern crystallography of helical macromolecules. In this section we describe the helical structure factors in the context of a generalized form of the CochranCrick-Vand CCV theory for x-ray scattering from helical macromolecules. In Secs. IV and V we utilize the same expressions in a different context and demonstrate that these structure factors and structural parameters also determine structure-dependent physical properties and interactions between macromolecules. The x-ray scattering intensity of an ensemble of macromolecules is given by Ik =
, i,j

FIG. 3. Convention used for x-ray analysis of helical molecules. The location of each molecule is given by the lateral R and axial coordinates Z in the laboratory frame X , Y , Z . The local r , z , coordinates of the atomic scattering centers in the molecule have their origin at the location R , Z . The molecular azimuthal orientation of a helix is determined by a cut through molecular origin R , Z parallel to the XY plane.

dependent variations in molecular structure, discussed in Sec. II.E and throughout the review . For helical macromolecules, it is convenient to describe the density of scattering centers within each molecule in cylindrical coordinates r , z , , which are coaxial with the main axis of this molecule and associated with a selected point of origin Z , R , which defines the lateral R and axial Z coordinates of the molecule Fig. 3 . It is also convenient to separate scattering centers into subhelices i, which are distinguished not only by the kind of centers they are composed of e.g., phosphate, carbon, oxygen, and other atoms but also by their radii ai, ~ ni r, z, ni r r - ai , 3

f if j F i k F j - k ,

1

where k kz , K , is the scattering vector Fig. 2 , fi is the scattering amplitude for each type of scattering center, and Fi k = 1 2
3/2

where x is the Dirac function. For straight, helical macromolecules, the structure factors are given by CCV as Fi k Fj - k = 1 2 i
n,m n-m

s

ni r exp ik · r d3r

, i,j

kz, n, m Jn Kai Jm Ka e
ikz Z -Z

j

2

e

-in

+im

e

iK R -R

.

4

is the Fourier transform of the density ni r of scattering centers i on the molecule . Hereafter we refer to Fi k as the structure amplitude and to Fi k Fj -k as the average structure factor, although some authors reserve the latter term for ifiFi k . The structure factor is averaged over both time and volume inside the x-ray beam, to account for dynamic thermal fluctuations and static, quenched disorder e.g., due to sequenceRev. Mod. Phys., Vol. 79, No. 3, July­September 2007

Here K K , Jn x is the cylindrical Bessel function of order n, is the azimuthal orientation of each molecule at the point of origin Fig. 3 , and we introduced the molecular structure factors s
, i,j

q, n, m =

i

q, n

j

- q,- m ,

5

based on the Fourier transforms in local coordinates associated with each molecule,


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i

q, n =

1 2

2

d
0 -

dz
0

~ rdrni r, z,

e

in

e

iqz

. 6

The advantage of utilizing the latter factors is that they are independent of the locations and orientations of the molecules and are determined only by the molecular structure.
C. Ideal helices

therefore seems reasonable to suppose that in structure B the structural units DNA are relatively free from the influence of neighboring molecules, each unit being shielded by a sheath of water" Franklin and Gosling, 1953 . Indeed, assuming uncorrelated molecular rotations we find e
-in +im ,

e
n,m

ikz Z -Z

e
,

iK· R -R kz,0 n,0 m,0

=

+ 1-

e

iK· R -R

.

10

The structure factors for ideal helical chains of atoms have a fairly simple form. Consider, e.g., a two-stranded, right-handed helix formed by regularly spaced points, which mimics the pattern of phosphates on DNA Figs. 1 b and 7 , np r =
l

Taking into account that the scattering amplitude of phosphates is much larger than those of other atoms or molecules, the contribution of the latter to the diffraction pattern from noncrystalline fibers can be neglected and Klug et al., 1958 Ik N
n,j=- kz,ng-jG 2 cos2 n ~ s Jn Ka iK· R -R

2z - ~s- + H r-a a z - lh ,

2z + ~s- H 7

+

2 kz,0J0

Ka

e

,

11

where l is the running index numbering the phosphates, 2 ~ s is the azimuthal angle between the strands, H is the pitch of the helix, h is the axial rise per residue, and a is the helix radius. The molecular structure factor of this helix is given by s
, p,p

k z, n , m =

N

2 p 2

cos n ~ s cos m ~

s

j,J=-

kz,Gj-gn m,n+JG/g

,

8

where Np is the total number of phosphates on each strand, G =2 / h, g =2 / H, and x,y is the Kronecker delta x,y =1 at x = y and x,y =0 at x y . Because of the helical symmetry, the structure factor is not zero only along the layer lines Cochran, Crick, and Vand, 1952 , kz = jG - ng , 9

whose separation is determined by the helical pitch and the axial rise per residue. In fact, the same symmetry rule also determines important properties of interactions between helical macromolecules see Sec. V .
D. Discovery of DNA structure

Soon after the CCV theory of helical structure factors was published, high-quality fiber diffraction patterns of DNA were reported by Franklin and Gosling 1953 see Fig. 2 and by Wilkins et al. 1953 . Interpretation of these patterns based on the CCV theory allowed Watson and Crick to decipher the structure of DNA Watson and Crick, 1953 . The classical interpretation of the DNA fiber diffraction pattern shown in Fig. 2 is based on the assumption most clearly formulated by Franklin and Gosling: "It
Rev. Mod. Phys., Vol. 79, No. 3, July­September 2007

where N is the number of DNA molecules in the x-ray beam. The first and second terms in Eq. 11 describe intramolecular and intermolecular scattering, respectively. The two strong spots on the equator kz =0 in the diffraction pattern shown in Fig. 2 correspond to the first order of intermolecular Bragg scattering on a hexagonally packed fiber described by the second term in Eq. 11 . They correspond to the smallest K at which K · R -R =2 M M =0, ±1, ±2, ... and are related to the interaxial spacing dint between the nearest neighbors as K =4 / 3dint. Higher-order diffraction peaks cannot be seen clearly because of imperfect hexagonal packing and the relatively small number of molecules in the fiber. The nonequatorial n 0 and/or j 0 diffraction spots originate from intramolecular scattering described by the first term in Eq. 11 . The cross formed by the diffraction spots on the n = ±1, ±2, ±3, ±5; j = 0 layer lines is in good agreement with the maxima of the corresponding Bessel functions Jn Ka , assuming that the radius of DNA is a 10 å. The distance between these layer lines 2 / H reveals the helical pitch of the molecule, H 34 å. The notable absence of diffraction spots at n =± 4, j = 0 suggests that cos 4 ~ s 0 or ~ s 0.38 . The two darkest spots on the meridian correspond to n =0, j =± 1, kz =2 / h and reveal the axial rise per base pair, h 3.4 å. Their smearing toward the center of the pattern is related to imperfect orientation of the molecules in the fiber, imperfections of the helical structure, thermal motions, and overlap with adjacent diffraction spots n = ±1, ±2; j =± 1 .
E. Nonideal helices: Helical coherence length

At the time of the discovery of DNA structure the ideal helix approximation seemed reasonable, and no further structural details were known. Later, atomicresolution structures of DNA oligomers short, 10­20


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base pair fragments were obtained by x-ray diffraction from crystals Dickerson and Drew, 1981; Dickerson, 1992 . They revealed significant deviations from the ideal helix due to sequence dependence of the structural parameters. It was argued that such nonideality of the DNA structure is important for its function Gorin et al., 1995; Rozenberg et al., 1998 . To illustrate possible effects of deviations from the ideal helix on the structure factors, we first modify the DNA model discussed above by incorporating a realistic sequence-dependent twist between adjacent base pairs. In the ideal helical conformation described by Eq. 7 , the twist angle between the adjacent base pairs, z z+h - z, 12

n

2

4

c

/H .

17

Not only the twist but the base pair tilt, roll, and axial rise depend on the sequence Bolshoy et al., 1991 . These, as well as thermally induced structural variations, may contribute to further deviations from the ideal helix. However, such variations still exhibit the simple random-walk behavior described by Eq. 15 at large distance scales. As a result, the deviation of the average structure factor from an ideal helix can generally be described by Eq. 16 with the single parameter of the helical coherence length c. Different independent contributions ci into the helical coherence length simply add up as Lee, Wynveen, and Kornyshev, 2004
-1 c

was assumed to be constant. In contrast, real DNA has ten distinct combinations of adjacent base pairs, all of which have different preferred values of . The axial pattern of this intrinsic twist angle l z = lh is a unique, sequence-dependent "fingerprint" of DNA structure. We now take into account that l have different values and that real DNA sequences have no long-range correlations in l Stanley et al., 1999 . The deviation from the average twist angle ,
l

=
i

1/

i c

,

18

=

l

-

,

13

2 is relatively small 4° ­6°, 34° l Kabsch et al., 1982; Gorin et al., 1995; Olson et al., 1998 , and denotes an ensemble average over all possible l . Nevertheless, this nonideality has important implications. In particular, it leads to a deviation of lh from the value expected for an ideal helix 0+ l,

reducing the total helical coherence length c. An estimate of the helical coherence length of DNA based on the known average structural parameters and elasticity constants Kabsch et al., 1982; Hagerman, 1988; Dickerson, 1992; Gorin et al., 1995; Olson et al., 1998 and from direct analysis of known NMR structures of B-DNA fragments in solution yields c 100 ­ 300 å and 4 c / H 30 ­ 100 or smaller. The diffraction pattern calculated from Eq. 16 shows a noticeable deviation from the ideal helix approximation at n = 3 and strong smearing and almost complete disappearance of the diffraction peaks at n = 5. In contrast, the n = 5 peaks in the observed diffraction patterns Fig. 2 are still quite sharp and consistent with the ideal helix model.
F. Effect of assembly on DNA structure

lh =

lh -

0-
2

l,

14

which follows the simple law of a random walk, lh - lh = l - l h/ c , 15

1 , where c at a large number of steps l - l = h / i l l+i may be referred to as the helical coherence length Kornyshev and Leikin, 2001; Cherstvy et al., 2004 . For such helices, the molecular structure factor has only diagonal components n = m but its general form is rather cumbersome Inouye, 1994; Mu et al., 1997 . For typical parameters of DNA, the dominant term in this expression is given by s
, p,p

k z, n , m

N

p n,m 2

cos n ~

s

2

n 2/ ch
j

kz + ng - jG 2 + n4/4

2 c

.

16

Similar to ideal helices, Eq. 16 has the maxima at kz given by Eq. 9 . However, the intensity of the maxima decreases as n-2 and their width increases as n2 Egelman et al., 1982; Egelman and DeRosier, 1982; Barakat, 1987 . It follows from Eq. 16 that the ideal helix approximation for the structure factor is valid only when
Rev. Mod. Phys., Vol. 79, No. 3, July­September 2007

Around the time Franklin and Gosling photographed the diffraction pattern shown in Fig. 2, the imperfectness of the helical structure of DNA was not known. It is the present knowledge of the sequence-dependent structure variations that makes the ideal appearance of the double helix in their pictures surprising. It suggests that long, natural DNA molecules in hydrated aggregates are closer to ideal helices than DNA in solution and short oligomers in crystals. In fact, the first indication that DNA molecules become more uniform and closer to ideal helices when packed into aggregates was obtained several decades ago. A significant variation in the twist angle per base pair bp with the average near 10.5 bp per helical turn was observed in solution Wang, 1979; Rhodes and Klug, 1980 , while nearly perfect, integral 10.0 bp/turn helices were observed in hydrated fibers regardless of packing density Zimmerman and Pheiffer, 1979; Rhodes and Klug, 1980 . In Sec. VI we detail further, direct evidence demonstrating that the simplifying assumption about DNA being "free from the influence of neighboring molecules" was incorrect. A more general interpretation of the classical diffraction patterns without this assumption turns out to be in better agreement with the established Watson and Crick model. The interactions between neigh-


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boring molecules are strong enough not only to affect their alignment but even to change their structure. Significantly, the variation of structural parameters of synthetic DNA oligomers in crystals is closer to that in natural DNA in solution than in hydrated fibers Dickerson, 1992 , suggesting that interactions between fragments with identical sequences in crystals are different from interactions between long DNA with uncorrelated sequences in fibers. In other words, intermolecular interactions between DNA depend not only on overall structure of the double helix, but also on the sequence of base pairs. In Sec. V we give a clear interpretation of this phenomenon.
G. Bent and supercoiled helices

In nature, DNA and other biological helices rarely exist as straight rods. Most of the time they bend, winding around other molecules and around each other. For instance, eukaryotic DNA is wound around a protein core in nucleosomes. The three left-handed helical chains of collagen triple helix are wound together in a right-handed helical supercoil. The two -helical chains in myosin filaments coil around each other forming an extended coiled coil. The latter term is typically used for -helices while supercoil is a part of DNA terminology, but otherwise they have similar meaning. Circular DNA fragments form a variety of supercoiled structures with the help of specialized enzymes topoisomerases . A schematic picture of a supercoil made by two molecules is shown in Fig. 4. Various aspects of the studies of such structures have been described in numerous reviews see, e.g., Vologodskii and Cozzarelli 1994 and Mason and Arndt 2004 . Surprisingly little attention, however, has been paid to the physics of the interactions governing the structural hierarchy in these objects. For instance, we still do not know how the pitch of the molecular helix affects the energy and the structure of the supercoil formed by two such helices tightly wound around each other. We return to a more detailed discussion of this issue in Sec. IV. Here, we point out that the first step toward a rigorous solution of this problem would be the calculation of the structure factor of a coiled coil. Such a calculation was actually reported by Crick 1953a, 1953b . The density of atoms in a cross section r = R , z of the strand j of a coiled coil with a large supercoli pitch P can be approximated as nj R, z
j ns R - R, z nj0 R , z d2R ,

FIG. 4. Sketch of a supercoil coiled coil