The geometrical origin of the strain-twist coupling in double helices
TThe geometrical origin of the strain-twist coupling in doublehelices
Kasper Olsen ∗ and Jakob Bohr † Department of Physics,Technical University of DenmarkBuilding 307 Fysikvej, DK-2800 Lyngby, Denmark
Abstract
The geometrical coupling between strain and twist in double helices is investigated. Overwinding,where strain leads to further winding, is shown to be a universal property for helices, which arestretched along their longitudinal axis when the initial pitch angle is below the zero-twist angle(39.4 ◦ ). Unwinding occurs at larger pitch angles. The zero-twist angle is the unique pitch angle atthe point between overwinding and unwinding, and it is independent of the mechanical propertiesof the double helix. This suggests the existence of zero-twist structures, i.e. structures that displayneither overwinding, nor unwinding under strain. Estimates of the overwinding of DNA, chromatin,and RNA are given. If one pulls a double helix structure by the end, one might think that it would unwind by the appliedtension. In this paper we show why this is not always the case: A helix can unwind, overwind, or itcan stay at its current twist (which we denote a zero-twist (ZT) structure). Overwinding is contraryto unwinding; unwinding is the de-twisting of the helices obtained by stretching the material. For thezero-twist structure there is no coupling from strain to twist. The existence of a twist-stretch couplingis a well-known phenomenon for helical steel wires [1] where it leads to unwinding, and design efforts gointo designing rotation resistant wire rope when desired [2, 3].The geometrical investigation presented below is based on the study of packed double helices modeledas two flexible tubes with hard walls. To be packed is defined by the constraint of the two tubes beingin contact. Does this mean that the helices are stretched? No, generally not, stretching is one way tosecure that a packed helix is obtained, however, for helices on the molecular size favorable molecularinteractions can also make it more preferable to be packed than not. A detailed analysis of packed helicesand their volume fractions showed that the helices with the highest volume fractions are noticeablysimilar to the molecular structure of DNA [4]; this suggests that close-packing is at work as a structureforming principle. For the description of compact strings and tube models, the importance of one kind ofoptimum shape has been discussed by Gonzalez and Maddocks [5] and Maritan et al. [6], and one relatedsuggestion for the best packing of proteins and DNA has been considered by Stasiak and Maddocks [7].A detailed analysis of the geometry of n -plies, and of their self-contacts, has been given by Neukirch andvan der Heijden [8]. The close-packed (CP) structure with an optimized volume fraction has a pitch angle of 32 . ◦ [4]: Thisstructure that has a central channel is shown in Figure 1a. Under a pull, the pitch angle is increased ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . b i o - ph ] M a y nd the diameter of the central channel gets smaller, and eventually, the inner channel disappears at apitch angle of 45 ◦ . Whether a helix overwinds or unwinds is then determined from the balance betweenthe gain in length from the reduction in the helical radius versus untwisting. The crossing point – whichwe denote as the zero-twist angle – is at 39 . ◦ (Figure 1b) and is smaller than the 45 ◦ , where the helicalradius becomes equal to the diameter of the tubes, and is maintained for all pitch angles above 45 ◦ . The45 ◦ motif, here denoted the tightly packed (TP) double helix, is shown in Figure 1c. a) b) c) Figure 1.
Figure 1:
Different geometries of a double helix of tubes of fixed diameter D . a) Close-packed (CP)structure of pitch angle . ◦ measured from horizontal. b) Zero-twist (ZT) structure with a pitch angleof . ◦ . It is at the point between overwinding/unwinding. c) Tightly-packed (TP) structure of pitchangle ◦ . Overwinding from stretching takes place from the a) to the b) confirmations; unwinding fromb) to c). Geometrically, the double helix is given by two tubes of diameter D , whose centerline defines twohelices with simple parametric equations. A helix is a curve of constant curvature, κ , and torsion, τ , andit can be specified by two parameters, for example a and H , where a is the helix radius (the radius ofthe cylinder hosting the helical lines) and H the helical pitch (the raise of the helix for each 2 π rotation).The tangent to each of the helical curves is at an angle v ⊥ ( the pitch angle ) with the horizontal axis,and it is determined by tan v ⊥ = h/a , where h = H/ π is the reduced pitch. We say that the doublehelix is packed when the shortest distance between the centerline of one helical tube to the next oneequals the diameter D of the tubes, i.e. the double helix is packed when the tubes are in contact. Thevolume fraction can be calculated using, as a reference volume, an enclosing cylinder of height H = 2 πh and volume V E = 2 π h ( a + D/ , and comparing it to the combined volume occupied by the twocircumscribed helical tubes, V H = π hD / sin v ⊥ . The volume fraction is the ratio of the two volumes,i.e. f V = V H /V E , which reads f V = 2(1 + ( ah ) ) / · ( 2 aD + 1) − (2.1)With this choice of reference volume the packing fraction depends only on the shape of the double helixstructure, which can be described by one parameter, e.g. the pitch angle, v ⊥ . The maximum of f V definesthe close-packed (CP) helix. For the double helix this maximum is at v ∗⊥ = 32 . ◦ , where f ∗ V = 0 . a [4]. Generally, the radius of the centralchannel, which is given by R i = a − D/
2, is a decreasing function of v ⊥ ; this can be seen from Figure 22hich shows 2 a/D depending on the pitch angle. For v ⊥ ≥ ◦ there is no central channel as 2 a/D = 1,see Figure 2. TP v ! a ! D Figure 2.
Figure 2:
Graph showing the ratio a/D as a function of pitch angle, v ⊥ [deg.], where a is the helix radiusand D the diameter of the helical tubes. The tightly packed double helix has a pitch angle of v T P = 45 ◦ ;it is the helix with the smallest pitch angle obeying the criterion that a = D . Consider a long straight segment of a double helix consisting of two long molecular strands each of length L M . The length of the double helix is H M = L M sin v ⊥ and the total twist is Θ M = L M cos v ⊥ /a . InFigure 3 the dimensionless ratio Dθ M / L M is shown as a function of the pitch angle. One can see thatfor v ⊥ < v ZT there is overwinding while for v ⊥ > v ZT there will be unwinding. We find numerically that v ZT = 39 . ◦ .We can determine the amount of overwinding and unwinding in the following way. If a long doublehelical segment is stretched a bit, the pitch angle, v ⊥ , will change by a small amount dv ⊥ , and hence H M changes by dH M = L M cos v ⊥ dv ⊥ (3.2)and Θ M by d Θ M = − L M sin v ⊥ a dv ⊥ − L M a cos v ⊥ dadv ⊥ dv ⊥ (3.3)so that d Θ M dH M = − a tan v ⊥ − a dadv ⊥ (3.4)If this derivative is positive, then the helix will overwind, and if it is negative, it will unwind. Thederivative in Eq. (3.4) has dimension of inverse length. From a geometrical viewpoint it is more naturalto look at the dimensionless function of v ⊥ , obtained by multiplying with the common radius of thetubes, ( D/ D d Θ M dH M = − D a tan v ⊥ + ddv ⊥ (cid:18) D a (cid:19) (3.5)3 T v ! D Θ M ! L M Figure 3.
Figure 3:
The total twist, θ M , for a long segment of the double helix; the dimensionless quantity Dθ M / L M is shown as a function of the pitch angle, v ⊥ [deg.]. The maximum value is obtained for the pitch angle v ZT = 39 . ◦ and mark the transition from overwinding to unwinding. At the ZT structure there is zerocoupling between twist and strain. This equation can be given a simple interpretation. The first term is negative and determines the amountof unwind, while the second term is positive and determines the amount of overwind. The graph of thisderivative, that dictates the coupling between strain and twist, is depicted in Figure 4. Notice that the CPdouble helix will always overwind since d Θ M /dH M >
0. At the close-packed structure, the overwind is( D/ d Θ M /dH M = 0 . D , of the tubes making up any close-packed double helix. At the zero-twist structure, v ZT = 39 . ◦ ,there is neither overwinding, nor unwinding. For larger pitch angles the overwind, ( D/ d Θ M /dH M ,is negative and the double helix will unwind under strain. It is therefore crucial, that the pitch angleis below that of the zero-twist (39 . ◦ ) for overwinding to be observed, but it also indicates that elasticproperties of the material are not essential to understanding the phenomenon. In the following we discuss some molecular examples. The phenomenon of overwinding in DNA wasfirst observed in 2006, see Lionnet et al. [9] and Gore et al. [10] using magnetic tweezers to control thewringing [9] and optical tweezers to control the pulling [10]: For small deformations, DNA overwindswhen stretched, i.e. it rotates counter to unwinding. During overwinding the extension of a long chain ofDNA-B has been reported to be 0 . ± . π rotation [9] and 0 . π rotation [10]. Veryrecently, it has been suggested that in the absence of tension DNA is an order of magnitude softer [11].Using the above mathematical solution for the double helical structure of DNA we find the change oflength ∆ H to be determined by ∆ H = dH M d Θ M ∆Θ (4.6)The diameter of the molecular tubes that make up the DNA helix is D = 1 .
15 nm, which is given fromour previous analysis of the close-packed structures [4]. We then estimate ∆ H per full 2 π turn to be π (0 . − × .
15 nm = 5 . P ZT ! ! v ! O v e r w i nd Figure 4.
Figure 4:
Graph showing the calculated overwind of double helices (solid line), i.e. Eq. (3.5) as a functionof v ⊥ [deg.]. A positive overwind means that the double helix will exhibit overwinding, while a negativeoverwind means that the double helix will exhibit unwinding. The zero-twist structure (ZT) is indicatedwith an arrow at v ZT = 39 . ◦ , the close-packed structure (CP) is indicated by an arrow at v CP = 32 . ◦ .The first derivative is discontinuous at v T P = 45 ◦ where the helix radius can not get smaller. The dashedline is the overwind for a triple helix, which has a zero-twist angle of . ◦ . The geometrical restriction imposed by base pairing and its influence on d Θ M /dH M has not been takeninto account. The numerical analysis has been performed for the symmetrical double helix where theclose-packed structure has a pitch angle of 32.5 ◦ . The asymmetrical DNA-B has a close-packed pitchangle of 38.3 ◦ and, as one can show, a zero-twist angle of 41.8 ◦ . Theoretical work on understandingthe overwinding of DNA has focused on constructing elastic models which show a negative twist-stretchcoupling [12] and on incorporating stochastic effects [13]. One elastic model was considered by Gore et al.[10], and consists of a rod with a stiff helical wire (analogous to the sugar-phosphate backbone) attachedto its surface. As this system is stretched, the inner rod decreases in diameter and the helix will overwind.Smith and Healey has argued that a linear material law is inadequate for the description and suggest anon-linear elastic rod [14].For chromatin, the above results can be related to recent experiments in twisting chromatin fibers,see e.g. [15, 16]. For a close-packed 30 nm chromatin fiber, in the so-called two-start geometry, weestimate a tube diameter of 30 / (2 a/D + 1) nm= 30 / (1 . . a/D is determinedfrom Figure 2. For the close-packed 30 nm chromatin structure we then estimate ∆ H per full 2 π turnto be π (0 . − × . H for Xenopus chromatin per turn at a pulling force of 0.3 pN. Using the depicteddata in ref. [16] we have estimated an average extension of ∼ ±
40 nm per turn. Here, we haveassumed the two-start helix to behave like a tubular packed double helix – that is a view which ignoresthe intricate details of the structure, details which are discussed for example by Barbi et al. [25], whereelaborate mechanical models are described, including one which maintain its twist while being stretched.We have presented a simple geometrical explanation for overwinding of helices – an effect which hasbeen observed before for the double helix of DNA and for chromatin, and which is contrary to usualunwinding. Our model of unwinding and overwinding can be applied to any symmetric double helixwhich is packed in the sense that the two helices touch each other, i.e. remain at the distance D from5ach other. Packed double helical structures will show an overwinding behavior similar to those alreadyobserved, as long as their initial pitch angle is sufficiently small. Perhaps, the analysis will be relevant forother helical structures such as nanofabricated quartz cylinders [17], fabricated twisted polymer nanofibers[18], and for the beautiful double helical structures formed from helical carbon nanotubes [19]. Further,the phenomenon may be important for some aspects of the working of molecular motors during geneexpression and regulation [20]. The analysis presented in this paper is straightforwardly applicable toRNA double helices [21], which we therefore predict will show overwinding. Using a value of 26 ˚A[22] forthe molecular diameter of the double helix, we estimate an overwinding of 5.6 nm. Necturus chromatinfibers [23] are known to pack as a double helix with a pitch angle of v ⊥ = 32 ± ◦ a value suggestive of beingclose-packed. Thus it follows that these chromatin double helices will overwind as well (other chromatinfibers with a different linker length would not necessarily overwind). Such predictions for overwindingand unwinding can nowadays be studied on single biomolecules using magnetic traps [24]. Furthermore,the derived geometrical expressions for overwinding are straightforwardly extended to helices with morethan two strands. In Figure 4 we have shown the solution for a triple helix (dashed line) which has azero-twist angle of 42 . ◦ .Maybe one will even find examples, where Nature has build zero-twist structures, i.e. structures thatdisplay neither overwinding, nor unwinding. Chromatin with an appropriate linker length, and collagenare possible candidates for structures with such properties. Acknowledgements
We would like to thank Jean-Marc Victor for helpful comments on a first version of the manuscript.6 eferences [1] W.S. Utting, N. Jones, The response of wire rope strands to axial tensile loads - Part I. Experimentalresults and theoretical predictions,
International Journal of Mechanical Sciences , 605 (1987).[2] D.L. Pellow, Rotation resistant wire rope, United States Patent 4365467 (1982).[3] R.B. Waterhouse, Fretting in steel ropes and cables - a review in ”Fretting fatigue: Advances in basicunderstanding and Applications”, Eds. Y. Mutol, S.E. Kinyon, D.W. Hoeppner, AST International,West Conshohochen, PA (2003).[4] K. Olsen, J. Bohr, The generic geometry of helices and their close-packed structures, Theor. Chem.Acc. , 207 (2010).[5] O. Gonzalez, J.H. Maddocks, Global curvature, thickness and the ideal shapes of knots,
PNAS ,4769 (1999).[6] A. Maritan, C. Micheletti, A. Trovato, R. Banavar, Optimal shapes of compact strings, Nature ,287 (2000).[7] A. Stasiak, J.H. Maddocks, Mathematics: Best packing in proteins and DNA,
Nature , 251 (2000).[8] S. Neukirch, G.H.M. van der Heijden, Geometry and mechanics of uniform n -plies: from engineeringropes to biological filaments, Journal of Elasticity , 41 (2002).[9] T. Lionnet, S. Joubaud, R. Lavery, D. Bensimon, V. Croquette, Wringing out DNA, Phys. Rev. Lett. , 178102 (2006).[10] J. Gore, Z. Bryant, M. N¨ollmann, M.U. Le, N.R. Cozzarelli, C. Bustamante, DNA overwinds whenstretched, Nature , 836 (2006).[11] R.S. Mathew-Fenn, R. Das, P.A.B. Harbury, Remeasuring the Double Helix,
Science , 446 (2008).[12] M.Y. Sheinin, M.D. Wang, Twist-stretch coupling and phase transition during DNA supercoiling,
Phys. Chem. Chem. Phys. , 4800 (2009).[13] C.C. Bernido, M.V. Carpio-Bernido, Overwinding in a stochastic model of an extended polymer, Physics Letters A , 1 (2007).[14] M.L. Smith, T.J. Healey, Predicting the onset of DNA supercoiling using a non-linear hemitropicelastic rod,
Int. Jour. of Non-Linear mechanics , 1020 (2008).[15] A. Bancaud, N. Conde e Silva, M. Barbi, G. Wagner, J.-F. Allemand, J. Mozziconacci, C. Lavelle,V. Croquette, J.-M. Victor, A. Prunell and J.-L. Viovy, Structural plasticity of single chromatin fibersrevealed by torsional manipulation, Nature structural & molecular biology , 444 (2006).[16] A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman and S. X.Sun, Magnetic tweezers measurement of single molecule torque, Nano Lett. , 1720 (2009).[17] C. Deufel, S. Forth, C.R. Simmons, S. Dejgosha, M.D. Wang, Nanofabricated quartz cylinders forangular trapping: DNA supercoiling torque detection, Nature Methods , 223 (2007).[18] B.K. Gu, M.K. Shin, K.W. Shon, S.I. Kim, S.J. Kim, S.K. Kim, H. Lee, J.S. Park, Direct fabricationof twisted nanofibers by electrospinning, Applied Physics Letters , 263902 (2007).[19] J. Liu, X. Zhang, Y. Zhang, X. Chen, J. Zhu, Nano-sized double helices and braids: interestingcarbon nanostructures, Materials Research Bulletin , 261 (2003).720] J. Michaelis, A. Muschielok, J. Andrecka, W. K¨ugel, J.R. Moffitt, DNA based molecular motors, Physics of Life Reviews , Nature Structural Biology , 56 (1995).[22] A. Varshavsky, Discovering the RNA double helix and hybridization, Cell , 1295 (2006).[23] S.P. Williams, B.D. Athey, L.J. Muglia, R.S. Schappe, A.H. Gough, J.P. Langmore, Chromatin fibersare left-handed double helices with diameter and mass per unit length that depend on linker length,
Biophysical Journal , 233 (1986).[24] A. Meglio, E. Praly, F. Ding, J.F. Allemand, D. Bensimon, V. Croquette, Single DNA/protein studieswith magnetic traps, Current Opinion in Structural Biology , 615 (2009).[25] M. Barbi, J. Mozziconacci, J.-M.Victor, How the chromatin fiber deals with topological constraints, Phys. Rev. E71