Twist and stretch of helices: All you need is Love

Bojan DurickovicProgram in Applied Mathematics, University of Arizona

Alain GorielyOCCAM, Mathematical Institute, University of Oxford

John H. MaddocksSection of Mathematics, Swiss Federal Institute of Technology Lausanne

In various single molecule experiments a chiral polymer, such as DNA, is simultaneously pulled andtwisted. We here address an elementary, but fundamental, question raised by various authors: doesthe molecule overwind or unwind when the tension loading is incremented? We show that within thecontext of the classic, mechanical Kirchhoff-Love rod model of elastic filaments both behaviors arepossible; the response depends on both the imposed loading and the precise constitutive relations ofthe polymer. More generally, our analysis provides an effective linear response theory for Kirchhoff-Love rods that relates changes of the two generalized displacements of axial translation and axialrotation to changes in the two generalized loadings of axial force and axial torque.

PACS numbers: 46.70.Hg,87.10.Pq,87.19.rd

Single-molecule tweezer experiments have revealedmany fascinating behaviors, some of which have been re-ported as counterintuitive. For instance, two differentgroups [? ? ] measured the twist response of single-molecules of DNA when pulled by optical tweezers. Inthese beautiful experiments a bead that is attached to theend of the DNA is pulled by optical tweezers and eitherthe angular rotation of the bead is tracked, or the angu-lar displacement is controlled via an applied torque. Theexperimental observations are that the DNA overwindswhen pulled, and extends when overwound. Both groupsdescribe these behaviors as surprising: “simple intuitionsuggests that DNA should unwind under tension”[? ],and “DNA should lengthen as it is unwound” [? ]. Weshall explain that this observed microscopic response isnot particular to DNA, but rather is generic for mostmacroscopic helical elastic filaments, and is captured bya simple 19th century model of helical wire that has beenverified in macroscopic experiments first performed morethan a century ago [? ? ].

Contemporary confusion in interpreting the behaviorof helical springs under tension is not restricted to theDNA literature; it also arises in discussions of the ba-sic problem of growth of stems and roots where cell wallanisotropy is thought to be responsible for the overall ro-tation, or handedness, that is observed. A typical argu-ment is that the cell wall is a cylindrical structure withreinforced right-handed helical microfibrils that shouldrotate clockwise during growth (when viewed from thetop) [? ? ], therefore unwinding, as is to be intu-itively expected because “helices unwind when they arestretched”[? ]. This explanation of handedness in plantgrowth by analogy with the purported simple behaviorof macroscopic helical springs is again falsidical.

The classic literature concerning helical rods has been

θ

Ny

z

xy

x

M

(A) (B)

Figure 1: A wrench applied to a helical spring gives rise toanother helical equilibrium. Two simple questions are to de-termine: (A) the sign of the increment of the rotation angle θwhen a pure axial extension force is applied with no torque,and (B) whether a spring extends or contracts as a result ofa pure axial torque loading with no applied force. Drawingsadapted from those of Robert Hooke’s 1678 Lectures De Po-tentia Restitutiva

largely confined to the study of uniform metal springswith helical unstressed shapes within the special case ofthe wire having a circular cross section, cf. Fig. ??. Suchsystems were extensively studied by Thomson and Tait [?, §605] for example, and were by the end of the 19th cen-tury already considered as a classic problem of elasticity.For circular cross-section wire the helical spring formula,which relates an applied wrench (with a prescribed ax-ial force and axial torque) to the pitch and radius of theresulting helical equilibrium, can be found in Love [? ,§271], which, as it happens, is a treatise often cited withinthe DNA modeling literature. This formula applies to thefollowing gedanken experiment. Apply an axial tensionand no axial torque to a helical spring standing along

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the z-direction, and follow the angular rotation of theprojection of an arbitrary, but fixed, material point ofthe spring onto the (x, y)-plane (cf. Fig. ??). The heli-cal spring formula implies that, for constitutive relationsappropriate for metal springs with circular cross section,as tension is increased monotonically from zero, the ro-tation first increases (overwinding the helical axis in thesame direction as the helix handedness) up to a maxi-mum, after which it starts to decrease, returning to itsinitial value and then unwinding (cf. Fig. ??). This sim-ple and quite general macroscopic behavior is no longerwidely known, and may therefore appear to be surprisingand counterintuitive.

Contemporary experiments on both molecular and bi-ological filamentary structures suggest that an accuratemodel of the data may require a rod theory with constitu-tive relations appropriate for filaments with non-circularcross section, and for materials with coupling betweenbending and twisting strains. The objective of this letteris to exploit a recent classification all helical equilibriaof such uniform filaments [? ? ] to extend the clas-sic helical spring formula to obtain an effective responsetheory relating the two generalized loadings of axial forceand axial torsion to the two generalized displacements ofaxial translation and axial rotation. For example withinthis more general class of helical elastic filaments we showthat helical equilibria may initially either overwind orunderwind when stretched, with the response dependentupon the detail of the loading and the constitutive rela-tions.

We first tersely review the Kirchhoff theory of inexten-sible, unshearable, uniform rods with quadratic strain-energy function [? ? ]. A rod with arc length s ∈ [0, L]is defined by a centreline r(s) and a unit vector field d1(s)perpendicular to the unit tangent r′(s) =: d3(s) (primedenotes the s-derivative). A local orthonormal basis ofdirectors is obtained by defining d2(s) := d3(s) × d1(s)so that d′i = u× di, i = 1, 2, 3 where u is the Darbouxvector. Any vector, and in particular u, can be writtenas u = u1d1 + u2d2 + u3d3 with the three componentsassembled into a triple u = (u1, u2, u3) (note the use ofsans-serif fonts in the director basis).

By defining a resultant force n(s) and moment m(s)acting across the cross section at r(s), and in the absenceof distributed body forces and couples, balance of forceand moment yields the equilibrium equations

n′+u× n = 0, (1)

m′+u×m + v × n = 0, v := (0, 0, 1)T . (2)

The general material response of a rod with quadraticenergy is given by a linear constitutive relation with eight

constants

m = K(u− u), K =

K1 0 K13

0 K2 K23

K13 K23 K3

, K1 ≤ K2,

(3)where u are the Darboux components of the unstressedshape, and K is positive definite.

Uniform configurations of a filament are solutions withu constant, so that u is itself also constant, which impliesthat uniform configurations have helical centrelines withaxis along u [? ] and with curvature, torsion and radiusgiven by κ =

√u21 + u22, τ = u3 and R = κ/|u|2. Let e =

εu/|u| be a unit vector along the axis such that e ·r′(s) ≥0, and ε = +1 for right-handed helices (and circular arcs)with u3 = τ ≥ 0 (ε = −1, for left-handed helices). Thepitch per unit arc length along e is p = |u3|/|u| while thecoiling angle per unit arc length is ρ = |u|, so that thenumber of helical periods is Lρ/2π for a helix segment ofarc length L. When one helical configuration is deformedinto another, it will overwind if θ := ρ2 − ρ1 > 0 andunwind if θ < 0 (respectively, increasing or decreasingthe number of helical repeats for a given arc length).

Uniform equilibria are characterized by constant triplesn, m and u that satisfy the equilibrium equations (??-??)and constitutive relations (??). The third component of(??) reads

Q(u) := u1m2 − u2m1 = 0. (4)

Together with the constitutive relations (??), Q(u) de-fines a quadric in Darboux space that contains the entireu3-axis and which, apart from degenerate cases (see be-low), is a one-sheeted hyperboloid that contains all val-ues of u corresponding to helical equilibria. The ori-gin u = 0 is one ‘helical’ equilibrium with a straightcentreline and force n of arbitrary magnitude alignedwith the centreline. All points on the hyperboloid offthe u3-axis represent helical equilibria with κ > 0 andunique associated stresses n and m. Indeed, for u 6= 0,(??) and Q(u) = 0 imply that n = µu with µ =m3−u3(u2m1+u1m2)/(2u1u2) (valid for u1 = 0 or u2 = 0,but not both). As the constitutive relations (??) providem as a function of u, we now have the basic quantitiesassociated with a helical equilibrium explicitly writtenas a function of points u in Darboux space lying on thehyperboloid Q(u) = 0 and off the u3-axis.

Conjugate to the two generalized displacements (p, ρ),we need two generalized loads given by the axial compo-nents of force and moment. These are the four accessibleeffective quantities in a single molecule tweezer exper-iment, and are also entirely natural variables in macro-scopic systems. The axial component of the force is N :=n · e = εµ|u| =: N (u), while the axial component of themoment is M := m · e = εu ·m/|u| =: M(u). Together,(N,M) forms a wrench: a pure axial force Ne = n(L)and pure axial moment Me = m(L)−Nr(L)×e appliedat the axis of the helical equilibrium (cf. Fig. ??).

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We now have a purely geometric problem; for a givenwrench (N,M) the observed response (p, ρ) is determinedby the intersection of the level sets of the functions N (u)and M(u) with the hyperboloid Q(u) = 0. In generalneither of these level sets, nor the resulting intersectionsare simple.

We start with a discussion of the response of a filamentgoverned by the classic, particular case of constitutiverelations (??) appropriate for a circular cross section andwith no bend-twist coupling, that is

K = diag(K1,K1,K3), u1 = 0, u2 > 0, (5)

and introduce Γ = K3/K1. Then the hyperboloid (??) ofequilibria reduces to the plane u1 = 0, κ = |u2|, and thetwo half-planes (κ, τ) and (u2 > 0, u3) can be identified.For the pure axial force loading illustrated in Fig. ??AM = 0, and for all Γ, the level set M(u) = 0 intersectsthe (κ, τ)-plane in the ellipse which passes through thetwo points (0, 0) and (κ, τ)

κ(κ− κ) + Γτ(τ − τ) = 0. (6)

It is easy to check (by evaluation of µ) that the he-lix always extends when pulled (that is |τ | > |τ | forN > 0). A point on the ellipse corresponds to over-wound configurations with respect to (κ, τ) if it lies out-side the circle centred at the origin and passing through(κ, τ). Such a situation occurs for Γ < 1 and we con-clude that a helix will initially overwind when pulledfrom its minimum energy state if and only if Γ < 1.Further, the maximum overwinding (point b on Fig. ??)is obtained when the ellipse (??) intersects the hyperbola2(1−Γ)κτ + Γτκ− κτ = 0, and the coiling angle returnsto its original value when the ellipse intersects the circleκ2 + τ2 = ρ2 (corresponding to point c). We remarkthat if the filament is formed from a three-dimensionalhomogeneous linearly elastic material with Poisson ratioin [0,1/2], e.g. standard metals, then Γ ∈ [2/3, 1], so thatinitial over-winding in response to positive tension alwaysarises for simple helical springs, a slightly surprising butyet completely classic effect. Note however that consti-tutive relations for rod models of molecules and plantsare not necessarily constrained by the bound on Γ.

Analogously, we can study the particular case N = 0when a helix is loaded with a pure axial torque, corre-sponding to the experiment in Fig. ??B. Again, an anal-ysis of the relative position of the level set curve N = 0with respect to both the circle passing through the un-stressed state (constant angle θ) and the line through theunstressed state and the origin (constant pitch p) revealsthat for Γ < 1, initially unstressed helices always extendwhen overwound and shorten when unwound, anotherconsequence of the classic theory of rods.

We now return to the general constitutive relations(??) and reconsider the pure axial force gedanken exper-iment close to the unstressed N = M = 0 helix u, and

Γ<1

Γ=1

θp c

onsta

nt

Γ>1

c

b

a N

κ

Γ<1

Γ<1,

Γ>1,

Γ>1,

τ

,(κ τ)^ ^

c b

N=0

N=0

M=0

0

a

Γ<1,M=0

θ constant

Figure 2: Left: For the particular constitutive relations (??),the set of M = 0 zero axial torque corresponding to exper-iments in Fig. ??A, helical equilibria lie on an ellipse in thecurvature-torsion plane determined by the value of Γ. Closeto the unstressed helix with N = 0 (point a), N > 0 equi-libria correspond to |τ | > |τ | (case τ > 0 shown). Initialoverwinding in extension occurs for Γ < 1, which is when thesection of the ellipse outside the circle centred at the originand passing through a has |τ | > |τ |. The set of N = 0 zeroforce equilibria (corresponding to experiments in Fig. ??B)is also shown. Depending on Γ, the helix either lengthenswhen wound (Γ < 1) or shortens (Γ > 1). Right: For M = 0the relative coiling angle as a function of the axial force. ForΓ < 1 and N > 0 the helix first overwinds, then unwinds.

again pose the question: does a helix over- or under-windforN > 0 and small? The required argument is in essencethe same as the one in the (κ, τ)-plane described in theprevious paragraph, but now on the quadric Q(u) = 0embedded in three-dimensional Darboux space. The setof possible equilibria under pure axial force are charac-terized by the intersection of the ellipsoid E = 0 withthe quadric Q = 0, which contains the point u. Startingfrom the unstressed configuration and applying a loadingof a pure tensile force N > 0, a one-parameter family ofhelices is obtained for small N . In other words the in-tersection of the two quadrics is a curve uN of equilibriain Darboux space parametrized by the force N ≥ 0 suchthat u0 = u. If |uN | > |u| close to u, then the helix ini-tially overwinds under applied tension and would unwindunder applied compression. Geometrically this impliesthat the loading curve on the hyperboiloid with N > 0lies outside the sphere S(u) := u.u− u.u = 0. Therefore,the criterion for deciding whether a given unstressed he-lix u will overwind under tensile stress is obtained bytaking the scalar product of the tangent tE to the curveuN at u oriented in the direction of increasing N withthe exterior normal to the sphere S(u) at u (see Fig. ??).Specifically, overwinding arises when εu.(Ku × h) > 0,where h = (−K1u2,K2u1,K23u1 − K13u2) is the vectornormal to the hyperboloid at u (cf. Fig. ??). Depen-dent on the details of the constitutive relation, the helixcan either overwind or unwind when pulled from the un-stressed state. Note also that since the ellipsoid E = 0 isbounded, the family of equilibria |uN | will always even-

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tually unwind for large enough N , just as before.

Figure 3: Left: The set of equilibria for a helix under pureaxial force (M = 0) is the intersection of the hyperboloidH (yellow) and the ellipsoid E (red). The set of equilibriawithout winding θ = 0 is the intersection of the hyperboloidH (yellow) and the sphere S (blue). Right: For the particularcase shown, the helix initially unwinds when pulled since thetangent tE to the curve of equilibria uN (red) oriented towardN > 0 lies within the sphere S, whose intersection with H isshown as the blue curve.

The fact that helices under pure axial force can eitherover- or under-wind is a simple manifestation of the com-plexity of the response of helical structures under loads.In turn this complexity is related to the nonlinearity ofthe problem. In particular, while the constitutive re-lations (??) are linear, both the geometry of the helices,and the balance laws (??-??) are nonlinear. Neverthelessit is possible to develop an effective linear response theoryclose to any given helical equilibrium via an appropriateTaylor expansion. Specifically, select a particular helicalequilibrium u∗ lying on the hyperboloid, and denote allassociated quantities at this equilibrium as M∗, N∗, etc.Then the linearization of the wrench (M,N) as a func-tion of the strains leads to a linear response relation ofthe form [

MN

]=

[M∗

N∗

]+

[A∗ B∗

B∗ C∗

] [θz

](7)

where θ = |u| − |u∗|, and z = u3/|u| − u∗3/|u∗|. A notablepoint is that the coefficient matrix is always symmetric,and is positive-definite at least in a neighborhood of theunstressed state u, and so can be regarded as an effec-tive stiffness matrix. The coefficients A∗, B∗, and C∗

are lengthy expressions of the material coefficients andthe equilibrium values and the general methods to ob-tain them is given explicitly in the Supplementary Ma-terial at [url to be inserted by editor]. Accordingly forsmall displacements around any equilibrium helical solu-tion, one can replace the helical structure by a simplified,effectively cylindrical, structure whose geometric config-uration is given by the two displacements θ and z, andwith the effective energy

E=E∗+(M∗−M)θ+(N∗−N)z+1

2(A∗θ2+2B∗zθ+C∗z2). (8)

The first term is the total energy of the system at theconfiguration u∗, the second and third terms are work as-sociated with the loads applied to deviations away fromu∗, and the quadratic form in the displacements is theelastic energy of the effective system. This effective en-ergy, which is derived here from the underlying helicalstructure, is an extension to general equilibria of the ef-fective energy postulated by various authors [? ? ? ]to describe DNA response under loads, with the maindifference being that the parameters in our effective en-ergy vary when the system is not close to the unloaded,absolute minimiser of energy. Indeed, the key here is torealise that the coefficients A∗, B∗ and C∗ depend on theload apply to the system. This behavior is routinely re-ported in experiments where it is observed that the effec-tive moduli of DNA varies with force [? ]. In the partic-ular experiment with no axial torque, the coefficients B∗

will change sign as the spring inverts its winding, leadingto the observed non-monotonic behavior. Note also thatthis effective energy can provide a reasonable approxi-mation to the statistical mechanics of the system closeto u∗ provided that the system is in the regime of rela-tively high tensile loads where fluctuations of the DNAdue to interaction with the solvent can reasonably be ne-glected. Finally we remark that provided the coefficientmatrix in equation (??) satisfies appropriate invertibilityconditions, the energy (??) can be regarded as a functionof any two of the four variables (M,N, θ, z), in order tomodel experiments with soft, hard, or hybrid loading.

This article provides a systematic way to constructtwo-dimensional effective energies for tweezer experi-ments when the underlying polymer constitutive rela-tions are known. In particular, it identifies the source ofnon-monotonic behavior in simple wrench experiments.Given the geometric framework constructed here, a nat-ural next step is to consider the inverse problem of ex-tracting approximations to the constitutive relations (??)from sets of measurements of the four effective coarse-grain axial load and displacement variables.

Acknowledgments: This publication is based in partupon work supported by Award No. KUK-C1-013-04,made by King Abdullah University of Science and Tech-nology (KAUST), and by the National Science Founda-tion under grants DMS-0907773 (BD). AG is a Wolf-son/Royal Society Merit Award Holder and acknowledgessupport from a Reintegration Grant under EC Frame-work VII. The work of JHM was supported by the SwissNational Science Foundation under grant nos 200021-126666/1&2, 200020-143613, and by an OCCAM SeniorVisiting Fellow award.

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