Equilibrium theory of elastic braids, with applications to DNA supercoiling

Gert van der Heijden

Department of Civil, Environmental & Geomatic Engineering University College London

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 1 / 40 Outline

Introduction – DNA supercoiling/braiding

Modelling elastic 2-strand braids

Solutions with hard contact (knots and links)

Buckling through interstrand interaction

DNA plectoneme – chiral interstrand interaction

Conclusions

(joint work with Eugene Starostin and Anthony Korte, with support from EPSRC and HFSP) (Journal of the Mechanics and Physics of Solids 64, 83–132, 2014)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 2 / 40 DNA supercoiling/braiding

Single-molecule DNA experiments and (closed) DNA plasmids:

Previous work: Benham (1983), Coleman & Swigon (2000), Thompson et al. (2002), Neukirch et al. (2008), Purohit et al. (2012) – assumed geometry and ‘simple’ electrostatic interaction

Moakher & Maddocks (2005): strongly linked rods (single-DNA strands) Goal: (i) braid with arbitrary axis (equilibria), (ii) include detailed DNA-DNA interaction (helical charge distributions) – spontaneous braiding?

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 3 / 40 Geometry of constant-distance curves r 1(s), r 2(σ) curves, with unit tangents t1(s), t2(σ)

2 Point-to-point squared distance function D2(s, σ) = ρ (s, σ), where ρ(s, σ) = r 2(σ) − r 1(s)

One-to-one mapping [0, L1] ↔ [0, L2]: s ↔ σ(s) such that

D2 is bicritical at corresponding points, i.e., ∂D2 t ∂D2 t ∂s (s, σ(s)) = 2ρ · 1 = 0 and ∂σ (s, σ(s)) = 2ρ · 2 = 0

2 Then D2(s, σ(s)) is constant (= ∆ ) (‘constant-distance curves’)

d 1 r r Unit chord vector 1(s) = ∆ [ 2(σ(s)) − 1(s)] (common parameter s)

First braid frame {t1, d 1, u1}

First material frame {t1, d 0, v 1} (rotation ξ1 about t1)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 4 / 40 Definition of braid strains

2 braid frames, 2 material frames

4 axial vectors: ω, Ω (braid), ω,e Ωe (material) (e.g.,   0 −ω3 ω2 −1 0 ωb =  ω3 0 −ω1  = R R , R rotation matrix first braid frame) −ω2 ω1 0 0 Ω = Ω + ξ0 ωe1 = ω1 − ξ1 e1 1 2 Ω1 = ω1 cos η − ω3 sin η 0 ωe2 = ω2 cos ξ1 − ω3 sin ξ1 Ωe2 = Ω2 cos ξ2 + Ω3 sin ξ2 Ω2 = ω2 + η ωe3 = ω2 sin ξ1 + ω3 cos ξ1 Ωe3 = −Ω2 sin ξ2 + Ω3 cos ξ2 Ω3 = ω1 sin η + ω3 cos η strand strains (ωe1, ωe2, ωe3, Ωe1, Ωe2, Ωe3) can be expressed in terms of the braid 0 strains and their derivatives (ω, ω , ξ1, ξ2)(unconstrained formulation) Cosserat theory with two extra directors

1
η braid angle: ω1 = ω3 − ∆ tan η π π For points at s and σ(s) to be closest-approach points we require η ∈ (− 2 , 2 )

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 5 / 40 Variational formulation in terms of braid strains

Elastic strain energy for the two strands (hyperelastic rods, but inextensible):

Z L1 Z L2 Z L1 0 0 U = U1 + U2, U1 = f1(ωe) ds, U2 = f2(Ω(e σ)) dσ = f2(Ωe/σ ) σ ds 0 0 0 Z L Interstrand (local) interaction potential (L := L1): E = φ(ω1, ω3, ξ1, ξ2) ds 0 r 0 t Isoperimetric constraint (fixed ends first strand, using inextensibility, 1 = 1): Z L −F · [r 1(L) − r 1(0)] = − F · t1 ds = const. 0   0 p 2 2 Inextensibility of the second strand σ = (∆ω1) + (∆ω3 − 1) :

Z L2 Z L Z L 0 L2 = dσ = σ ds =: fσ(ω1, ω3) ds 0 0 0 (Second-order) variational problem (F and h (constant) Lagrange multipliers): Z L 0 0 0 0 0 F t δ L ds = 0 where L = LF (ω, ω , ξ1, ξ1, ξ2, ξ2) = f1(ωe)+f2(Ωe/σ ) σ +φ+hfσ− · 1 0

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 6 / 40 Dynamic analogy – eqs of motion in rigid-body dynamics

Two common approaches (L Lagrangian): Coordinates (e.g., Euler angles q = (θ, ψ, φ))

L = L(q, q0) = L(θ, ψ, φ, θ0, ψ0, φ0) variations: δq(a) = 0 = δq(b)

Standard Euler-Lagrange equations: ∂L d ∂L − 0 = 0 ∂qi dt ∂qi Euler-Poincar´ereduction: 1 L = L(R, R0) = l(ω) = ωT I ω, ωˆ = R−1R0 2 Variations: δω = η0 + ω × η ηˆ = R−1δR, δR(a) = δR(b) = 0 Euler equation: I ω0 = I ω × ω (L0 = 0, L = I ω)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 7 / 40 Pure Euler-Poincar´ereduction – symmetric case

(Marsden, Holm, Ratiu, Bloch and co-workers) Left-invariant Lagrangian: L : TG → R (Lie group G, Lie algebra g) Reduced Lagrangian: l : g → R, l(ξ) := L(e, ξ) = L(g −1g, g −1g˙ ) = L(g, g˙ ) Then the following four statements are equivalent: Z b 1 The variational principle δ L(g(t), g˙ (t))dt = 0 ((g(t), g˙ (t)) ∈ TG) a holds, as usual, for variations δg(t) of g(t) vanishing at the end points.

2 The curve g(t) satisfies the Euler-Lagrange equations for L on G. Z b −1 3 The variational principle δ l(ξ(t))dt = 0 (ξ(t) = g(t) g˙ (t) ∈ g) a holds, using (induced) variations of the form δξ =η ˙ + [ξ, η], where η(t) is an arbitrary path in g that vanishes at the end points, i.e., η(a) = η(b) = 0. 4 The (pure) Euler-Poincar´eequations hold: d δl δl = ad∗ (ad (η) = [ξ, η]) dt δξ ξ δξ ξ −1 Reconstruction:g ˙ (t) = g(t)ξ(t), g(0) = g0 (ξ(0) = ξ0 := g0 v0, v0 =g ˙ (0)) Gert van der Heijden (UCL) Equilibrium theory of elastic braids 8 / 40 Extensions for quasi-rods

Rigid body case (G = SO(3)):

1  T  1 1 L(R, R˙ ) = tr R˙ JR˙ = tr ωˆJωˆT
= ωT I ω = l(ˆω) 2 2 2 δl = I ω, [ˆω, ηˆ] = ω\× η =⇒ I ω˙ = I ω × ω (Euler equation) δω Need two extensions: (i) forced (symmetry-broken) case, (ii) second-order case (i) Forced case: Various views:

1 Euler-Poincar´ewith advected parameters (L = LF (R, R˙ )): extended configuration space to SO(3)x(R3)∗ (semidirect extension) 2 Lagrange-d’Alembert principle, generalised forces (also covers dissipative systems; e.g., ferromagnetic rigid body (Bloch et al., 1996)) 3 Generalised Euler-Poincar´e: L = L(g(t), g˙ (t)) = L¯(g(t), ξ(t)) (special case of Hamel equations)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 9 / 40 Either way, for SO(3) we get (RF = F ):

d  δl  δl δl = × ω + × F dt δω δω δF

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 10 / 40 Euler-Poincar´ereduction – extension to second-order case

(ii) Second-order case: Consider a left-invariant Lagrangian: L : T (2)G → R, L = L(g, g˙ , g¨) = l(ξ, ξ˙) Z b Hamilton’s principle: δ L(g, g˙ , g¨)dt = 0, δg = δg˙ = 0 at end points a δl d δl Let M := − δξ dt δξ˙ Then, taking (induced) variations (now η andη ˙ and hence δξ zero at end points), Z b Z b  δl δl  Z b  δl d δl  δ l(ξ, ξ˙)dt = δξ + δξ˙ dt = − δξ dt a a δξ δξ˙ a δξ dt δξ˙ Z b Z b Z b ˙ ∗ = Mδξ dt = M(η ˙ + adξη) dt = (−M + adξ M)η dt a a a ˙ ∗ Stationarity gives M = adξ M Thus, for SO(3) (rigid body/quasi-rod; M generalises angular momentum I ω): δl d δl M˙ + ω × M = 0, M = − δω dt δω˙

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 11 / 40 Equilibrium equations (Euler-Poincar´eform)

0 0 0 Reduced (body) Lagrangian: l = l(ω, ω , F, ξ1, ξ1, ξ2, ξ2))

System of equilibrium equations:

F0 + ω × F = 0 (F 0 = 0)

0 0 0 M + ω × M + t1 × F = 0 (M + r × F = 0) ∂l d ∂l Mj = − 0 , j = 1, 2, 3 (‘constitutive relations’) ∂ωj ds ∂ωj ∂l d ∂l − 0 = 0, i = 1, 2 ∂ξi ds ∂ξi

(standard variations δξi (0) = δξi (L) = 0 on the ξi )

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 12 / 40 Standard braid equations (linear elasticity, isotropic)

0 0 0 Transform (ω1, ω2, ω3) −→ (η, ω2, ω3): l = g(η, η , ω2, ω3, F, ξ1, ξ1, ξ2, ξ2) " # ∆ cos2 η ∂g  ∂g 0 ∂g M1 = − 0 , M2 = ∆ω3 − 1 ∂η ∂η ∂ω2 " # ∂g ∆ sin η cos η ∂g  ∂g 0 M3 = − − 0 ∂ω3 ∆ω3 − 1 ∂η ∂η

0 0 0 g = ge (η, η , ω2, ω3, ξ1, ξ2) + φ + h(1 − ∆ω3)/ cos η − F1

 2 1 2 2 B2 2 2 2 C2  0  ge = B1(ω + ω ) + (Ωe + Ωe ) + C1(ω1 − ω01) + Ωe1 − σ ω02 2 e2 e3 σ0 2 3 e σ0

0 0 System of 13 equations (2 algebraic since g does not depend on ω2 and ω3) −1 0 t d u r 0 t Reconstruction of centrelines:ω ˆ = R R , R = ( 1, 1, 1), 1 = 1 4 Euler parameters (quaternions), second centreline r 2 = r 1 + ∆d 1 −→ 20D ODE

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 13 / 40 Hard interstrand contact

Given braid solution, solve inverse problem for individual strands: 0 F(1) + ω × F(1) = p(1) (1)0 (1) (1) (1) M + ω × M + t1 × F = m Strand forces F (i), moments M(i) and distributed loads (reactions) p(i), m(i) F = F (1) + F (2) (1) (2) (2) M = M + M + ∆d 1 × F Statically indeterminate with one degree of freedom (constitutive choice) (1) p(1) p(2) 0 Hard frictionless contact: p2 ≥ 0 (note + σ = 0) First integrals (valid for arbitrary hyperelastic rods): 1 h = h + (φ sin η + φ cos η) (Lagrange multiplier) 2 ∆ ω1 ω3 σ0 H = h − φ − (φ sin η + φ cos η) (Hamiltonian) 1 ∆ ω1 ω3 h1 and h2 strand energy integrals in absence of interaction φ Gert van der Heijden (UCL) Equilibrium theory of elastic braids 14 / 40 Special analytical cases

Double elastica Stacked rings Idealised open/closed starting solutions for parameter continuation: symmetric straight-axis braid (2-ply) and Villarceau circles (Hopf link) 2 02 2-ply: Br θ − V (θ) = const, θ = −η/2, r = ∆/2, B1 = B2 = B 2 4 2 V (θ) := r φ(θ) + B sin θ + Cr(τ − τ0) sin 2θ − Fpr cos θ − Mpr sin θ (Thompson et al., 2002; Coleman & Swigon, 2000) Fixed points θ < π/4 give helical solutions

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 15 / 40 Two-strand ply – energy minimisation

j d2

y+f y = 0 k i

j d2 f

y k i

Total potential energy (F and M external ply force and moment): R L 1 2 1 2
1 1 V = 0 2 Bκ + 2 Cτ ds + 2 FD − 2 MR 2 02 1 4 κ = θ + r 2 sin θ (geodesic + normal curvature) 0 0 τ = φ + ψ cos θ (Euler angles for {d 1, d 2, d 3}) R L D = 0 (1 − cos θ) ds (end shortening) R L 0 R = 0 ψ ds (end rotation, ‘ply link’) 0 1 ψ = r sin θ (cylindrical constraint)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 16 / 40 Equilibrium equation for loaded ply (twist Cτ = constant):

00 2B 3 Cτ 1 M Bθ = r 2 sin θ cos θ + r cos 2θ + 2 F sin θ − 2r cos θ Fixed points θ = const. represent uniform helical plies (F = M = 0):

2B sin3 θ cos θ Cτ Cτ = − , V = , T = pr = V tan θ r cos 2θ 2r

All forces and moments → ∞ as θ → 45◦ (lock-up angle!)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 17 / 40 2-ply

10

x 5

0 arclength shift -5

-10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 helical angle θ (rad)

Trivial branch (x = 0); pitchfork at θ = 45◦ (lock-up). x 6= 0: contact curve is a helix; central hole (∀ n). The net contact force is normal to the cylinder (∀ n). Optimal packing: densest coiling at θ = 45◦, or, p/r = 2π (p: pitch, r: radius) DNA data: r = 5.92 A,˚ p = 35.7 A˚ (10.5 base pairs)

Gert van der Heijden (UCL) Equilibriump theory/r of= elastic 6.03 braids 18 / 40 Straight-axis 2-ply revisited

Equilibrium if either

0 1 η = 0 (uniform braid)

Can allow interaction φ = φ(η, ω3, ξ1, ξ2)

2 B1 = B2 (equal bending stiffnesses) Can allow η0 6= 0, but require φ = 0

Conclusion: straight axis and η0 6= 0 generally inconsistent if φ 6= 0

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 19 / 40 Torus links

Lk = −n (n = 1, 2, 3, 4, 5), left-handed (∆/L = 0.02); note D2n-symmetry

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 20 / 40 4-link (symmetry properties)

Contact pressure (> 0) and braid angle Colouring according to contact pressure p

3900 0.50935 3800 0.5093 3700p η 3600 0.50925 3500 0.5092 3400 0.50915 3300 3200 0.5091 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 s 0.6 0.8 1 s

Symmetry: p is period-n, η is period-2n (not imposed by boundary conditions!)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 21 / 40 Torus knots (n, 2)

Lk = −n/2 (n = 1, 3, 5, 7, 9), left-handed (∆/L = 0.02); note Dn-symmetry

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 22 / 40 pentafoil knot (symmetry properties)

Contact pressure (> 0) and curvature of the contact curve (L2 6= L):

6.8

550 6.6 κ 500 c p 6.4 450 6.2

400 0.0 0.250.5 0.75 1.0 0 0.5 1s 1.5 2 s

Symmetry: p is period-n on [0, L + L2], κc is period-n on [0, L] (not imposed by boundary conditions!)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 23 / 40 Bifurcation diagram for twisted knots (3-, 7-, 11-, 15-foil)

T = T1 = −T2, common strand torque (knot cut and twist inserted); F = |F | Gert van der Heijden (UCL) Equilibrium theory of elastic braids 24 / 40 Bifurcation diagram for a twisted loop

(Hoffman, Proc. R. Soc. A 461, 2005) (isotropic rod)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 25 / 40 Buckling modes for twisted knots

1st, 2nd and 3rd mode for 11-foil knot (‘i’, ‘s’, ‘u’):

Cable trefoil knot (after 2nd-mode triple self-intersection 7-foil → 13-foil, ‘q’):

All have positive pressure

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 26 / 40 Cable knots along bifurcating branches of (7, 2) torus knot

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 27 / 40 Buckling of 6-turn straight-axis braid under compression

29.6 29.5 29.4 29.3 T1 29.2 29.1 29 28.9 -6950 -6900 -6850 -6800 -6750 -6700 -6650 -6600

Fp

0.7

0.6 0.7 0.5 0.6

0.5 0.4 z z 0.4 0.3 0.3

0.2 0.2 0.1 0.1 0 0.008 0.006 0.01 0.012 0.002 0 0 x ±0.002 0.014 y ±0.006 x y

Localised buckling modes (like torsional modes of a single rod)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 28 / 40 Torsional buckling of straight-axis braid

Bifurcation diagram (constant torque Mp = −8.9 and controlled tensile load Fp) Torque applied about the end-to-end vector of the braid axis Overwinding right-handed 6-turn braid with θ = 30◦

1.2

1.15

2 1.1 L

1.05

1 0 50 100 150 200 250 300 350 400

Fp Symmetry-breaking bifurcation (diamond), p > 0 everywhere

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 29 / 40 Approaching ideal shapes (trefoil knot)

Ideal knots are the tighest knots within a given knot class (or shortest ropelength) Equilibria!

0.5

0.4

s/Ltot 0.3 σ/Ltot 0.5 0 0.25 0.5 κ∆/2 0 0.2

−2.5 0.1 τ∆/2 0.45 0.0 −5.0 0 0.25 0.5 0.0 0.1 0.2 0.3 0.4 0.5 s/L tot s/Ltot 2 Shortest ropelength= (L1 + L2)/R = 32.8556 Local non-self-intersection: κ ≤ ∆ Best estimate: 32.742934477 (Przybyl & Pieranski, 2014) Ideal contact curve is not a topological circle (and 1-to-1 mapping broken)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 30 / 40 Pentafoil

Symmetry-breaking bifurcation (D5-symmetry −→ C2-symmetry):

1.4 1.3 1.2

1 1.1 /L 2

L 1 0.9 0.8 0.7 40 60 80 100 120 140 160 180 200 Ropelength

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 31 / 40 Buckling of two linked rings due to interstrand interaction

Model for a DNA minicircle with simplified chiral electrostatic potential η φ = α tan 2 (Kornyshev et al., 2007) (α > 0 tends locally to decrease η) 0 12 3

a b d -2 e

-4 c (d)

(b) T1 -6 -1 d -1.2 b -8 -1.4

T 1 -1.6

-10 -1.8 e -2 (c) 1000 1500 2000 2500 3000 3500 f α (e) -12 -1000 -500 0 500 1000 1500 2000 2500 3000 3500 α

Material frames closed with linking numbers Lki = 5 Transition from left-handed (Lk = −1) to right-handed link (Lk = 1); Lki = 5 + 2 = 7 Gert van der Heijden (UCL) Equilibrium theory of elastic braids 32 / 40 Different contact models:

Normal (pn, dashed) and tangential (pt , solid) pressure components of 2nd-mode solution (c)

pt , p n pt , pn

p p t along contact curve t normal to contact curve

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 33 / 40 Chiral effects in DNA-DNA interaction

Chiral torque due to helical charge distribution (Kornyshev et al., 2007)

(Cortini et al., Biophysical Journal 101, 2011) Strong enough to induce spontaneous braiding?

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 34 / 40 Chiral effects in single-DNA experiments

(Charvin et al., Biophysical Journal 88, 2005) A positive number of turns n corresponds to a right-handed braid (Small but consistent) chiral effect: right-handed braids more stable (at high salt)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 35 / 40 Theory vs. experiment (Soft Matter 9, 9833, 2013):

(a) F = 4 pN (10 mM), (b) F = 2 pN (10 mM), (c) F = 2 pN (100 mM):

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 36 / 40 Protein-free brading in single-molecule experiments

(Lee, Kornyshev, Prentiss et al., Proc. R. Soc. A 472, 2016)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 37 / 40 Model predictions and comparison against experimental data:

Salt concentrations: (a) 0.15 M, (b) 0.5 M, (c) 1 M, (d) 3 M

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 38 / 40 Spin-off: curves on surfaces – invariant characterisation

Cylindrical curve: In terms of intrinsic curve properties κ (curvature) and τ (torsion)

κ cos ξ = R[(τ + ξ0)2 + κ2 cos2 ξ], κ0 cos ξ = κ(2τ + 3ξ0) sin ξ for some function ξ and some constant (radius) R

In terms of geometric properties (τg , κg , κn) of the curve as it lies in the surface:

2 2 0 κn = R(κn + τg ), κn = 2κg τg

Spherical curve: " #0 1  1 0 τ + = 0 τ κ κ

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 39 / 40 Conclusions

We developed a new theory for equilibria of elastic 2-braids, i.e., two rods winding around each other in continuous contact and allowing for interstrand interaction (or only hard frictionless contact).

Computed both open and closed (knotted or linked) structures.

Found buckling of linked braid due to interstrand interaction.

Predictions consistent with historical data showing chiral effects.

(reinterpreted solution, r1 + r2 = ∆)

Gert van der Heijden (UCL) Equilibrium theory of elastic braids 40 / 40