DNA topology, geometry, and mechanics David Swigon University of Pittsburgh September 2007 DNA • Identical copy in each cell • Single molecule, long and thin: E. coli – length/thickness = 106 • Features different from regular polymers • Relatively rigid on small scales • Torsionally constrained Nature needed to solve problems with • Topology of closed molecules • Influence of mechanics on function • DNA compaction • Accessibility for processing What are the associated mathematical problems? "Since the two chains in our model are intertwined, it is essential for them to untwist if they are to separate. ...... Although it is difficult at the moment to see how these processes occur without everything getting tangled, we do not feel that this objection would be insuperable.“ J. D. Watson and F. H. C. Crick, 1953 Mathematics of closed DNA Closed DNA >>> closed curve in space DNA knots Two closed DNAs are of the same knot type if and only if one can be deformed into the other without the curve passing through itself ? = Reidemeister moves ? = III. I. II. II. Problems: How to show a sequence exists (does not exist)? How to find the sequence? I. KNOT THEORY Knot invariants : Alexander polynomial, (Jones, Conley, Vassiliev,….) Prime knots Prime catenanes (links) Knots in biology Catenanes occur during replication 1 closed DNA => 2 closed DNA, interlinked Separation requires knot removal Knots and catenanes occur during recombination, or artificially during DNA closure Unregulated catenation and knotting leads to cell death Type II topoisomerases = enzymes regulating DNA topology – change knot type by cutting 2 strands and performing strand passage 31 41 52 Mathematics of closed DNA Closed DNA >>> 2 curves in space The topology of DNA Linking number Lk = 1/2 the number of signed crossings in a planar projection + – Gauss formula (with t = dx/ds): 1 t (s ) × t (s ) ⋅ (x (s ) − x (s )) Lk(C ,C ) = 1 1 2 2 1 1 2 2 ds ds 1 2 ∫∫ 3 1 2 4π x (s ) − x (s ) CC12 1 1 2 2 – + + – + + Lk = –1 Properties: Lk = 2 Lk = 8 • Attains only integer values • Is a topological invariant Writhe + – Wr = average, over all planar projections, of the number of signed self-crosings Wr < 0 for left-handed helix Wr = 0 for planar curves (no crossings) Wr ~ 1 + Plectonemic toroidal Wr ~ –2 – – Writhe is a measure of helicity 1 t (s) × t (~s ) ⋅ (x (s) − x (~s )) Wr(C ) = 1 1 1 1 dsd~s 1 4π ∫∫ ~ 3 CC x (s ) − x (s ) Wr ~ 0 11 1 1 1 + – [Fuller, PNAS 68 (1971) 815-819; Fuler PNAS 75 (1978) 3557-3561; Aldinger et al, J Knot Theor Ram 4 (1995) 343-372] Twist Tw = number of turns of one curve 1 Tw(C2 ,C1) = d(s1) × d′(s1) ⋅ t(s1) ds1 about the other (d = x – x ) 2π ∫ 2 1 C1 Tw = 0 Tw = 0.5 Tw = 1 Theorem [Vinograd et al, PNAS 53 (1965) 1104-1111; Calugareanu, Czech. Math. J 11 (1961) 588-625; White, Amer. J. Math. 91 (1969) 693-728] Lk (C1,C2 ) = Wr(C1 ) + Tw(C2 ,C1 ) Supercoiling of DNA = deformation accompanied by an increase in |Wr| • Untwisting of DNA in a closed plasmid leads to increase in Wr ∆Wr(C1) = −∆Tw(C2 ,C1) • An increase in Lk causes an increase in both Wr and Tw (as DNA prefers twist of 1 turn per 10.5 bases) ∆Wr(C1) + ∆Tw(C2 ,C1) = ∆Lk(C1,C2 ) Supercoiling occurs during transcription, … during replication, … Regulated supercoiling is necessary for or due to action of untwisting survival proteins and drugs. Type I and II topoisomerases = enzymes that adjust Lk Type I: cuts 1 strand + rotates 360º ∆Tw = 36° DNA geometry (continuous) Axial curve: smooth function x ( s ), 0 ≤ s ≤ L , s = arc-length of the curve Tangent: t(s) = x′(s) t(s) = 1 Curvature: κ (s) = t′(s) Normal: ′ n(s) ⊥ t(s) n(s) = t (s) κ(s) b Binormal: b(s) = t(s) × n(s) b′(s) || n(s) t n Torsion: τ (s) = −b′(s) ⋅n(s) Serret-Frenet equations for space curve: • A 3D curve is uniquely determined by giving its curvature and torsion. • x(s) is the solution of the following system: t′ = κ n ′ • Curves with constant κ and τ are helices n = −κ t +τ b b′ = −τ n 1 Twist density Ω(s) = d(s) × d′(s) ⋅ t(s) Tw(C ,C ) = Ω(s)ds 2 1 2π ∫ C1 DNA geometry (base-pair level) Atoms Base pairs Tilt θ1 Roll θ2 Twist θ3 Shift ρ1 Slide ρ2 Rise ρ3 Reversible parametrization dn ⋅ dn+1 = Z 1 θ n − γ n Y κ n Z 1 θ n + γ n i j ik (2 3 ) kl ( ) lj (2 3 ) dn ⋅ rn = Z 1 θ n − γ n Y 1 κ n Z γ n ρ n i ik ()2 3 kl ()()2 lj j n n n n n n θ1 = κ sinγ , θ2 = κ cosγ Parameters are almost invariant under a change in DNA direction n n n n n n n n n n n n (θ1 ,θ2 ,θ3 , ρ1 , ρ2 , ρ3 )↔ (− θ1 ,θ2 ,θ3 ,−ρ1 , ρ2 , ρ3 ) [El Hassan & Calladine, J Mol Biol 251 (1995) 648] Mechanics of DNA Basic model structure • Continuum •Discrete Physics • Elasticity (short range atomic interactions) • Electrostatics (long range interactions) • Secondary structure changes (melting, kinking, phase transitions) Environment • Counterions • Solvent • DNA-binding proteins A simple continuum model for DNA Kirchhoff’s theory of elastic rods applied to DNA [Benham, PNAS 74 (1977) 2397-2401] t Assumptions – intrinsically straight – homogeneous d – isotropic – inextensible Configuration – axial curve x = x(s) – twist density Ω = Ω(s) = d(s) × d´(s) · t(s) Elastic energy Balance equation for moments l l Ψ = 1 Aκ (s)2 ds + 1 C(Ω(s) − Ωu )2 ds At × t′′ + C ∆Ω t′ = F × t 2 ∫ 2 ∫ 0 0 A – bending modulus F and ∆Ω are Lagrange multipliers C – twisting modulus Explicit solutions can be found Ωu – intrinsic twist First Integral At × t′ + C ∆Ω t = F × r + λF In dimensionless units x′ × x′′ + Tx′ = 2(k × x + λk) Solution in cylindrical coordinates: 2 2 sinψ = sn(s u − u ) r = u3 − (u3 − u2 )sin ψ 3 1 (T / 2 − λa) u3 − u2 φ = λs + Π(n;ψ | m) m = , u − u u3 u3 − u1 3 1 u3 − u2 z = (a − u1)s − u3 − u1 E(n;ψ | m) n = u3 Where sn is Jacobi elliptic function, E and Π are Jacobi elliptic integrals, and u1, u2, and u3 are the roots of P(u) = u3 + (λ2 − 2a)u2 + (a2 − 2aλ2 + Tλ −1)u + (T / 2 − aλ)2 [see Landau & Lifshitz, Theory of Elasticity, 1986; Tobias, Coleman, Olson, J Chem Phys 101 (1994) 10990-10996] Constraints • End conditions l Closed DNA: t(0) = t(l), ∫ t(s)ds = 0 0 ∆Lk = α/2π • Excess Linking number ∆L = W(x) + T(Ω) – T(Ωu) – topological invariant that can be varied continuously by changing T(Ωu) or by cutting and rotation of ends t(s) Configurations with self-contact Contact conditions x(s) x(s) − x(s*) = D x(s*) t(s) ⋅ ()x(s) − x(s*) = 0 Balance of forces [F](s) = (x(s*) − x(s)) f •Solutions are composed of contact-free segments [Coleman & Swigon, J. Elasticity 60 (2000) 173-221] Bifurcation diagram for knot-free DNA ring l/D = 122, C/A = 1.5 Primary branch Conditions for stability Necessary condition I: d∆LdW ≥ 0 Sufficient condition: d∆LdW > 0 and x locally uniformly minimizes ΨB at fixed W Necessary condition II: d∆LdW ≥ 0 holds with any subsegment held rigid Theorem: Condition II is sufficient for differential stability (δ2Ψ≥0). Secondary branches Observations • Stability requires self- contact • Regions of continuous self-contact along lines • Higher-order branches are unstable Bistability metastable stable [Coleman, Swigon, Tobias, Phys Rev E 61 (2000) 759-770] Contact-free DNA knots Observations • Contact-free configurations have the topology of torus knots • All contact-free knots are unstable [Langer & Singer, J Lond Math Soc 30 (1984) 512-530] [Coleman & Swigon, Proc Roy Soc Lond A, 362 (2004) 1281-1299] DNA trefoil knot with self-contact Observations • Stability requires self- contact • Regions of continuous self-contact along curves [Coleman & Swigon, Proc Roy Soc Lond A, 362 (2004) 1281-1299] A general rod model for DNA • Axial curve r(s) • Directors d (s) i d3 d2 • Curvature vector di,s = κ × di • Shears r = v d s ∑i i i d1 • Equilibrium equations Fs = f Ms + rs × F = m • Constitutive equations (hyperelastic, quadratic, no coupling) u u u F = A1(v1 − v1 ) d1 + A2 (v2 − v2 ) d2 + A3(v3 − v3 ) d3 u u u M = K1(κ1 − κ1 ) d1 + K2 (κ 2 − κ 2 ) d2 + K3(κ3 − κ3 ) d3 • Describes potentially nonhomogeneous, anisotropic, extensible, shearable DNA with intrinsic curvature and shear • No explicit solutions: must be solved numerically, e.g., by numerical integration using Euler parameters [Dichman, Li, Maddocks, IMA Vol Math Appl 68 (1996) 71] Intrinsically curved DNA minicircles can have multiple equilibrium configurations: Two configurations with identical Lk Two locally minimizing configurations of a nicked minicircle [Furrer, Manning, Maddocks, Biophys J 79 (2000) 116-136] Base-pair level elastic model for DNA Dinucleotide model N n n n n n n n Tilt θ Roll θ Twist θ Ψ=∑ψ (θ1 ,θ2 ,θ3 , ρ1 , ρ2 ,ρ3 ) 1 2 3 n=1 Shift ρ1 Slide ρ2 Rise ρ3 Quadratic approximation: • FXY, GXY, HXY are elastic moduli of the base-pair step XY XY XY •θ , ρ are intrinsic values of kinematic parameters ψ n = 1 ∆θ n ⋅ F XY ∆θ n + ∆θ n ⋅ G XY ∆ρ n + 1 ∆ρ n ⋅ H XY ∆ρ n 2 2 ∆θ n = θ n − θ XY ∆ρ n = ρ n − ρ XY Higher order models: trinucleotide, tetranucleotide, … ψ n =ψ n (θ n ,θ n−1,..., ρ n , ρ n−1,...) [Packer, Dauncey, Hunter, J Mol Biol 295 (2000) 85-103] Sequence-dependent properties Intrinsic curvature AA straight o GG bent with θ 2 ~5 (AAAAACGGGC) n A-tracts = intrinsic curvature (~13˚/10bp) Twist-roll coupling 0.13 < F23/F22 < 0.58 (bending induces untwisting) Twist-stretch coupling –0.80 < G33 < –0.25 (stretching induces overtwisting) [Trifonov, Trends Biochem.