DNA topology, geometry, and mechanics

David Swigon

University of Pittsburgh

September 2007

• 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?

DNA

"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

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

?=

DNA knots

Reidemeister moves ?=

Problems:How to show a sequence exists (does not exist)? How to find the sequence?

KNOT THEORY

Knot invariants : Alexander polynomial, (Jones, Conley, Vassiliev,….)

III.

II.

I.

II.

I.

Prime knots

Prime catenanes (links)

Catenanes occur during replication1 closed DNA => 2 closed DNA, interlinked

Separation requires knot removal

Knots in biology

Knots and catenanes occur during recombination, or artificially during DNA closure

31 41 52

Unregulated catenation and knotting leads tocell death

Type II topoisomerases = enzymes regulating DNA topology – change knot type by cutting 2 strands and performing strand passage

Mathematics of closed DNA

Closed DNA >>> 2 curves in space

The topology of DNA Linking numberLk = 1/2 the number of signed crossings in a planar projection

Gauss formula (with t = dx/ds):+ –

Lk = –1

–

–

+

+ +

+

Lk = 2Properties:• Attains only integer values• Is a topological invariant

Lk = 8

( )213

2211

2211221121

1 2)()(

)()()()(41),( dsds

ss

ssssCCLk

C C∫ ∫

−

−⋅×=

xx

xxttπ

Writhe

Wr = average, over all planar projections, of the number of signed self-crosings

Wr = 0 for planar curves(no crossings)

Wr ~ 1 +

Wr ~ –2 ––

Plectonemic toroidal

Wr < 0 for left-handed

helix

Writhe is a measure of helicity

Wr ~ 0 –+

+ –

( )sdsd

ss

ssssCWr

C C

~)~()(

)~()()~()(41)(

1 1

3111

11111 ∫ ∫

−

−⋅×=

xx

xxttπ

[Fuller, PNAS 68 (1971) 815-819; Fuler PNAS 75 (1978) 3557-3561;Aldinger et al, J Knot Theor Ram 4 (1995) 343-372]

TwistTw = number of turns of one curve

about the other (d = x2 – x1)111112

1

)()()(21),( dssssCCTw

C∫ ⋅′×= tdd

π

Tw = 0 Tw = 0.5 Tw = 1

Theorem

),()(),( 12121 CCTwCWrCCLk +=

[Vinograd et al, PNAS 53 (1965) 1104-1111;Calugareanu, Czech. Math. J 11 (1961) 588-625;White, Amer. J. Math. 91 (1969) 693-728]

Supercoiling of DNA = deformation accompanied by an increase in |Wr|

• Untwisting of DNA in a closed plasmid leads to increase in Wr

• An increase in Lk causes an increase in both Wr and Tw(as DNA prefers twist of 1 turn per 10.5 bases)

),()( 121 CCTwCWr ∆−=∆

),(),()( 21121 CCLkCCTwCWr ∆=∆+∆

Supercoiling occurs during transcription, …

or due to action of untwisting proteins and drugs.

∆Tw = 36°

during replication, …

Regulated supercoiling is necessary for survival Type I and II topoisomerases = enzymes that adjust Lk

Type I:cuts 1 strand +rotates 360º

DNA geometry (continuous)

Axial curve: smooth function , s = arc-length of the curve

Tangent:

Curvature:

Normal:

Binormal:

Torsion:

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:

• Curves with constant κ and τ are helices

Twist density

)()( ss xt ′=

Lss ≤≤0),(x

1)( =st

)()( ss t′=κ

)()()( sss κtn ′= )()( ss tn ⊥

)()()( sss ntb ×= tn

b)(||)( ss nb′

)()()( sss nb ⋅′−=τ

nbbtn

nt

ττκ

κ

−=′+−=′

=′

dssCCTwssssC∫ Ω=⋅′×=Ω

1

)(21),()()()()( 12 π

tdd

Atoms Base pairs

DNA geometry (base-pair level)

Tilt θ1 Roll θ2 Twist θ3

Shift ρ1 Slide ρ2 Rise ρ3

( ) ( ) ( )( ) ( ) ( ) n

jn

ljn

klnn

iknn

i

nnlj

nkl

nnik

nj

ni

ZYZ

ZYZ

ργκγθ

γθκγθ

21

321

321

3211

−=⋅

+−=⋅ +

rd

dd

nnnnnn γκθγκθ cos,sin 21 ==

Reversible parametrization

( ) ( )nnnnnnnnnnnn321321321321 ,,,,,,,,,, ρρρθθθρρρθθθ −−↔

Parameters are almost invariant under a change in DNA direction

[El Hassan & Calladine, J Mol Biol 251 (1995) 648]

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

Mechanics of DNA

Kirchhoff’s theory of elastic rods applied to DNA [Benham, PNAS 74 (1977) 2397-2401]

Assumptions – intrinsically straight– homogeneous– isotropic– inextensible

Configuration – axial curve x = x(s)– twist density

Ω = Ω(s) = d(s) × d´(s) · t(s)

t

d

A simple continuum model for DNA

Elastic energy Balance equation for moments

tFttt ×=′∆Ω+′′× CA

F and ∆Ω are Lagrange multipliersExplicit solutions can be found

A – bending modulusC – twisting modulusΩu – intrinsic twist

∫∫ Ω−Ω+=Ψl

ul

dssCdssA0

221

0

221 ))(()(κ

First Integral

FrFttt λ+×=∆Ω+′× CA

( )kxkxxx λ+×=′+′′×′ 2T

In dimensionless units

Solution in cylindrical coordinates:

)|;()(

)|;()2/(

sin)(

131

133

2233

2

mnEuusuaz

mnuuuaTs

uuur

ψ

ψλλφ

ψ

−−−=

Π−

−+=

−−=

3

23

13

23

13

,

)(snsin

uuun

uuuum

uus

−=

−−

=

−=ψ

Where sn is Jacobi elliptic function, E and Π are Jacobi elliptic integrals, and u1, u2, and u3 are the roots of

222223 )2/()12()2()( λλλλ aTuTaauauuP −+−+−+−+=

[see Landau & Lifshitz, Theory of Elasticity, 1986;Tobias, Coleman, Olson, J Chem Phys 101 (1994) 10990-10996]

Constraints• End conditions

Closed DNA:

• 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

0ttt == ∫l

dssl0

)(),()0(∆Lk = α/2π

t(s)

x(s)x(s*)

Configurations with self-contactContact conditions

Balance of forces

•Solutions are composed of contact-free segments

( ) 0*)()()(*)()(

=−⋅=−

sssDss

xxtxx

( ) fsss )(*)()]([ xxF −=

[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:

Sufficient condition:and x locally uniformly minimizes ΨB at fixed W

Necessary condition II:holds with any subsegment held rigid

Theorem: Condition II is sufficient for differential stability (δ2Ψ ≥ 0).

d∆L dW ≥ 0

d∆L dW > 0

d∆L dW ≥ 0

Secondary branches

Observations

• Stability requires self-contact• Regions of continuous self-contact along lines• Higher-order branches are unstable

metastable stable

[Coleman, Swigon, Tobias, Phys Rev E 61 (2000) 759-770]

Bistability

Contact-free DNA knots

Observations

• Contact-free configurations have the topology of torus knots

• All contact-free knots are unstable

[Coleman & Swigon, Proc Roy Soc Lond A, 362 (2004) 1281-1299]

[Langer & Singer, J LondMath Soc 30 (1984) 512-530]

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

d3

d1

d2

• Axial curve• Directors• Curvature vector• Shears • Equilibrium equations

• Constitutive equations (hyperelastic, quadratic, no coupling)

• 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]

)(sr

)(sid

isi dd ×= κ,

mFrMfF

=×+=

ss

s

333322221111

333322221111

)()()(

)()()(

dddM

dddFuuu

uuu

KKK

vvAvvAvvA

κκκκκκ −+−+−=

−+−+−=

∑=i iis v dr

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]

Ψ = ψ n(θ1n,θ2

n,θ3n, ρ1

n, ρ2n ,ρ3

n )n=1

N

∑

Base-pair level elastic model for DNA

Quadratic approximation:• FXY, GXY, HXY are elastic moduli of the base-pair step XY• are intrinsic values of kinematic parameters

Tilt θ1 Roll θ2 Twist θ3

Shift ρ1 Slide ρ2 Rise ρ3

Dinucleotide model

Higher order models: trinucleotide, tetranucleotide, …

nXYnnXYnnXYnn ρρρθθθ ∆⋅∆+∆⋅∆+∆⋅∆= HGF 21

21ψ

XYnn θθθ −=∆ XYnn ρρρ −=∆

XYXY ρθ ,

,...),,...,,( 11 −−= nnnnnn ρρθθψψ

[Packer, Dauncey, Hunter, J Mol Biol 295 (2000) 85-103]

Intrinsic curvatureAA straightGG bent with(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)

Bending anisotropy 1.3 < F11/F22 < 3.0

Shear

[Trifonov, Trends Biochem. Sci. 16 (1991) 467-470; Calladine & Drew, J. Mol. Biol. 178 (1984) 773-782; Bolshoy et al. PNAS 88 (1991) 2312-2316; Gorin et al., J. Mol. Biol. 247, (1995) 34-48;Dlakic & Harrington, PNAS 93 (1996) 3847-3852; Olson et al., PNAS 95 (1998) 11163-11168;Gore et al., Nature 442 (2006) 836-840]

θ 2 ~ 5o

514.3 2/12

2/12≈

>∆<>∆<

θρ

Sequence-dependent properties

Extraction of moduli and intrinsic parameters from MD simulations[Gonzales & Maddocks, Theor Chem Acc. 106 (2001) 76-82; Dixit et al, Biophys J. 89 (2005) 3721]

Variational equations

1−= nn ffnnnn rfmm ×=− −1 n

j

nnnn

ijni

n Qρψθθθ

∂∂

=⋅ ),,( 321df

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Λ

∂∂

+∂∂

Γ=⋅ nl

nnnkljn

k

n

nj

nnnn

ijni

n ρθθθρψ

θψθθθ ),,(),,( 321321dm

End conditions• closure• strong anchoring fixed

Nii

N ddxx == 11 ,Nii

N ddxx ,,, 11

( ) ( ) 0,,...,,0,,..., 131

131 == −− n

innn

in

innn

i mmff ρθρθ

( ) ( ) iNii

Ni ρρθρθρθθ ~,...,,~,..., 1

311

13

11 ==

Open problems:• Uniqueness of IVP• Spurious solutions of BVP• Choice of parametrization, energy function, generality of results

Solution: IVP – recursive solutionBVP – shooting method

Multiple equilibria of DNA O-ringA

Ψ = 0

C

α

β

γ

δ

Ψ = 61.0

B

Ψ = 73.6

D

Ψ = 61.4 Kinetoplast DNA from Leishmania tarentolae

Sequence dependent effects

[Coleman, Olson, Swigon, J. Chem. Phys. 118 (2003) 7127-7140]

-300 -200 -100 0 100 200 300

Ψ

α

10

15

20

25

30

35

40

45

50I

Isym

Iall

Iper

Effect of roll-twist coupling on twist softening or hardening

• I – ideal DNA, no coupling• Iper – periodically distributed coupling• Isym – symmetrically distributed coupling• Iall – all coupling

Results• Reduction of effective bending and twisting moduli

C C C C C C C C C C C C C C C

C C C C C C C C C C C C C C

Aeff Ceff/Aeff

I 0.0427 1.4

Iper 0.035 0.004

Isym 0.036 0.5

Iall 0.030 1.2

O-ring

O-ring + coupling

S-shaped

S-shaped + coupling

α = –330º –240º –120º 0º 120º 240º 330º

• Collapse of loops under small twisting• Localization of twisting deformation

Summary

• Understanding of DNA behavior was greatly enhanced by new concepts and results in topology (Lk, Wr, Tw formula)

• Knot theory helped in deciphering the mode of action of topoisomerasesand recombinases

• Elasticity theory was employed to study DNA supercoiling and loop formation

• General continuum model or base-pair level discrete model are needed to account for base-pair variability of elastic properties

Challenges• Accurate model of DNA elasticity accounting for sequence dependence,

salt dependence, and higher order effects (kinking)• New algorithms for solving equilibrium equations, locating metastable and

unstable (transition) states