david swigon university of pittsburgh september 2007dna topology, geometry, and mechanics david...
TRANSCRIPT
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