predicting local geometric properties of dna from ...arbogast/cam397/gonzalez1.pdf · introduction...

Introduction Geometric model Diffusion model Numerical method Preliminary results

Predicting local geometric properties of DNAfrom hydrodynamic diffusion data

O. Gonzalez and J. Li

UT-Austin

27 October 2008

O. Gonzalez and J. Li UT-Austin

Hydrodynamic modeling of DNA


Introduction




Goal

To obtain improved estimates of the material parameters thatdescribe sequence-dependent shape of DNA.




Approach

Refine parameters through hydrodynamic diffusion modeling.

GGTAATGCTTAACACCTGTACGTTAATTCGTAGGA

TACCCGTA

geometric

modeldiffusion

step modelrefinement

diffusionexperiment predicted

behavior

test shapesmeasuredbehavior

parameters




Outline

1 Geometric model

2 Diffusion model

3 Numerical method

4 Preliminary results




Geometric model




Basepair model

DNA can be represented as a tube formed by oriented discs.

T A

CG

CG

GC

AT r




Model parameters

XX

YYXY XY

X X

Y Y

η θ

T

A

C

G

(*,*,*)

X

(*,*,*) (*,*,*) (*,*,*)

CGA T

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

(*,*,*)

Y

ηXY = relative displacement for dimer step XY.

θXY = relative rotation for dimer step XY.

r = radius of tubular surface.

η = ηXY ∈ R26, θ = θXY ∈ R26, r ∈ R.




Parameter estimates

Tilt | Roll | Twist×10−1 (deg)†

X X

Y Y

Twist

RollTilt Slide

Rise

Shift

r

T A C G

A 0.0|1.1|2.9 −1.4|0.7|3.5 −0.1|0.7|3.2 −1.7|4.5|3.2T −1.4|0.7|3.5 0.0|3.3|3.8 −1.5|1.9|3.6 0.5|4.7|3.7G −0.1|0.7|3.2 −1.5|1.9|3.6 0.0|0.3|3.4 −0.1|3.6|3.3C −1.7|4.5|3.2 0.5|4.7|3.7 −0.1|3.6|3.3 0.0|5.4|3.6

†From x-ray data: Olson et al, PNAS 95 (1998) 11163.




Parameter estimates

Shift×10 | Slide×10 | Rise (A)†

X X

Y Y

Twist

RollTilt Slide

Rise

Shift

r

T A C G

A 0.0|−5.9|3.3 −0.3|−0.8|3.3 1.3|−5.8|3.4 0.9|−2.5|3.3T −0.3|−0.8|3.3 0.0| 0.5|3.4 −2.8| 0.9|3.4 0.9| 5.3|3.3G 1.3|−5.8|3.4 −2.8| 0.9|3.4 0.0|−3.8|3.4 0.5|−2.2|3.4C 0.9|−2.5|3.3 0.9| 5.3|3.3 0.5|−2.2|3.4 0.0| 4.1|3.4

†From x-ray data: Olson et al, PNAS 95 (1998) 11163.




Parameter estimates

Hydrated radius (A)‡

X X

Y Y

Twist

RollTilt Slide

Rise

Shift

r

r.= 13.

‡Using straight model: Tirado et al, J Chem Phys 81 (1984) 2047.




Model construction

X X X XX X X XS = X X X X

1 2 3 n1 2 3 n1 2 3 n

Points qa(S , η, θ), Curve γ(S , η, θ), Surface Γ(S , η, θ, r)∗.

* Assumed rigid.




Example 40-bp sequences

A=aqua, T=blue, G=gold, C=red




Example 120-bp sequences

A=aqua, T=blue, G=gold, C=red




Diffusion model




Single molecule drift dynamics

Setup. Consider a single molecule in a fluid subject to externalloads.

v(t)

(t)

DNA

fluid

f τ

ω

ext ext

(v , ω)(t) = linear, angular velocities.

(f , τ)ext(t) = external force, torque.




Single molecule drift dynamics

Drift velocitiesωv ,

t

v , ω

t

v(t), ω(t) = 1T

∫ t+Tt v(s), ω(s) ds.

T = window size.

Governing eqns L

[vω

]=

[fτ

]ext

or

[vω

]= M

[fτ

]ext

.

L,M ∈ R6×6 Hydrodynamic drag,mobility matrices of molecule.




Definition of hydrodynamic matrices

Stokes eqns

µ∆u = ∇p in R3\B∇ · u = 0 in R3\B

u = U[v , ω] on ∂Bu, p → 0 as |x | → ∞.

BCs, loadsU[v , ω] = v + ω × (x − c)

σ[u, p] = −pI + µ(∇u +∇uT

)f [σ] =

∫∂B σn dA

τ [σ] =∫∂B(x − c)× σn dA.




Definition of hydrodynamic matrices

Linearity result. The map from (v , ω) ∈ R6 to (f , τ) ∈ R6 islinear and invertible:

(v , ω)Stokes−→ (u, p)

∂−→ σ

∫−→ (f , τ).

Thus there exists L,M ∈ R6×6 such that[fτ

]= −

[L1 L3

L2 L4

]︸︷︷︸

L

[vω

],

[vω

]= −

[M1 M3

M2 M4

]︸︷︷︸

M

[fτ

].

L,M are called the Stokes drag, mobility matrices.




Properties of hydrodynamic matrices

L

[vω

]=

[fτ

]ext

or

[vω

]= M

[fτ

]ext

L,M are symmetric, positive-definite.

L,M depend on shape of ∂B and and ref point c .

Must solve Stokes equations in R3\B to determine.

Characterize velocities (v , ω) for given loads (f , τ)ext.




Collective drift dynamics

Setup. Consider a dilute solution of identical molecules in a fluidsubject to external loads.

Dom x SO3 Ω x A

Physicaldomain

(r,Q)

Configurationspace

Local coordspace

(q,η )

Dom

ext

extτ

f

σ = mass concentration in config space.

hext = (f , τ)ext = external loads.

M = Stokes mobility matrix.





Governing eqns

Dom x SO3 Ω x A

(r,Q) (q,η )

Dom

ext

extτ

f

σt + g−1∇ · (gJ) = 0 in Ω× A, t > 0J · n = 0 on (∂Ω)× A, t ≥ 0J · n periodic on Ω× (∂A), t ≥ 0σ = σ0 in Ω× A, t = 0.

J = −D∇σ + σChext, D = βbMbT , C = bMc .β = Boltzmann factor. g , b, c = geometric factors.

D,C ∈ R6×6 local diffusion, convection matrices.





Length, time scales

Dom x SO3Dom Molecule

ext

extτ

f

L

L

1

Trans: ttrn = time to drift across Dom

Rot: trot = time to drift across SO3

trotttrn

∼(`

L

)2





Experimentsext

extτ

f

L

M =

»M1 M3M2 M4

–, trot

ttrn∼

(`L

)2

type scale measureable

ultracentrifugedyn light scattering

ttrn Dtrn =1

3tr(M1)

birefringencecircular dichroism

trot Drot =1

2tr(P⊥

d M4P⊥d )∗

* d=molecular axis of polarization




Numerical method




Numerical method

Stokes eqns

Γ= B

Bµ∆u = ∇p in R3\B∇ · u = 0 in R3\B

u = U[v , ω] on ∂Bu, p → 0 as |x | → ∞.

Gu,p(x , y),Hu,p(x , y) = stokeslet, stresslet (Green’s) functions.

Surface potentials

u(x) = λ∫

γGu(x , ξ)ψ(y(ξ)) dAξ + (1− λ)

∫ΓHu(x , y)ψ(y) dAy

p(x) = λ∫

γGp(x , ξ)ψ(y(ξ)) dAξ + (1− λ)

∫ΓHp(x , y)ψ(y) dAy .

ψ(y) = potential density. λ, φ = conditioning parameters.

(u, p) satisfy Stokes eqns w/o BC for any ψ, λ, φ.




Numerical method

Stokes eqns

ξ γ

y

n(y) φ

Γ= B

µ∆u = ∇p in R3\B∇ · u = 0 in R3\B

u = U[v , ω] on ∂Bu, p → 0 as |x | → ∞.

Gu,p(x , y),Hu,p(x , y) = stokeslet, stresslet (Green’s) functions.

Surface potentials

u(x) = λ∫

γGu(x , ξ)ψ(y(ξ)) dAξ + (1− λ)

∫ΓHu(x , y)ψ(y) dAy

p(x) = λ∫

γGp(x , ξ)ψ(y(ξ)) dAξ + (1− λ)

∫ΓHp(x , y)ψ(y) dAy .

ψ(y) = potential density. λ, φ = conditioning parameters.

(u, p) satisfy Stokes eqns w/o BC for any ψ, λ, φ.




Numerical method

Integral eqn for ψξ γ

y

n(y) φ

Γ= Bλ

∫γ

Gu(x , ξ)ψ(y(ξ)) dAξ

+ (1− λ)∫ΓHu(x , y)[ψ(y)− ψ(x)] dAy

= U[v , ω](x), ∀x ∈ Γ.

Operator has bounded inverse, integrands ∀λ ∈ (0, 1), φ ∈ (0, φΓ).

Approximation∑Nb=1 Kabψb = Ua[v , ω], a = 1, . . . ,N.

(v , ω) −→ ψ −→ (f , τ)︸︷︷︸6 times

−→ L −→ M −→ (Dtrn,Drot).




Numerical method

Conditioning vs (λ, φ)

0 0.2 0.4 0.6 0.8 10

1

σ min

λ

1

7

13

σ max

0.1 0.3 0.5 0.7 0.90.5

1

1.5

2

2.5

3

λ

log 10

(σm

ax/σ

min

)φ/φΓ =

1

8(dots),

2

8(crosses),

3

8(pluses),

4

8(stars), . . ..

Condition numberσmax

σmin≤ 101.5 for (λ, φ/φΓ) near (

1

2,1

2).




Numerical method

Accuracy vs h

4 8 12 16 2028.45

28.5

28.55

28.6

1/h

|F|

5 10 15 2065.3

65.4

65.5

65.6

1/h

|T|

|∆f ||f | ,

|∆τ ||τ | ≤ 0.1% over range of geometries of interest.




Preliminary results




Application: radius estimation

Given: Measured values of Dtrn and Drot on various sequences.

Find: Radius r of hydrated DNA using two different geometricalmodels:

Classic straight model.

Sequence-dependent model.




Results for classic model: Dtrn vs length

0 20 40 60 80 100 120 140 1601

1.5

2

2.5

3

3.5

4

4.5

5

basepairs n

dim

ensi

onle

ss D

t x

n

Symbols: experiments (ultracentrifuge, light scattering,electrophoresis).

Curves: numerics w/r = 10, 11, . . . , 15A (top to bottom).

Results predict DNA radius of r = 10− 15A.




Results for sequence-dependent model: Dtrn vs length

0 20 40 60 80 100 1201

1.5

2

2.5

3

3.5

4

4.5

5

basepairs n

dim

ensi

onle

ss D

t x

n


Crosses, pluses: random sequences w/r = 10, 15A.

Results show systematic offset.

Results now predict DNA radius of r = 12− 17A.




Results for classic model: Drot vs length

0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

3

basepairs n

dim

ensi

onle

ss D

r x

n^3

Assume: polarization axis is long principal axis.

Symbols: experiments (birefringence, light scattering).


Results predict DNA radius of r = 13− 17A.




Results for sequence-dependent model: Drot vs length

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

basepairs n

dim

ensi

onle

ss D

r x

n^3

Assume: polarization axis is long principal axis.


Crosses, pluses: random sequences w/r = 12, 18A.

Results show dramatic dependence on curvature.




Concluding remarks

Simplest model of sequence-dependent DNA geometry involves

many parameters (η ∈ R26, θ ∈ R26, r ∈ R).

Estimates for all parameters exist, but there is little consensus.

Hydrodynamic modeling offers promising approach to validate

and refine parameters.

Successful realization of approach poses many challenges.

Support

NSF



predicting local geometric properties of dna from ...arbogast/cam397/gonzalez1.pdf · introduction...

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