THE MECHANICS OF DNA LOOPING AND

THE INFLUENCE OF INTRINSIC CURVATURE

EngineeringSachin GoyalSachin GoyalTodd Lillian Todd Lillian Noel PerkinsNoel PerkinsEdgar MeyhoferEdgar Meyhofer

Seth BlumbergSeth BlumbergDavid WilsonDavid WilsonChris MeinersChris MeinersAlexei Tkachenko Alexei Tkachenko Ioan Andricioaei Ioan Andricioaei

NSF, LLNLNSF, LLNL

Physics/Biophysics & Chemistry – Univ. Michigan

Chemistry Univ. Maryland Jason Kahn

Engineering Structural Mechanics

Drs. C.L. Lu, C. Gatti-Bono, S. Goyal

Evolution of loops and tangles in cables

DNA supercoiling and looping

OutlineOutline

1. Background

2. Computational Rod Model

3. Quick Example - Plectoneme Formation

4. Looping of Highly Curved DNA (Kahn’s Sequences)

5. Looking Forward - New Hypotheses

2. Computational Rod Model2. Computational Rod Model

ChallengesChallenges

Nonlinearity (large bending & torsion)

Non-isotropy

Non-homogeneity

Non-trivial stress-free shapes

Self-contact / Excluded volume

Structural ModelingStructural Modeling Multi-Physical InteractionsMulti-Physical Interactions

Elasticity

Hydrodynamics( Drag / Coupling )

Thermal Kinetics

Electrostatics

),( tsai

),( tsR

Computational Rod ModelComputational Rod ModelGoyal et al., Goyal et al., Comp. PhysicsComp. Physics, 2005, 2005

( , ) ( , )q s t s t

moment/curvature relation

internal force

internal moment

f

q

velocity

angular velocity

v

0( , ) ( )[ ( , ) ( )]q s t B s s t s example constitutive law

)(00

0)(0

00)(

)( 2

1

sC

sA

sA

sB

1

2

( ) 0 0

( ) 0 ( ) 0

0 0 ( )

A s

B s A s

C s

intrinsic or stress-free curvature

Computational Rod Model

c c

f vf A v F

s t

Linear Momentum

3( )q

q I I f as t

Angular Momentum

s t

Compatibility Condition

3

vv a

s

Inextensibility Constraint

Field Variables: {v, ω, f, κ}

0( , ) ( )[ ( , ) ( )]q s t B s s t s

Constitutive Law

1

2

( ) 0 0

( ) 0 ( ) 0

0 0 ( )

A s

B s A s

C s

3. Quick Example3. Quick ExamplePlectoneme FormationPlectoneme Formation

EnergyEnergy

Work

Elastic Energy

Torsional Energy

Bending Energy

En

erg

y

1

2

3

4

Twist Tw

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time, t(s)

LinkingNumber,

Lk

Twist,Tw

Writhe,Wr

Linking NumberLinking Number

Wr

Tw

, Wr,

Lk

Tw

Time

1

2

3

4

L k

Known Lac crystal structure (loop boundary conditions)

Courtesy: Courtesy: http://www.ks.uiuc.edu/Research/pro_DNA/elastic

4. Protein-Mediated Looping of DNA4. Protein-Mediated Looping of DNA(LacR protein - regulating expression of LacZ,Y,A in (LacR protein - regulating expression of LacZ,Y,A in E. coliE. coli))

Highly Curved SequencesHighly Curved SequencesJ. Kahn, Univ. MarylandJ. Kahn, Univ. Maryland

J. Mol. Biol., 1999

plstraight ‘linker’

straight ‘linker’

curved A-tract

PDB files for sequences (zero temperature in aqueous solution) generated by webtool: http://hydra.icgeb.trieste.it/~kristian/dna/index.html, [Gabrielian and Pongor FEBS Letters, 1996]

Unbent Control

11C12

9C14

7C1670°

11C12

9C14

7C16

Unbent Control

Highly Curved SequencesHighly Curved SequencesJ. Kahn, Univ. MarylandJ. Kahn, Univ. Maryland

J. Mol. Biol., 1999

(a) Input 1: Sequence of Substrate DNA

Operator “Oid” at location L1 - - Inter-Operator sequence - - Operator “Oid” at location L2

5’ … GGTAATTGTGAGC-GCTCACAATTAGA … … … … … GCTAATTGTGAGC-GCTCACAATTCGT … 3’3’ … ccattaacactcg-cgagtgttaatct … … … … … cgattaacactcg-cgagtgttaagca … 5’

(d) Output: Topology and energetics of loop formation

Simulate Dynamic Kirchhoff Rod Model

LacR

(c) Input 2: DNA-Operator Crystal Structure

Oid Oid

Compute Boundary Conditions

(b) Compute Stress-Free Shape Based on Consensus Tri-nucleotide Model

+Input 3: Constitutive Law(e.g., Bending and Torsional Persistence Lengths)

Multiple Binding TopologiesMultiple Binding Topologies(Multiple Boundary Conditions)(Multiple Boundary Conditions)

Most “Compact” Loop

Example Calculation

Minimum Energy Conformations

Control

11C12

7C16

9C14

E=12kTR=8.4nm

E=7.5kTR=8.0nm

E=8.5kTR=7.5nm

E=11kTR=7.7nm

A2F

A2F

A2R

P1F

11C12 Unbent Control

Intrinsic Curvature Lowers Energetic Cost of Looping

kT/bp

A Survey of the Experimental Data

for the Highly Curved Sequences

Binding Topology of 9C14 via SM-FRETMorgan, et al., Biophysical J., 2005.,

“The LacI-9C14 loop exists exclusively in a single closed form exhibiting

essentially 100% ET” (~3.4 nm)

Lowest 11kT P1F

Second 11.5kT P1R

Binding Topology of 11C12 via Bulk FRETEdelman, et al., Biophysical J., 2003.,

FRET efficiency 10% Lowest 7.5kT A2R

Second 10.5kT A1R

8 nm

11C12

Most Stable Sequence(63% labeled remaining)

Competition Assays & Loop Stability and EnergyMehta and Kahn, J. Mol. Bio., 1999.,

Least Stable Sequence(3.8% labeled remaining)

Control

E=12kTGreatest Energy

E=7.4kTLeast Energy

The relative stability of the two intermediate cases (7C16 and 9C14) are not correctly predicted by the rod elastic energies

(Labeled looped DNA with a 50-fold concentration of unlabeled DNA)

Gel Mobility Assays & Loop SizeMehta and Kahn, J. Mol. Bio., 1999.,

Control

11C12

slowest

fastest

9C14

7C16

8.4

8.0

7.5

7.7

largest

smallest

gR

5. Looking Forward - New Hypotheses5. Looking Forward - New Hypotheses

1bp

2bp

Phasing of A-tract determined by: &1bp 2bp

Possible Minimum Loop Energies Possible Minimum Loop Energies and Preferred Binding Topologiesand Preferred Binding Topologies

1bp1bp

2bp 2bp

10

5

12

5

5

Energy kT

Possible Loop Sizes and Topologies (*)Possible Loop Sizes and Topologies (*)

1bp1bp

2bp 2bp

(*) See cyclization assays in Mehta and Kahn, J. Mol. Bio., 1999.,

Radius of Gyration Change in Link

ConclusionsConclusions

Established Predictive Ability of Rod Model for Highly-Curved Sequences Preferred (P1) Binding Topology of 9C14 (SM-FRET) Preferred (A1, A2) Binding Topology of 11C12 (Bulk FRET) Max and Min Loop Stabilities (Competition Assays) Relative Loop Sizes (Gel Mobility Assays)

Intrinsic Curvature May Have a Pronounced Effect on Preferred Binding Topology Loop Elastic Energy Loop Size Loop Topology (Tw, Wr, Lk)