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Sensitivity of DNA Looping to Sequence-dependent Stiffness Sachin Goyal 1 , Todd Lillian 2 , David Wilson 2 , Edgar Meyhofer 2 , Jens-Christian Meiners 2 and Noel Perkins 2 . 1 Woods Hole Oceanographic Institution, Woods Hole, MA, USA, 2 University of Michigan, Ann Arbor, MI, USA.

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Sensitivity of DNA Looping to Sequence-dependent StiffnessSachin Goyal1, Todd Lillian2, David Wilson2, Edgar Meyhofer2,Jens-Christian Meiners2 and Noel Perkins2. 1Woods Hole Oceanographic Institution, Woods Hole, MA, USA, 2University of Michigan, Ann Arbor, MI, USA.

Protein-mediated DNA looping plays an important role in gene regulation. Empirically it is known that sequence-dependent mechanical effect such as intrinsic bends or softer regions in the substrate DNA affect loop formation, but quantitative models are lacking. We employa continuum rod model to simulate protein-mediated DNA looping as a means to explore how the sequence maps to the overall structural properties of the duplex. The model includes sequence-dependent intrinsic curvature, chirality, and stiffness. We address the fundamental question of how sequence-dependent stiffness influences the looping of DNA bound to regulatory proteins like the lactose repressor. We report two major findings: First, any non-uniform stiffness tends to lower the energetic cost of looping. Second, the deformation tends to localize in ‘softer’ regions which in turn affects the loop topology as characterized by twist and writhe.The model also offers the capability to calibrate and benchmark experimental measurements of sequence-dependent stiffness.

slide2

Protein-Mediated Looping of DNA

RNAP

CAP

LacA

O3

O1

LacZ

LacY

3 structural genes

Schematic of Lac Operon (E. coli):

5’ to 3’

DNA loop computed from “rod” model

  • Introduction:
  • The genes LacZ, LacY and LacA are repressed when the Lac-R protein binds to “operator” sites O3 and O1 of DNA and the intervening DNA is deformed into a loop.
  • DNA looping is a common mechanism in gene regulation.
  • Mechanics of loop formation and how it influences gene regulation is an area of research.

O1

O3

Known crystal structure of Lactose-Repressor protein bound to O3 and O1 sites of DNA

[Lewis et al., 1995]

slide3

-

+

Free Energy

LacR-DNA Complex(Genes “off”)

Free DNA(Genes “on”)

LacR Protein

Protein-Mediated Looping of DNALac Operon (E. coli)

Mechano-Chemistry:

Gene Repression Level 

Dominant component for sub-persistence length DNA

Strain energy of DNA loop

Entropy [2574-Plat, Wilson et al.]

Free Energy Budget

Protein Flexibility

Other (e.g. surface binding, electrostatics)

slide4

-

+

Free energy

LacR-DNA complex

Free DNA

LacR protein

  • Energy
  • Topology(twist and writhe)
  • Reaction moment & force on protein

Rod Model

[Goyal et al. 2005]

  • Stiffness
  • Intrinsic curvature
  • Chirality (right-handedness)

Protein-Mediated Looping of DNALac Operon (E. coli)

Mechano-Chemistry:

Known

[Gabrielian et al., 1996]

Known

[Lewis et al., 1995]

Unknown ?

Unknown ?

Boundary conditions

Initial conditions,structural properties

(material law)

Loop properties

rod model
Rod Model

(Captures stiffness in two-axes bending and torsion)

Cross-section fixed reference frame

Material Law:

where:

Restoring Moment

Curvature and Torsion

formulation of nonlinear rod dynamics goyal et al 2005
Formulation of Nonlinear Rod Dynamics[Goyal et al., 2005]

Field Variables: {v, ω, f, κ}

Field Equations:

(curvature & twist)

(internal force)

(velocity)

Linear Momentum Equation:

(angular velocity)

(internal moment)

Angular Momentum Equation:

Free Body Diagram:

Inextensibility & Unshearability Constraint:

Compatibility Condition:

linear material constitutive law
Linear Material (Constitutive) Law

[Pos/B209, Lillian et al.]

Materialproperties

  • Stiffnesstensor
  • Intrinsiccurvature

Restoringmoment

Curvature ofdeformed state

Note: 3.Chirality (right-handedness of the molecule) can also be captured in the rod-constitutive law [Goyal et al., J. Comp. Phys., 2005].

linear material constitutive law8
Linear Material (Constitutive) Law
  • The material properties are sequence-dependent and hence are non-uniform along the rod-length.
  • The stiffness tensor includes two-axes bending and torsional stiffness.
  • Bending stiffness is effectively isotropic on long length-scales due to high intrinsic twist of the molecule [Maddocks and co-workers].

Question:

How does the non-uniform stiffness affect DNA looping?

strain energy of dna loop
Strain Energy of DNA Loop

0

0

Non-uniformstiffness

To analyze the influence of non-uniform stiffness on looping, we set intrinsic curvature to zero in the rod model.

non uniform stiffness

F

Pure Torsion

Untwist localized in soft zone

Stiff

Soft

Stiff

Non-uniform Stiffness

Insight:

k1

k2

Strain and Strain Energy tends to concentrate in soft regions (Both distribute in the inverse proportion of stiffness)

Problem Set-up:

Computed Result:

two computed lac r dna loops under twisted and over twisted

Strain energy of DNA loop shown in

= Boltzmann constant and = absolute room temperature in Kelvin

Two Computed Lac-R DNA Loops(under-twisted and over-twisted)

Twist surplus(+)/

deficit(-)

deg./bp

(uniform stiffness)

(uniform stiffness)

(non-uniform stiffness)

(non-uniform stiffness)

LacR Protein

LacR Protein

Under-twisted DNA loop

Over-twisted DNA loop

figure description
Figure Description

The figure shows a simulation example pertaining to LacR-DNA loops where the stiffness is lowered by an order of magnitude at a specified location (see next slide). The results are contrasted with those predicted by the rod with uniform stiffness. The color scale shows the distribution of twist surplus (+) or deficit (-) over the nominal twist of 34.6° per base-pair step.

Observations

  • Twist/ untwist and bending localizes in the softer region.
  • Strain energy of the loop is lowered with non-uniform stiffness.
description of the rod with non uniform stiffness

An order of magnitude softer than the rest of the domain

Length L = 77 base-pairs ≈ 26 nm

0.2 L

0.2 L

0.6 L

  • Average stiffness (= ) is same as that of the uniform rod.
  • Average Bending stiffness = [Hagerman, 1988].
  • Average Torsional stiffness = [Strick et al., 1996].
Description of the Rod with Non-Uniform Stiffness:
possible sources of non uniform stiffness
Possible Sources of Non-uniform Stiffness
  • Sequence-dependence:2 H-bonds in A-T base-pairs vs 3 H-bonds in G-C base-pairs. A-T rich regions are expected to be softer.
  • Base-pair flipping(kink-ability):

Base-pair flipping unconstrains the two strands of DNA and might lower the stiffness by more than an order of magnitude. (The net stiffness of two independent strands is the sum of their individual stiffness. For example, the bending stiffness of individual strand is 0.75 nm-kT [Smith et al., 1996]. The total bending stiffness of the two unconstrained strands would be 1.5 nm-kT (imagine two bending springs in parallel), which is << 50 nm-KT stiffness of double-stranded DNA.)

  • Melting:Melting also unconstrains the two strands of DNA. Local melting may occur at RNAP binding site.
conclusions insights from rod model simulations
Conclusions/ Insights from Rod Model Simulations
  • Non-uniform stiffness reduces energetic cost of looping.
  • Non-uniform stiffness alters loop topology by localizing deformations (twist and bending) in soft regions.

Additional Thoughts

  • Softer regions of DNA might be more prone to melting/ kinking due to strain energy concentration.

Please also visit:

  • 1981-Pos/B209: Computational rod theory predicts experimental characteristics of DNA looping by the Lac repressor, Todd D. Lillian, Sachin Goyal, Noel C. Perkins, Jens-Christian Meiners, Jason D. Kahn.
  • 9:30 am, Wed, Mar 7, 2574-Plat, Modeling the Entropic Cost of DNA Looping, David P. Wilson, Todd D. Lillian, Bachelors, Sachin Goyal, Noel C. Perkins, Alexei Tkachenko, Jens C. Meiners.
online reference
Online Reference

Acknowledgements:(NSF, ONR, LLNL)

Special thanks to:(Andricioaei et al, Tkachenko et al.)

Questions/ comments e.mail to:Sachin Goyalsgoyal@umich.edu

Todd Lillian

tlillian@umich.edu

David Wilson

doros@umich.edu

Edgar Meyhofer

meyhofer@umich.edu

Jens-Christian Meiners

meiners@umich.edu

Noel Perkins

ncp@umich.edu

Website contents:

(handout (PPT), publications)

http://www.whoi.edu/sites/sgoyal

(Go paperless, go blue!)