1 Sensitivity of DNA Looping to Sequence-dependent Stiffness 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. 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 employ a 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.

2 Protein-Mediated Looping of DNA DNA loop computed from “rod” model O3O3 O1O1 Introduction: The genes LacZ, LacY and LacA are repressed when the Lac-R protein binds to “operator” sites O 3 and O 1 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. Known crystal structure of Lactose-Repressor protein bound to O 3 and O 1 sites of DNA [Lewis et al., 1995] CAP RNAP O3O3 O1O1 LacZLacY LacA 3 structural genes Schematic of Lac Operon (E. coli): 5’ to 3’

3 Strain energy of DNA loop Protein-Mediated Looping of DNA Lac Operon (E. coli) Free Energy Budget Entropy [2574-Plat, Wilson et al.] Protein Flexibility Other (e.g. surface binding, electrostatics) Mechano-Chemistry: LacR Protein Free DNA (Genes “on”) LacR-DNA Complex (Genes “off”) +  - Free Energy Gene Repression Level  Dominant component for sub-persistence length DNA

4 StiffnessStiffness Intrinsic curvatureIntrinsic curvature Chirality (right- handedness)Chirality (right- handedness) EnergyEnergy Topology (twist and writhe)Topology (twist and writhe) Reaction moment & force on proteinReaction moment & force on protein Protein-Mediated Looping of DNA Lac Operon (E. coli) Mechano-Chemistry: LacR protein Free DNA LacR-DNA complex +  - Free energy Known [Gabrielian et al., 1996] Known [Lewis et al., 1995] Unknown ? Initial conditions, structural properties (material law) Boundary conditions Loop properties Unknown ? Rod Model [Goyal et al. 2005]

5 Rod Model (Captures stiffness in two-axes bending and torsion) Cross-section fixed reference frame Material Law: where: Restoring MomentCurvature and Torsion

6 Formulation of Nonlinear Rod Dynamics [Goyal et al., 2005] Linear Momentum Equation: Angular Momentum Equation: Compatibility Condition: Inextensibility & Unshearability Constraint: Field Variables: {v, ω, f, κ} Field Equations: Free Body Diagram: (velocity) (angular velocity) (internal force) (internal moment) (curvature & twist)

7 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) Law 1.Stiffness tensor 2.Intrinsic curvature Restoring moment Curvature of deformed state Material properties  [Pos/B209, Lillian et al.]

8 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]. Linear Material (Constitutive) Law Question: How does the non-uniform stiffness affect DNA looping?

9 Strain Energy of DNA Loop 00 Non-uniform stiffness To analyze the influence of non-uniform stiffness on looping, we set intrinsic curvature to zero in the rod model.

10 Non-uniform Stiffness StiffStiffSoft Pure Torsion Untwist localized in soft zone StiffnessStretchEnergy k1k1 F/ k 1 F 2 / 2k 1 k2k2 F/ k 2 F 2 / 2k 2  Strain and Strain Energy tends to concentrate in soft regions (Both distribute in the inverse proportion of stiffness) F k1k1 k2k2 Problem Set-up: Insight: Computed Result:

11 Under-twisted DNA loop Two Computed Lac-R DNA Loops (under-twisted and over-twisted) Twist surplus(+)/ deficit(-) deg./bp Over-twisted DNA loop LacR Protein (non-uniform stiffness) (uniform stiffness) Strain energy of DNA loop shown in = Boltzmann constant and = absolute room temperature in Kelvin

12 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.

13 Description of the Rod with Non-Uniform Stiffness: Length L = 77 base-pairs ≈ 26 nm 0.2 L 0.6 L An order of magnitude softer than the rest of the domain Average stiffness (= ) is same as that of the uniform rod. Average Bending stiffness = [Hagerman, 1988]. Average Torsional stiffness = [Strick et al., 1996].

14 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.

15 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: 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. 2.9:30 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.

16 Online Reference Website contents: (handout (PPT), publications) Acknowledgements: (NSF, ONR, LLNL) Special thanks to: (Andricioaei et al, Tkachenko et al.) Questions/ comments e.mail to: Sachin Goyal Todd Lillian David Wilson Edgar Meyhofer Jens-Christian Meiners Noel Perkins (Go paperless, go blue!)