Effects of DNA structure on its micromechanical properties Yuri Popov University of California, Santa Barbara Alexei Tkachenko University of Michigan, Ann Arbor June 2007

Mechanical properties of a single DNA molecule Single DNA stretching experiments Smith et al (1992) Marko and Siggia (1995): Elastic properties of a single DNA molecule are best described by the wormlike chain model, with bending energy: where persistence length l p = 53 nm

Effects of sequence disorder on DNA looping and cyclization Phys. Rev. E, in press arXiv:cond-mat/

DNA looping Protein-mediated looping in regulation of gene expression (transcription control e.g. by lac repressor) DNA packaging into nucleosomes (wrapping around histones) Understanding spontaneous looping is a prerequisite to understanding protein-mediated one

Cyclization probability (J-factor) J-factor: DNA loopingcyclization

Classical theory Theory (Shimada and Yamakawa, 1984): bending energy

Classical theory vs. experiment Theory (Shimada and Yamakawa, 1984): Experiment (Cloutier and Widom, 2004): cyclization For L = 94 bp:

Effective energy of a chain with a random sequence: Here k(s) is the intrinsic curvature; for a random sequence it is Guassian with zero average: 1. Effect of intrinsic curvature

Exact result for the ensemble-averaged J-factor: with renormalized persistence length in the original result: For consensus-scale data of Gabrielian, Vlahovicek, and Pongor (1998): k 0 l p ≈ Effect of intrinsic curvature

Effective energy of a chain with a random sequence: J-factor for a particular sequence: instead of 2. Effect of random bending rigidity with

Effective energy of a chain with a random sequence: J-factor for a particular sequence: instead of 2. Effect of random bending rigidity with Gaussian non-Gaussian

Probability distribution of J

2. Effect of random bending rigidity

Overall effect of sequence disorder

Summary Effect of sequence disorder is strong and always present. Sequence disorder gives rise to orders-of- magnitude variation in cyclization probability for completely random sequences Most importantly, there is no self-averaging in DNA looping or cyclization. The J-factor is not a well- defined function of the chain length L, not even to the first approximation. No “typical” DNA. Effects of random bending rigidity and intrinsic curvature provide comparable contribution Boundary conditions may be the key to explaining the experimental results

Effects of kinks on DNA elasticity Phys. Rev. E 71, (2005) arXiv:cond-mat/

Model Theoretical study of the effects of localized structural singularities on the elastic behavior of double- and single-stranded DNA Model: wormlike chain with reversible kinks Want to know the elastic response (extension vs. force) K K K K F F

What systems are described by this model? Protein-induced bending and looping in dsDNA Elastic description of ssDNA/RNA (trans-gauche rotations) Need a hybrid model where the finite bond elasticity is combined with structural defects (i.e. WLC with discrete features)

Theoretical approach Chain is stretched by force F in the z direction Effective energy of a chain segment between two kinks: Here t = ∂r/∂s is tangent vector, s i is location of kink i, L = Σ(s i+1 -s i ) is total length of the chain Constraint at each kink: Here K is opening angle of each kink, κ is average line density of kinks without force

Analogy with Quantum Rotator Schrodinger-like (“diffusion”) equation for evolution along the chain: Here ψ(t) is the chain propagator (distribution function of chain ends orientations) and Ĥ is the effective Hamiltonian: The lowest eigenvalue determines the free energy

Solution Analytical: solve the eigenvalue problem by variational method with trial function where ω is variational parameter Numerical: solve the original evolution problem directly

Results: K=135° numerical analytical

Results: K=90° numerical analytical

Results: K=45° numerical analytical

Small stretching forces Elastic response is characterized by the renormalized persistence length: Upon proper geometrical identification, exactly reproduces the Flory model for trans-gauche rotational isomers in the limit of high bending rigidity and rare kinks Flory our model WLC K K where

Large stretching forces Main order: pure wormlike chain result Exponential corrections due to the ideal gas of kinks K K K K F F where

Results: K=45° numerical analytical

Inadequate variational trial function ψ(t) for smaller angles: Secondary peak = kink pairs Favorable: little bending and short “non-aligned” portion Why the difference between numerical and analytical results? K = 135°K = 45° K K z

Summary Hybrid model, with both the discrete defects and bending worm-like rigidity Small forces: renormalized persistence length. Crossover between WLC and rotational isomer model Large forces: WLC with the ideal gas of kinks High rigidity, small kink angle: kink pairing

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References J. F. Marko and E. D. Siggia, Macromolecules, 28, 8759 (1995) J. Shimada and H. Yamakawa, Macromolecules, 17, 689 (1984) T. E. Cloutier and J. Widom, Molecular Cell, 14, 355 (2004) A. Gabrielian, K. Vlahovicek, and S. Pongor, DNA tools, M. G. Munteanu, K. Vlahovicek, S. Parthasarathy, I. Simon, and S. Pongor, TIBS, 23, 341 (1998) P. A. Wiggins, R. Phillips, and P. C. Nelson, arXiv:cond-mat/