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DNA looping

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DNA looping

  1. 1. STATISTICAL MECHANICS OF DNA LOOPING FELIX X.-F. YE 1,2 , NATHAN BAKER 1 , PANOS STINIS 1 , HONG QIAN 2 1 PNNL,2 UNIVERSITY OF WASHINGTON INTRODUCTION The formation of DNA loops is an important part in the biological processes, including gene expres- sion, genetic recombination, DNA replication and repair. Some of binding sites are located many thousands of base pair apart so it is required for a DNA molecule to form a loop. In this presen- tation, we are interested in the thermodynamic cost of DNA looping in terms of the statistical distribution of polymer con- formations in different models. Lisa G. DeFazio et al. EMBO J. 2002; 21:3192-3200 Figure 1: Each end of DNA molecules is bound to a single protein. The one in the lower right has been circularized with what two proteins bound to the ends of the DNA. J FACTOR Figure 2: Intra- and Intermolecular synapsis reactions. The cyclization reaction is also a part of two step mechanism. The J factor is a quantification of the free energy cost of cyclization in terms of the ratio of respec- tive equilibrium constants Kc and Kd for two dif- ferent reactions. J = 8π2 k1/k−1 kd = 8π2 W(0) NAv = 8π2 Zc/Z NAv (1) where W(0) is the probability density for end- to-end distance ree evaluated at 0, Zc and Z are the canonical ensemble partition functions for the loop chain and the chain without any constraints. FREELY JOINTED MODEL Consider the DNA molecule is homoge- neous and the dynamics is the local jump process. The probability density W(0) can be solved easily. The J factor is J = 8π2 (3/2πNl2 )3/2 NAv (2) Figure 3: A DNA molecule con- sists of N links, each of length l and every bond vector rn = Rn − Rn−1 are independent of each other. ROUSE MODEL We have the stochas- tic differential equation of motion for a freely draining polymer with both end are free in dilute solution. It is the mechanically over- damping spring model. Figure 4: A DNA molecule con- sists of N beads and each bead is connected with a homogeneous spring with spring constant k. The position of beads is a 3D vector and we can study each component separately for simplicity. dR dt = 1 ζ AR + 2kbT ζ dW(t) dt (3) where ζ the frictional coefficient, W(t) is the Wiener process and the matrix A is A = k       −1 1 0 . . . 0 0 1 −2 1 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 1 −2 1 0 0 . . . 0 1 −1       (4) The corresponding forward Fokker-Planck equa- tion for P(R, t) is ∂P ∂t = · (Dr P − 1 ζ ARP) (5) The diffusion constant Dr = kbTI/ζ, that is fluctuation-dissipation relation. The stationary distribution is the equilibrium Boltzmann distri- bution, Ps(R) ∝ exp(1 2 RT AR/kbT). ROUSE MODEL CONT. If we consider 3D model, the dimension of R ex- tends to 3(N + 1). The J factor is J ∗ NAv = 8π2 ( k 2πkbT N + 1 N2 )3/2 (6) If we choose the spring constant k = 3kbT l2 , then the J factor will be J ∗ NAv = 8π2 ( 3 2πl2 N + 1 N2 )3/2 (7) It is consistent with the result in freely joint model as N → +∞. ZIMM MODEL For long polymer, the hydrodynamic interaction must be included. The stochastic differential equation will be dR dt = HAR + 2kbTH dW(t) dt (8) where this mean hydrodynamic interaction ma- trix H is H = 1 ζ        1 1√ 2 . . . 1√ N 1 1 . . . 1√ N−1 1√ 2 1 . . . 1√ N−2 . . . . . . . . . . . . . . . 1√ N 1√ N−1 1√ N−2 . . .        (9) The corresponding forward Fokker-Planck equa- tion for P(R, t) is ∂P ∂t = · (−HARP + Dz P) (10) The diffusion constant Dz = kbTH. The stationary distribution is the same as the Rouse model. So the J factor for Zimm model is the same as Rouse model. Figure 5: Non-local interactions are included. The further two beads apart, the weaker the in- teractions are. REACTION RATE CONSTANT Although all J factors are the same for these three models, the re- action rate constants are different. It is believed the hydrody- namic interaction ac- celerates the reaction rate. Figure 6: The capture radius of DNA looping here is α. The looping time tL is the mean first passage time for the length of ree smaller than the capture radius. In the normal model analysis, ree = −4 p:odd Xp(t) (11) where Xp is the normal coordinate and each is governed by decoupled linear stochastic differen- tial equation. The reaction rate constant k1 is the reciprocal of the first passage time. It is now an 1D diffusion model and the passage time is given by SSS theory t(x0) = x0 a 1 DeePs(ree) dree L x Ps(r )dr (12) It can be further simplified by Kramer rate theory, tL ≈ 1 DeePs(α) (13) INHOMOGENEOUS BOND The DNA sequence is not homogeneous so it is necessary to con- sider this effect. If one of the springs has far greater spring con- stant k than others k. It is possible to exhibit different time scale behaviors. Figure 8: In the simplest case, only one bond is inhomogeneous. REFERENCES [1] M. Doi and S. Edwards, The Theory of Polymer Dynamics, The Clarendon Press, Oxford Uni- versity Press, New York, 1986, 331 pp. [2] R. Afra and B. Todd, Kinetics of loop formation in worm-like chain polymers The Journal of Chemical Physics, 138, 174908 (2013)

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