Flexible polymer chain dynamics

1. Flexible polymer chain dynamics Yong-Gu Lee

2. Statistical mechanics of chain molecules

3. (p.3)

4. End-to-End, Radius of gyration Pg. 8

5. The freely jointed chain • No constraint on bond angles • No constraint on bond rotations • Bond length can be considered to be on “average”

6. Average values

8. The freely rotating chain (The Porod-Kratky Chain) • Yes constraint on bond angles • No constraint on bond rotations • Bond length can be considered to be on “average”

11. Bead-spring chain model Pg 29 Adouble-stranded DNA molecule is described by a bead-spring chain model composed of Nbbeads of hydrodynamic radius a = 77 nm connected by Ns = Nb− 1 entropic springs. Each bead represents a DNA segment of 4850 base pairs, i.e., Nb = 11 corresponds to a stained -DNA, which has a contour length of 22 μmand radius of gyration of 730 nm. The springs connecting the beads obey a worm-like chain force law. Here, bk is the Kuhn length for DNA and Nk,s= 20 is the number of Kuhn length per spring. Increased from natural 16.3um by TOTO-1 dye Pg 5 Yu Zhang, “Brownian dynamics simulation of DNA in complex geometries,” PhD thesis, University of Wisconsin – Madison, 2011

12. Diffusion coefficient • The projection of the Brownian motion in the x-y plane was tracked using video fluorescent microscopy and the increase in the mean square displacments with time t were recorded for an ensemble of molecule paths. Douglas E. Smith, Thomas T. Perkins, and Steven Chu, “Dynamical scaling of DNA diffusion coefficents,” Macromolecules, 1996, pp. 1372-1373

13. Zimm model for the radius of Gyration >> 22^(3/5) ans = 6.3893

14. Persistent length • Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional random walk. • Formally, the persistence length, P, is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain. • Let us define the angle θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0, along the contour of the chain. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance http://en.wikipedia.org/wiki/Persistence_length

15. Persistent length • For a chain possessing some degree of stiffness, the tangents to two segments of the chain contour will tend to be pointed in the same direction, provided that the segments are sufficiently close to each other, In other words, the local contour tends to persist in a given direction (segment directions are correlated). If the chain contour is represented by a virtual chain comprising a string of segment vectors, each of length b, the stiffness of the chain can be represented in a quantitative fashion as the sum , of the average projections of each segment vectors on the first segment vector. Paul J. Haggerman, “Flexibility of DNA,” Annual Review of Biophysics, Vol 17. pp. 265-286, 1988

16. Radius of gyration • It is defined as the root-mean-square distance of the collection of atoms, or groups, from their common center of gravity. Paul J. Flory, “Statistical mechanics of chain molecules,” pp. 4, Hanser Publishers, 1988

17. Kuhn length • The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with each other. Each segment in a freely jointed chain can randomly orient in any direction without the influence of any forces, independent of the directions taken by other segments. Instead of considering a real chain consisting of n bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with N connected segments, now called Kuhn segments, that can orient in any random direction.

18. Kramers freely jointed bead-rod chain model P. M. Saville and E. M. Sevick, “collision of a field-driven polymer with a finite-sized obstacle: A Brownian dynamics simulation,” Macromolecules, Vol. 32, pp. 892-899, 1999

19. Length based sorting P. M. Saville and E. M. Sevick, “collision of a field-driven polymer with a finite-sized obstacle: A Brownian dynamics simulation,” Macromolecules, Vol. 32, pp. 892-899, 1999

20. Experimental parameters for Kramers freely joined bead-rod chain model • As in previous work, fluorescently tagged Lambda-DNA is modeled using the Kramers freely jointed bead-rod chain model. This model has demonstrated quantitative agreement with experiments in single molecule flow studies of DNA even though the model does not reproduce the correct bending forces or effective entropic spring force of the wormlike chain or Kratky-Porodmodel. To simulate the Kratky-Porod model, however, in the present application, 10 sub-Kuhn step level discretization points are needed in order to produce the correct persistence length in a discrete model appropriate for Brownian dynamics. Thus, the number of dynamics variables becomes prohibitively large for the long time, large ensemble simulations considered in this work (see Ensemble Size section below). Therefore, we choose to focus on the qualitative comparisons to the experimental data with our simulations. As we shall discuss below, errors associated with ensemble size in these comparisons are at least as important as those associated with the shortcomings of the model. In the Kramers model, the polyelectrolyte is divided into N beads, at which the mass and hydrodynamic drag are concentrated, connected by N - 1 massless rods. The entire molecule has a contour length of 21 um and is modeled using 150 beads, yielding a Kuhn step size, bk, of 0.142 um. The drag is assumed to be isotropic, and hydrodynamic interactions (HI) between segments of the chain are ignored. >> 150* 0.142 ans = 21.3000 Nerayo P. Teclemariam, Victor A. Beck, Eric S. G. Shaqfeh,,§ and Susan J. Muller, “Dynamics of DNA Polymers in Post Arrays: Comparison of Single Molecule Experiments and Simulations,” Macromolecules, Vol. 40, pp. 3848-3859, 2007

21. Polymer chain dynamics Tony W. Liu. “Flexible polymer chain dynamics and rheological properties in steady flows,” Journal of Chemical Physics, Vol 90. No. 5826, 1989

22. /* Rouse matrix *** Aij*/ • MatrixNminus1d* getRouseMatrix() • /* Modified Rouse matrix *** AHatij*/ • MatrixNminus1d* getModifiedRouseMatrix(Matrix3Nminus1d& umatrix) • /* Kramers matrix *** Cij*/ • MatrixNminus1d* getKramersMatrix() • /* Kramers tensors matrix is second rank tensor *** Kij*/ • Matrix3d* getKramersTensorMatrix(int j, int k, Matrix3Nminus1d& umatrix) • /* B_bar_ij*/ • MatrixNminus1_Nd* getB_BarMatrix() • const double physical_coefficient = 1.0 / a * sqrt(2.0 * boltzman_constant * temperature/friction_coefficient); • Matrix3d kappa;//Transpose of the velocity gadient tensor del(v)