We study the dynamics of a double-stranded DNA (dsDNA) segment, as a semiflexible polymer, in a shear flow, the strength of which is customarily expressed in terms of the dimensionless Weissenberg number Wi. Polymer chains in shear flows are well-kno
wn to undergo tumbling motion. When the chain lengths are much smaller than the persistence length, one expects a (semiflexible) chain to tumble as a rigid rod. At low Wi, a polymer segment shorter than the persistence length does indeed tumble as a rigid rod. However, for higher Wi the chain does not tumble as a rigid rod, even if the polymer segment is shorter than the persistence length. In particular, from time to time the polymer segment may assume a buckled form, a phenomenon commonly known as Euler buckling. Using a bead-spring Hamiltonian model for extensible dsDNA fragments, we first analyze Euler buckling in terms of the oriented deterministic state (ODS), which is obtained as the steady-state solution of the dynamical equations by turning off the stochastic (thermal) forces at a fixed orientation of the chain. The ODS exhibits symmetry breaking at a critical Weissenberg number Wi$_{text c}$, analogous to a pitchfork bifurcation in dynamical systems. We then follow up the analysis with simulations and demonstrate symmetry breaking in computer experiments, characterized by a unimodal to bimodal transformation of the probability distribution of the second Rouse mode with increasing Wi. Our simulations reveal that shear can cause strong deformation for a chain that is shorter than its persistence length, similar to recent experimental observations.
Using a recently developed bead-spring model for semiflexible polymers that takes into account their natural extensibility, we report an efficient algorithm to simulate the dynamics for polymers like double-stranded DNA (dsDNA) in the absence of hydr
odynamic interactions. The dsDNA is modelled with one bead-spring element per basepair, and the polymer dynamics is described by the Langevin equation. The key to efficiency is that we describe the equations of motion for the polymer in terms of the amplitudes of the polymers fluctuation modes, as opposed to the use of the physical positions of the beads. We show that, within an accuracy tolerance level of $5%$ of several key observables, the model allows for single Langevin time steps of $approx1.6$, 8, 16 and 16 ps for a dsDNA model-chain consisting of 64, 128, 256 and 512 basepairs (i.e., chains of 0.55, 1.11, 2.24 and 4.48 persistence lengths) respectively. Correspondingly, in one hour, a standard desktop computer can simulate 0.23, 0.56, 0.56 and 0.26 ms of these dsDNA chains respectively. We compare our results to those obtained from other methods, in particular, the (inextensible discretised) WLC model. Importantly, we demonstrate that at the same level of discretisation, i.e., when each discretisation element is one basepair long, our algorithm gains about 5-6 orders of magnitude in the size of time steps over the inextensible WLC model. Further, we show that our model can be mapped one-on-one to a discretised version of the extensible WLC model; implying that the speed-up we achieve in our model must hold equally well for the latter. We also demonstrate the use of the method by simulating efficiently the tumbling behaviour of a dsDNA segment in a shear flow.
We study the dynamical properties of semiflexible polymers with a recently introduced bead-spring model. We focus on double-stranded DNA. The two parameters of the model, $T^*$ and $ u$, are chosen to match its experimental force-extension curve. The
bead-spring Hamiltonian is approximated in the first order by the Hessian that is quadratic in the bead positions. The eigenmodels of the Hessian provide the longitudinal (stretching) and transverse (bending) eigenmodes of the polymer, and the corresponding eigenvalues match well with the established phenomenology of semiflexible polymers. Using the longitudinal and transverse eigenmodes, we obtain analytical expressions of (i) the autocorrelation function of the end-to-end vector, (ii) the autocorrelation function of a bond (i.e., a spring, or a tangent) vector at the middle of the chain, and (iii) the mean-square displacement of a tagged bead in the middle of the chain, as sum over the contributions from the modes. We also perform simulations with the full dynamics of the model. The simulations yield numerical values of the correlation functions (i-iii) that agree very well with the analytical expressions for the linearized dynamics. We also study the mean-square displacement of the longitudinal component of the end-to-end vector that showcases strong nonlinear effects in the polymer dynamics, and we identify at least an effective $t^{7/8}$ power-law regime in its time-dependence. Nevertheless, in comparison to the full mean-square displacement of the end-to-end vector the nonlinear effects remain small at all times --- it is in this sense we state that our results demonstrate that the linearized dynamics suffices for dsDNA fragments that are shorter than or comparable to the persistence length. Our results are consistent with those of the wormlike chain (WLC) model, the commonly used descriptive tool of semiflexible polymers.
The conditions are investigated under which a row of increasing dominoes is able to keep tumbling over. The analysis is restricted to the simplest case of frictionless dominoes that only can topple not slide. The model is scale invariant, i.e. domino
es and distance grow in size at a fixed rate, while keeping the aspect ratios of the dominoes constant. The maximal growth rate for which a domino effect exist is determined as a function of the mutual separation.
We present a model for semiflexible polymers in Hamiltonian formulation which interpolates between a Rouse chain and worm-like chain. Both models are realized as limits for the parameters. The model parameters can also be chosen to match the experime
ntal force-extension curve for double-stranded DNA. Near the ground state of the Hamiltonian, the eigenvalues for the longitudinal (stretching) and the transversal (bending) modes of a chain with N springs, indexed by p, scale as lambda_lp ~ (p/N)^2 and lambda_tp ~ p^2(p-1)^2/N^4 respectively for small p. We also show that the associated decay times tau_p ~ (N/p)^4 will not be observed if they exceed the orientational time scale tau_r ~ N^3 for an equally-long rigid rod, as the driven decay is then washed out by diffusive motion.
Using extensive computer simulations, the behavior of the structural modes --- more precisely, the eigenmodes of a phantom Rouse polymer --- are characterized for a polymer in the three-dimensional repton model, and are used to study the polymers dyn
amics at time scales well before the tube renewal. Although these modes are not the eigenmodes for a polymer in the repton model, we show that numerically the modes maintain a high degree of statistical independence. The correlations in the mode amplitudes decay exponentially with $(p/N)^2A(t)$, in which $p$ is the mode number, $N$ is the polymer length and $A(t)$ is a single function shared by all modes. In time, the quantity $A(t)$ causes an exponential decay for the mode amplitude correlation functions for times $<1$; a stretched exponential with an exponent 1/2 between times 1 and $tau_Rsim N^2$, the time-scale for diffusion of tagged reptons along the contour of the polymer; and again an exponential decay for times $t>tau_R$. Having assumed statistical independence and the validity of a single function $A(t)$ for all modes, we compute the temporal behavior of three structural quantities: the vectorial distance between the positions of the middle monomer and the center-of-mass, the end-to-end vector, and the vector connecting two nearby reptons around the middle of the polymer. Furthermore, we study the mean-squared displacement of the center-of-mass and the middle repton, and their relation with the temporal behavior of the modes.
The two-dimensional cage model for polymer motion is discussed with an emphasis on the effect of sideways motions, which cross the barriers imposed by the lattice. Using the Density Matrix Method as a solver of the Master Equation, the renewal time a
nd the diffusion coefficient are calculated as a function of the strength of the barrier crossings. A strong crossover influence of the barrier crossings is found and it is analyzed in terms of effective exponents for a given chain length. The crossover scaling functions and the crossover scaling exponents are calculated.
A simple one-dimensional model is constructed for polymer motion. It exhibits the crossover from reptation to Rouse dynamics through gradually allowing hernia creation and annihilation. The model is treated by the density matrix technique which permi
ts an accurate finite-size-scaling analysis of the behavior of long polymers.
We discuss the exact solution for the properties of the recently introduced ``necklace model for reptation. The solution gives the drift velocity, diffusion constant and renewal time for asymptotically long chains. Its properties are also related to
a special case of the Rubinstein-Duke model in one dimension.
We consider an arbitrarily charged polymer driven by a weak field through a gel according to the rules of the Rubinstein-Duke model. The probability distribution in the stationary state is related to that of the model in which only the head is charge
d. Thereby drift velocity, diffusion constant and orientation of any charged polymers are expressed in terms of those of the central model. Mapping the problem on a random walk of a tagged particle along a one-dimenional chain, leads to a unified scaling expression for the local orientation. It provides also an elucidation of the role of corrections to scaling.
