Study of Polymers Nonlinear Dynamics

Central features of these dynamical phenomena are the inextensibility and finite bending elasticity of the polymers. A natural continuum model with these features, the Kratky-Porod or "wormlike" model [4], derives from the elastic theory of thin rods with an energy quadratic in the local curvature. The study of equilibrium aspects of this model is highly developed [5,6], but its dynamical properties in viscously dominated flow are far less well understood. Dynamical formulations that address inexten- sibility date back to the important work on stiff polymers of Harris and Hearst [7], and others [8] who have emphasized the fundamentally nonlocal nature of this constraint. It has been touched upon as well in more recent studies of electrophoresis [9], hairpin defect motion in polymeric liquid crystals [10], supercoiled DNA [11], and motility assays [12], but no general method has been proposed to answer the basic question: What is the motion of a nonstretching flexible polymer in a viscous medium? Here we develop a unifying formalism for the dynamics of flexible but inextensible polymers within the simplest hydrodynamic model in which the polymer is subject to local viscous forces. The methods are completely general, capable of incorporating both local and nonlocal energetic contributions including elasticity with intrinsic curvature, pair interactions among polymer segments, and external forces. They are also completely intrinsic, making no reference to any idealized reference shape. This is particularly important since from the foundations of elasticity theory [5] we know that the mathematical problem ofthe equilibrium configurations of such an object isintrinsically nonlinear, and we must expect the samefor the dynamics [10]. These nonlinearities arise fromthe fact that the arc- length parametrization s of aspace curve is not independent of its position vectorr(s), and while unimportant for weakly curvedconfigurations these nonlinearities are essential for themany complex experimentally observed conformations.Utilizing geometrical methods we reduce the intrinsicnonlinear shape evolution to an extremely compactform as a pair of coupled partial differential equations(PDE's) of a type familiar in the field of patternformation [13], and for which there are highlydeveloped computational methods. The complex dynamical processes that may bedescribed by these methods are illustrated with themodel problem of a competition between bendingelasticity and a pair potential having a short-rangerepulsion (preventing self- crossing) and an attractiveminimum. These elastic and potential forces aremutually frustrating; in order for segments widelyseparated along the curve to be in the attractiveminimum there must be energetically costly bends inthe chain. This deterministic "folding" problem is one of the simplest in which to address such issues as theuniqueness of ground states and the pathways to them[14]. In this regard, we expect these methods to beuseful in theoretical studies of gene regulation andtopoisomerase activity. The equations of motion derive from an action principle

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used often in polymer physics [15] and recently for interfaces and membranes [16]. The polymer is parametrized bywith generalized coordinates and velocities Ifis the Lagrangian and R is the Rayleigh dissipation function, the equations of motion are R measures the rate of energy dissipation by the viscous forces, and is quadratic in the velocities. Assuming local, isotropic drag with friction coefficient where is the metric. Since the motions of interest occur at extremely small Reynolds numbers, we neglect inertial terms, the dynamics becomes first order in time derivatives, and the Lagrangian is simply the negative of the general potential energy functional Rewriting (1) in terms of functional derivatives, where are the unit normal, bi-normal, and tangent vectors and U, V, and W are the associated forces. Equation (3) is like the Rouse model [17], but allows for an arbitrary energy functionalrather than the simple collection of Hookean springs originally considered in that context. Generalizations of (3) to stochastic dynamics will be described elsewhere. The no-stretching or local arc length conservation con-dition (also clearly enforcing fixed total length) is imposed with a Lagrange multiplier function Consider first the geometrically simplest case of a polymer confined to the plane, and let U0 and Wo be those forces derived via Eq. (3) from the intrinsic energy functional alone. Functional differentiation of (4) yields total normal and tangential forces The curvature k(s) isdetermined by the Frenet-Serret equations In Eq. (4) we seethat —A plays the role of a locally varying line tension[9], and thus its contributions to (5) can be interpretedas a Young-Laplace force in the normal direction and aMarangoni force in the tangential direction. Local inextensibility requires a time-independent met- ric. This implies which from (3) leads to known also in the context of integrable nonstretchingcurve dynamics [18,19]. Using Eqs. (5) and (6), we find that A obeys an elliptic ordinary differential equation at each instant of time [20]: Equations (3)-(7) constitute a complete dynamical de-scription ofthe polymer shape evolution once an energy functional is given. As a sample problem, consider a polymer with bendingelasticity and a monomer pair interaction. The elastic contribution to the energy is in which we have included an intrinsic curvature k0(s),and where A is an elastic constant. Eq. (8) leads tononlinear forces Note that for closed polymers an s-independent k0does not enter the dynamics since both of itscontributions to the energy (8) are constant if thepolymer length and topology are fixed. If the polymerexperiences an external potential or a pair interaction (e.g., from electrostatic, dispersion,or steric forces), it has energy functionals where These produce purely normalforces

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The invariance of the dynamics under redefinitions of the zero of and is easily verified by noting from (11) that if, for instance, the bare normal force transforms as Equation (7) shows that this corresponds to the shift so the total force is unchanged. In the absence of potentials and F,and with k0 0, the normal velocity is For constant A this gives the "curve-straightening equation" [21], and U = 0 defines Euler elastica in the plane [5]. It resembles "geometric" models of interface motion [22] used for dendrite growth, which invoke expansions of U in powers of the curvature and its derivatives, but it arises here from a variational formulation not envisioned in those nonequilibrium processes. The intrinsic dynamical evolution is completed by following the time dependence of the tangent angle or curvature k(s) =according to the PDE's [22], + ksW. Apart from the clear simplification of considering scalar rather than vector PDE's, these intrinsic dynamics allow for a natural treatment of the inherent numerical stiffness associated with elastic forces. Note that the and k evolutions both have the form +..., where the ellipsis stands for nonlinear terms and terms of lower order in s derivatives. The fourth-order derivative severely limits the acceptable time steps in standard finite-difference schemes, but by its linearity may be treated exactly in pseudospectral methods utilizing integrating factors [23]. The treatment of space curves involves both the curva- ture and the torsion , obeying the Frenet-Serret equations The curve dynamics may be studied directly at the level of the evolution equations for k and where Theseare unnecessarily complicated. Indeed, the forces thatarise from the simplest elastic energy and We = 0, conspire with (12) to present a formidablecomputational problem. Moreover, the torsiondynamics is problematical at inflection points, where k= 0. The curve dynamics can be drastically simplified byutilizing Hasimoto's transformation [24] relating vortexfilament motion to the nonlinear Schrodinger equation.Define the complex curvature and the complex velocityperpendicular to the curve, Then for general U, V, and W, obeys [19] Three important featuresarise from this formulation. (i) The form of is remarkably com-pact: (As noted previously [25], nonplanar elastica [26] aredefined by the time-independent nonlinear Schrodinger equation (ii) The dynamics of inflectionpoints is mathematically well defined, behaving muchlike phase slips in one-dimensional superconducting wires [27]. (iii) The dynamics isagain amenable to integrating factor methods. Finally, by defining the complex vector exp , the Frenet-Serret equations become and the curve may be re- constructed fromandalone.

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We turn finally to the folding problem described in the introduction. The issue we address by simulations is how the presence of preferentially curved segments may determine the chosen folded configurations of an elastic polymer. In the context of DNA supercoiling there is experimental [28] and theoretical evidence [29] for the localization of regions of high intrinsic curvature at hairpin loops, an effect that may be relevant to structural regulation. Figure 1 shows the succession of straightening pro-cesses that takes a highly distorted initial configuration to a ground state having three hairpin loops [30]. These shapes have length L = 10 X 2 , with a Lennard- Jones potential of strength and a spontaneous curvature k0(s) having three plateaus (as shown in the lower panel). The three regions along the chain in which k0 > 0.5 are indicated by heavy lines, and it is apparent that the relaxation process indeed localizes the regions of intrinsic bend at the hairpin loops. The two sequences shown have the same initial condition, but the second has the peaks in k0(s) shifted along the chain. We see that this same ground state (modulo rotations and reflections) may be obtained even when the peaks in k0 do not correspond to those in the initial condition; these and other results show in at least a limited sense that this ground state is reached independent of initial conditions. As an aside, we note that these structures bear an intriguing resemblance to those of transfer RNA. FIG. 1. Folding of a closed elastic polymer in two dimensions. Time evolution proceeds from upper left to lower right. Regions of the polymer for which k0(s) > 0.5 (lower panel) are indicated by heavy lines. Twotemporal evolutions are shown ( black and gray),corresponding to the same initial condition, but with adisplacement of the function k0(s) along the chain. We expect that the methods outlined will be useful inthe context of the full hydrodynamic problem of stiffpolymer dynamics with its associated long-rangeinteractions [17,31], and related studies of the motionof defect lines in liquid crystals [32]. They may also be generalized to include an internal twist degree offreedom relevant to supercoiling, and to allow for localstretching, relevant to recent experiments on DNA"combing" by a moving meniscus [33]. Otherapplications include the more abstract problem ofnonlocal relaxational dynamics of knotted space curves[34]. We are indebted to M. B. Schmid for important discus-sions, and to A. I. Pesci, D. M. Petrich, and M. J. Shel-ley for suggestions. R. E. G. was supported in part byan NSF Presidential Faculty Fellowship (Grant No.DMR 93-50227) and the Alfred P. Sloan Foundation.

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