Free energy measurements by the generalized fluctuation theorems: Theory and numerical study of a model filament.

We measure the free energy of a model filament, which undergoes deformations and structural transitions, as a function of its extension, in silico. We perform Brownian Dynamics (BD) simulations of pulling experiments at various speeds, following a protocol close to experimental ones. The results from the fluctuation theorems are compared with the estimates from Monte Carlo (MC) simulation, where the rugged free energy landscape is produced by the density of states method. The fluctuation theorems (FT) give accurate estimates of the free energy up to moderate pulling speeds. At higher pulling speeds, the work distributions do not efficiently sample the domain of small work and FT slightly overestimates free energy. In order to comprehend the differences, we analyze the work distributions from the BD simulations in the framework of trajectory thermodynamics and propose the generalized fluctuation theorems that take into account the information (relative entropy) evaluated in the expanded phase space. The measured work - free energy relation is consistent with the results obtained from the generalized fluctuation theorems. We discuss operational methods to improve the estimates at high pulling speed.

Adsorption of a Helical Filament Subject to Thermal Fluctuations.

We consider semiflexible chains governed by preferred curvature and twist and their flexural and twist moduli. These filaments possess a helical rather than straight three-dimensional (3D) ground state and we call them helical filaments (H-filament). Depending on the moduli, the helical shape may be smeared by thermal fluctuations. Secondary superhelical structures are expected to form on top of the specific local structure of biofilaments, as is documented for vimentin. We study confinement and adsorption of helical filaments utilizing both a combination of numerical simulations and analytical theory. We investigate overall chain shapes, transverse chain fluctuations, loop and tail distributions, and energy distributions along the chain together with the mean square average height of the monomers 〈 z 2 〉 . The number fraction of adsorbed monomers serves as an order parameter for adsorption. Signatures of adsorbed helical polymers are the occurrence of 3D helical loops/tails and spiral or wavy quasi-flat shapes. None of these arise for the Worm-Like-Chain, whose straight ground state can be embedded in a plane.

Translocation of polyampholytes and intrinsically disordered proteins⋆.

Polyampholytes are polymers carrying electrical charges of both signs along their backbone. We consider synthetic polyampholytes with a quenched random charge sequence and intrinsically disordered proteins, which have a well-defined charge sequence and behave like polyampholytes in the denaturated state. We study their translocation driven by an electric field through a pore. The role of disorder along the charge sequence of synthetic polyampholytes is analyzed. We show how disorder slows down the translocation dynamics. For intrinsically disordered proteins, the translocation vs. rejection rates by the pore depends on which end is engaged in the translocation channel. We discuss the rejection time, the blockade time distributions and the translocation speed for the charge sequence of two specific intrinsically disordered proteins differing in length and structure.

Direct Measurement of Lateral Correlations under Controlled Nanoconfinement.

Lateral correlations along hydrophobic surfaces whose separation can be varied continuously are measured by x-ray scattering using a modified surface force apparatus coupled with synchrotron radiation, named SFAX. A weak isotropic diffuse scattering along the equatorial plane is revealed for mica surfaces rendered hydrophobic and charge neutral by immersion in cationic surfactant solutions at low concentrations. The peak corresponds to a lateral surface correlation length ξ≈12 nm, without long-range order. These findings are compatible with the atomic force microscopy imaging of a single surface, where adsorbed surfactant stripes appear surrounded by bare mica zones. Remarkably, the scattering patterns remain stable for gap widths D larger than the lateral period but change in intensity and shape (to a lesser extent) as soon as D<ξ. This evolution codes for a redistribution of counterions (counterion release from antagonistic patches) and the associated new x-ray labeling of the patterns. The redistribution of counterions is also the key mechanism to the long-range electrostatic attraction between similar, overall charge-neutral walls, reported earlier.

Marginally compact hyperbranched polymer trees.

Assuming Gaussian chain statistics along the chain contour, we generate by means of a proper fractal generator hyperbranched polymer trees which are marginally compact. Static and dynamical properties, such as the radial intrachain pair density distribution ρpair(r) or the shear-stress relaxation modulus G(t), are investigated theoretically and by means of computer simulations. We emphasize that albeit the self-contact density diverges logarithmically with the total mass N, this effect becomes rapidly irrelevant with increasing spacer length S. In addition to this it is seen that the standard Rouse analysis must necessarily become inappropriate for compact objects for which the relaxation time τp of mode p must scale as τp ∼ (N/p)5/3 rather than the usual square power law for linear chains.

Shape of adsorbed supercoiled plasmids: an equilibrium description.

Inspired by recent atomic force microscope (AFM) images of plasmids deposited on oppositely charged supported lipid bilayers from salt free solution, we propose a model for strongly adsorbed supercoiled cyclic stiff polyelectrolytes. We discuss how the excess linking number Lk of the deposited cycle is shared between writhe Wr and twist Tw at equilibrium and obtain the typical number of self-crossings in the deposited cycle as a function of surface charge density. The number of crossings at equilibrium is simply determined by the crossing penalty which is a local quantity and by the excess linking number. The number of crossings is well defined despite versatile plasmid shapes. For moderate numbers of crossings the loops are rather small and localized along the primary cycle, as expected from entropic loops. In the regime of many crossings, the cycle takes the shape of a regular flat ply ruled by local stiffness. The model allows for a semiquantitative comparison with the AFM images of deposited plasmids which are strongly charged.

Mode-coupling approach to polymer diffusion in an unentangled melt. I. The effect of density fluctuations.

We quantitatively assess the effect of density fluctuation modes on the dynamics of a tagged polymer in an unentangled melt. To this end, we develop a density-based mode-coupling theory (dMCT) using the Mori-Zwanzig approach and projecting the fluctuating force onto pair-density fluctuation modes. The effect of dynamical density fluctuations on the center-of-mass (c.m.) dynamics is also analyzed based on a perturbative approach and we show that dMCT and perturbation techniques yield identical results. The c.m. velocity autocorrelation function (c.m. VAF) exhibits a slow power law relaxation in the time range between the monomer time t_{1} and the Rouse relaxation time t_{N}. We obtain an analytical expression for the c.m. VAF in terms of molecular parameters. In particular, the c.m. VAF scales as -N^{-1}t^{-5/4} (where N is the number of monomer units per chain) in the relevant time regime. The results are qualitatively accounted for by the dynamical correlation hole effect. The predicted -t^{-5/4} dependence of the c.m. VAF is supported by data of non-momentum-conserving computer simulations. However, the comparison shows that the theory significantly underestimates the amplitude of the effect. This issue is discussed and an alternative approach is addressed in the second part of this series [Farago et al., Phys. Rev. E 85, 051807 (2012), the following paper].

Scale-free center-of-mass displacement correlations in polymer melts without topological constraints and momentum conservation: a bond-fluctuation model study.

By Monte Carlo simulations of a variant of the bond-fluctuation model without topological constraints, we examine the center-of-mass (COM) dynamics of polymer melts in d = 3 dimensions. Our analysis focuses on the COM displacement correlation function C(N)(t)≈∂(t) (2)h(N)(t)/2, measuring the curvature of the COM mean-square displacement h(N)(t). We demonstrate that C(N)(t) ≈ -(R(N)∕T(N))(2)(ρ∗/ρ) f(x = t/T(N)) with N being the chain length (16 ≤ N ≤ 8192), R(N) ∼ N(1/2) is the typical chain size, T(N) ∼ N(2) is the longest chain relaxation time, ρ is the monomer density, ρ(*)≈N/R(N) (d) is the self-density, and f(x) is a universal function decaying asymptotically as f(x) ∼ x(-ω) with ω = (d + 2) × α, where α = 1/4 for x ≪ 1 and α = 1/2 for x ≫ 1. We argue that the algebraic decay NC(N)(t) ∼ -t(-5/4) for t ≪ T(N) results from an interplay of chain connectivity and melt incompressibility giving rise to the correlated motion of chains and subchains.

Polymer-brush lubrication in the limit of strong compression.

By means of molecular dynamics simulations we demonstrate power laws for macroscopic transport properties of strongly compressed polymer-brush bilayers to stationary shear motion beyond the Newtonian response. The corresponding exponents are derived from a recently developed scaling theory, where the interpenetration between the brushes is taken as the relevant length scale. This allows to predict the dependence of the critical shear rate, which separates linear and non-linear behavior, on compression and molecular parameters of the bilayer. We present scaling plots for chain extension (R), viscosity (η) , and shear force (F over a wide range of Weissenberg numbers, W . In agreement with our theory, the simulation reveals simple power laws, R ∼ W (0.53), η ∼ W (-0.46), and F ∼ W (0.54), for the non-Newtonian regime.

Algebraic displacement correlation in two-dimensional polymer melts.

Dense self-avoiding polymer chains in strictly two dimensions are compact objects with fractal contours. Using scaling arguments and molecular dynamics simulations (with negligible momentum conservation) it is shown that correlated amoebalike fluctuations of the (sub)chain contours dominate the relaxation dynamics on all scales. The incompressibility of the melt and the compactness of (sub)chains impose a scale-free constraint on the contour fluctuations. This leads to strong long range spatiotemporal correlations of the displacement field as shown, e.g., by the (negative) algebraic decay of the center-of-mass velocity correlation function C(t)∼-1/t(6/5) with time t.

Drift and diffusion of a confined semiflexible chain.

We study the transverse and longitudinal linear response function of rigid chains subjected to an external force. Our main concern are stiff polymers confined in narrow pores with diameter less than their persistence length. We explicitly consider confinement in a transverse harmonic potential and generalize results by scaling arguments. Our results describe the drift of the filament under an external force, time evolution of the filament shape, and filament diffusion. Diffusion of a confined filament resembles the celebrated reptation process for flexible chains, albeit with distinct kinetic exponents. The limiting case of stiff free filaments is also mentioned.

Non-extensivity of the chemical potential of polymer melts.

Following Flory's ideality hypothesis, the chemical potential of a test chain of length n immersed into a dense solution of chemically identical polymers of length distribution P(N) is extensive in n . We argue that an additional contribution deltamu(c)(n) approximately +1/rho (sqrt[n]) arises (rho being the monomer density) for all P(N) if n " which can be traced back to the overall incompressibility of the solution leading to a long-range repulsion between monomers. Focusing on Flory-distributed melts, we obtain deltamu(c)(n) approximately equal to (1 - 2n/ ) / rho (sqrt[n]) for n " (2) , hence, deltarho(c)(n) approximately equal to -1/rho (sqrt[n]) if n is similar to the typical length of the bath . Similar results are obtained for monodisperse solutions. Our perturbation calculations are checked numerically by analyzing the annealed length distribution P(N) of linear equilibrium polymers generated by Monte Carlo simulation of the bond fluctuation model. As predicted we find, e.g., the non-exponentiality parameter K (p) = 1 - (p)! (p) to decay as K (p) approximately equal to 1/ (sqrt[]) for all moments p of the distribution.

Depletion of ideal polymer chains near a spherical colloid particle beyond the Dirichlet boundary conditions.

We reconsider the depletion interaction of an ideal polymer chain, characterized by the gyration radius R(G) and bond length a , and an impenetrable spherical colloid particle of radius R . Forbidding the polymer-colloid penetration explicitly (by the use of Mayer functions) without any other requirement we derive and solve analytically an integral equation for the chain partition function of a long ideal polymer chain for the spherical geometry. We find that the correction to the solution of the Dirichlet problem depends on the ratios R/R (G) and R/a . The correction vanishes for the continuous chain model (i.e. in the limit R/R (G) --> 0 and R/a --> infinity but stays finite (even for an infinite chain) for the discrete chain model. The correction can become substantial in the case of nano-colloids (the so-called protein limit).

Frictional forces between strongly compressed, nonentangled polymer brushes: molecular dynamics simulations and scaling theory.

By means of molecular dynamics simulations and scaling theory we study the response of opposing polymer brushes to constant shear motion under good solvent conditions. Model systems that contain explicit solvent molecules (Lennard-Jones dimers) are compared to solvent-free systems while varying of the distance between the grafted layers and their molecular parameters, chain length and grafting density. Our study reveals a power-law dependence of macroscopic transport properties on the Weissenberg number, W, beyond linear response. For instance, we find that the kinetic friction constant scales as mu approximately W(0.57) for large values of W. We develop a scaling theory that describes our data and previous numerical data including recent experiments.

A finite excluded volume bond-fluctuation model: static properties of dense polymer melts revisited.

The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo algorithm for coarse-grained polymer chains where each monomer occupies exclusively a certain number of lattice sites. In this paper we propose a generalization of the BFM where we relax this constraint and allow the overlap of monomers subject to a finite energy penalty epsilon. This is done to vary systematically the dimensionless compressibility g of the solution in order to investigate the influence of density fluctuations in dense polymer melts on various static properties at constant overall monomer density. The compressibility is obtained directly from the low-wave vector limit of the static structure factor. We consider, e.g., the intrachain bond-bond correlation function P(s) of two bonds separated by s monomers along the chain. It is shown that the excluded volume interactions are never fully screened for very long chains. If distances smaller than the thermal blob size are probed (s<>g) like swollen chains of incompressible blobs with P(s) approximately g(0)s(-3/2).

Perimeter length and form factor in two-dimensional polymer melts.

Self-avoiding polymers in two-dimensional (d=2) melts are known to adopt compact configurations of typical size R(N) approximately N;{1/d} , with N being the chain length. Using molecular-dynamics simulations we show that the irregular shapes of these chains are characterized by a perimeter length L(N) approximately R(N);{d_{p}} of fractal dimension d_{p}=d-Theta_{2}=5/4 , with Theta_{2}=3/4 being a well-known contact exponent. Due to the self-similar structure of the chains, compactness and perimeter fractality repeat for subchains of all arclengths s down to a few monomers. The Kratky representation of the intramolecular form factor F(q) reveals a strong nonmonotonous behavior with q;{2}F(q) approximately 1/(qN;{1/d});{Theta_{2}} in the intermediate regime of the wave vector q . Measuring the scattering of labeled subchains the form factor may allow to test our predictions in real experiments.

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions.

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G (D)(r,r', N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r'. The expression for G(r,r', s) is found to be the sum of the commonly accepted Dirichlet result G (D)(r,r', N) = G (0)(r,r', N) - G (0)(r,r'', N) , where r'' is the mirror image of r', and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/[Formula: see text]is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.

Measuring the kinetics of biomolecular recognition with magnetic colloids.

We introduce a general methodology based on magnetic colloids to study the recognition kinetics of tethered biomolecules. Access to the full kinetics of the reaction is provided by an explicit measure of the time evolution of the reactant densities. Binding between a single ligand and its complementary receptor is here limited by the colloidal rotational diffusion. It occurs within a binding distance that can be extracted by a reaction-diffusion theory that properly accounts for the rotational Brownian dynamics. Our reaction geometry allows us to probe a large diversity of bioadhesive molecules and tethers, thus providing a quantitative guidance for designing more efficient reactive biomimetic surfaces, as required for diagnostic, therapeutic, and tissue engineering techniques.

Static Rouse modes and related quantities: corrections to chain ideality in polymer melts.

Following the Flory ideality hypothesis intrachain and interchain excluded-volume interactions are supposed to compensate each other in dense polymer systems. Multichain effects should thus be neglected and polymer conformations may be understood from simple phantom chain models. Here we provide evidence against this phantom chain, mean-field picture. We analyze numerically and theoretically the static correlation function of the Rouse modes. Our numerical results are obtained from computer simulations of two coarse-grained polymer models for which the strength of the monomer repulsion can be varied, from full excluded volume ("hard monomers") to no excluded volume ("phantom chains"). For nonvanishing excluded volume we find the simulated correlation function of the Rouse modes to deviate markedly from the predictions of phantom chain models. This demonstrates that there are nonnegligible correlations along the chains in a melt. These correlations can be taken into account by perturbation theory. Our simulation results are in good agreement with these new theoretical predictions.