Stretching semiflexible polymer chains: evidence for the importance of excluded volume effects from Monte Carlo simulation

Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice (d = 3 dimensions) and square lattice (d = 2 dimensions), varying chain stiffness by an energy penalty ε(b) for chain bending. In the absence of excluded volume interactions, the persistence length l(p) of the polymers would then simply be l(p) = l(b)(2d - 2)(-1)q(b) (-1) with q(b) = exp(-ε(b)/k(B)T), the bond length l(b) being the lattice spacing, and k(B)T is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both q(b) and the chain length N are varied over a wide range (0.005 ≤ q(b) ≤ 1, N ≤ 50,000), and also a stretching force f is applied to one chain end (fixing the other end at the origin). In the absence of this force, in d = 2 a single crossover from rod-like behavior (for contour lengths less than l(p)) to swollen coils occurs, invalidating the Kratky-Porod model, while in d = 3 a double crossover occurs, from rods to Gaussian coils (as implied by the Kratky-Porod model) and then to coils that are swollen due to the excluded volume interaction. If the stretching force is applied, excluded volume interactions matter for the force versus extension relation irrespective of chain stiffness in d = 2, while theories based on the Kratky-Porod model are found to work in d = 3 for stiff chains in an intermediate regime of chain extensions. While for q(b) ? 1 in this model a persistence length can be estimated from the initial decay of bond-orientational correlations, it is argued that this is not possible for more complex wormlike chains (e.g., bottle-brush polymers). Consequences for the proper interpretation of experiments are briefly discussed.     

Interrogation of single synthetic polymer chains and polysaccharides by AFM-based force spectroscopy.

This contribution reviews selected mechanical experiments on individual flexible macromolecules using single-molecule force spectroscopy (SMFS) based on atomic force microscopy. Focus is placed on the analysis of elasticity and conformational changes in single polymer chains upon variation of the external environment, as well as on conformational changes induced by the mechanical stress applied to individual macromolecular chains. Various experimental strategies regarding single-molecule manipulation and SMFS testing are discussed, as is theoretical analysis through single-chain elasticity models derived from statistical mechanics. Moreover, a complete record, reported to date, of the parameters obtained when applying the models to fit experimental results on synthetic polymers and polysaccharides is presented.     

Deformation,Stretching, and Relaxation of Single‐Polymer Chains: Fundamentals and Examples#

Abstract

We review the basic theory for the static and dynamic properties of ideal and real chains. We describe the stretching of single ideal chains in both the strain and the stress ensembles, give scaling arguments for the deformation of real chains, and also consider the impact of chain rigidity by discussing the stretching of worm-like chains. We complement this theoretical outlook with specific examples that highlight the relevance of the single-molecule deformation in a practical perspective: the translocation of a macromolecule over a potential barrier, the collision of a single chain with fixed obstacles or other chains, the dynamics of a thethered chain in a strong shear flow, and some remarkable predictions concerning the deformation of composite chain molecules during free-solution electrophoresis.     

Molecular dynamics simulation of semirigid polymers

The chain rigidity of poly(p-hydroxybenzoate) was estimated through the theoretical evaluation of its persistence length (L_p). A non-Brownian molecular dynamics (MD) simulation of an isolated chain with 20 monomeric units was performed. The sampled conformational population was analyzed and the orientational correlation function between monomeric units along the chain was calculated. An algorithm based on the worm-like chain model was applied to evaluate the persistence length. The results were compared with those obtained from equilibrium models like the freely-rotating-chain and the rotational-matrix method with fluctuations. Equilibrium models give different results depending on the degree of accuracy used in describing the monomeric unit. The inclusion of thermal fluctuations is crucial to obtain realistic results. These coincide with those given by MD simulation when only nearest-neighbour orientational correlations are taken into account: inclusion of higher-order correlation terms leads to lower values of the persistence length. The origin of this discrepancy was investigated. The MD simulation results are characterized by an overrepresentation of conformations with a short end-to-end distance resulting from an anomalous energy concentration in the first bending mode of the chain. In analogy with previous simulation results from systems characterized by a week coupling amoung their degrees of freedom, failure in the energy equipartition is proposed as a likely explanation of the anomalous dynamical behaviour.     

The molecular kink paradigm for rubber elasticity: numerical simulations of explicit polyisoprene networks at low to moderate tensile strains

Based on recent molecular dynamics and ab initio simulations of small isoprene molecules, we propose a new ansatz for rubber elasticity. We envision a network chain as a series of independent molecular kinks, each comprised of a small number of backbone units, and the strain as being imposed along the contour of the chain. We treat chain extension in three distinct force regimes: (Ia) near zero strain, where we assume that the chain is extended within a well defined tube, with all of the kinks participating simultaneously as entropic elastic springs, (II) when the chain becomes sensibly straight, giving rise to a purely enthalpic stretching force (until bond rupture occurs) and, (Ib) a linear entropic regime, between regimes Ia and II, in which a force limit is imposed by tube deformation. In this intermediate regime, the molecular kinks are assumed to be gradually straightened until the chain becomes a series of straight segments between entanglements. We assume that there exists a tube deformation tension limit that is inversely proportional to the chain path tortuosity. Here we report the results of numerical simulations of explicit three-dimensional, periodic, polyisoprene networks, using these extension-only force models. At low strain, crosslink nodes are moved affinely, up to an arbitrary node force limit. Above this limit, non-affine motion of the nodes is allowed to relax unbalanced chain forces. Our simulation results are in good agreement with tensile stress vs. strain experiments.     

Statics and dynamics of dense polymer systems studied by monte carlo simulation

Monte Carlo simulations of coarse–grained models of macromolecules offer a unique tool to study the interplay between coil conformations, thermodynamic properties, and chain configurational relaxation and diffusion. Two examples are discussed where the chain conformation strongly differs from a gaussian coil: (i) collapsed chains in a bad solvent, where anomalous diffusion occurs in the Rouse limit and the relaxation time increases at least with the third power of chain length. (ii) Expulsion of a chain from a semidilute polymer brush. The initially stretched chain contracts to a gaussian coil and the center of mass moves outward with constant velocity until it reaches the region of the “last blob” where crossover to diffusive behavior occurs.     

Calculations on the force constants of triatomic group II metal halides

A comparison between various polarizable ion models for calculating bending force constants of linear MX₂ molecules is presented. A convergence test is applied to the models. Further, the contribution of an induced dipole-induced quadrupole interaction term to the force constant, is examined. The criteria for a useful comparison between calculated and experimental values of force constants are discussed.     

Kinetics of a semiflexible chain under external force

The kinetic properties of a semiflexible chain subject to an external force are investigated using scaling arguments and computer simulations. By monitoring the mean square displacements in principal axes, the authors found that the anisotropic dynamic fluctuations go through several distinct kinetic regimes characterized by two different exponents corresponding to transverse and longitudinal fluctuations. When a force is applied at one chain end, the tension propagates gradually to the other end, leading to nonuniform tension profiles. At short times, they observe sublinear relaxation of the mean square fluctuations in both longitudinal and transverse directions. At intermediate times, the kinetics is dominated by tension driven straightening with smaller kinetic exponents. Nonuniform tension profiles lead to the superlinear dependence of the longitudinal mean square displacement. In contrast, the late stage relaxation is diffusive again once the tension profile becomes uniform. The detailed tension profiles are reported for constant force measurement as well as constant pulling speed measurement.     

Depletion Interactions Produced by Nonadsorbing Charged and Uncharged Spheroids

The effect of macromolecule shape on the depletion attraction between two hard spherical particles in a solution with nonadsorbing hard spheroidal macromolecules of arbitrary size and aspect ratio was investigated using a modified form of the force-balance model of J. Y. Walz and A. Sharma (1994, J. Colloid Interface Sci. 168, 495). The macromolecules were represented as general spheroids, which could be either charged or uncharged. For the uncharged case, a set of analytical expressions describing the depletion attraction, valid for particles much larger than the characteristic macromolecule size, was developed. Comparisons with the case of spherical macromolecules were made under the condition of either constant macromolecule number density, rho(b), or constant volume fraction, phi. It was found that increasing the spheroidal macromolecule aspect ratio (major axis length/minor axis length) decreases the depletion attraction at constant rho(b), but increases the interaction at constant phi. In the latter case, the interaction produced by prolate macromolecules is greater than that produced by oblate macromolecules of equal axis lengths, while the opposite is true at constant rho(b). A simple scaling analysis is used to explain these trends. Surface charge is found to increase both the range and the magnitude of the depletion attraction; however, the general trends are the same as those found in the uncharged systems. Finally, the effect of the depletion attraction produced by spherical and spheroidal macromolecules on the stability of a dispersion of charged particles was examined. It was found that charged spheroids at concentrations of order 1% volume can produce secondary energy wells of sufficient magnitude to induce flocculation in a dispersion of charged spherical particles. Copyright 2000 Academic Press.     

Computer simulations of the translocation and unfolding of a protein pulled mechanically through a pore

Protein degradation by ATP-dependent proteases and protein import into the mitochondrial matrix involve the unfolding of proteins upon their passing through narrow constrictions. It has been hypothesized that the cellular machinery accomplishes protein unfolding by pulling mechanically at one end of the polypeptide chain. Here, we use Langevin dynamics simulations of a minimalist off-lattice model to examine this hypothesis and to study the unfolding of a protein domain pulled mechanically through a long narrow pore. We compute the potential of mean force (PMF) experienced by the domain as a function of its displacement along the pore and identify the unfolding intermediates corresponding to the local minima of the PMF. The observed unfolding mechanism is different from that found when the two termini are pulled apart, as in single-molecule mechanical unfolding experiments. It depends on the pore diameter, the magnitude of the pulling force, and on whether the force is applied at the N- or the C-terminus of the chain. Consequently, the translocation time exhibits a pulling force dependence that is more complex than a simple exponential function expected on the basis of simple phenomenological models of translocation.     

Molecular conformations,hydrodynamics and optics of ladder polymers

Hydrodynamic (diffusion, sedimentation, viscosity), dynamo-optical and electro-optical properties of several ladder polysiloxanes with different substituents in the side groups have been investigated.In accordance with the theories of hydrodynamic properties of worm-like chains, the equilibrium rigidity of the main chain of these macromolecules was evaluated quantitatively. High equilibrium rigidity of ladder polysiloxanes is a direct consequence of the double-chain structure of their molecular chains.Values of reduced birefringence in electrical and mechanical fields in solutions of ladder polysiloxanes increase with molecular weight but tend to a limit, which is characteristic for semirigid macromolecules.Birefringence in an electrical field for solutions of all ladder polymers investigated is by two orders higher than the value of electro-optical effect in polymers with flexible chains and its sign (negative) coincides with that of flow birefringence. In a variable (sinusoidal) field in the region of high frequencies, strong frequency dependence of birefringence is characteristic for all samples. Unique electrooptical properties in constant and variable fields prove that, in contrast to polymers with flexible chains, highly organized orientational long-range order exists in ladder macromolecules, moreover, it is not only an axial order but also a polar one. The occurrence of long-range correlation in the orientation of polar groups and bonds of the molecular chain gives rise to high total moment of the macromolecule, which is responsible for its rotation in an electrical field. The direction of this dipole moment coincides with the long axis of the molecule since the Kerr effect is of the same sign as flow birefringence.     

Molecular tensile machines: intrinsic acceleration of disulfide reduction by dithiothreitol

Significant tension on the order of 1 nN is self-generated along the backbone of bottlebrush macromolecules due to steric repulsion between densely grafted side chains. The intrinsic tension is amplified upon adsorption of bottlebrush molecules onto a substrate and increases with grafting density, side chain length, and strength of adhesion to the substrate. These molecules were employed as miniature tensile machines to study the effect of mechanical force on the kinetics of disulfide reduction by dithiothreitol (DTT). For this purpose, bottlebrush macromolecules containing a disulfide linker in the middle of the backbone were synthesized by atom transfer radical polymerization (ATRP). The scission reaction was monitored through molecular imaging by atomic force microscopy (AFM). The scission rate constant increases linearly with the concentration of DTT and exponentially with mechanical tension along the disulfide bond. Moreover, the rate constant at zero force is found to be significantly lower than the reduction rate constant in bulk solution, which suggests an acidic composition of the water surface with pH = 3.7. This work demonstrates the ability of branched macromolecules to accelerate chemical reactions at specific covalent bonds without applying an external force.     

STATISTICAL MODELS FOR SEMI-RIGID NEMATIC POLYMERS

Semi-rigid liquid crystal polymer is a class of liquid crystal polymers different from long rigidrod liquid crystal polymer to which the well-known Onsager and Flory theories are applied. Inthis paper, three statistical models for the semi-rigid nematic polymer were addressed. They arethe elastically jointed rod model, worm-like chain model, and non--homogeneous chain model.The nematic-isotropic transition temperature was examined. The pseudo-second transition tem-perature is expressed analytically. Comparisons with the experiments were made and the agree-ments were found.     

Simulating the collapse transition of a two-dimensional semiflexible lattice polymer

It has been revealed by mean-field theories and computer simulations that the nature of the collapse transition of a polymer is influenced by its bending stiffness epsilon(b). In two dimensions, a recent analytical work demonstrated that the collapse transition of a partially directed lattice polymer is always first order as long as epsilon(b) is positive [H. Zhou et al., Phys. Rev. Lett. 97, 158302 (2006)]. Here we employ Monte Carlo simulation to investigate systematically the effect of bending stiffness on the static properties of a two-dimensional lattice polymer. The system's phase diagram at zero force is obtained. Depending on epsilon(b) and the temperature T, the polymer can be in one of the three phases: crystal, disordered globule, or swollen coil. The crystal-globule transition is discontinuous and the globule-coil transition is continuous. At moderate or high values of epsilon(b) the intermediate globular phase disappears and the polymer has only a discontinuous crystal-coil transition. When an external force is applied, the force-induced collapse transition will either be continuous or discontinuous, depending on whether the polymer is originally in the globular or the crystal phase at zero force. The simulation results also demonstrate an interesting scaling behavior of the polymer at the force-induced globule-coil transition.     

Force spectroscopy of quadruple H-bonded dimers by AFM: dynamic bond rupture and molecular time-temperature superposition

We report on the application of the time-temperature superposition principle to supramolecular bond-rupture forces on the single-molecule level. The construction of force-loading rate master curves using atomic force microscopy (AFM)-based single-molecule force spectroscopy (SMFS) experiments carried out in situ at different temperatures allows one to extend the limited range of the experimentally accessible loading rates and hence to cross from thermodynamic nonequilibrium to quasi-equilibrium states. The approach is demonstrated for quadruple H-bonded ureido-4[1H]-pyrimidinone (UPy) moieties studied by variable-temperature SMFS in organic media. The unbinding forces of single quadruple H-bonding (UPy)2 complexes, which were identified based on a polymeric spacer strategy, were found to depend on the loading rate in the range of 5 nN/s to 500 nN/s at 301 K in hexadecane. By contrast, these rupture forces were independent of the loading rate from 5 to 200 nN/s at 330 K. These results indicate that the unbinding behavior of individual supramolecular complexes can be directly probed under both thermodynamic nonequilibrium and quasi-equilibrium conditions. On the basis of the time-temperature superposition principle, a master curve was constructed for a reference temperature of 301 K, and the crossover force (from loading-rate independent to -dependent regimes) was determined as approximately 145 pN (at a loading rate of approximately 5.6 nN/s). This approach significantly broadens the accessible loading-rate range and hence provides access to fine details of potential energy landscape of supramolecular complexes based on SMFS experiments.     

Excluded volume effects in macromolecular forces and ion-interface interactions

A charged Yukawa liquid confined in a slit nanopore is studied in order to understand excluded volume effects in the interaction force between the pore walls. A previously developed self-consistent scheme [S. Buyukdagli, C. V. Achim, and T. Ala-Nissila, J. Stat. Mech. 2011, P05033] and a new simpler variational procedure that self-consistently couple image forces, surface charge induced electric field, and pore modified core interactions are used to this aim. For neutral pores, it is shown that with increasing pore size, the theory predicts a transition of the interplate pressure from an attractive to a strongly repulsive regime associated with an ionic packing state, an effect observed in previous Monte Carlo simulations for hard core charges. We also establish the mean-field theory of the model and show that for dielectrically homogeneous pores, the mean-field regime of the interaction between the walls corresponds to large pores of size d > 4 ?. The role of the range of core interactions in the ionic rejection and interplate pressure is thoroughly analyzed. We show that the physics of the system can be split into two screening regimes. The ionic packing effect takes place in the regime of moderately screened core interactions characterized with the bare screening parameter of the Yukawa potential b ? 3/l(B), where l(B) is the Bjerrum length. In the second regime of strongly screened core interactions b ? 3/l(B), solvation forces associated with these interactions positively contribute to the ionic rejection driven by electrostatic forces and enhance the magnitude of the attractive pressure. For weakly charged pores without a dielectric discontinuity, core interactions make a net repulsive contribution to the interplate force and also result in oscillatory pressure curves, whereas for intermediate surface charges, these interactions exclusively strengthen the external pressure, thereby reducing the magnitude of the net repulsive interplate force. The pronounced dependence of the interplate pressure and ionic partition coefficients on the magnitude and the range of core interactions indicates excluded volume effects as an important ion specificity and a non-negligible ingredient for the stability of macromolecules in electrolyte solutions.     

The theory of viscoelastic characteristics of a highly stretched macromolecule in single molecule AFM

The theory for the deformation of a model macromolecule stretched by its ends under the action of high constant and low periodic forces is constructed. The macromolecule is composed of monomer units in three conformational states. The proposed theory describes the regime of a severe stretching of a macromolecule extended to a length close to its contour length, when its extension proceeds via conformational transitions between different states of monomer units. The structural parameters of the monomer unit are found to correlate with viscoelastic characteristics, which are calculated from the experimental results on the deformation of an individual macromolecule obtained by the frequency atomic force microscopy. For a monomer unit with three conformations, the force dependences of viscoelastic characteristics (effective coefficients of elasticity and friction) can show one or two minima. When the experimental dependences of the above parameters show two minima, the monomer unit can have three or more equilibrium states. With the knowledge of the viscoelastic characteristics of a macromolecule, it is possible to unequivocally estimate all structural parameters of a monomer unit for its three-state conformational model. When the force dependence of viscoelastic characteristics show only one minimum, the monomer unit can have two or more states and analysis of the corresponding viscoelastic characteristics at the minimum makes it possible to select between two- and three-state conformational models. Then, for the three-state model, experimental data allow the prediction of only equilibrium parameters of the monomer unit (position of the minima and energy); dynamic parameters (positions and height of barriers between equilibrium states) remain indeterminate. The proposed theory is used for the interpretation of the viscoelastic characteristics of dextran obtained by single-molecule AFM experiments. The three-state conformational model of a dextran unit is shown to agree better with the experimental data than with the two-state conformational model.     

On the origin of the unusual behavior in the stretching of single-stranded DNA

Force-extension curves (FECs), which quantify the response of a variety of biomolecules subject to mechanical force (f), are often quantitatively fit using worm-like chain (WLC) or freely jointed chain (FJC) models. These models predict that the chain extension, x, normalized by the contour length increases linearly at small f and at high forces scale as x ~ (1 - f(-α)), where α = 0.5 for WLC and unity for FJC. In contrast, experiments on single-stranded DNA (ssDNA) show that over a range of f and ionic concentration, x scales as x ~ ln f, which cannot be explained using WLC or FJC models. Using theory and simulations we show that this unusual behavior in FEC in ssDNA is due to sequence-independent polyelectrolyte effects. We show that the x ~ ln f arises because in the absence of force the tangent correlation function, quantifying chain persistence, decays algebraically on length scales on the order of the Debye length. Our theory, which is most appropriate for monovalent salts, quantitatively fits the experimental data and further predicts that such a regime is not discernible in double-stranded DNA.     

New theoretical considerations in polymer rheology: elastic breakdown of chain entanglement network

Recent experimental evidence has motivated us to present a set of new theoretical considerations and to provide a rationale for interpreting the intriguing flow phenomena observed in entangled polymer solutions and melts [P. Tapadia and S. Q. Wang, Phys. Rev. Lett. 96, 016001 (2006); 96, 196001 (2006); S. Q. Wang et al., ibid. 97, 187801 (2006)]. Three forces have been recognized to play important roles in controlling the response of a strained entanglement network. During flow, an intermolecular locking force f(iml) arises and causes conformational deformation in each load-bearing strand between entanglements. The chain deformation builds up a retractive force f(retract) within each strand. Chain entanglement prevails in quiescence because a given chain prefers to stay interpenetrating into other chains within its pervaded volume so as to enjoy maximum conformational entropy. Since each strand of length l(ent) has entropy equal to k(B)T, the disentanglement criterion is given by f(retract)>f(ent) approximately k(B)Tl(ent) in the case of interrupted deformation. This condition identifies f(ent) as a cohesive force. Imbalance among these forces causes elastic breakdown of the entanglement network. For example, an entangled polymer yields during continuous deformation when the declining f(iml) cannot sustain the elevated f(retract). This opposite trend of the two forces is at the core of the physics governing a "cohesive" breakdown at the yield point (i.e., the stress overshoot) in startup flow. Identifying the yield point as the point of force imbalance, we can also rationalize the recently observed striking scaling behavior associated with the yield point in continuous deformation of both shear and extension.