Gaussian Polymer Chain Research Articles

Statistics of Gaussian polymer chains in harmonic applied fields.

The model of an ideal polymer chain in a harmonic applied field has broad applicability in situations involving polymer confinement and deformation due to applied stress. In this work we (1) formulate a general analytical model for a continuous Gaussian chain under a harmonic applied potential and (2) evaluate the statistical mechanics of this model given the potential, obtaining partition functions and moment generating functions (MGFs) that describe the chain configurations. Closed-form expressions for the squared radius of gyration, potential energy, partition function, and MGF for the center of mass are obtained for a general and multidimensional harmonic field. The expressions are compared with results of Monte Carlo simulations of a discrete Gaussian chain as well as results for related systems obtained from the literature. The theory derived here is used to test the applicability of the current model assumptions to relations from the literature describing polymer confinement and deformation in experiment, theory, and simulations.

Upper critical solution temperature polymer assemblies via variable temperature liquid phase transmission electron microscopy and liquid resonant soft X-ray scattering

Here, we study the upper critical solution temperature triggered phase transition of thermally responsive poly(ethylene glycol)-block-poly(ethylene glycol) methyl ether acrylate-co-poly(ethylene glycol) phenyl ether acrylate-block-polystyrene nanoassemblies in isopropanol. To gain mechanistic insight into the organic solution-phase dynamics of the upper critical solution temperature polymer, we leverage variable temperature liquid-cell transmission electron microscopy correlated with variable temperature liquid resonant soft X-ray scattering. Heating above the upper critical solution temperature triggers a reduction in particle size and a morphological transition from a spherical core shell particle with a complex, multiphase core to a micelle with a uniform core and Gaussian polymer chains attached to the surface. These correlated solution phase methods, coupled with mass spectral validation and modeling, provide unique insight into these thermoresponsive materials. Moreover, we detail a generalizable workflow for studying complex, solution-phase nanomaterials via correlative methods.

Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion.

In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the nonequilibrium stationary state. Here we highlight a new facet of the TUR by showing that it also bounds the timescale on which a finite system can exhibit anomalous kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a single file confined to a ring as well as a dragged Gaussian polymer chain even when detailed balance is satisfied. Conversely, the TUR bounds the onset of superdiffusion in the active comb model. Remarkably, the fluctuations in a comb model evolving from a steady state behave anomalously as soon as detailed balance is broken. Our work establishes a link between stochastic thermodynamics and the field of anomalous dynamics that will fertilize further investigations of thermodynamic consistency of anomalous diffusion models.

Modeling short-chain branched polyethylenes in dilute solution under variable solvent quality conditions: Basic configurational properties

We perform molecular dynamics (MD) simulations of a coarse-grained model of linear low-density polyethylenes (LLDPEs) having short-chain branching in an implicit solvent and calculate basic configurational properties (radius of gyration Rg, hydrodynamic radius Rh, and intrinsic viscosity [η]) using the path-integration program ZENO. Solvent quality effects in our model are accounted for through the introduction of an attractive polymer-polymer interaction potential with solvent quality “strength” λ that is tuned from a “good” solvent limit, where the inter-polymer interactions are purely repulsive, to a “very poor” solvent regime, where the polymers collapse due to their self-attractive intermolecular interactions. We also identify a compensation condition or “θ point” in our model between these limits, where the polymers are “ideal” in the sense in which attractive interactions effectively nullify inter-polymer repulsions so that the chains behave akin to Gaussian polymer chains in their average conformational properties. In order to compare with experiments performed under variable solvent conditions, we define a dimensionless measure of excluded volume interaction parameter δ in terms of the mass scaling exponent ν for Rg or νh for Rh, which can be determined from both experiment and simulation. We illustrate our variable solvent approach to estimate polymer solution properties in the case of polyethylenes (PEs) and evaluate the δ value that corresponds to the experimental solvent conditions. Our combined use of MD and ZENO allows for the numerical estimation of polymer solution properties for polymers having general monomer structures and solvent qualities in a computationally tractable fashion and should be useful as a general computational tool for polymer structural characterization.

Aggregation of a macromolecule in a nano cube

We model a macromolecule as an infinitely long Gaussian semi-flexible polymer chain and the conformations of the chain were realized in the nano cube using a cubic lattice. A modified version of the recursion relations is used to calculate the grand canonical partition function of the chain to investigate distinct thermo-dynamical properties of the nano polymer aggregate than corresponding bulk behaviour of the macromolecule. Our analytical estimates on the thermo-dynamical properties of the macromolecule clearly show that the nano aggregate of the polymer chain has interesting and distinct conformational statistics than its corresponding bulk state, and the method described in the present report can be easily extend to investigate the thermodynamics of the self-avoiding polymer chain in the nano dimensions.

Gaussian polymer chains in a harmonic potential: the path integral approach

We study conformations of the Gaussian polymer chains in d-dimensional space in the presence of an external field with the harmonic potential. We apply a path integral approach to derive an explicit expression for the probability distribution function of the gyration radius. We calculate this function using Monte Carlo simulations and show that our numerical and theoretical results are in a good agreement for different values of the external field.

Shape analysis of random polymer networks

We propose the model of a random polymer network, formed on the base on Erdös–Rényi random graph. In the language of mathematical graphs, the chemical bonds between monomers can be treated as vertices, and their chemical functionalities as degrees of these vertices. We consider graphs with fixed number of vertices N = 5 and variable parameter c (connectedness), defining the total number of links L = cN(N − 1)/2 between vertices. Each link in such graphs is treated as a Gaussian polymer chain. The universal rotationally invariant size and shape characteristics, such as averaged asphericity and size ratio of such structures are obtained both numerically by application of Wei’s method and analytically within the continuous chain model. In particular, our results quantitatively indicate an increase of asymmetry of polymer network structure when its connectedness c decreases.

PHYSICAL-CHEMICAL PROPERTIES OF CHONDROITIN SULFATE IN AGGRECAN

In current work theoretical study of physical-chemical properties of chondroitin sulfate in aggrecan within the self-consistent field theory was developed. As a model for aggrecan a model of cylindrical polymer brush was chosen. Aggrecan is considered as a system of n gaussian polymer chains consisting of N segments tethered to cylindrical surface of core protein. Basing on the results of previous work where using gaussian equivalent representation for functional integrals equation of state for chondroitin sulfate in aqueous solution was obtained and also osmotic pressure, dissociation degree, radius of gyration and renormalized persistent length were calculated with respect to monomers electrostatic interactions and effect of counterion condensation, one can obtain configuration integral for cylindrical polymer brush. Persistent length is introduced instead of Kuhn segment length which supposedly helps one to take into account rigidity of polymer chain determined by electrostatic repulsion of monomers (disaccharides groups). Expression for free energy of a polymer brush modeled as chondroitin sulfate chains tethered to a cylindrical surface was obtained. Dependence of chondroitin sulfate in aggrecan dissociation degree as a function of distance from the cylindrical surface is represented. For given system it varies insignificantly from the one for free chains in solution. Results for a polymer density as a function of distance from the cylindrical surface obtained within mean-field approximation are in qualitative agreement with those obtained using density functional method. However, calculations within a mean field framework are understable, thus it is necessary to use regularization methods.Forcitation:NogovitsynE.A., KalikinN.N., ZheleznyakN.I. Physical-chemical properties of chondroitin sulfate in aggrecan. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018. V. 61. N 2. P. 35-39

On a new application of the path integrals in polymer statistical physics

We propose a new approach based on the path integral formalism to the calculation of the probability distribution functions of quadratic quantities of the Gaussian polymer chain in d-dimensional space, such as the radius of gyration and potential energy in the parabolic well. In both cases we obtain the exact relations for the characteristic function and cumulants. Using the standard steepest-descent method, we evaluate the probability distribution functions in two limiting cases of the large and small values of corresponding variables.

Folding of Protein L with Implications for Collapse in the Denatured State Ensemble.

A fundamental question in protein folding is whether the coil to globule collapse transition occurs during the initial stages of folding (burst phase) or simultaneously with the protein folding transition. Single molecule fluorescence resonance energy transfer (FRET) and small-angle X-ray scattering (SAXS) experiments disagree on whether Protein L collapse transition occurs during the burst phase of folding. We study Protein L folding using a coarse-grained model and molecular dynamics simulations. The collapse transition in Protein L is found to be concomitant with the folding transition. In the burst phase of folding, we find that FRET experiments overestimate radius of gyration, Rg, of the protein due to the application of Gaussian polymer chain end-to-end distribution to extract Rg from the FRET efficiency. FRET experiments estimate ≈6 Å decrease in Rg when the actual decrease is ≈3 Å on guanidinium chloride denaturant dilution from 7.5 to 1 M, thereby suggesting pronounced compaction in the protein dimensions in the burst phase. The ≈3 Å decrease is close to the statistical uncertainties of the Rg data measured from SAXS experiments, which suggest no compaction, leading to a disagreement with the FRET experiments. The transition-state ensemble (TSE) structures in Protein L folding are globular and extensive in agreement with the Ψ-analysis experiments. The results support the hypothesis that the TSE of single domain proteins depends on protein topology and is not stabilized by local interactions alone.

Cyclization kinetics of Gaussian semiflexible polymer chains.

We consider the dynamics and the cyclization kinetics of Gaussian semiflexible chains, in which the interaction potential tends to align successive bonds. We provide asymptotic expressions for the cyclization time, for the eigenvalues and eigenfunctions, and for the mean square displacement at all time and length scales, with explicit dependence on the capture radius, on the positions of the reactive monomers in the chain, and on the finite number of beads. For the cyclization kinetics, we take into account non-Markovian effects by calculating the distribution of reactive conformations of the polymer, which are not taken into account in the classical Wilemski-Fixman theory. Comparison with numerical simulations confirms the accuracy of this non-Markovian theory.

Relationships between the winding angle, the characteristic radius, and the torque for a long polymer chain wound around a cylinder: implications for RNA winding around DNA during transcription.

Long polymer chains are ubiquitous in biological systems and their mechanical properties have significant impact upon biological processes. Of particular interest is the situation in which polymer chains are wound around each other or around other objects. We have analyzed the parameters of a long Gaussian polymer chain wound around a cylinder as a function of the torque applied to the ends of the chain. We have shown that for sufficiently long polymer chains, an average winding angle and a characteristic radius of the chain can be determined from a modified Bessel function of purely imaginary order, in which the value of the order is equivalent to the applied torque, normalized to the product of the absolute temperature and the Boltzmann constant. The obtained results are consistent with a simplified interpretation in terms of "torsional blobs," and this could be extended to nonideal chains with excluded volumes. We have also extended our results to the case of a polymer chain rotating in viscous medium. Our results could be used to estimate the mechanical strains that appear in DNA and RNA during transcription, as these might initiate formation of unusual DNA structures, invasion of RNA into the DNA duplex (R-loop formation), and modulation of the interactions of DNA and RNA with proteins.

Deformation dependent dielectric permittivity and its effect on actuator performance and stability

We utilize a model for birefringence/permittivity based on the statistical mechanics of a Gaussian polymer chain to construct a relationship for the dependence of the dielectric permittivity of an elastomer on a general 3-dimensional state of deformation. The model, due to Kuhn and Grün (1942 [1]), expresses the birefringence/permittivity of a Gaussian polymer chain elastomer as a function of the end-to-end distance of the chains, and assumes that the motions of the chains are affine to the overall deformation. The outcome is an expression for the permittivity tensor of the elastomer as a function of its stretch ratios. The permittivity is isotropic in the undeformed state and under pure dilatation, but otherwise becomes anisotropic during deformation. With this model, we use the free energy of the elastomer to compute the response of a neo-Hookean thin film in an actuator configuration subject to electric and mechanical loading for conditions where the permittivity in the through thickness direction is allowed to increase or decrease with the in-plane extension of the thin film. With such an approach, we study the deformation characteristics of the actuator and its stability under through thickness electric fields. Our calculations show that the deformation dependent permittivity can hasten or postpone an electromechanical instability that can cause a sudden thinning of the dielectric, accompanied by in-plane stretching, when the through thickness electric field is raised above a critical magnitude. Specifically, we consider the case of an actuator exhibiting a through thickness permittivity that decreases with in-plane extension. We observe that in such an actuator the instability is delayed to a higher electric field than would be the case if the dielectric permittivity were independent of strain. Furthermore, we establish that upon removal of the electric field the system follows a different path in terms of potential versus charge, and so develops a hysteresis loop, similar to that identified by Zhao et al. (2007 [2]) for dielectric elastomers with constant isotropic permittivity, but that stiffen during straining.

Why and how does native topology dictate the folding speed of a protein?

Since the pioneering work of Plaxco, Simons, and Baker, it is now well known that the rates of protein folding strongly correlate with the average sequence separation (absolute contact order (ACO)) of native contacts. In spite of multitude of papers, our understanding to the basis of the relation between folding speed and ACO is still lacking. We model the transition state as a gaussian polymer chain decorated with weak springs between native contacts while the unfolded state is modeled as a gaussian chain only. Using these hamiltonians, our perturbative calculation explicitly shows folding speed and ACO are linearly related when only the first order term in the series is considered. However, to the second order, we notice the existence of two new topological metrics, termed COC(1) and COC(2) (COC stands for contact order correction). These additional correction terms are needed to properly account for the entropy loss due to overlapping (nested or linked) loops that are not well described by simple addition of entropies in ACO. COC(1) and COC(2) are related to fluctuations and correlations among different sequence separations. The new metric combining ACO, COC(1), and COC(2) improves folding speed dependence on native topology when applied to three different databases: (i) two-state proteins with only α∕β and β proteins, (ii) two-state proteins (α∕β, β and purely helical proteins all combined), and (iii) master set (multi-state and two-state) folding proteins. Furthermore, the first principle calculation provides us direct physical insights to the meaning of the fit parameters. The coefficient of ACO, for example, is related to the average strength of the contacts, while the constant term is related to the protein folding speed limit. With the new scaling law, our estimate of the folding speed limit is in close agreement with the widely accepted value of 1 μs observed in proteins and RNA. Analyzing an exhaustive set (7367) of monomeric proteins from protein data bank, we find our new topology based metric (combining ACO, COC(1), and COC(2)) scales as N(0.54), N being the number of amino acids in a protein. This is in remarkable agreement with a previous argument based on random systems that predict protein folding speed depends on exp (-N(0.5)). The first principle calculation presented here provides deeper insights to the role of topology in protein folding and unifies many parallel arguments, seemingly disconnected, demonstrating the existence of universal mechanism in protein folding kinetics that can be understood from simple polymer physics based principles.

How to understand the ensemble equivalence during stretching of a single macromolecule

In this paper, we discuss the elastic behavior of an isolated macromolecule in conjugated and non-conjugated ensembles during different modes of applying a mechanical force to the chain ends, namely, the mechanical effects corresponding to the methods of atomic force microscopy, magnetic levitation, and stretching of a macromolecule in an isotropic force field. Recently published results on the Langevin dynamics computer simulation of the stretching of a Gaussian polymer chain in different ensembles are analyzed. An analytical description of all the results of this simulation is given to show that the conclusion made by the authors of those studies about the nonequivalence of conjugated ensembles for Gaussian chains is not quite correct. A theoretical examination of the stretching of individual semirigid chains, which exhibit interesting behavior because of manipulations with actin molecules and microtubules, which are components of the cytoskeleton of most cells, is performed. We present rigorous analytical results about the stretching of chains composed of two or three freely jointed rods of equal length without consideration for excluded-volume inter-actions. It is shown that the stress-strain curves of these chains differ not only quantitatively but also qualitatively in different ensembles and that these curves for a chain of two freely jointed rods have an anomalous shape. These results on the different behavior of stress-strain for different modes of external mechanical impacts can be useful for interpreting experiments on the stretching of individual polymer molecules with different structures obtained via different methods, such as atomic-force microscopy and magnetic levitation.

Synchrotron SAXS to probe cross-linked network of polyamide ‘reverse osmosis’ and ‘nanofiltration’ membranes

Abstract A typical ‘reverse osmosis’ (RO) or ‘nanofiltration’ (NF) polyamide cross-linked structure is formed by interfacial reaction between an aqueous solution of m -phenylene diamine (or piperazine) and a trimesoyl chloride solution in n -hexane. Using synchrotron SAXS, it was observed that a little change in the reactant concentrations resulted to different cross-linked networks of the polyamides. A clear difference between the cross-linking networks of RO and NF polyamides was observed even though there were some similarities in terms of the nature of polymer chain compaction and the cluster formation by aggregation. In both the polyamides, there was strong compaction of polymer chains forming globular like objects having smooth interface which was well deviated from the Gaussian polymer chain nature ( I ∞ Q − 2 ) wherein the NF polyamide had relatively more compaction with Porod scattering I ∞ Q −3.8 than I ∞ Q −3.4 of the RO polyamide. The size of the compacted globules containing about 1400 trimesamide monomeric units for the NF polyamide was slightly smaller than that of the RO polyamide. These globules were very strongly aggregated to form polymer nodules. The nodular cluster size for the NF polyamide was ∼2000–3000 A which was ∼6–7 times larger than that of RO polyamide. These differences in the nature of polymer chain compaction and the cluster size agreed with the differences in effective cross-link density measured by equilibrium swelling in dimethyl acetamide and the polymer nodule sizes observed by the AFM.

Equilibrium statistics of weakly slip-linked Gaussian polymer chains

We calculate the free energy and the pressure of a weakly slip-linked Gaussian polymer chains. We show that the equilibrium statistics of a slip-linked system is different from one of the corresponding ideal chain system without any constraints by slip-links. It is shown that the pressure of a slip-linked system decreases compared with the ideal system, which implies that slip-linked chains spontaneously form aggregated cluster-like compact structures. These are qualitatively consistent with previous theoretical analyses or multichain simulations. We also show that repulsive potentials between chains, which have been phenomenologically utilized in simulations, can cancel the artificial pressure decrease. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011

Dynamics of a polymer chain confined in a membrane

We present a Brownian dynamics theory with full hydrodynamics (Stokesian dynamics) for a Gaussian polymer chain embedded in a liquid membrane which is surrounded by bulk solvent and walls. The mobility tensors are derived in Fourier space for the two geometries, namely, a free membrane embedded in a bulk fluid, and a membrane sandwiched by the two walls. Within the preaveraging approximation, a new expression for the diffusion coefficient of the polymer is obtained for the free-membrane geometry. We also carry out a Rouse normal mode analysis to obtain the relaxation time and the dynamical structure factor. For large polymer size, both quantities show Zimm-like behavior in the free-membrane case, whereas they are Rouse-like for the sandwiched membrane geometry. We use the scaling argument to discuss the effect of excluded-volume interactions on the polymer relaxation time.

Modeling the Chain Entropy of Biopolymers: Unifying Two Different Random Walk Models under One Framework

Entropy plays a critical role in the long range structure of biopolymers. To model the coarse-grained chain entropy of the residues in biopolymers, the lattice model or the Gaussian polymer chain (GPC) model is typically used. Both models use the concept of a random walk to find the conformations of an unstructured polymer. However, the entropy of the lattice model is a function of the coordination number, whereas the entropy of the GPC is a function of the root-mean square separation distance between the ends of the polymer. This can lead to inconsistent predictions for the coarse-grained entropy. Here we show that the GPC model and the lattice model both are consistent under transformations using the cross-linking entropy (CLE) model and that the CLE model generates a family of equations that include these two models at important limits. We show that the CLE model is a unifying approach to the thermodynamics of biopolymers that links these incompatible models into a single framework, elicits their similarities and differences, and expands beyond the models allowing calculation of variable flexibility and incorporating important corrections such as the worm-like-chain model. The CLE model is also consistent with the contact-order model and, when combined with existing local pairing potentials, can predict correct structures at the minimum free energy.

An analytic expression for the double quantum 1H nuclear magnetic resonance build-up and decay from a Gaussian polymer chain with dynamics governed by a single relaxation time

An analytic result for the double quantum (1)H nuclear magnetic resonance build-up and decay is presented for a polymer chain with Gaussian statistics. The analytic expression is obtained for the particular case of bond vector dynamics governed by a single relaxation time. The result is used to check the accuracy of the commonly used second moment approximation applied to the same problem. This approximation is shown to differ significantly from our exact result over the entire range of relaxation times, set by the relevant interaction constant. We show that this measurement technique is peculiarly sensitive to slow dynamics, and that these slow motions reveal themselves in an experimental time window t that is, curiously, much smaller than the timescale tau of the dynamics being probed. We found t approximately tau(1/3), which enables a wide range of bond vector dynamics to be measured.