Kinetics of loop formation in polymer chains

We investigate the kinetics of loop formation in ideal flexible polymer chains (the Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is tau(c) approximately tau(0)N(2) (where tau(0) is a microscopic time scale and N is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when a, the capture radius for contact formation, exceeds b, the average distance between two connected beads. Simulations also show that when a < b, tau(c) approximately N(alpha)(tau), where 1.5 < alpha(tau) < or = 2 in the range 7 < N < 200 used in the simulations. By using a diffusion coefficient that is dependent on the length scales a and b (with a < b), which captures the two-stage mechanism by which looping occurs when a < b, we obtain an analytic expression for tauc that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly effected when interactions between monomers are taken into account. Remarkably, for N < 100, the values of tau(c) decrease by more than 2 orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for tau(c) to a power law in N (tau(c) approximately N(alpha)(tau)) show that alpha(tau) varies from about 2.4 in a good solvent to about 1.0 in poor solvents. The effective exponent alpha(tau) decreases as the strength of the attractive monomer-monomer interactions increases. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain. The time for contact formation between beads that are interior to the chain in good solvents changes nonmonotonically as the loop length varies. In contrast, the variation in interior loop closure time is monotonic in poor solvents. The implications of our results for contact formation in polypeptide chains, RNA, and single-stranded DNA are briefly outlined.     

End-to-end vs interior loop formation kinetics in unfolded polypeptide chains

The conformational search for favorable intramolecular interactions during protein folding is limited by intrachain diffusion processes. Recent studies on the dynamics of loop formation in unfolded polypeptide chains have focused on loops involving residues near the chain ends. During protein folding, however, most contacts are formed between residues in the interior of the chain. We compared the kinetics of end-to-end loop formation (type I loops) to the formation of end-to-interior (type II loops) and interior-to-interior loops (type III loops) using triplet-triplet energy transfer from xanthone to naphthylalanine. The results show that formation of type II and type III loops is slower compared to type I loops of the same size and amino acid sequence. The rate constant for type II loop formation decreases with increasing overall chain dimensions up to a limiting value, at which loop formation is about 2.5-fold slower for type II loops compared to type I loops. Comparing type II loops of different loop size and amino acid sequence shows that the ratio of loop dimension over total chain dimension determines the rate constant for loop formation. Formation of type III loops is 1.7-fold slower than formation of type II loops, indicating that local chain motions are strongly coupled to motions of other chain segments which leads to faster dynamics toward the chain ends. Our results show that differences in the kinetics of formation of type I, type II, and type III loops are mainly caused by differences in internal flexibility at the different positions in the polypeptide chain. Interactions of the polypeptide chain with the solvent contribute to the kinetics of loop formation, which are strongly viscosity-dependent. However, the observed differences in the kinetics of formation of type I, type II, and type III loops are not due to the increased number of peptide-solvent interactions in type II and type III loops compared to type I loops as indicated by identical viscosity dependencies for the kinetics of formation of the different types of loops.     

Phase behavior of semiflexible polymer chains

Monte Carlo simulations were performed on semiflexible polymer chains with the goal of delineating their isotropic-nematic (IN) and gas-liquid coexistence envelopes. The chain monomers are spherical beads that interact via a square-well potential with all other beads. Bonded beads are connected by strings chosen so that bond length varies between 1.01sigma and 1.05sigma (where sigma is the hard sphere diameter). The stiffness of the molecules is controlled via a potential between beads separated by two bonds; this potential restricts the distance between these beads to be between 2.02sigma and 2.1sigma. The vapor-liquid coexistence and IN coexistence curves are obtained using computer simulations. An IN transition is found for 10相似文献   

Complexation of semiflexible chains with oppositely charged cylinder

We study the complexation of long thin semiflexible polymer chains with an oppositely charged cylinder. Starting from the linear Poisson-Boltzmann equation, we calculate the electrostatic potential and the energy of such a charge distribution. We find that sufficiently flexible chains prefer to wrap around the cylinder in a helical manner, when their charge density is smaller than that of the cylinder. The optimal value of the helical pitch is found by minimization of the sum of electrostatic and bending energies. The dependence of the pitch on the number of chains, their rigidity, and salt concentration in solution is analyzed. We discuss our results in the light of recent experiments on DNA complexation with cylindrical dendronized polymers.     

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.     

Phase transitions in lattice solutions of semiflexible chains of finite length

Lattice solutions of finite chains with reduced flexibility are shown to form phase equilibria characterized by broad phase gaps of width increasing with molecular weight. DiMarzio's treatment of excluded volume also gives predictions on bulk transitions. Both in solution and in the bulk, the orientational order parameter and fraction of straight configurations of the anisotropic phase at the transition increase with chain length. Calculations on infinite chain systems fail to detect phase equilibrium in solution at different concentrations.     

Theoretical analysis of polydispersity in the nematic phase of self-assembled semiflexible chains

In recent simulations of semiflexible equilibrium polymers [X. Lu and J. T. Kindt, J. Chem. Phys. 120, 10328 (2004)] a roughly biexponential distribution of chain lengths was observed in the nematic phase. In this study, we show that a theory representing the nematic phase as an equilibrium mixture of randomly oriented chains below a critical length Lc and chains above Lc obeying a Gaussian orientational distribution reproduces this distribution qualitatively. The agreement between predicted and simulated phase boundaries is improved, with a narrower isotropic-nematic coexistence region, compared with a monoexponential chain distribution. We find, furthermore, that the critical length Lc scales as the inverse of monomer concentration, irrespective of the bond strength of the chains, but that at the phase boundary, the fraction of monomers found in disordered chains peaks at a certain bond strength.     

Persistence lengths of semiflexible chains — methods and approximations

Several models and methods for stiff polymer chains are discussed. The basic idea is to develop approximate solutions to the problem of the presistence length of stiff polymers. It turns out that the persistence length can be regarded as a measure for the quality of approximations. Mean-field methods for field theoretical calculations of the persistence length show similarities of 1/d expansions in statistical physics (d being the space dimension) and saddle point approximations become reliable in various limits. Gaussian approximations become — as well known for the Ising model — simple extensions of random walks as trivial renormalisations of the Wiener-Edwards model for bosonic strings.     

Influence of surface interactions on folding and forced unbinding of semiflexible chains

We have investigated the folding and forced unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe, at an elementary level, a number of biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface, and the wormlike chain (WLC) is used for the semiflexible chain (SC). Using Langevin simulations, in the overdamped limit we examine the ordering kinetics of racquet-like and toroidal structures in the presence of an attractive interaction between the surface and the polymer chain. For a range of interactions, temperature, and the persistence length, l(p), we obtained the monomer density distribution, n(x), (x is the perpendicular distance of a tagged chain end from the surface) for all of the relevant morphologies. There is a single peak in n(x) inside the range of attractive forces, b, for chains in the extended conformations, whereas in racquet and toroidal structures there is an additional peak at x approximately b. The simulated results for n(x) are in good agreement with theory. The formation of toroids on the surface appears to be a first-order transition as evidenced by the bimodal distribution in n(x). The theoretical result underestimates the simulated n(x) for x < b and follows n(x) closely for x >/= b; the calculated density agrees exactly with n(x) in the range x < b. The chain-surface interaction is probed by subjecting the surface structures to a pulling force, f. The average extension, x( f), as a function of f exhibits a sigmoidal profile with sharp all-or-none transition at the unfolding force threshold f = f(c) which increases for more structured states. Simulated x(f) compare well with the theoretical predictions. The critical force, f(c), is a function of l(s)/l(c) for a fixed temperature, where l(c) and l(s) are the length scales that express the strength of the intramolecular and SC-surface attraction, respectively. For a fixed l(s), f(c) increases as l(p) decreases.     

Effect of proline and glycine residues on dynamics and barriers of loop formation in polypeptide chains

Glycine and proline residues are frequently found in turn and loop structures of proteins and are believed to play an important role during chain compaction early in folding. We investigated their effect on the dynamics of intrachain loop formation in various unstructured polypeptide chains. Loop formation is significantly slower around trans prolyl peptide bonds and faster around glycine residues compared to any other amino acid. However, short loops are formed fastest around cis prolyl bonds with a time constant of 6 ns for end-to-end contact formation in a four-residue loop. Formation of short loops encounters activation energies in the range of 15 to 30 kJ/mol. The altered dynamics around glycine and trans prolyl bonds can be mainly ascribed to their effects on the activation energy. The fast dynamics around cis prolyl bonds, in contrast, originate in a higher Arrhenius pre-exponential factor, which compensates for an increased activation energy for loop formation compared to trans isomers. All-atom simulations of proline-containing peptides indicate that the conformational space for cis prolyl isomers is largely restricted compared to trans isomers. This leads to decreased average end-to-end distances and to a smaller loss in conformational entropy upon loop formation in cis isomers. The results further show that glycine and proline residues only influence formation of short loops containing between 2 and 10 residues, which is the typical loop size in native proteins. Formation of larger loops is not affected by the presence of a single glycine or proline residue.     

Structure of flexible and semiflexible polyelectrolyte chains in confined spaces of slit micro/nanochannels

Understanding the behavior of a polyelectrolyte in confined spaces has direct relevance in design and manipulation of microfluidic devices, as well as transport in living organisms. In this paper, a coarse-grained model of anionic semiflexible polyelectrolyte is applied, and its structure and dynamics are fully examined with Brownian dynamics (BD) simulations both in bulk solution and under confinement between two negatively charged parallel plates. The modeling is based on the nonlinear bead-spring discretization of a continuous chain with additional long-range electrostatic, Lennard-Jones, and hydrodynamic interactions between pairs of beads. The authors also consider the steric and electrostatic interactions between the bead and the confining wall. Relevant model parameters are determined from experimental rheology data on the anionic polysaccharide xanthan reported previously. For comparison, both flexible and semiflexible models are developed accompanying zero and finite intrinsic persistence lengths, respectively. The conformational changes of the polyelectrolyte chain induced by confinements and their dependence on the screening effect of the electrolyte solution are faithfully characterized with BD simulations. Depending on the intrinsic rigidity and the medium ionic strength, the polyelectrolyte can be classified as flexible, semiflexible, or rigid. Confined flexible and semiflexible chains exhibit a nonmonotonic variation in size, as measured by the radius of gyration and end-to-end distance, with changing slit width. For the semiflexible chain, this is coupled to the variations in long-range bond vector correlation. The rigid chain, realized at low ionic strength, does not have minima in size but exhibits a sigmoidal transition. The size of confined semiflexible and rigid polyelectrolytes can be well described by the wormlike chain model once the electrostatic effects are taken into account by the persistence length measured at long length scale.     

Folding transition of a single semiflexible polyelectrolyte chain through toroidal bundling of loop structures

We consider how the DNA coil-globule transition progresses via the formation of a toroidal ring structure. We formulate a theoretical model of this transition as a phenomenon in which an unstable single loop generated as a result of thermal fluctuation is stabilized through association with other loops along a polyelectrolyte chain. An essential property of the chain under consideration is that it follows a wormlike chain model. A toroidal bundle of loop structures is characterized by a radius and a winding number. The statistical properties of such a chain are discussed in terms of the free energy as a function of the fraction of unfolded segments. We also present an actual experimental observation of the coil-globule transition of single giant DNA molecules, T4 DNA (165.5 kbp), with spermidine (3+), where intrachain phase segregation appears at a NaCl concentration of more than 10 mM. Both the theory and experiments lead to two important points. First, the transition from a partially folded state to a completely folded state has the characteristics of a continuous transition, while the transition from an unfolded state to a folded state has the characteristics of a first-order phase transition. Second, the appearance of a partially folded structure requires a folded structure to be less densely packed than in the fully folded compact state.     

Comb copolymer cylindrical brushes containing semiflexible side chains: a Monte Carlo study

The off‐lattice Monte Carlo method is applied to investigate the equilibrium conformations of isolated comb copolymer cylindrical brushes in an athermal solution. The molecules considered consist of a flexible backbone, which is densely grafted with semiflexible side chains. The study focuses on the influence of the degree of intrinsic stiffness, λ_side, of the side chains on the conformational behavior of the molecules. It is demonstrated that with a fixed side chain length, M, the local length scale conformational fluctuations of the backbone increase as a function of λ_side. However, the persistence length, λ, of the cylindrical brush increases considerably with the side chain stiffness, indicating that the backbone becomes more extended at the large length scale. Moreover, as a function of λ_side, there is an increase in the ratio λ/D of the persistence length and the diameter, D, of the brush. This behavior is in good agreement with recent theoretical predictions and provides important new insight in the latest experimental observations.     

Gelation in semiflexible polymers

Molecular dynamics simulations have been employed to study the formation of a physical gel by semiflexible polymer chains. The formation of a geometrically connected network of these chains is investigated as a function of temperature and rate of cooling. The stiffness of the molecules is controlled via a potential between beads separated by two bonds. As the temperature is lowered, a percolated homogeneous solution phase separates to form a high-density, non-percolated nematic fluid and a low-density gas phase. On further decreasing the temperature, the chains are dynamically arrested preventing the completion of the vapor-liquid (VL) phase separation. As a result, the chains are stuck in a three-dimensional network of nematic bundles forming a percolated gel. Apart from temperature, the rate of cooling also plays an important role in the formation of the gel. Cooling the system at a faster rate yields gel while slower rates result in complete VL phase separation.     

Kinetics of styrylquinolinc formation

A comprehensive kinetic investigation of reactions occurring in the formation of styryl-quinolines has been conducted. Specific rate data such as rate equations, rate constants, and thermodynamic activation values have been determined and utilized in a study of which factors are of greatest importance in the reactions forming 2-styrylquinolines. A mechanism has been proposed for the condensation reaction which agrees with rate relationships found. Gas-liquid partition chromatography was used to follow the kinetics of the condensation reactions. A rate constant of 5.41 × 10^?2M^?1min^?1 was found for the reaction of benzaldehyde with 2-methyl-quinoline using zinc chloride as a catalyst at 104.0°. Rate constants of 1.28 × 10^?2 MT^?1 min^?1 and 1.05 × 10^?2 M^?1 min^?1 were found for the reactions of p-methylbenzaldehyde and p-methoxybenzaldehyde with quinaldine to form 2-(p-methylstyryl)quinoline and 2-(p-methoxystyryl)-quinoline, respectively at 92.4°. A linear relationship was found using the Hammett equation. An Arrhenius plot was constructed from rate constants determined at five different temperatures for the reaction of benzaldehyde and quinaldine to form 2-styrylquinoline, using zinc chloride as a catalyst. The energy of activation, E_a, was found to be 22.2 kcal/mole for this reaction. The enthalpy of activation, ΔH?, free energy of activation, ΔF?, and entropy of activation, ΔS?, were found to be 21.4 kcal/mole, 27.7 kcal/mole and -16.7 eu/mole, respectively, at 104.0°. The mechanism proposed in the formation of 2-styrylquinoline involves the fast formation of a carbanion-zinc chloride complex, which then attacks, in the rate determining step, the aldehyde utilized in the reaction. The lack of reaction of certain methylquinolines is attributed to the inadequacy of the carbanion formed and not to the difficulty involved in the initial formation of the carbanion.     

Kinetics of vesicle formation

 The kinetics of vesicle formation from a hydrotrope (sodium xylenesulfonate) solution of a surfactant (Laureth 4) is studied by the use of a stopped-flow apparatus combined with a dynamic light scattering device to determine vesicle size in the system. The hydrotrope system studied presents a system with a high surfactant solubilization combined with vesicle formation simply by dilution with water. The kinetic results show a single exponential decay time. The kinetic analysis indicates that the vesicles are formed from a molecular solution which resulted from the shear in the stopped-flow device and grow by monomeric association. Received: 1 October 1996 Accepted: 22 November 1996     

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.     

Kinetics of soot formation in tetrachloromethane pyrolysis

Kinetics of soot formation is studied in tetrachloromethane pyrolysis behind shock waves. The time dependences of macrokinetic characteristics of soot particle growth (the induction period, the soot yield, and the apparent rate constant of soot particle growth) are determined. Based on the experimental data, the quantitative model of soot formation is developed for tetrachloromethane pyrolysis behind shock waves. Special attention is paid to the thermal effects in CC1⁴ pyrolysis.     

Intramolecular relaxation dynamics in semiflexible dendrimers

The intramolecular relaxation dynamics of semiflexible dendrimers in dilute solutions are theoretically investigated in the framework of optimized Rouse-Zimm formalism. Semiflexibility is implemented by modeling topological restrictions on the bond directions and orientations of the respective bond-vectors. Based on our recently developed approach for semiflexible dendrimers [A. Kumar and P. Biswas, Macromolecules 43, 7378 (2010)], the mechanical and dielectric relaxation moduli are studied as functions of local flexibility parameters and branching topology. It is rather interesting to observe that semiflexibility affects the local modes of G'(ω) and Δε'(ω), which have lower relaxation rate with increasing bond restrictions, while the collective modes with small relaxation rate remain almost constant. The relaxation dynamics of the flexible dendrimer is similar to that of the semiflexible dendrimer with unrestricted bond orientations (Φ = 0) and is flanked by the compressed (Φ = 30°) and expanded (Φ = 150°) conformations, respectively. The effect of semiflexibility is typically reflected in the intermediate frequency regime. The expanded conformations of semiflexible dendrimers display a power-law behavior in the intermediate frequency regime for both loss and storage modulus resembling fractal structures, while the compressed and unrestricted bond orientation conformations exhibit an approximately logarithmic dependence. The power-law exponent is found to be similar to the flexible dendrimers with excluded volume interactions. Thus, by tuning Φ, a spectrum of dynamic relaxation pattern is obtained spanning a broad range of conformations from a power-law fractal network to a non-fractal one. In certain limits, this highly generalized model captures the characteristics of flexible dendrimers and also resembles La Ferla's model semiflexible dendrimers. The influence of hydrodynamic interactions reduces the dynamical range and the width of the intermediate domain by decreasing the smaller relaxation rates and increasing the higher relaxation rates correspondingly.