Based on the Zimm-Bragg model we study cooperative helix-coil transition driven by a finite-speed change of temperature. There is an asymmetry between the coil-to-helix and helix-to-coil transition: the latter is displayed already for finite speeds, and takes shorter time than the former. This hysteresis effect has been observed experimentally, and it is explained here via quantifying system's stability in the vicinity of the critical temperature. A finite-speed cooling induces a non-equilibrium helical phase with the correlation length larger than in equilibrium. In this phase the characteristic length of the coiled domain and the non-equilibrium specific heat can display an anomalous response to temperature changes. Several pertinent experimental results on the kinetics helical biopolymers are discussed in detail.
We present the exact solution of a microscopic statistical mechanical model for the transformation of a long polypeptide between an unstructured coil conformation and an $α$-helix conformation. The polypeptide is assumed to be adsorbed to the interface between a polar and a non-polar environment such as realized by water and the lipid bilayer of a membrane. The interfacial coil-helix transformation is the first stage in the folding process of helical membrane proteins. Depending on the values of model parameters, the conformation changes as a crossover, a discontinuous transition, or a continu…
The problem of the helix-coil transition of biopolymers in explicit solvents, like water, with the ability for hydrogen bonding with solvent is addressed analytically using a suitably modified version of the Generalized Model of Polypeptide Chains. Besides the regular helix-coil transition, an additional coil-helix or reentrant transition is also found at lower temperatures. The reentrant transition arises due to competition between polymer-polymer and polymer-water hydrogen bonds. The balance between the two types of hydrogen bonding can be shifted to either direction through changes not only…
The extension elasticity of rod-coil mutliblock copolymers is analyzed for two experimentally accessible situations. In the quenched case, when the architecture is fixed by the synthesis, the force law is distinguished by a sharp change in the slope. In the annealed case, where interconversion between rod and coil states is possible, the resulting force law is sigmoid with a pronounced plateau. This last case is realized, for example, when homopolypeptides capable of undergoing a helix-coil transition are extended from a coil state. Both scenarios are relevant to the analysis and design of exp…
The helix-coil transition is modified by grafting to a surface. This modification is studied for short peptides capable of forming $α$-helices. Three factors are involved: (i) the grafting can induced change of the boundary free energy of the helical domain (ii) the van der Waals attraction between the helices and (iii) the crowding induced stretching of the coils. As a result the helix-coil transition acquires ``all or nothing'' characteristics. In addition the transition temperature is elevated and the transition itself sharpens as the grafting density increases.
We discuss the effects of the solvent composition on the helix-coil transition of a polypeptide chain. We use a simple model to demonstrate that improving the hydrogen-bonding ability of the solvent can make the transition less cooperative, without affecting the transition temperature. This effect is very different from other solvent effects which primarily influence the melting transition rather than the cooperativity.
Most helix-coil transition theories can be characterized by a set of three parameters: energetic, describing the (free) energy cost of forming a helical state in one repeating unit; entropic, accounting for the decrease of entropy due to the helical state formation; and geometric, indicating how many repeating units are affected by the formation of one helical state. Depending on their effect on the helix-coil transition, solvents or co-solutes can be classified with respect to their action on these parameters. Solvent interactions that alter the entropic cost of helix formation by their osmot…
In this paper we summarize some results of our theoretical investigations of helix-coil transition both in single-strand (polypeptides) and two-strand (polynucleotides) macromolecules. The Hamiltonian does not contain any parameter designed especially for helix-coil transition and uses pure molecular microscopic parameters. To calculate averages we evaluate the partition function using transfer-matrix approach. The GMPC allowed to describe the influence of a number of factors, affecting the transition, basing on a unified microscopic approach. Thus we obtained, that solvents change transition …
A self-consistent field theory was developed in the grand-canonical ensemble formulation to study transitions in a helix-coil multiblock globule. Helical and coil parts are treated as stiff rods and self-avoiding walks of variable lengths correspondingly. The resulting field-theory takes, in addition to the conventional Zimm-Bragg (B.H. Zimm, I.K. Bragg, J. Chem. Phys. 31, 526 (1959)) parameters, also three-dimensional interaction terms into account. The appropriate differential equations which determine the self-consistent fields were solved numerically with finite element method. Three diffe…
We explore the coupling between the helix-coil and coil-globule transitions of a helical polymer using Monte Carlo simulations. A very rich state diagram is found. Each state is characterized by a specific configuration of the chain which could be a helix, a random coil, an amorphous globule or one of various other globular states which carry residual helical strands. We study the boundaries between states and provide further insight into the physics of the system with a detailed analysis of the order parameter and other properties.
We determine the probability that a partially melted heteropolymer at the melting temperature will either melt completely or return to a helix state. This system is equivalent to the splitting probability for a diffusing particle on a finite interval that moves according to the Sinai model. When the initial fraction of melted polymer is f, the melting probability fluctuates between different realizations of monomer sequences on the polymer. For a fixed value of f, the melting probability distribution changes from unimodal to a bimodal as the strength of the disorder is increased.
A microscopic spin model is proposed for the phenomenological Zimm-Bragg model for the helix-coil transition in biopolymers. This model is shown to provide the same thermophysical properties of the original Zimm-Bragg model and it allows a very convenient framework to compute statistical quantities. Physical origins of this spin model are made transparent by an exact mapping into a one-dimensional Ising model with an external field. However, the dependence on temperature of the reduced external field turns out to differ from the standard one-dimensional Ising model and hence it gives rise to d…
I propose a variation of the standard worm--like chain model to account for internal order parameter (helix/coil) fields on the polymer chain. This internal order parameter field influences polymer conformational statistics by locally modifying the persistence length of the chain. Using this model, I make predictions for the bending and stretching response of an alpha-helical domain of a protein. In particular, I show that alpha-helical protein domains will buckle under applied torque. This highly nonlinear elastic behavior may be important in the understanding of allosteric control of biochem…
A careful analysis of measurements of circular dichroism of poly(L-glutamic acid) (PGA) shows that the data can be very accurately described by introducing a third state for the PGA configuration, in addition to the helix and coil ones, and considering a simple equilibrium between these three states, without cooperativity. The third state is more conspicuous when high molecular weight polyethyleneglycol (PEG) is added. Excluded volume effects shown by differences in presence of short and long PEG chains indicate a direct interaction of PEG and PGA rather than an osmotic effect.
The theory of transition between $α$-helix, $β$-sheet and random coil conformation of a protein is discussed through a simple model, that includes both short and long-range interactions. Besides the bonding parameter and helical initiation factor in Zimm-Bragg model, three new parameters are introduced to describe beta structure: the local constraint factor for a single residue to be contained in a $β$-strand, the long-range bonding parameter that accounts for the interaction between a pair of bonded $β$-strands, and a correction factor for the initiation of a $β$-sheet. Either increasing loca…
The quasi-equilibrium evolution of the helical fraction occurring in a biopolymer network (gelatin gel) under an applied stress has been investigated by observing modulation in its optical activity. Its variation with the imposed chain extension is distinctly non-monotonic and corresponds to the transition of initially coiled strands to induced left-handed helices. The experimental results are in qualitative agreement with theoretical predictions of helices induced on chain extension. This new effect of mechanically stimulated helix-coil transition has been studied further as a function of the…
We study the stress response to a step strain of covalently bonded gelatin gels in the temperature range where triple helix reversible crosslink formation is prohibited. We observe slow stress relaxation towards a $T$-dependent finite asymptotic level. We show that this is assignable to the strain-induced coil $\rightarrow$ helix transition, previously evidenced by S. Courty, J.L. Gornall and E.M. Terentjev (PNAS, {\bf 102}, 13453 (2005)), of a fraction of the polymer strands. Relaxation proceeds, in a first stage, according to a stretched exponential dynamics, then crosses over to a terminal …
We report circular dichroism measurements on the helix-coil transition of poly(L-glutamic acid) in solution with polyethylene glycol (PEG) as a crowding agent. Using small angle neutron scattering, PEG solutions have been characterized and found to be well described by the picture of a transient network of mesh size $ξ$, usual for semi-diluted chains in good solvent. We show that the increase of PEG concentration stabilizes the helices and increases the transition temperature. But more unexpectedly we also notice that the increase of crowding agent concentration reduces the mean helix extent a…
We analyze the problem of the helix-coil transition in explicit solvents analytically by using spin-based models incorporating two different mechanisms of solvent action: explicit solvent action through the formation of solvent-polymer hydrogen bonds that can compete with the intrinsic intra-polymer hydrogen bonded configurations (competing interactions) and implicit solvent action, where the solvent-polymer interactions tune biopolymer configurations by changing the activity of the solvent (non-competing interactions). The overall spin Hamiltonian is comprised of three terms: the background \…
The conformations of semiflexible (bio)polymers are studied in flow through geometrically structured microchannels. Using mesoscale hydrodynamics simulations, we show that the polymer undergoes a rod-to-helix transition as it moves from the narrow high-velocity region into the wide low-velocity region of the channel. The transient helix formation is the result of a non-equilibrium and non-stationary buckling transition of the semiflexible polymer, which is subjected to a compressive force originating from the fluid-velocity variation in the channel. The helix properties depend on the diameter …
We study the kinetics of helix-coil transition in water solutions of gelatin (collagen protein) by optical rotation techniques combined with thermal characterization. By examining the rates of secondary helix folding, and covering a very wide range of solution concentrations, we are able to identify a universal exponential dependence of folding rate on concentration and quench temperature. We demonstrate a new concentration-temperature superposition of data at all temperatures and concentrations, and build the corresponding master curve. The results support the concept of a diffuse helix-coil …
The force laws governing the extension behavior of homopolypeptides are obtained from a phenomenological free energy capable of describing the helix-coil transition. Just above the melting temperature of the free chains, T*, the plot of force, f, vs. end-to-end distance, R, exhibits two plateaus associated with coexistence of helical and coil domains. The lower plateau is due to tension induced onset of helix-coil transition. The higher plateau corresponds to the melting of the helices by overextension. Just below T* the f-R plot exhibits only the upper plateau. The f-R plots, the helical frac…
A self-consistent-field theory (SCFT) in the grand canonical ensemble formulation is used to study transitions in a helix-coil multiblock copolymer globule. The helices are modeled as stiff rods. In addition to the established coil-globule transition we show for the first time that, even without explicit rod-rod alignment interaction, the system undergoes a transition to a nematic liquid-crystalline (LC) globular state. The LC-globule formation is driven by the hydrophobic helical segment attraction and the anisotropy of the globule surface energy. The full phase diagram of the copolymer was c…
We study the nonequilibrium kinetics during the coil-globule transition of a flexible polymer chain with active beads after a quench from good to poor solvent condition using molecular dynamics simulation. Activity for each bead is introduced via the well-known Vicsek-like alignment rule due to which the velocity of a bead tries to align towards the average direction of its neighbors. We investigate the role of quenching temperature with varying activity during collapse of this polymer. We find that although for lower activities the kinetics remains qualitatively similar for different temperat…
By means of continuous space Monte Carlo simulation we study conformational structures formed by star and comb heteropolymers during kinetics of folding from the coil to the globule, as well as the corresponding equilibrium states on going from the good to the poor solution. Particular examples of combs with hydrophobic backbone and hydrophilic side-groups (and vice versa), as well as stars with flexible and semi-stiff arms are studied. It is interesting to note that star-like conformations naturally appear for a comb polymer with a strongly hydrophobic backbone. We emphasise the crucial diffe…
We elucidate the structural transitions in a helical off-lattice homopolymer induced by crowding agents, as function of the number of monomers ($N$) and volume fraction ($ϕ_c$) of crowding particles. At $ϕ_c = 0$, the homopolymer undergoes transitions from a random coil to a helix, helical hairpin \textbf{HH}, and helix bundle \textbf{HB} structures depending on $N$, and temperature. Crowding induces chain compaction that can promote \textbf{HH} or \textbf{HB} formation depending on $ϕ_c$. Typically, the helical content decreases which is reflected in the decrease in the transition temperature…
We present a novel Monte Carlo simulation of protein folding, in which all heavy atoms are represented as interacting hard spheres. This model includes all degrees of freedom relevant to folding - all sidechain and backbone torsions - and uses a Go potential. In this study, we focus on the 46 residue alpha-beta protein crambin and two of its structural components, the helix and helix hairpin. For a wide range of temperatures, we have recorded multiple folding events of these three structures from random coils to native conformations that differ by less than 1 A dRMS from their crystal structur…
Recent experiments on unzipping of RNA helix-loop structures by force have shown that about 40-base molecules can undergo kinetic transitions between two well-defined `open' and `closed' states, on a timescale = 1 sec [Liphardt et al., Science 297, 733-737 (2001)]. Using a simple dynamical model, we show that these phenomena result from the slow kinetics of crossing large free energy barriers which separate the open and closed conformations. The dependence of barriers on sequence along the helix, and on the size of the loop(s) is analyzed. Some DNAs and RNAs sequences that could show dynamics …
In this paper we extend the Gaussian self-consistent method to permit study of the equilibrium and kinetics of conformational transitions for heteropolymers with any given primary sequence. The kinetic equations earlier derived by us are transformed to a form containing only the mean squared distances between pairs of monomers. These equations are further expressed in terms of instantaneous gradients of the variational free energy. The method allowed us to study exhaustively the stability and conformational structure of some periodic and random aperiodic sequences. A typical phase diagram of a…
In this article, a novel cluster analysis algorithm was employed in the study of polymer coil to globule transition via single chain Monte Carlo simulations. The algorithm, which has been recently applied in Molecular Dynamics simulations of atomistic systems that tend to phase separate [arXiv: 1307.7366 [cond-mat.soft]], provides us with a convenient means to map out the dynamics of "pearls" formation along the backbone chain together with extracting meaningful quantitative information about their shape and size distribution. Preliminary findings tend to favour a two-stage model of collapse k…
The basic models of helix-coil transitions in biomolecules are introduced. These include phenomenological, zipper (Bragg-Zimm) models of polypeptides, loop-entropy (Poland-Scheraga) and Hamiltonian (Peyrard-Bishop) models of homogeneous DNA denaturation. The transfer integral approach to one-dimensional thermodynamics is presented in some detail, including the necessary extensions to deal with the singular integral equations arising in the case of on-site potentials with a flat top. The (non-)applicability of the theorems which prohibit phase transitions in one-dimensional systems is discussed…
The Poland-Scheraga (PS) model for the helix-coil transition of DNA considers the statistical mechanics of the thermally induced binding of two complementary strands of DNA. In this paper, we show how to modify the PS model when a torque is applied to the extremities of DNA: We propose a simple model for the energy of twisted DNA and compute the entropy of a loop, subject to angular constraints (supercoiling). The denaturation curves are shifted towards lower or higher temperatures depending on the sign of the torque, and the UV absorption peaks are softened. The properties of supercoiled DNA …
Understanding the role of self-propulsion on the conformational properties of active filamentous objects has relevance in biology. In this context, we consider a flexible bead-spring model polymer for which along with both attractive and repulsive interactions among the non-bonded monomers, activity for each bead works along its intrinsic direction of self-propulsion. We study its kinetics in the overdamped limit, following a quench from good to poor solvent condition. We observe that with low activities, though the kinetic pathways remain similar, the scaling exponent for the relaxation time …
We analyze the conformational dynamics and statistical properties of an active polymer model. The polymer is described as a freely-jointed bead-rod chain subject to stochastic active force dipoles that act on the suspending solvent where they drive long-ranged fluid flows. Using Langevin simulations of isolated chains in unconfined domains, we show how the coupling of active flows with polymer conformations leads to emergent dynamics. Systems with contractile dipoles behave similarly to passive Brownian chains with enhanced fluctuations due to dipolar flows. In systems with extensile dipoles, …
We extend to nonequilibrium processes our recent theory for the long time dynamics of flexible chain molecules. While the previous theory describes the equilibrium motions for any bond or interatomic separation in (bio)polymers by time correlation functions, the present extension of the theory enables the prediction of the nonequilibrium relaxation that occurs in processes, such as T-jump experiments, where there are sudden transitions between, for example, different equilibrium states. As a test of the theory, we consider the ``unfolding'' of pentadecane when it is transported from a constrai…
The coil-globule transition of an energy polydisperse chain, a model heteropolymer system where the number of monomer species is as large as the total number of monomers, is studied by means of computer simulations. In this study, we systematically explore the consequences of having different functional form and variance of the energy distribution on the coil-globule transition in general. In particular, considering Gaussian (G) and uniform (U) distributions, the effect of varying polydispersity index, $δ$, on the transition temperature $θ^\ast$, chain size, internal structure and spatial orga…
A statistical mechanical distance constraint model (DCM) is presented that explicitly accounts for network rigidity among constraints present within a system. Constraints are characterized by local microscopic free energy functions. Topological re-arrangements of thermally fluctuating constraints are permitted. The partition function is obtained by combining microscopic free energies of individual constraints using network rigidity as an underlying long-range mechanical interaction -- giving a quantitative explanation for the non-additivity in component entropies exhibited in molecular systems…
To what extent do general features of folding/unfolding kinetics of small globular proteins follow from their thermodynamic properties? To address this question, we investigate a new simplifed protein chain model that embodies a cooperative interplay between local conformational preferences and hydrophobic burial. The present four-helix-bundle 55mer model exhibits proteinlike calorimetric two-state cooperativity. It rationalizes native-state hydrogen exchange observations. Our analysis indicates that a coherent, self-consistent physical account of both the thermodynamic and kinetic properties …
Thermoresponsive hydrogels have been studied intensively for creating smart drug carriers and controlled drug delivery. Understanding the drug release kinetics and corresponding transport mechanisms of nanoparticles (NPs) in a thermoresponsive hydrogel network is the key to the successful design of a smart drug delivery system. We construct a mesoscopic model of rigid NPs entrapped in a hydrogel network in an aqueous solution, where the hydrogel network is formed by cross-linked semiflexible polymers of PNIPAM. By varying the environmental temperature crossing the lower critical solution tempe…
Dynamics of various biological filaments can be understood within the framework of active polymer models. Here we consider a bead-spring model for a flexible polymer chain in which the active interaction among the beads is introduced via an alignment rule adapted from the Vicsek model. Following a quench from the high-temperature coil phase to a low-temperature state point, we study the coarsening kinetics via molecular dynamics (MD) simulations using the Langevin thermostat. For the passive polymer case the low-temperature equilibrium state is a compact globule. Results from our MD simulation…
