Helix-coil transition in homopolypeptides under stretching

Polymer crystallization under external flow

The general aspects of polymer crystallization under external flow, i.e., flow-induced crystallization (FIC) from fundamental theoretical background to multi-scale characterization and modeling results are presented. FIC is crucial for modern polymer processing, such as blowing, casting, and injection modeling, as two-third of daily-used polymers is crystalline, and nearly all of them need to be processed before final applications. For academics, the FIC is intrinsically far from equilibrium, where the polymer crystallization behavior is different from that in quiescent conditions. The continuous investigation of crystallization contributes to a better understanding on the general non-equilibrium ordering in condensed physics. In the current review, the general theories related to polymer nucleation under flow (FIN) were summarized first as a preliminary knowledge. Various theories and models, i.e., coil-stretch transition and entropy reduction model, are briefly presented together with the modified versions. Subsequently, the multi-step ordering process of FIC is discussed in detail, including chain extension, conformational ordering, density fluctuation, and final perfection of the polymer crystalline. These achievements for a thorough understanding of the fundamental basis of FIC benefit from the development of various hyphenated rheometer, i.e., rheo-optical spectroscopy, rheo-IR, and rheo-x-ray scattering. The selected experimental results are introduced to present efforts on elucidating the multi-step and hierarchical structure transition during FIC. Then, the multi-scale modeling methods are summarized, including micro/meso scale simulation and macroscopic continuum modeling. At last, we briefly describe our personal opinions related to the future directions of this field, aiming to ultimately establish the unified theory of FIC and promote building of the more applicable models in the polymer processing.

Tensile elasticity of a freely jointed chain with reversible hinges

Many biopolymers exhibit reversible conformational transitions within the chain, which affect their bending stiffness and their response to a stretching force. For example, double stranded DNA may have denatured "bubbles" of unzipped single strands which open and close randomly. In other polymers, the transitions may be due to the reversible attachment and detachment of ligands on ligand-receptor complexes along the backbone. Semiflexible bundles under tension formed by the reversible attachment of cross-linkers, on a coarse-grained level, exhibit similar behaviour. The simplest theoretical model which captures what the above mentioned systems have in common is a freely jointed chain (FJC) with reversible hinges. Each hinge can be open, as in the usual FJC, or closed forcing the adjacent segments to align (stretch). In this article, we analyse it in the Gibbs ensemble. Remarkably, even though the usual FJC in the thermodynamic limit exhibits ensemble equivalence, the reversible FJC exhibits ensemble inequivalence. Even though a mean field treatment suggests a continuous phase transition to a fully hinged state at a certain force, the generating function method ("necklace model") shows that there is no phase transition. However, there is a crossover between the two states with clearly different responses. In the low force (linear response) regime, the reversible FJC has higher tensile compliance than its usual counterpart. In contrast, in the strong force regime, the tensile compliance of the reversible FJC is much lower than that of the usual FJC.

Frustrating Strain-Induced Crystallization of Natural Rubber with Biaxial Stretch

The supreme mechanical performance of natural rubber (NR) is commonly attributed to strain-induced crystallization (SIC). The SIC of NR during uniaxial stretch has been extensively investigated, whereas that under multiaxial deformation has been rarely reported, which is close to real service conditions (i.e., tire). In this work, the crystallization behavior of NR under biaxial stretch was studied with in situ synchrotron radiation wide-angle X-ray diffraction in combination with a custom-built biaxial stretch machine. It is observed that biaxial stretch frustrates the SIC of NR: within λ_x/λ_y < 1.6, where λ_x and λ_y are stretch ratios of two mutually perpendicular axes, no crystallization emerges even under large drawing ratio until sample fracture at ambient temperature. This finding challenges the common wisdom of the self-reinforcement mechanism of SIC in NR under multiaxial deformation in real service conditions. A theoretical SIC model is proposed, which can decouple the contributions of conformational entropy reduction ΔS_f and amorphous chain orientation f to final Gibbs free energy change (ΔG) during multiaxial deformation. This model quantitatively renders a reproduction of the crystallinity during the biaxial stretch, which is well consistent with experimental results and can be further generalized for flow-induced crystallization of semicrystalline polymers.

Soft-Mechanochemistry: Mechanochemistry Inspired by Nature

Cells and bacteria use mechanotransduction processes to transform a mechanical force into a chemical/biochemical response. The area of chemistry where chemical reactions are induced by mechanical forces is called mechanochemistry. Over the last few years, chemists developed force-induced reactions affecting covalent bonds in molecules under tension which requires high energy input and/or high intensity forces. In contrast, in nature, mechanotransduction processes take place with forces of much weaker intensity and much less demanding energy. They are mainly based on protein conformational changes or changes in supramacromolecular architectures. Mechanochemistry based on such low-energy-demanding processes and which does not affect chemical bonds can be called soft-mechanochemistry. In this feature article, we first discuss some examples of soft-mechanochemistry processes encountered in nature, in particular, cryptic sites, allowing us to define more precisely the concepts underlying soft-mechanochemistry. A series of examples, developed over the past few years, of chemomechanoresponsive systems based on soft-mechanochemistry principles are given. We describe, in particular, cryptic site surfaces, enzymatically active films whose activity can be modulated by stretching and films where stretching induces changes in their fluorescence properties. Finally, we give our view of the future of soft-mechanochemistry.

Stretch-Induced Helical Conformations in Poly(l-lysine)/Hyaluronic Acid Multilayers

We investigate the effect of stretching on the secondary structure of cross-linked poly(l-lysine)/hyaluronic acid (PLL/HA) multilayers. We show that stretching these films induces changes in the secondary structure of PLL chains. Our results suggest that not only α- but also 310-helices might form in the film under stretching. Such 310-helices have never been observed for PLL so far. These changes of the secondary structure of PLL are reversible, i.e., when returning to the nonstretched state one recovers the initial film structure. Using molecular dynamics simulations of chains composed of 20 l-lysine residues (PLL20), we find that these chains never adopt a helical conformation in water. In contrast, when the end-to-end distance of the chains is restrained to values smaller than the mean end-to-end distance of free chains, a distance domain rarely explored by the free chains, helical conformations become accessible. Moreover, the formation of not only α- but also 310-helices is predicted by the simulations. These results suggest that the change of the end-to-end distance of PLL chains in the stretched film is at the origin of the helix formation.

Molecular basis of the mechanical hierarchy in myomesin dimers for sarcomere integrity

Myomesin is one of the most important structural molecules constructing the M-band in the force-generating unit of striated muscle, and a critical structural maintainer of the sarcomere. Using molecular dynamics simulations, we here dissect the mechanical properties of the structurally known building blocks of myomesin, namely α-helices, immunoglobulin (Ig) domains, and the dimer interface at myomesin's 13th Ig domain, covering the mechanically important C-terminal part of the molecule. We find the interdomain α-helices to be stabilized by the hydrophobic interface formed between the N-terminal half of these helices and adjacent Ig domains, and, interestingly, to show a rapid unfolding and refolding equilibrium especially under low axial forces up to ∼ 15 pN. These results support and yield atomic details for the notion of recent atomic-force microscopy experiments, namely, that the unique helices inserted between Ig domains in myomesin function as elastomers and force buffers. Our results also explain how the C-terminal dimer of two myomesin molecules is mechanically outperforming the helices and Ig domains in myomesin and elsewhere, explaining former experimental findings. This study provides a fresh view onto how myomesin integrates elastic helices, rigid immunoglobulin domains, and an extraordinarily resistant dimer into a molecular structure, to feature a mechanical hierarchy that represents a firm and yet extensible molecular anchor to guard the stability of the sarcomere.

Coil-helix transition of polypeptide at water-lipid interface

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 continuous transition with helicity in the role of order parameter. Our model is constructed as a system of statistically interacting quasiparticles that are activated from the helix pseudo-vacuum. The particles represent links between adjacent residues in coil conformation that form a self-avoiding random walk in two dimensions. Explicit results are presented for helicity, entropy, heat capacity, and the average numbers and sizes of sboth coil and helix segments.

Thermoresponsive Gelatin Nanogels

We present a novel tunable thermoresponsive gelatin nanogel that shows a volume transition at ∼32 °C. A thermally induced volume reduction of more than 30× is observed due to the helix to random coil transition of gelatin chains confined in the nanogels. The physical process and key factors influencing thermoresponsive properties are investigated using dynamic light scattering (DLS), transmission electron microscopy (TEM), and polarimetry. The thermoresponsive properties of this nanogel can be exploited in the development of new types of stimuli-responsive, biomedically relevant materials based on natural polymers.

Competition for hydrogen-bond formation in the helix-coil transition and protein folding

The problem of the helix-coil transition of biopolymers in explicit solvents, such as water, with the ability for hydrogen bonding with a 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 in temperature, but also by pressure, mechanical force, osmotic stress, or other external influences. Both polypeptides and polynucleotides are considered within a unified formalism. Our approach provides an explanation of the experimental difficulty of observing the reentrant transition with pressure and underscores the advantage of pulling experiments for studies of DNA. Results are discussed and compared with those reported in a number of recent publications with which a significant level of agreement is obtained.

A theoretical model for the mechanical unfolding of repeat proteins

We consider the mechanical stretching of a polypeptide chain formed by multiple interacting repeats. The folding thermodynamics and the interactions among the repeats are described by the Ising model. Unfolded repeats act as soft entropic springs, whereas folded repeats respond to a force as stiffer springs. We show that the resulting force-extension curve may exhibit a pronounced force maximum corresponding to the unfolding of the first repeat. This event is followed by the unfolding of the remaining repeats, which takes place at a lower force. As the protein extension is increased, the force-extension curve of a sufficiently long repeat protein displays a plateau, where the force remains nearly constant and the protein unfolds sequentially so that the number of unfolded repeats is proportional to the extension. Such a sequential mechanical unfolding mechanism is displayed even by the repeat proteins whose thermal denaturation is highly cooperative, provided that they are long enough. By contrast, the unfolding of short repeat progressions can be cooperative.

Free energies and forces in helix-coil transition of homopolypeptides under stretching

We show here that constant velocity steered molecular dynamics (SMD) simulations of alpha-helices in a vacuum present a well defined plateau in the force-extension relationship for homopolypeptides having more than (approximately) twenty residues. With the processes being far away from equilibrium, the energies strongly depend on the stretching velocity. Importantly, for a given velocity variation, the energy variation depends also on the helix sequence. Additionally, our observations show that homopolypeptides made of ten different amino acids (Ala, Cys, Gln, Ile, Leu, Met, Phe, Ser, Thr and Val) present a linear helix-coil transition.

Stretching of homopolymeric RNA reveals single-stranded helices and base-stacking

We have found strong supporting evidence for the helical structures of single-stranded nucleic acids by stretching individual molecules of polyadenylic acid [poly(A)] and polycytidylic acid [poly(C)]. Analyzing the force versus extension data using a two-state elastic model in which random-coil domains alternate with rigid helical domains allows one to extract the thermodynamic and structural properties. In addition, it also yields moderate to low cooperativity of the helix-coil transition for poly(A) and poly(C), respectively.

Globular structures of a helix-coil copolymer: Self-consistent treatment

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, [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 different phase states are found: open chain, amorphous globule, and nematic liquid-crystalline (LC) globule. The LC-globule formation is driven by the interplay between the hydrophobic helical segment attraction and the anisotropic globule surface energy of an entropic nature. The full phase diagram of the helix-coil copolymer was calculated and thoroughly discussed. The suggested theory shows a clear interplay between secondary and tertiary structures in globular homopolypeptides.

Stretching globular polymers. I. Single chains

We review the force-extension behavior of polymers collapsed in poor solvent, modified to include the effects of semiflexibility and considered for globules with "ordered" and "disordered" internal structures. A series of ordered globules is used as a model for the unbinding of a disordered globule beneath its glass transition and for multiple-repeat proteins such as the poly-Ig-domain titin used in atomic force microscopy studies. These single-chain results form the foundation for the treatment of cross-linked networks of globular polymers.

Nonlinear elasticity of an alpha-helical polypeptide: Monte Carlo studies

We report on Monte Carlo studies of the elastic properties of the helix-coil wormlike chain model of alpha-helical polypeptides. In this model the secondary structure enters as a scalar (Ising-like) variable that controls the local chain bending modulus. We characterize the nonlinear elastic properties of these molecules including their response to applied tensile forces and bending torques both individually and in combination. We find a pronounced effect of applied torque on the extensional compliance of the molecule and a similar effect of tension on the bending compliance. Finally we speculate that the strongly nonlinear response of alpha-helical polypeptides to combinations of torque and force plays a role in allosteric transitions in proteins.

Structural Transitions in the Polyalanine α-Helix under Uniaxial Strain

We analyzed the response to strain of an infinite polyalanine chain in the alpha-helical conformation using density functional theory. Under compressive strain the alpha-helix is found to undergo structural transitions to a pi-helix when the length of the helix is reduced by more than 10%. Under tensile strain the structure changes into a 3(10)-helix when the length is stretched by more than 10%. Our analysis of these transitions shows that they proceed essentially in two steps: At first there is mainly a length change, and only with some delay the helix twist adjusts.

Mechanically induced helix-coil transition in biopolymer networks

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 nonmonotonic 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 elastic properties of the polymer network: crosslink density and network aging.

Induced helicity in biopolymer networks under stress

By combining dynamic mechanical and optical measurements in probing the internal structure of a biopolymer network (gelatin gel), we studied the quasi-equilibrium evolution of helical content as a function of the applied stress. Assuming that the net optical activity is proportional to the concentration of secondary helices of collagen chains, and assuming that affine mechanical deformation, we find a nonmonotonic relationship between the helical domains and an imposed deformation. The results are in qualitative agreement with theoretical predictions of alpha-helices induced by chain end-to-end stretching, and give a consistent picture of mechanically stimulated helix-coil transition in networks of denatured polypeptides.

Nonlinear elasticity of an alpha -helical polypeptide

We study a minimal extension of the wormlike chain model to describe polypeptides having alpha-helical secondary structure. In this model the presence or absence of secondary structure enters as a scalar variable that controls the local chain bending modulus. Using this model we compute the extensional compliance of an alpha-helix under tensile stress, the bending compliance of the molecule under externally imposed torques, and the nonlinear interaction of such torques and forces on the molecule. We find that, due to coupling of the "internal" secondary structure variables to the conformational degrees of freedom of the polymer, the molecule has a highly nonlinear response to applied stress and force couples. In particular we demonstrate a sharp lengthening transition under applied force and a buckling transition under applied torque. Finally, we speculate that the inherent bistability of the molecule may underlie protein conformational change in vivo.

A coarse-grained model and associated lattice Monte Carlo simulation of the coil–helix transition of a homopolypeptide

A new coarse-grained lattice model neglecting atomic details is proposed for the coil-helix transition and a new physical parameter is suggested to characterize a helical structure. In our model, each residue is represented by eight lattice sites, and side groups are not considered explicitly. Chirality and hydrogen bonding are taken into consideration in addition to chain connectivity and the excluded volume effect. Through a dynamic Monte Carlo simulation, the physical properties of the coil-helix transition of a single homopolypeptide have been produced successfully within a short computing time on the PC. We also examined the effects of the variation of chain configurations including chain size and chain shape, etc. A spatial correlation function has been introduced in order to characterize periodicity of a helical chain in a simple way. A propagation parameter and a nucleation parameter have also been calculated, which compares favorably with the results of the Zimm-Bragg theory for the coil-helix transition.

Theory of high-force DNA stretching and overstretching

Single-molecule experiments on single- and double-stranded DNA have sparked a renewed interest in the force versus extension of polymers. The extensible freely jointed chain (FJC) model is frequently invoked to explain the observed behavior of single-stranded DNA, but this model does not satisfactorily describe recent high-force stretching data. We instead propose a model (the discrete persistent chain) that borrows features from both the FJC and the wormlike chain, and show that it resembles the data more closely. We find that most of the high-force behavior previously attributed to stretch elasticity is really a feature of the corrected entropic elasticity; the true stretch compliance of single-stranded DNA is several times smaller than that found by previous authors. Next we elaborate our model to allow coexistence of two conformational states of DNA, each with its own stretch and bend elastic constants. Our model is computationally simple and gives an excellent fit through the entire overstretching transition of nicked, double-stranded DNA. The fit gives the first value for the bend stiffness of the overstretched state. In particular, we find the effective bend stiffness for DNA in this state to be about 12 nm k(B)T, a value quite different from either the B-form or single-stranded DNA.