Extension of rod-coil multiblock copolymers and the effect of the helix-coil transition

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.

Energetics and structure of alanine-rich α-helices via adaptive steered molecular dynamics

The energetics and hydrogen bonding profiles of the helix-to-coil transition were found to be an additive property and to increase linearly with chain length, respectively, in alanine-rich α-helical peptides. A model system of polyalanine repeats was used to establish this hypothesis for the energetic trends and hydrogen bonding profiles. Numerical measurements of a synthesized polypeptide Ac-Y(AEAAKA)kF-NH2 and a natural α-helical peptide a2N (1-17) provide evidence of the hypothesis's generality. Adaptive steered molecular dynamics was employed to investigate the mechanical unfolding of all of these alanine-rich polypeptides. We found that the helix-to-coil transition is primarily dependent on the breaking of the intramolecular backbone hydrogen bonds and independent of specific side-chain interactions and chain length. The mechanical unfolding of the α-helical peptides results in a turnover mechanism in which a 310-helical structure forms during the unfolding, remaining at a near constant population and thereby maintaining additivity in the free energy. The intermediate partially unfolded structures exhibited polyproline II helical structure as previously seen by others. In summary, we found that the average force required to pull alanine-rich α-helical peptides in between the endpoints-namely the native structure and free coil-is nearly independent of the length or the specific primary structure.

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.

Single-molecule mechanical unfolding experiments reveal a critical length for the formation of α-helices in peptides

α-Helix is the most predominant secondary structure in proteins and supports many functions in biological machineries. The conformation of the helix is dictated by many factors such as its primary sequence, intramolecular interactions, or the effect of the close environment. Several computational studies have proposed that there is a critical maximum length for the formation of intact compact helical structures, supporting the fact that most intact α-helices in proteins are constituted of a small number of amino acids. To obtain a detailed picture on the formation of α-helices in peptides and their mechanical stability, we have synthesized a long homopolypeptide of about 90 amino acids, poly(γ-benzyl-l-glutamate), and investigated its mechanical behaviour by AFM-based single-molecule force spectroscopy. The characteristic plateaus observed in the force-extension curves reveal the unfolding of a series of small helices (from 1 to 4) of about 20 amino acid residues connected to each other, rather than a long helix of 90 residues. Our results suggest the formation of a tertiary structure made of short helices with kinks, instead of an intact compact helical structure for sequences of more than 20 amino acid residues. To our knowledge, this is the first experimental evidence supporting the concept of a helical critical length previously proposed by several computational studies.

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.

The non-equilibrium phase diagrams of flow-induced crystallization and melting of polyethylene

Combining extensional rheology with in-situ synchrotron ultrafast x-ray scattering, we studied flow-induced phase behaviors of polyethylene (PE) in a wide temperature range up to 240 °C. Non-equilibrium phase diagrams of crystallization and melting under flow conditions are constructed in stress-temperature space, composing of melt, non-crystalline δ, hexagonal and orthorhombic phases. The non-crystalline δ phase is demonstrated to be either a metastable transient pre-order for crystallization or a thermodynamically stable phase. Based on the non-equilibrium phase diagrams, nearly all observations in flow-induced crystallization (FIC) of PE can be well understood. The interplay of thermodynamic stabilities and kinetic competitions of the four phases creates rich kinetic pathways for FIC and diverse final structures. The non-equilibrium flow phase diagrams provide a detailed roadmap for precisely processing of PE with designed structures and properties.

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.

An energetic model for macromolecules unfolding in stretching experiments

We propose a simple approach, based on the minimization of the total (entropic plus unfolding) energy of a two-state system, to describe the unfolding of multi-domain macromolecules (proteins, silks, polysaccharides, nanopolymers). The model is fully analytical and enlightens the role of the different energetic components regulating the unfolding evolution. As an explicit example, we compare the analytical results with a titin atomic force microscopy stretch-induced unfolding experiment showing the ability of the model to quantitatively reproduce the experimental behaviour. In the thermodynamic limit, the sawtooth force-elongation unfolding curve degenerates to a constant force unfolding plateau.

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.

Force-driven separation of short double-stranded DNA

Short double-stranded DNA is used in a variety of nanotechnological applications, and for many of them, it is important to know for which forces and which force loading rates the DNA duplex remains stable. In this work, we develop a theoretical model that describes the force-dependent dissociation rate for DNA duplexes tens of basepairs long under tension along their axes ("shear geometry"). Explicitly, we set up a three-state equilibrium model and apply the canonical transition state theory to calculate the kinetic rates for strand unpairing and the rupture-force distribution as a function of the separation velocity of the end-to-end distance. Theory is in excellent agreement with actual single-molecule force spectroscopy results and even allows for the prediction of the rupture-force distribution for a given DNA duplex sequence and separation velocity. We further show that for describing double-stranded DNA separation kinetics, our model is a significant refinement of the conventionally used Bell-Evans model.

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.

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

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.

Effects of stacking on the configurations and elasticity of single-stranded nucleic acids

Stacking interactions in single-stranded nucleic acids give rise to configurations of an annealed rod-coil multiblock copolymer. Theoretical analysis identifies the following resulting signatures for long homopolynucleotides: a nonmonotonic dependence of size on temperature, the corresponding effects on cyclization and a plateau in the extension force law. Explicit numerical results for polydeoxyadenylate [poly(dA)] and polyriboadenylate [poly(rU)] are presented.

Force-induced melting of the DNA double helix 1. Thermodynamic analysis

The highly cooperative elongation of a single B-DNA molecule to almost twice its contour length upon application of a stretching force is interpreted as force-induced DNA melting. This interpretation is based on the similarity between experimental and calculated stretching profiles, when the force-dependent free energy of melting is obtained directly from the experimental force versus extension curves of double- and single-stranded DNA. The high cooperativity of the overstretching transition is consistent with a melting interpretation. The ability of nicked DNA to withstand forces greater than that at the transition midpoint is explained as a result of the one-dimensional nature of the melting transition, which leads to alternating zones of melted and unmelted DNA even substantially above the melting midpoint. We discuss the relationship between force-induced melting and the B-to-S transition suggested by other authors. The recently measured effect on T7 DNA polymerase activity of the force applied to a ssDNA template is interpreted in terms of preferential stabilization of dsDNA by weak forces approximately equal to 7 pN.

Helix-coil transition in homopolypeptides under stretching

We consider the effect of an external applied force on the alpha-helix-coil transition of a single-stranded homopolypeptide chain. An annealed scenario is assumed, where the building amino acid monomers may interconvert between random-coiled and ordered alpha-helical configurations. By exact evaluation of the partition function of the freely jointed chain with helix-coil internal degrees of freedom in the thermodynamic limit, we obtain the result that the stress-strain characteristic has an asymmetrical sigmoid shape with a prominent pseudoplateau. Because of the one-dimensional nature of this system, fluctuations dominate over the mean-field approximation, which incorrectly predicts a second-order phase transition.