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

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.

Unfolding mechanism and free energy landscape of single, stable, alpha helices at low pull speeds

Single alpha helices (SAHs) stable in isolated form are often found in motor proteins where they bridge functional domains. Understanding the mechanical response of SAHs is thus critical to understand their function. The quasi-static force-extension relation of a small number of SAHs is known from single-molecule experiments. Unknown, or still controversial, are the molecular scale details behind those observations. We show that the deformation mechanism of SAHs pulled from the termini at pull speeds approaching the quasi-static limit differs from that of typical helices found in proteins, which are stable only when interacting with other protein domains. Using molecular dynamics simulations with atomistic resolution at low pull speeds previously inaccessible to simulation, we show that SAHs start unfolding from the termini at all pull speeds we investigated. Unfolding proceeds residue-by-residue and hydrogen bond breaking is not the main event determining the barrier to unfolding. We use the molecular simulation data to test the cooperative sticky chain model. This model yields excellent fits of the force-extension curves and quantifies the distance, xE = 0.13 nm, to the transition state, the natural frequency of bond vibration, ν0 = 0.82 ns-1, and the height, V0 = 2.9 kcal mol-1, of the free energy barrier associated with the deformation of single residues. Our results demonstrate that the sticky chain model could advantageously be used to analyze experimental force-extension curves of SAHs and other biopolymers.

Bubble-raft collapse and the nonequilibrium dynamics of two-state elastica

We report on the collapse of bubble rafts under compression in a closed rectangular geometry. A bubble raft is a single layer of bubbles at the air-water interface. A collapse event occurs when bubbles submerge beneath the neighboring bubbles under compression, causing the structure of the bubble raft to go from single-layer to multilayer. We studied the collapse dynamics as a function of compression velocity. At higher compression velocity we observe a more uniform distribution of collapse events, whereas at lower compression velocities the collapse events accumulate at the system boundaries. We propose that this system can be understood in terms of a linear elastic sheet coupled to a local internal (Ising) degree of freedom. The two internal states, which represent one bubble layer versus two, couple to the elasticity of the sheet by locally changing the reference state of the material. By exploring the collapse dynamics of the bubble raft, one may address the basic nonlinear mechanics of a number of complex systems in which elastic stress is coupled to local internal variables.

Buckling transition in long α-helices

The treatment of bending and buckling of stiff biopolymer filaments by the popular worm-like chain model does not provide adequate understanding of these processes at the microscopic level. Thus, we have used the atomistic molecular-dynamic simulations and the Amber03 force field to examine the compression buckling of α-helix (AH) filaments at room temperature. It was found that the buckling instability occurs in AHs at the critical force f(c) in the range of tens of pN depending on the AH length. The decrease of the force f(c) with the contour length follows the prediction of the classic thin rod theory. At the force f(c) the helical filament undergoes the swift and irreversible transition from the smoothly bent structure to the buckled one. A sharp kink in the AH contour arises at the transition, accompanied by the disruption of the hydrogen bonds in its vicinity. The kink defect brings in an effective softening of the AH molecule at buckling. Nonbonded interactions between helical branches drive the rearrangement of a kinked AH into the ultimate buckled structure of a compact helical hairpin described earlier in the literature.

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.

Cavity approach for modeling and fitting polymer stretching

The mechanical properties of molecules are today captured by single molecule manipulation experiments, so that polymer features are tested at a nanometric scale. Yet devising mathematical models to get further insight beyond the commonly studied force-elongation relation is typically hard. Here we draw from techniques developed in the context of disordered systems to solve models for single and double-stranded DNA stretching in the limit of a long polymeric chain. Since we directly derive the marginals for the molecule local orientation, our approach allows us to readily calculate the experimental elongation as well as other observables at wish. As an example, we evaluate the correlation length as a function of the stretching force. Furthermore, we are able to fit successfully our solution to real experimental data. Although the model is admittedly phenomenological, our findings are very sound. For single-stranded DNA our solution yields the correct (monomer) scale and yet, more importantly, the right persistence length of the molecule. In the double-stranded case, our model reproduces the well-known overstretching transition and correctly captures the ratio between native DNA and overstretched DNA. Also in this case the model yields a persistence length in good agreement with consensus, and it gives interesting insights into the bending stiffness of the native and overstretched molecule, respectively.

Calmodulin readily switches conformation upon protonating high pKa acidic residues

We investigate protonation as a possible route for triggering conformational change in proteins by focusing on the calmodulin (CaM) example. Two hundred nanosecond molecular dynamics (MD) simulations are performed on both the extended and compact forms of calcium loaded CaM. The stability of both structures is confirmed under prevailing conditions. Protonation of nine acidic residues with upshifted pK(a) values leads to a large conformational change in less than 100 ns. The structure attained is consistent with fluorescence resonance energy transfer experimental results as well as structures from an ensemble compatible with NMR data. Analysis of the MD trajectories summing up to one microsecond implies that the key events leading to the completion of the conformational change begins with an initial formation of a salt bridge between the N-lobe and the linker, followed by the bending of the C-lobe and the organization of a stabilizing hydrophobic patch between the lobes. We find that CaM utilizes its Ca(2+) ions to harden/soften different regions so as to achieve various conformations. Thus, barrier crossing between extended and compact forms of CaM which is normally a rare event due to the repulsive electrostatic interactions between the two lobes is facilitated by protonation of high pK(a) residues. The results delineate how pH changes might be utilized in the cell to achieve different conformation-related functions.

A rigidity-enhanced antimicrobial activity: a case for linear cationic α-helical peptide HP(2-20) and its four analogues

Linear cationic α-helical antimicrobial peptides are referred to as one of the most likely substitutes for common antibiotics, due to their relatively simple structures (≤ 40 residues) and various antimicrobial activities against a wide range of pathogens. Of those, HP(2-20) was isolated from Helicobacter pylori ribosomal protein. To reveal a mechanical determinant that may mediate the antimicrobial activities, we examined the mechanical properties and structural stabilities of HP(2-20) and its four analogues of same chain length by steered molecular dynamics simulation. The results indicated the following: the resistance of H-bonds to the tensile extension mediated the early extensive stage; with the loss of H-bonds, the tensile force was dispensed to prompt the conformational phase transition; and Young's moduli (N/m(2)) of the peptides were about 4 ∼ 8 × 10(9). These mechanical features were sensitive to the variation of the residue compositions. Furthermore, we found that the antimicrobial activity is rigidity-enhanced, that is, a harder peptide has stronger antimicrobial activity. It suggests that the molecular spring constant may be used to seek a new structure-activity relationship for different α-helical peptide groups. This exciting result was reasonably explained by a possible mechanical mechanism that regulates both the membrane pore formation and the peptide insertion.

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.

Tensile mechanics of alanine-based helical polypeptide: force spectroscopy versus computer simulations

In nature, an alpha-helix is commonly used to build thermodynamically stable and mechanically rigid protein conformations. In view of growing interest in the mechanical rigidity of proteins, we measured the tensile profile of an alanine-based alpha-helical polypeptide on an atomic-force microscope to investigate the basic mechanics of helix extension with minimal interference from side-chain interactions. The peptide was extended to its maximum contour length with much less force than in reported cases of poly-L-Glu or poly-L-Lys, indicating that chain stiffness strongly depended on the physicochemical properties of side chains, such as their bulkiness. The low tensile-force extension originated presumably in locally unfolded parts because of spontaneous structural fluctuations. In 50% trifluoroethanol, the well-known helix-promoting agent, the rigidity of the sample polypeptide was markedly increased. Computer simulations of the peptide-stretching process showed that a majority of constituent residues underwent a transition from an alpha-helical to an extended conformation by overcoming an energy barrier around psi approximately 0 degrees on the Ramachandran plot. The observed lability of an isolated helix signified the biological importance of the lateral bundling of helices to maintain a rigid protein structure.