Mechanical unfolding and refolding of proteins: an off-lattice model study

Translocation of Hydrophobic Polyelectrolytes under Electrical Field: Molecular Dynamics Study

We studied the translocation of polyelectrolyte (PE) chains driven by an electric field through a pore by means of molecular dynamics simulations of a coarse-grained HP model mimicking high salt conditions. Charged monomers were considered as polar (P) and neutral monomers as hydrophobic (H). We considered PE sequences that had equally spaced charges along the hydrophobic backbone. Hydrophobic PEs were in the globular form in which H-type and P-type monomers were partially segregated and they unfolded in order to translocate through the narrow channel under the electric field. We provided a quantitative comprehensive study of the interplay between translocation through a realistic pore and globule unraveling. By means of molecular dynamics simulations, incorporating realistic force fields inside the channel, we investigated the translocation dynamics of PEs at various solvent conditions. Starting from the captured conformations, we obtained distributions of waiting times and drift times at various solvent conditions. The shortest translocation time was observed for the slightly poor solvent. The minimum was rather shallow, and the translocation time was almost constant for medium hydrophobicity. The dynamics were controlled not only by the friction of the channel, but also by the internal friction related to the uncoiling of the heterogeneous globule. The latter can be rationalized by slow monomer relaxation in the dense phase. The results were compared with those from a simplified Fokker-Planck equation for the position of the head monomer.

Quantitative analysis of amyloid-integrated biofilms formed by uropathogenic Escherichia coli at the air-liquid interface

Bacterial biofilms are complex multicellular assemblies, characterized by a heterogeneous extracellular polymeric matrix, that have emerged as hallmarks of persistent infectious diseases. New approaches and quantitative data are needed to elucidate the composition and architecture of biofilms, and such data need to be correlated with mechanical and physicochemical properties that relate to function. We performed a panel of interfacial rheological measurements during biofilm formation at the air-liquid interface by the Escherichia coli strain UTI89, which is noted for its importance in studies of urinary tract infection and for its assembly of functional amyloid fibers termed curli. Brewster-angle microscopy and measurements of the surface elasticity (G(s)') and stress-strain response provided sensitive and quantitative parameters that revealed distinct stages during bacterial colonization, aggregation, and eventual formation of a pellicle at the air-liquid interface. Pellicles that formed under conditions that upregulate curli production exhibited an increase in strength and viscoelastic properties as well as a greater ability to recover from stress-strain perturbation. The results suggest that curli, as hydrophobic extracellular amyloid fibers, enhance the strength, viscoelasticity, and resistance to strain of E. coli biofilms formed at the air-liquid interface.

Evolution of the potential energy landscape with static pulling force for two model proteins

The energy landscape is analyzed for off-lattice bead models of protein L and protein G as a function of a static pulling force. Two different pairs of attachment points (pulling directions) are compared in each case, namely, residues 1/56 and 10/32. For the terminal residue pulling direction 1/56, the distinct global minimum structures are all extended, aside from the compact geometry that correlates with zero force. The helical turns finally disappear at the highest pulling forces considered. For the 10/32 pulling direction, the changes are more complicated, with a variety of competing arrangements for beads outside the region where the force is directly applied. These alternatives produce frustrated energy landscapes, with low-lying minima separated by high barriers. The calculated folding pathways in the absence of force are in good agreement with previous work. The N-terminal hairpin folds first for protein L and the C-terminal hairpin for protein G, which exhibits an intermediate. However, for a relatively low static force, where the global minimum retains its structure, the folding mechanisms change, sometimes dramatically, depending on the protein and the attachment points. The scaling relations predicted by catastrophe theory are found to hold in the limit of short path lengths.

Discrete breathers in a realistic coarse-grained model of proteins

We report the results of molecular dynamics simulations of an off-lattice protein model featuring a physical force-field and amino-acid sequence. We show that localized modes of nonlinear origin, discrete breathers (DBs), emerge naturally as continuations of a subset of high-frequency normal modes residing at specific sites dictated by the native fold. DBs are time-periodic, space-localized vibrational modes that exist generically in nonlinear discrete systems and are known for their resilience and ability to concentrate energy for long times. In the case of the small β-barrel structure that we consider, DB-mediated localization occurs on the turns connecting the strands. At high energies, DBs stabilize the structure by concentrating energy on a few sites, while their collapse marks the onset of large-amplitude fluctuations of the protein. Furthermore, we show how breathers develop as energy-accumulating centres following perturbations even at distant locations, thus mediating efficient and irreversible energy transfers. Remarkably, due to the presence of angular potentials, the breather induces a local static distortion of the native fold. Altogether, the combination of these two nonlinear effects may provide a ready means for remotely controlling local conformational changes in proteins.

Monte Carlo simulation of mechanical unfolding of proteins based on a simple two-state model

Single molecule methods are becoming routine biophysical techniques for studying biological macromolecules. In mechanical unfolding of proteins, an externally applied force is used to induce the unfolding of individual protein molecules. Such experiments have revealed novel information that has significantly enhanced our understanding of the function and folding mechanisms of several types of proteins. To obtain information on the unfolding kinetics and the free energy landscape of the protein molecule from mechanical unfolding data, a Monte Carlo simulation based on a simple two-state kinetic model is often used. In this paper, we provide a detailed description of the procedure to perform such simulations and discuss the approximations and assumptions involved. We show that the appearance of the force versus extension curves from mechanical unfolding of proteins is affected by a variety of experimental parameters, such as the length of the protein polymer and the force constant of the cantilever. We also analyze the errors associated with different methods of data pooling and present a quantitative measure of how well the simulation results fit experimental data. These findings will be helpful in experimental design, artifact identification, and data analysis for single molecule studies of various proteins using the mechanical unfolding method.

Unfolding times for proteins in a force clamp

The escape process from the native valley for proteins subjected to a constant stretching force is examined using a model for a beta barrel. For a wide range of forces, the unfolding dynamics can be treated as one-dimensional diffusion, parametrized in terms of the end-to-end distance. In particular, the escape times can be evaluated as first passage times for a Brownian particle moving on the protein free-energy landscape, using the Smoluchowski equation. At strong forces, the unfolding process can be viewed as a diffusive drift away from the native state, while at weak forces thermal activation is the relevant mechanism. An escape-time analysis within this approach reveals a crossover from an exponential to an inverse Gaussian escape-time distribution upon passing from weak to strong forces. Moreover, a single expression valid at weak and strong forces can be devised both for the average unfolding time as well as for the corresponding variance. The analysis offers a possible explanation of recent experimental findings for the proteins ddFLN4 and ubiquitin.

Free-energy landscape of mechanically unfolded model proteins: extended Jarzinsky versus inherent structure reconstruction

The equilibrium free-energy landscape of off-lattice model heteropolymers as a function of an internal coordinate, namely the end-to-end distance, is reconstructed from out-of-equilibrium steered molecular dynamics data. This task is accomplished via two independent methods: By employing an extended version of the Jarzynski equality and the inherent structure formalism. A comparison of the free energies estimated with these two schemes with equilibrium results obtained via the umbrella sampling technique reveals a good quantitative agreement among all the approaches in a range of temperatures around the "folding transition" for the two examined sequences. In particular, for the sequence with good foldability properties, the mechanically induced structural transitions can be related to thermodynamical aspects of folding. Moreover, for the same sequence the knowledge of the landscape profile allows for a good estimation of the lifetimes of the native configuration for temperatures ranging from the folding to the collapse temperature. For the random sequence, mechanical and thermal unfolding appear to follow different paths along the landscape.

Reconstructing the free-energy landscape of a mechanically unfolded model protein

The equilibrium free-energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the "folding transition" we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce equilibrium properties of the examined system. Moreover, for the studied model the structural transitions induced by pulling can be related to thermodynamical aspects of folding.

Energy landscape distortions and the mechanical unfolding of proteins

Molecular simulations and an energy landscape analysis are used to examine the stretching of a model protein. A mapping of the energy landscape shows that stretching the protein causes energy minima and energy barriers to flatten out and disappear, and new energy minima to be created. The implications of these landscape distortions depend on the timescale regime under which the protein is stretched. When the timescale for thermally activated processes is longer than the timescale of stretching, the disappearances of energy barriers provide the mechanism for protein unfolding. When the timescale for thermally activated processes is shorter than the timescale of stretching, the landscape distortions influence the stretching process by changing the number and types of energy minima in which the system can exist.

Spectral density analysis of nonlinear hysteretic systems

A method for the analysis of spectral densities of hysteretic nonlinearities driven by diffusion processes is presented. This method is based on the Preisach formalism for the description of hysteresis and the mathematical machinery of diffusion processes on graphs. The calculations are appreciably simplified by the introduction of the "effective" distribution function. The implementation of the method for the case of the Ornstein-Uhlenbeck input process is presented in detail, and analytical expressions for spectral noise densities for various hysteretic systems are obtained. The general qualitative features of these spectral densities are examined and their dependence on various parameters is discussed. Because of the universality of the Preisach model, this approach can be used to compute spectra in hysteresis nonlinearities of various physical origins.

Comment on “Comment on ‘Constant temperature molecular dynamics simulations by means of a stochastic collision model. II. The harmonic oscillator’ [J. Chem. Phys. , 3732 (1996)]” [J. Chem. Phys. , 1646 (1997)]

An argument brought forward by Sholl and Fichthorn against the stochastic collision-based constant temperature algorithm for molecular dynamics simulations developed by Kast et al. is refuted. It is demonstrated that the large temperature fluctuations noted by Sholl and Fichthorn are due to improperly chosen initial conditions within their formulation of the algorithm. With the original form or by suitable initialization of their variant no deficient behavior is observed.

Pulling geometry defines the mechanical resistance of a beta-sheet protein

Proteins show diverse responses when placed under mechanical stress. The molecular origins of their differing mechanical resistance are still unclear, although the orientation of secondary structural elements relative to the applied force vector is thought to have an important function. Here, by using a method of protein immobilization that allows force to be applied to the same all-beta protein, E2lip3, in two different directions, we show that the energy landscape for mechanical unfolding is markedly anisotropic. These results, in combination with molecular dynamics (MD) simulations, reveal that the unfolding pathway depends on the pulling geometry and is associated with unfolding forces that differ by an order of magnitude. Thus, the mechanical resistance of a protein is not dictated solely by amino acid sequence, topology or unfolding rate constant, but depends critically on the direction of the applied extension.