Fast chain contraction during protein folding: "foldability" and collapse dynamics

Theory indicates that at least some proteins will undergo a rapid and unimpeded collapse, like a disorganized hydrophobic chain, prior to folding. Yet experiments continue to find signs of an organized, or barrier-limited, collapse in even the fastest (approximately mus) folding proteins. Does the kinetic barrier represent a signature of the equilibrium "foldability" of these molecules? We have measured the rate of chain contraction in two nonfolding analogs of a very fast-collapsing protein. We find that these chains contract on the same time scale (approximately 10(-5)s) as the natural protein, and both pass over an energetic barrier at least as large as that encountered by the protein. The equilibrium foldability of the native structure therefore does not alone determine the dynamics of collapse; even the disordered chains contract approximately 1000x slower than expected for an ideal chain.     

Critical Casimir effects in 2D Ising model with curved defect lines

This work is aimed at studying the influence of critical Casimir effects on energetic properties of curved defect lines in the frame of 2D Ising model. Two types of defect curves were investigated. We start with a simple task of globule formation from four-defect line. It was proved that an exothermic reaction of collapse occurs and the dependence of energy release on temperature was observed. Critical Casimir energy of extensive line of constant curvature was also examined. It was shown that its critical Casimir energy is proportional to curvature that leads to the tendency to radius decreasing under Casimir forces. The results obtained can be applied to proteins folding problem in polarized liquid.     

A new approach for protein design based on the relative entropy

A new effective and fast minimization approach completely based on the physical theory is proposed for protein design. The sequence space is essentially searched according to the Boltzmann distribution. In this approach, the relative entropy is used as a minimization object function. The method has been tested on an off-lattice model of proteins and the results are better than those obtained from other similar work. Therefore, it can be applied as a uniform frame for both folding and inverse folding of proteins.     

A network of conformational transitions in an unfolding process of HP-35 revealed by high-temperature MD simulation and a Markov state model

An understanding of protein folding/unfolding processes has important implications for all biological processes, including protein degradation, protein translocation, aging, and diseases. All-atom molecular dynamics(MD) simulations are uniquely suitable for it because of their atomic level resolution and accuracy. However, limited by computational capabilities, nowadays even for small and fast-folding proteins, all-atom MD simulations of protein folding still presents a great challenge. An alternative way is to study unfolding process using MD simulations at high temperature. High temperature provides more energy to overcome energetic barriers to unfolding, and information obtained from studying unfolding can shed light on the mechanism of folding. In the present study, a 1000-ns MD simulation at high temperature(500 K)was performed to investigate the unfolding process of a small protein, chicken villin headpiece(HP-35). To infer the folding mechanism, a Markov state model was also built from our simulation, which maps out six macrostates during the folding/unfolding process as well as critical transitions between them, revealing the folding mechanism unambiguously.     

Conformations of proteins in equilibrium

We introduce a simple theoretical approach for an equilibrium study of proteins with known native-state structures. We test our approach with results on well-studied globular proteins, chymotrypsin inhibitor (2ci2), barnase, and the alpha spectrin SH3 domain, and present evidence for a hierarchical onset of order on lowering the temperature with significant organization at the local level even at high temperatures. A further application to the folding process of HIV-1 protease shows that the model can be reliably used to identify key folding sites that are responsible for the development of drug resistance.     

A Nuclear Magnetic Resonance study of the reversible denaturation of hydrated lysozyme

The understanding of the physical processes that occur below the threshold of protein thermal denaturation is of fundamental importance. In this thermal region proteins undergo a reversible folding/unfolding process whose evolution depends upon temperature and time. When the kinetics of the folding is altered, the specific biological activity of the protein is altered as well and aggregation phenomena usually intervene. The most important role in driving these processes is played by the solvent and water is certainly the solvent par excellence. It is well known that proteins become biologically active with no less than a water monolayer covering their surface. The knowledge of the physical properties of this monolayer is of basic importance to prevent folding alterations. We present a proton Nuclear Magnetic Resonance study at very high resolution of the thermodynamic properties of lysozyme hydration water as a function of temperature and time in the thermal region of the reversible denaturation.     

Folding kinetics of two-state proteins based on the model of general random walk in native contact number space

We investigated the folding kinetics of a series of two-state proteins by using the model of general random walk in native contact number space, and derive the observed linear relationship between the logarithms of the folding rate constants and the numbers of native contacts from “kinetic viewpoint”. The protein folding speed limit and stability in this model are consistent with experimental observations.     

A study of Barstar folding events using boundary value simulations

From extensive biophysical studies of protein folding, two competing mechanisms emerged: hydrophobic collapse and the framework model. Our protein of choice is Barstar—a barnase inhibitor. The approximation algorithm we used to study Barstar folding trajectories is called SDEL—stochastic difference equation in length. Using the native structure as the final boundary value and a collection of unfolded structures as the varying initial boundary value, SDEL calculates an ensemble of least action pathways between these boundaries. The results are atomically detailed folding pathways, with as many intermediate structures as you request in the input. We generated 12 pathways, starting from a structurally wide selection of unfolded conformations. Using the protein's radius of gyration as our primary reaction coordinate, we tracked H-bonds, dihedral angles, native and non-native contacts, and energy along the folding pathways. This paper will follow our findings, with special emphasis on pinpointing hydrophobic collapse as a more appropriate mechanism for Barstar. Comparison with pathway predictions for Barstar using experimental techniques will also be discussed.     

Quantum theory on protein folding

The conformational change of biological macromolecule is investigated from the point of quantum transition. A quantum theory on protein folding is proposed. Compared with other dynamical variables such as mobile electrons, chemical bonds and stretching-bending vibrations the molecular torsion has the lowest energy and can be looked as the slow variable of the system. Simultaneously, from the multi-minima property of torsion potential the local conformational states are well defined. Following the idea that the slow variables slave the fast ones and using the nonadiabaticity operator method we deduce the Hamiltonian describing conformational change. It is shown that the influence of fast variables on the macromolecule can fully be taken into account through a phase transformation of slow variable wave function. Starting from the conformation-transition Hamiltonian the nonradiative matrix element was calculated and a general formulas for protein folding rate was deduced. The analytical form of the formula was utilized to study the temperature dependence of protein folding rate and the curious non-Arrhenius temperature relation was interpreted. By using temperature dependence data the multi-torsion correlation was studied. The decoherence time of quantum torsion state is estimated. The proposed folding rate formula gives a unifying approach for the study of a large class problems of biological conformational change.     

Diffusive model of protein folding dynamics with Kramers turnover in rate

We study the folding kinetics of a three-helix bundle protein using a coarse polymer model. The folding dynamics can be accurately represented by one-dimensional diffusion along a reaction coordinate selected to capture the transition state. By varying the solvent friction, we show that position-dependent diffusion coefficients are determined by microscopic transitions on a rough energy landscape. A maximum in the folding rate at intermediate friction is explained by "Kramers turnover" in these microscopic dynamics that modulates the rate via the diffusion coefficient; overall folding remains diffusive even close to zero friction. For water friction, we find that the "attempt frequency" (or "speed limit") in a Kramers model of folding is about 2 micros-1, with an activation barrier of about 2kBT, and a folding transition path duration of approximately equal to 100 ns, 2 orders of magnitude less than the folding time of approximately equal to 10 micros.     

Early stages of homopolymer collapse

Interest in the protein folding problem has motivated a wide range of theoretical and experimental studies of the kinetics of the collapse of flexible homopolymers. In this paper, a phenomenological model is proposed for the kinetics of the early stages of homopolymer collapse following a quench from temperatures above to below the straight theta temperature. In the first stage, nascent droplets of the dense phase are formed, with little effect on the configurations of the bridges that join them. The droplets then grow by accreting monomers from the bridges, thus causing the bridges to stretch. During these two stages, the overall dimensions of the chain decrease only weakly. Further growth of the droplets is accomplished by the shortening of the bridges, which causes the shrinking of the overall dimensions of the chain. The characteristic times of the three stages scale as N0, N(1/5), and N(6/5), respectively, where N is the degree of polymerization of the chain.     

Time resolved collapse of a folding protein observed with small angle x-ray scattering

High-intensity, "pink" beam from an undulator was used in conjunction with microfabricated rapid-fluid mixing devices to monitor the early events in protein folding with time resolved small angle x-ray scattering. This Letter describes recent work on the protein bovine beta-lactoglobulin where collapse from an expanded to a compact set of states was directly observed on the millisecond time scale. The role of chain collapse, one of the initial stages of protein folding, is not currently understood. The characterization of transient, compact states is vital in assessing the validity of theories and models of the folding process.     

Simple off-lattice model to study the folding and aggregation of peptides

We present a numerical study of a new protein model. This off-lattice model takes into account both the hydrogen bonds and the amino-acid interactions. It reproduces the folding of a small protein (peptide): morphological analysis of the conformations at low temperature shows two well-known substructures α-helix and β-sheet depending on the chosen sequence. The folding pathway in the scope of this model is studied through a free-energy analysis. We then study the aggregation of proteins. Proteins in the aggregate are mainly bound via hydrogen bonds. Performing a free-energy analysis we show that the addition of a peptide to such an aggregate is not favourable. We qualitatively reproduce the abnormal aggregation of proteins in prion diseases.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Conformational temperature characterizing the folding of a protein

The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.     

Thermodynamics of heat-shock response

Production of heat-shock proteins is induced when a living cell is exposed to a rise in temperature. The heat-shock response of protein DnaK synthesis in E.coli for temperature shifts T-->T+DeltaT and T-->T-DeltaT is measured as a function of the initial temperature T. We observe a reversed heat shock at low T. The magnitude of the shock increases when one increases the distance to the temperature T0 approximately 23 degrees C, thereby mimicking the nonmonotonous stability of proteins at low temperature. This suggests that stability related to hot as well as cold unfolding of proteins is directly implemented in the biological control of protein folding.     

Protein Folding under Mediation of Ordering Water： an Off-Lattice Goe-Like Model Study

Water plays an important role in the structure and function of biomolecules. Water confined at the nanoscale usually exhibits phenomena not seen in bulk water, including the ice-like ordering structure on the surfaces of many substrates. We investigate the behaviour of protein folding in which the proteins are assumed in an environment with ordering water by using of an off-lattice GO-like model, It is found that in the physiological temperature, both the folding rate and the thermodynamic stability of the protein are greatly promoted by the existence of ordering of water.     

Folding/unfolding kinetics of lattice proteins studied using a simple statistical mechanical model for protein folding, I: Dependence on native structures and amino acid sequences

The folding/unfolding kinetics of a three-dimensional lattice protein was studied using a simple statistical mechanical model for protein folding that we developed earlier. We calculated a characteristic relaxation rate for the free energy profile starting from a completely unfolded structure (or native structure) that is assumed to be associated with a folding rate (or an unfolding rate). The chevron plot of these rates as a function of the inverse temperature was obtained for four lattice proteins, namely, proteins a1, a2, b1, and b2, in order to investigate the dependency of the folding and unfolding rates on their native structures and amino acid sequences. Proteins a1 and a2 fold to the same native conformation, but their amino acid sequences differ. The same is the case for proteins b1 and b2, but their native conformation is different from that of proteins a1 and a2. However, the chevron plots of proteins a1 and a2 are very similar to each other, and those of proteins b1 and b2 differ considerably. Since the contact orders of proteins b1 and b2 are identical, the differences in their kinetics should be attributed to the amino acid sequences and consequently to the interactions between the amino acid residues. A detailed analysis revealed that long-range interactions play an important role in causing the difference in the folding rates. The chevron plots for the four proteins exhibit a chevron rollover under both strongly folding and strongly unfolding conditions. The slower relaxation time on the broad and flat free energy surfaces of the unfolding conformations is considered to be the main origin of the chevron rollover, although the free energy surfaces have features that are rather complicated to be described in detail here. Finally, in order to concretely examine the relationship between changes in the free energy profiles and the chevron plots, we illustrate some examples of single amino acid substitutions that increase the folding rate.     

Interpreting the influence of liquid temperature on cavitation collapse intensity through bubble dynamic analysis

The violent collapse of inertial bubbles generates high temperature inside and emits strong impulsive pressure. Previous tests on sonoluminescence and cavitation erosion showed that the influence of liquid temperature on these two parameters is different. In this paper, we conducted a bubble dynamic analysis to explore the mechanism of the temperature effect and account for the above difference. The results show that the increase of vapor at higher liquid temperatures changes both the external compression pressure and the internal cushion and is responsible for the variation of bubble collapse intensity. The different trends of the collapsing temperature and emitted sound pressure are caused by the energy distribution during the bubble collapse. Moreover, a series of simulations are conducted to establish the distribution map of the optimum liquid temperature where the collapse intensity is maximized. The relationship between the collapse intensity and the radial dynamics of the bubble is discussed and the reliable indicator is identified. This study provides a clear picture of how the thermodynamic process changes cavitation aggressiveness and enriches the understanding of this complex thermal-hydrodynamic phenomenon.