Cooperativity in thermal and force-induced protein unfolding: integration of crack propagation and network elasticity models

Exploring the unfolding pathways of protein families using Elastic Network Model

We explore how a protein's native structure determines its unfolding process. We examine how the local structural features, like shear, and the global structural properties, like the number of soft modes, change during unfolding. Simulations are performed using a Gaussian Network Model (GNM) with bond breaking for both thermal and force-induced unfolding scenarios. We find that unfolding starts in areas of high shear in the native structure and progressively spreads to the low shear regions. Interestingly, analysis of single domain protein families (Chymotrypsin inhibitor and Barnase) reveal that proteins with distinct unfolding pathways exhibit divergent behavior of the number of soft modes during unfolding. This suggests that the number of soft modes might be a valuable tool for understanding thermal unfolding pathways. Additionally, we found a strong link between a protein's overall structural similarity (TM-score) and its unfolding pathways, highlighting the importance of the native structure in determining how a protein unfolds.

Appearance of de Gennes length in force-induced transitions

Using Langevin dynamic simulations, a simple coarse-grained model of a DNA protein construct is used to study the DNA rupture and the protein unfolding. We identify three distinct states: (i) zipped DNA and collapsed protein, (ii) unzipped DNA and stretched protein, and (iii) unzipped DNA and collapsed protein. Here, we find a phase diagram that shows these states depending on the size of the DNA handle and the protein. For a less stable protein, unfolding is solely governed by the size of the linker DNA, whereas if the protein's stability increases, complete unfolding becomes impossible because the rupture force for DNA has reached a saturation regime influenced by the de Gennes length. We show that unfolding occurs via a few intermediate states by monitoring the force-extension curve of the entire protein. We extend our study to a heterogeneous protein system, where similar intermediate states in two systems can lead to different protein unfolding paths.

Force-induced melting of DNA hairpin: Unfolding pathways and phase diagrams

Using the exact enumeration technique, we have studied the force-induced melting of a DNA hairpin on the face centered cubic lattice for two different sequences which differ in terms of loop closing base pairs. The melting profiles obtained from the exact enumeration technique is consistent with the Gaussian network model and Langevin dynamics simulations. Probability distribution analysis based on the exact density of states revealed the microscopic details of the opening of the hairpin. We showed the existence of intermediate states near the melting temperature. We further showed that different ensembles used to model single-molecule force spectroscopy setups may give different force-temperature diagrams. We delineate the possible reasons for the observed discrepancies.

Can one detect intermediate denaturation states of DNA sequences by following the equilibrium open-close dynamic fluctuations of a single base pair?

Melting of DNA sequences may occur through a few major intermediate states, whose influence on the melting curve has been discussed previously, while their effect on the kinetics has not been explored thoroughly. Here, we chose a simple DNA sequence, forming a hairpin in its native (zipped) state, and study it using molecular dynamic (MD) simulations and a model integrating the Gaussian network model with bond-binding energies-the Gaussian binding energy (GBE) model. We find two major partial denaturation states, a bubble state and a partial unzipping state. We demonstrate the influence of these two states on the closing-opening base pair dynamics, as probed by a tagged bond auto-correlation function (ACF). We argue that the latter is measured by fluorescence correlation spectroscopy experiments, in which one base of the pair is linked to a fluorescent dye, while the complementary base is linked to a quencher, similar to the experiment reported by Altan-Bonnet et al. [Phys. Rev. Lett. 90, 138101 (2003)]. We find that tagging certain base pairs at temperatures around the melting temperature results in a multi-step relaxation of the ACF, while tagging other base pairs leads to an effectively single-step relaxation, albeit non-exponential. Only the latter type of relaxation has been observed experimentally, and we suggest which of the other base pairs should be tagged in order to observe multi-step relaxation. We demonstrate that this behavior can be observed with other sequences and argue that the GBE can reliably predict these dynamics for very long sequences, where MD simulations might be limited.

Heat-induced conversion of multiscale molecular structure of natural food nutrients: A review

Thermal process is the most important way of treating foods. Heat energy inputted into the natural food system induces the depolymerization of multi-scale structures of matrix, and causes the intramolecular and intermolecular interactions of different nutrients. It attacks and breaks the original polymeric molecule structures and the functional properties of macronutrients such as carbohydrates, proteins and lipids. Micronutrients such as vitamins and other novel functional ingredients are also thermally converted. The heat-induced conversions of nutrients are slightly or totally with discrepancy in simple-, simulated- and real-food systems, respectively. Thus, this review aims to extensively summarize the heat-induced structural characteristics, thermal conversion pathways and pyrolysis mechanism of nutrients both in simple and complex food matrices. The structural change of each nutrient and its thermal reaction kinetics depend on the molecule structure and polymeric characteristic of the unit substances in the system.

Geometry and Flexibility of Optimal Catalysts in a Minimal Elastic Model

We have general knowledge of the principles by which catalysts accelerate the rate of chemical reactions but no precise understanding of the geometrical and physical constraints to which their design is subject. To analyze these constraints, we introduce a minimal model of catalysis based on elastic networks where the implications of the geometry and flexibility of a catalyst can be studied systematically. The model demonstrates the relevance and limitations of the principle of transition-state stabilization: optimal catalysts are found to have a geometry complementary to the transition state but a degree of flexibility that nontrivially depends on the parameters of the reaction as well as on external parameters such as the concentrations of reactants and products. The results illustrate how simple physical models can provide valuable insights into the design of catalysts.

Enhancement of protein mechanical stability: Correlated deformations are handcuffed by ligand binding

As revealed by previous experiments, protein mechanical stability can be effectively regulated by ligand binding with the binding site distant from the force-bearing region. However, the mechanism for such long-range allosteric control of protein mechanics is still largely unknown. In this work, we use protein topology-based elastic network model (ENM) and all-atomic steered molecular dynamics (SMD) simulations to study the impact of ligand binding on protein mechanical stability in two systems, i.e., GB1 and CheY-binding P2-domain of CheA (CBDCheA). Both ENM and SMD results show that the ligand binding has considerable and negligible effects on the mechanical stability of these two proteins, respectively. These results are consistent with the experimental observations. A physical mechanism for the enhancement of protein mechanical stability was then proposed: the correlated deformations of the force-bearing region and the binding site are handcuffed by the binding of ligand. The handcuff effect suppresses the propagation of internal force in the force-bearing region, thus improving the resistance to the loading force. Our study indicates that ENM method can effectively identify the structure motifs allosterically related to the deformation in the force bearing region, as well as the force propagation pathway within the structure of the studied proteins. Hence, it should be helpful to understand the molecular origin of the different mechanical properties in response to ligand binding for GB1 and CBDCheA.

Soft Vibrational Modes Predict Breaking Events during Force-Induced Protein Unfolding

We investigate the correlation between soft vibrational modes and unfolding events in simulated force spectroscopy of proteins. Unfolding trajectories are obtained from molecular dynamics simulations of a Gō model of a monomer of a mutant of superoxide dismutase 1 protein containing all heavy atoms in the protein, and a normal mode analysis is performed based on the anisotropic network model. We show that a softness map constructed from the superposition of the amplitudes of localized soft modes correlates with unfolding events at different stages of the unfolding process. Soft residues are up to eight times more likely to undergo disruption of native structure than the average amino acid. The memory of the softness map is retained for extensions of up to several nanometers, but decorrelates more rapidly during force drops.

Gaussian network model can be enhanced by combining solvent accessibility in proteins

Gaussian network model (GNM), regarded as the simplest and most representative coarse-grained model, has been widely adopted to analyze and reveal protein dynamics and functions. Designing a variation of the classical GNM, by defining a new Kirchhoff matrix, is the way to improve the residue flexibility modeling. We combined information arising from local relative solvent accessibility (RSA) between two residues into the Kirchhoff matrix of the parameter-free GNM. The undetermined parameters in the new Kirchhoff matrix were estimated by using particle swarm optimization. The usage of RSA was motivated by the fact that our previous work using RSA based linear regression model resulted out higher prediction quality of the residue flexibility when compared with the classical GNM and the parameter free GNM. Computational experiments, conducted based on one training dataset, two independent datasets and one additional small set derived by molecular dynamics simulations, demonstrated that the average correlation coefficients of the proposed RSA based parameter-free GNM, called RpfGNM, were significantly increased when compared with the parameter-free GNM. Our empirical results indicated that a variation of the classical GNMs by combining other protein structural properties is an attractive way to improve the quality of flexibility modeling.

Sufficient minimal model for DNA denaturation: Integration of harmonic scalar elasticity and bond energies

We study DNA denaturation by integrating elasticity - as described by the Gaussian network model - with bond binding energies, distinguishing between different base pairs and stacking energies. We use exact calculation, within the model, of the Helmholtz free-energy of any partial denaturation state, which implies that the entropy of all formed "bubbles" ("loops") is accounted for. Considering base pair bond removal single events, the bond designated for opening is chosen by minimizing the free-energy difference for the process, over all remaining base pair bonds. Despite of its great simplicity, for several known DNA sequences our results are in accord with available theoretical and experimental studies. Moreover, we report free-energy profiles along the denaturation pathway, which allow to detect stable or meta-stable partial denaturation states, composed of bubble, as local free-energy minima separated by barriers. Our approach allows to study very long DNA strands with commonly available computational power, as we demonstrate for a few random sequences in the range 200-800 base-pairs. For the latter, we also elucidate the self-averaging property of the system. Implications for the well known breathing dynamics of DNA are elucidated.

Identification of Hot Spots in Protein Structures Using Gaussian Network Model and Gaussian Naive Bayes

Residue fluctuations in protein structures have been shown to be highly associated with various protein functions. Gaussian network model (GNM), a simple representative coarse-grained model, was widely adopted to reveal function-related protein dynamics. We directly utilized the high frequency modes generated by GNM and further performed Gaussian Naive Bayes (GNB) to identify hot spot residues. Two coding schemes about the feature vectors were implemented with varying distance cutoffs for GNM and sliding window sizes for GNB based on tenfold cross validations: one by using only a single high mode and the other by combining multiple modes with the highest frequency. Our proposed methods outperformed the previous work that did not directly utilize the high frequency modes generated by GNM, with regard to overall performance evaluated using F1 measure. Moreover, we found that inclusion of more high frequency modes for a GNB classifier can significantly improve the sensitivity. The present study provided additional valuable insights into the relation between the hot spots and the residue fluctuations.

Statistical mechanics of the Huxley-Simmons model

The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.

The Intrinsic Dynamics and Unfolding Process of an Antibody Fab Fragment Revealed by Elastic Network Model

Antibodies have been increasingly used as pharmaceuticals in clinical treatment. Thermal stability and unfolding process are important properties that must be considered in antibody design. In this paper, the structure-encoded dynamical properties and the unfolding process of the Fab fragment of the phosphocholine-binding antibody McPC603 are investigated by use of the normal mode analysis of Gaussian network model (GNM). Firstly, the temperature factors for the residues of the protein were calculated with GNM and then compared with the experimental measurements. A good result was obtained, which provides the validity for the use of GNM to study the dynamical properties of the protein. Then, with this approach, the mean-square fluctuation (MSF) of the residues, as well as the MSF in the internal distance (MSFID) between all pairwise residues, was calculated to investigate the mobility and flexibility of the protein, respectively. It is found that the mobility and flexibility of the constant regions are higher than those of the variable regions, and the six complementarity-determining regions (CDRs) in the variable regions also exhibit relative large mobility and flexibility. The large amplitude motions of the CDRs are considered to be associated with the immune function of the antibody. In addition, the unfolding process of the protein was simulated by iterative use of the GNM. In our method, only the topology of protein native structure is taken into account, and the protein unfolding process is simulated through breaking the native contacts one by one according to the MSFID values between the residues. It is found that the flexible regions tend to unfold earlier. The sequence of the unfolding events obtained by our method is consistent with the hydrogen-deuterium exchange experimental results. Our studies imply that the unfolding behavior of the Fab fragment of antibody McPc603 is largely determined by the intrinsic dynamics of the protein.

Multidimensional persistence in biomolecular data

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high-dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness, and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryoelectron microscopy data, and the scale dependence of nanoparticles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.

Spatial distribution of thermal energy in equilibrium

The equipartition theorem states that in equilibrium, thermal energy is equally distributed among uncoupled degrees of freedom that appear quadratically in the system's Hamiltonian. However, for spatially coupled degrees of freedom, such as interacting particles, one may speculate that the spatial distribution of thermal energy may differ from the value predicted by equipartition, possibly quite substantially in strongly inhomogeneous or disordered systems. Here we show that for systems undergoing simple Gaussian fluctuations around an equilibrium state, the spatial distribution is universally bounded from above by 1/2k(B)T. We further show that in one-dimensional systems with short-range interactions, the thermal energy is equally partitioned even for coupled degrees of freedom in the thermodynamic limit and that in higher dimensions nontrivial spatial distributions emerge. Some implications are discussed.

Protein unfolding from free-energy calculations: integration of the Gaussian network model with bond binding energies

Motivated by single molecule experiments, we study thermal unfolding pathways of four proteins, chymotrypsin inhibitor, barnase, ubiquitin, and adenylate kinase, using bond network models that combine bond energies and elasticity. The protein elasticity is described by the Gaussian network model (GNM), to which we add prescribed bond binding energies that are assigned to all (nonbackbone) connecting bonds in the GNM of native state and assumed identical for simplicity. Using exact calculation of the Helmholtz free energy for this model, we consider bond rupture single events. The bond designated for rupture is chosen by minimizing the free-energy difference for the process, over all (nonbackbone) bonds in the network. Plotting the free-energy profile along this pathway at different temperatures, we observe a few major partial unfolding, metastable or stable, states, that are separated by free-energy barriers and change role as the temperature is raised. In particular, for adenylate kinase we find three major partial unfolding states, which is consistent with single molecule FRET experiments [Pirchi et al., Nat. Commun. 2, 493 (2011)] for which hidden Markov analysis reveals between three and five such states. Such states can play a major role in enzymatic activity.

Sequence-based Gaussian network model for protein dynamics

Motivation: Gaussian network model (GNM) is widely adopted to analyze and understand protein dynamics, function and conformational changes. The existing GNM-based approaches require atomic coordinates of the corresponding protein and cannot be used when only the sequence is known.

Results: We report, first of its kind, GNM model that allows modeling using the sequence. Our linear regression-based, parameter-free, sequence-derived GNM (L-pfSeqGNM) uses contact maps predicted from the sequence and models local, in the sequence, contact neighborhoods with the linear regression. Empirical benchmarking shows relatively high correlations between the native and the predicted with L-pfSeqGNM B-factors and between the cross-correlations of residue fluctuations derived from the structure- and the sequence-based GNM models. Our results demonstrate that L-pfSeqGNM is an attractive platform to explore protein dynamics. In contrast to the highly used GNMs that require protein structures that number in thousands, our model can be used to study motions for the millions of the readily available sequences, which finds applications in modeling conformational changes, protein–protein interactions and protein functions.

Contact:zerozhua@126.com

Supplementary information:Supplementary data are available at Bioinformatics online.

Rigidity loss of protein macromolecule induced by force--effective field theory

In the framework of the effective field theory for the order parameter, which characterizes the degree of deviating the protein globule structure from its native state, the phase transition of the protein macromolecule from the elastic state into the plastic one under its mechanical stretching is considered. Elastic properties of a protein are studied as a function of the applied force, temperature, and the mean coordination number of the protein "network."