Network structure of protein folding pathways

Proceedings of the National Academy of Sciences, 2012

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δ E , the energetic roughness Δ E , and the scale of landscape measured by the entropy S . We show that the dimensionless ratio between the gap, roughness, and entropy of the system accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated wi...

2005

Die folgende Dissertation befasst sich mit der Entwicklung und Anwendung dreier neur Ansatze fur die Erzeugung und Untersuchung von Energy Landscapes mit dem Ziel, die Proteinfaltung zu charakterisieren und besser zu verstehen. Das Paradigma der Energy Landscapes hat sich in den letzten zehn Jahren bei der Untersuchung des Faltungsverhaltens von Proteinen bewahrt. Dieses Paradigma fuhrt dazu, dass die Energy Landscape eines Proteins vereinfacht als Trichter dargestellt werden kann. Projektionen dieser Energy Landscape auf Ordnungsparameter, wie zum Beispiel die Projektion auf die freie Energie zur Untersuchung von Stabilitat, sollten eine vereinfachte Sicht der Zustande und Energiebarrieren wiedergeben, wobei oftmals versteckte Annahmen getroffen werden, die nicht erfullt sind. Basierend auf der Theorie komplexer Netzwerke wird als erstes ein neur Ansatz zur Untersuchung der Proteinzustande vorgestellt. Dabei sind die Konformationen, welche im Verlauf einer molecular dynamics (MD) Simulation angenommen werden, die Knoten und die zeitlichen Uebergange die Verbindungen. Dieses Netzwerk reprasentiert die vollstandige multi-dimensionale Energy Landscape ohne Projektion auf willkurlich gewahlte Ordnungsparameter. Die Anwendung von Netzwerken enthullt ein komplexes Zusammenspiel von Minima, Basins und Super-Basins, die in der Energy Landscape die Rolle von Attraktoren spielen. Als erstes wird die MD-Simulation eines strukturierten Peptides (Beta3s) untersucht. Dabei werden Basins mit kleiner Entropie/ kleiner Enthalpie, sowie Basins mit grosser Entropie/grosser Enthalpie gefunden. Hierbei ist interessant, dass Beta3s im gefalteten Zustand (im Unterschied zu einer kinetischen Trap) nicht nur enthalpisch, sondern auch entropisch stabilisiert wird. Free Energy Basins entsprechen Subgraphen (Communities) des gesamten Netzwerks, wobei die Knoten in einen Basin dicht miteinander verbunden sind. Das Finden dieser Communities und die Validierung einer Unterteilung des gesamten Netzwerkes in solche ist trotz der Fulle von existierenden Algorithmen noch nicht zufriedenstllend gelost. In dieser Dissertation wird dazu ein Mass, die Goodness Deviation, vorgeschlagen. Zweitens ist die Kinetik der Proteinfaltung ebenfalls eng mit der Topographie von Energy Landscapes verknupft. Die schwer fassbare Natur des Transition State Ensemble (TSE) erlaubt kein einfaches Bild fur Faltunsubergange. Zur Charakterisierung des TSE wird oft die Faltungswahrscheinlichkeit pfold einer Proteinstruktur benutzt. Obwohl g fur die Analyse von Faltungsubergangen sehr nutzlich ist, erfordert dessen Berechnung einen grossen rechnerischen Aufwand, was die praktische Anwendungen limitiert. Die kinetische Homogenitat von strukturell ahnlichen Snapshots erlaubt jedoch einen statistischen Zugang fur die Analyse einzelner Konformationen. Die Idee des sogenannten Cluster-pfold ermoglicht es in dieser Dissertation, g fur Strukturen einer MD-simulation approximativ zu berechnen. Die Anwendung dieser Methode auf Beta3s zeigt ein breites, heterogenes TSE, sowie zwei Haupt-Pathways. Diese Ergebnisse decken sich interessanterweise mit der Netzwerk-Analyse aus vorhergehenden Untersuchungen. Der marginale Rechenaufwand fur den Cluster-pfold ermoglichte die Analyse des TSE von vielen Beta3s-Mutanten und die Berechnung von Phi-Werten. Mit herkommlichen Methoden ware dies fur unsere riesige Menge Simulationsdaten (0.65ms) unmoglich gewesen. Drittens sind in dieser Dissertation alle Simulationen mit konstanter Temperatur bei einer Temperatur T gemacht worden, welche hoher ist als die physiologische Temperatur. Es sei erwahnt, dass die Hohe der Energie-Barrieren proportional zu exp(-(DeltaE/kT) ansteigt, wobei k die Boltzmann-Konstante ist. Daher bleiben MD-Simulationen bei physiologischer Temperatur leicht in lokalen Minima stecken und ergeben somit kein korrektes Sampling der Energy Landscape innert nutzlicher Rechenzeit. In dieser Dissertation wird die Replica-exchange Methode (REM) benutzt, um die fruhen Stadium der Peptid-Aggregation eines amyloidogenetischen Peptides des Prionen-Proteins Sup35

Polymer, 2004

Two different computational methods are employed to predict protein folding nuclei from native state structures, one based on an elastic network (EN) model and the other on a constraint network model of freely rotating rods. Three sets of folding cores are predicted with these models, and their correlation against the slow exchange folding cores identified by native state hydrogen -deuterium exchange (HX) experiments is used to test each method. These three folding core predictions rely on differences in the underlying models and relative importance of global or local motions for protein unfolding/folding reactions. For non-specific residue interactions, we use the Gaussian Network Model (GNM) to identify folding cores in the limits of two classes of motions, shortly referred to as global and local. The global mode minima from GNM represent the residues with the greatest potential for coordinating collective motions and are explored as potential folding nuclei. Additionally, the fast mode peaks that have previously been labeled as the kinetically hot residues are identified as a second folding core set dependent on local interactions. Finally, a third folding core set is defined by the most stable residues in a simulated thermal denaturation procedure of the FIRST software. This method uses an all-atomic analysis of the rigidity and flexibility of protein structures, which includes specific hydrophobic, polar and charged interactions. Comparison of the three folding core sets to HX data indicate that the fast mode peak residues determined by the GNM and the rigid folding cores of FIRST provide statistically significant enhancements over random correlation. The role of specific interactions in protein folding is also investigated by contrasting the differences between these two network-based computational methods. q

Biophysical Journal, 2005

Given the three-dimensional structure of a protein, its thermodynamic properties are calculated using a recently introduced distance constraint model (DCM) within a mean-field treatment. The DCM is constructed from a free energy decomposition that partitions microscopic interactions into a variety of constraint types, i.e., covalent bonds, salt-bridges, hydrogen-bonds, and torsional-forces, each associated with an enthalpy and entropy contribution. A Gibbs ensemble of accessible microstates is defined by a set of topologically distinct mechanical frameworks generated by perturbing away from the native constraint topology. The total enthalpy of a given framework is calculated as a linear sum of enthalpy components over all constraints present. Total entropy is generally a nonadditive property of free energy decompositions. Here, we calculate total entropy as a linear sum of entropy components over a set of independent constraints determined by a graph algorithm that builds up a mechanical framework one constraint at a time, placing constraints with lower entropy before those with greater entropy. This procedure provides a natural mechanism for enthalpy-entropy compensation. A minimal DCM with five phenomenological parameters is found to capture the essential physics relating thermodynamic response to network rigidity. Moreover, two parameters are fixed by simultaneously fitting to heat capacity curves for histidine binding protein and ubiquitin at five different pH conditions. The three free parameter DCM provides a quantitative characterization of conformational flexibility consistent with thermodynamic stability. It is found that native hydrogen bond topology provides a key signature in governing molecular cooperativity and the folding-unfolding transition.

Physical Review E, 2009

A method for reconstructing the energy landscape of simple polypeptidic chains is described. We show that we can construct an equivalent representation of the energy landscape by a suitable directed graph. Its topological and dynamical features are shown to yield an effective estimate of the time scales associated with the folding and with the equilibration processes. This conclusion is drawn by comparing molecular dynamics simulations at constant temperature with the dynamics on the graph, defined by a temperature dependent Markov process. The main advantage of the graph representation is that its dynamics can be naturally renormalized by collecting nodes into "hubs", while redefining their connectivity. We show that both topological and dynamical properties are preserved by the renormalization procedure. Moreover, we obtain clear indications that the heteropolymers exhibit common topological properties, at variance with the homopolymer, whose peculiar graph structure stems from its spatial homogeneity. In order to obtain a clear distinction between a "fast folder" and a "slow folder" in the heteropolymers one has to look at kinetic features of the directed graph. We find that the average time needed to the fast folder for reaching its native configuration is two orders of magnitude smaller than its equilibration time, while for the bad folder these time scales are comparable. Accordingly, we can conclude that the strategy described in this paper can be successfully applied also to more realistic models, by studying their renormalized dynamics on the directed graph, rather than performing lengthy molecular dynamics simulations.

Proceedings of the National Academy of Sciences, 2008

Physical Review E

In this paper, a geometrical and thermodynamical analysis of the global properties of the potential energy landscape of a minimalistic model of a polypeptide is presented. The global geometry of the potential energy landscape is supposed to contain relevant information about the properties of a given sequence of amino acids, that is, to discriminate between a random heteropolymer and a protein. By considering the SH3 and PYP protein-sequences and their randomized versions it turns out that in addition to the standard signatures of the folding transition-discriminating between protein sequences of amino acids and random heteropolymer sequences-also peculiar geometric signatures of the equipotential hypersurfaces in configuration space can discriminate between proteins and random heteropolymers. Interestingly, these geometric signatures are the "shadows" of deeper topological changes that take place in correspondence with the protein folding transition. The protein folding transition takes place in systems with a small number of degrees of freedom (very far from the Avogadro number) and in the absence of a symmetry-breaking phenomenon. Nevertheless, seen from the deepest level of topology changes of equipotential submanifolds of phase space, the protein folding transition fully qualifies as a phase transition.

Physical Review Letters, 1997

In a statistical approach to protein structure analysis, Miyazawa and Jernigan (MJ) derived a 20 × 20 matrix of inter-residue contact energies between different types of amino acids. Using the method of eigenvalue decomposition, we find that the MJ matrix can be accurately reconstructed from its first two principal component vectors as Mij = C0 + C1(qi + qj) + C2qiqj , with constant C's, and 20 q values associated with the 20 amino acids. This regularity is due to hydrophobic interactions and a force of demixing, the latter obeying Hildebrand's solubility theory of simple liquids.

Proceedings of the National Academy of Sciences, 1999

Recently, the funnel shape energy landscape theory has been successfully utilized to describe the folding (1-4) and binding (5) behavior in proteins. While in general the free energy landscape is depicted by a funnel-like shape, the details of the landscape surface of a folding funnel will be affected by changes in the surrounding environment. On the basis of Freire's work in this issue of the Proceedings (6), we describe a shift in the energy landscape of a folding funnel, caused by a binding event. Shifts in energy landscapes portray shifts in the populations of the substates.

Physical Review Letters, 2007

We develop a theoretical approach to the protein folding problem based on out-of-equilibrium stochastic dynamics. Within this framework, the computational difficulties related to the existence of large time scale gaps in the protein folding problem are removed and simulating the entire reaction in atomistic details using existing computers becomes feasible. In addition, this formalism provides a natural framework to investigate the relationships between thermodynamical and kinetic aspects of the folding. For example, it is possible to show that, in order to have a large probability to remain unchanged under Langevin diffusion, the native state has to be characterized by a small conformational entropy. We discuss how to determine the most probable folding pathway, to identify configurations representative of the transition state and to compute the most probable transition time. We perform an illustrative application of these ideas, studying the conformational evolution of alanine di-peptide, within an all-atom model based on the empiric GROMOS96 force field.