Distinguishing Biomolecular Pathways and Metastable States

The Journal of Chemical Physics, 2009

The rate of protein folding is governed by the transition state so that a detailed characterization of its structure is essential for understanding the folding process. In vitro experiments have provided a coarse-grained description of the folding transition state ensemble ͑TSE͒ of small proteins. Atomistic details could be obtained by molecular dynamics ͑MD͒ simulations but it is not straightforward to extract the TSE directly from the MD trajectories, even for small peptides. Here, the structures in the TSE are isolated by the cut-based free-energy profile ͑cFEP͒ using the network whose nodes and links are configurations sampled by MD and direct transitions among them, respectively. The cFEP is a barrier-preserving projection that does not require arbitrarily chosen progress variables. First, a simple two-dimensional free-energy surface is used to illustrate the successful determination of the TSE by the cFEP approach and to explain the difficulty in defining boundary conditions of the Markov state model for an entropically stabilized free-energy minimum. The cFEP is then used to extract the TSE of a ␤-sheet peptide with a complex free-energy surface containing multiple basins and an entropic region. In contrast, Markov state models with boundary conditions defined by projected variables and conventional histogram-based free-energy profiles are not able to identify the TSE of the ␤-sheet peptide.

Proceedings of the National Academy of Sciences, 2006

Minding your p's and q's has become as important to proteinfolding theorists as it is for those being instructed in the rules of etiquette. To assess the quality of structural reaction coordinates in predicting the transition-state ensemble (TSE) of protein folding, we benchmarked the accuracy of four structural reaction coordinates against the kinetic measure P fold, whose value of 0.50 defines the stochastic separatrix for a two-state folding mechanism. For two proteins that fold by a simple two-state mechanism, c-src SH3 and CI-2, the ⌽-values of the TSEs predicted by native topologybased reaction coordinates (including Q, the fraction of native contacts) are almost identical to those of the TSE based on P fold, with correlation coefficients of >0.90. For proteins with complex folding mechanisms that have especially broad, asymmetrical free energy barriers such as the designed 3-ankyrin repeating protein (3ANK) or proteins with distinct intermediates such as cyanovirin-N (CV-N), we show that the ensemble of structures with P fold ‫؍‬ 0.50 generally does not include the chemically relevant transition states. This weakness of P fold limits its usefulness in protein folding studies. For such systems, elucidating the essential features of folding mechanisms requires using multiple reaction coordinates, although the number is still rather small. At the same time, for simple folding mechanisms, there is no indication of superiority for P fold over structurally chosen and thermodynamically relevant reaction coordinates that correctly measure the degree of nativeness. energy landscape theory ͉ minimal frustration ͉ transition-state ensemble ͉ P fold ͉ intermediates

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

Biophysical journal, 2016

A complete description of the pathways and mechanisms of protein folding requires a detailed structural and energetic characterization of the conformational ensemble along the entire folding reaction coordinate. Simulations can provide this level of insight for small proteins. In contrast, with the exception of hydrogen exchange, which does not monitor folding directly, experimental studies of protein folding have not yielded such structural and energetic detail. NMR can provide residue specific atomic level structural information, but its implementation in protein folding studies using chemical or temperature perturbation is problematic. Here we present a highly detailed structural and energetic map of the entire folding landscape of the leucine-rich repeat protein, pp32 (Anp32), obtained by combining pressure-dependent site-specific (1)H-(15)N HSQC data with coarse-grained molecular dynamics simulations. The results obtained using this equilibrium approach demonstrate that the mai...

The Journal of Chemical Physics, 2013

The thermodynamics and kinetics of protein folding and protein conformational changes are governed by the underlying free energy landscape. However, the multidimensional nature of the free energy landscape makes it difficult to describe. We propose to use a weighted-graph approach to depict the free energy landscape with the nodes on the graph representing the conformational states and the edge weights reflecting the free energy barriers between the states. Our graph is constructed from a molecular dynamics trajectory and does not involve projecting the multi-dimensional free energy landscape onto a low-dimensional space defined by a few order parameters. The calculation of free energy barriers was based on transition-path theory using the MSMBuilder2 package. We compare our graph with the widely used transition disconnectivity graph (TRDG) which is constructed from the same trajectory and show that our approach gives more accurate description of the free energy landscape than the TRDG approach even though the latter can be organized into a simple tree representation. The weighted-graph is a general approach and can be used on any complex system.

Proceedings of the National Academy of Sciences, 2008

arXiv: Biological Physics, 2017

Protein folding processes are generally described statistically with the help of multidimensional free energy landscape, typically reduced to a 1-D free energy profile along good reaction co-ordinate. There are many physical parameters which are responsible for protein molecule to hop between the native and unfolded states. The transition path region across the barrier is the region corresponding to the minimum fluctuation. The extent to which this transition region can extend beyond the obvious 1/2 kBT region has been a question of interest for a long time. We propose a new method to locate this transition path region and to study its dependence on the asymmetry of the transition state for a given free energy landscape. We have performed Brownian dynamics simulations with Gaussian white noise and Monte-Carlo simulation by sampling ten thousand successful transitions across the barrier for three different energy landscape having fixed barrier height with asymmetry in their curvature...

We review recent advances in determining and characterizing protein folding kinetics from crystal structures using computational techniques. We also describe a new protein model and show that it reproduces the experimentally observed folding thermodynamics and kinetics of SH3 domain, a small protein, that has been experimentally studied in detail. We verify a nucleation mechanism as a scenario for the folding of SH3 domain. We identify the transition state ensemble of the SH3 domain and dissect it by quantifying the protein topology using concepts of graph theory.

Macromolecular Crystallography, Pt D, 2003

We review recent advances in determining and characterizing protein folding kinetics from crystal structures using computational techniques. We also describe a new protein model and show that it reproduces the experimentally observed folding thermodynamics and kinetics of SH3 domain, a small protein, that has been experimentally studied in detail. We verify a nucleation mechanism as a scenario for the folding of SH3 domain. We identify the transition state ensemble of the SH3 domain and dissect it by quantifying the protein topology using concepts of graph theory.

The Journal of Chemical Physics, 2000

The kinetic data from the coarse representation of polypeptide torsional dynamics described in the preceding paper [Fernandez and Berry, J. Chem. Phys. 112, 5212 (2000), preceding paper] is inverted by using detailed balance to obtain a topographic description of the potential-energy surface (PES) along the dominant folding pathway of the bovine pancreatic trypsin inhibitor (BPTI). The topography is represented as a sequence of minima and effective saddle points. The dominant folding pathway displays an overall monotonic decrease in energy with a large number of staircaselike steps, a clear signature of a good structure-seeker. The diversity and availability of alternative folding pathways is analyzed in terms of the Shannon entropy σ(t) associated with the time-dependent probability distribution over the kinetic ensemble of contact patterns. Several stages in the folding process are evident. Initially misfolded states form and dismantle revealing no definite pattern in the topograp...