Exploring the Free Energy Landscape: From Dynamics to Networks and Back

Falo F (2009) Exploring the Free Energy Landscape: From Dynamics to Networks and Back. PLoS Comput Biol 5(6): e1000415. doi:10.1371/journal.pcbi.1000415 Exploring the Free Energy Landscape: From Dynamics to Networks and Back Diego Prada-Gracia 0 Jesu s Go mez-Garden es 0 Pablo Echenique 0 Fernando Falo 0 Michael Levitt, Stanford University, United States of America 0 1 Departamento de F sica de la Materia Condensada, Universidad de Zaragoza , Zaragoza , Spain , 2 Instituto de Biocomputaci o n y F sica de Sistemas Complejos (BIFI), Universidad de Zaragoza , Zaragoza , Spain , 3 Departamento de Matema tica Aplicada, ESCET, Universidad Rey Juan Carlos , Mo stoles (Madrid), Spain , 4 Departamento de F sica Teo rica, Universidad de Zaragoza , Zaragoza , Spain Knowledge of the Free Energy Landscape topology is the essential key to understanding many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers there are, what the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times and rate constants, or hierarchical relationships among basins, completes the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides. - Funding: This work has been financed by grants FIS2005-00337 and FIS2008-01240 of the Spanish Ministry of Education. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. Polymers and, more specifically, proteins, show complex behavior at the cellular system level, e.g. in protein-protein interaction networks [1], and also at the individual level, where proteins show a large degree of multistability: a single protein can fold in different conformational states [24]. As a complex system [5,6], the dynamics of a protein cannot be understood by studying its parts in isolation, instead, the system must be analyzed as a whole. Tools able to represent and handle the information of the entire picture of a complex system are thus necessary. Complex network theory [7,8] has proved to be a powerful tool used in seemingly different biologically-related fields such as the study of metabolic reactions, ecological and food webs, genetic regulatory systems and the study of protein dynamics [7]. In this latter context, diverse studies have analyzed the conformational space of polymers and proteins making use of network representations [912], where nodes account of polymer conformations. Additionally, some studies have tried to determine the common and general properties of these conformational networks [13,14] looking at magnitudes such as clustering coefficient, cyclomatic number, connectivity, etc. Recently, trying to decompose the network in modules corresponding to the free energy basins, the use of community algorithms over these conformational networks have been proposed [15]. Although this approach has opened a promising path for the analysis of Free Energy Landscapes (FEL), the community based description of the network leads to multiple characterizations of the FEL and thus it is difficult to establish a clear map from the communities found to the basins of the FEL. A similar approach, commonly used to analyze the complex dynamics, is the construction of Markovian models. Markovian state models let us treat the information of one or several trajectories of molecular dynamics (MD) as a set of conformations with certain transition probabilities among them [9,16,17]. Therefore, the time-continuous trajectory turns into a transition matrix, offering global observables as relaxation times and modes. In [1618] the use of Markovian models is proposed with the aim of detecting FEL meta-stable states. However, the above approaches to analyze FELs of peptides involves extremely large computational cost: either general community algorithms or large transition matrices. Finally, other strategies to characterize the FEL that have successfully helped to understand the physics of biopolymers, are based on the study of the Potential Energy Surface (PES) [3,4,19 21]. The classical transition-state theory [22] allows us to project the behavior of the system at certain temperature from the knowledge of the minima and transition states of the PES. This approach entails some feasible approximations, such as harmonic approximation to the PES, limit of high damping, assumption of high barriers, etc. These approximations could be avoided working directly from the MD data. In this article we make a novel study of the FEL capturing its mesoscopic structure and hence characterizing conformational states and the transitions between them. Inspired by the A complete description of complex polymers, such as proteins, includes information about their structure and their dynamics. In particular it is of utmost importance to answer the following questions: What are the structural conformations possible? Is there any relevant hierarchy among these conformers? What are the transition paths between them? These and other questions can be addressed by analyzing in an efficient way the Free Energy Landscape of the system. With this knowledge, several problems about biomolecular reactions (such as enzymatic activity, protein folding, protein deposition diseases, etc.) can be tackled. In this article we show how to efficiently describe the Free Energy Landscape for small and large peptides. By mapping the trajectories of molecular dynamics simulations into a graph (the Conformational Markov Network) and unveiling its structural organization, we obtain a coarse grained description of the protein dynamics across the Free Energy Landscape in terms of the relevant kinetic magnitudes of the system. Therefore, we show the way to bridge the gap between the microscopic dynamics and the macroscopic kinetics by means of a mesoscopic description of the associated Conformational Markov Network. Along this path the compromise between the physical nature of the process and the magnitudes that characterize the network is carefully kept to assure the reliability of the results sho (...truncated)