Complex Energy Landscapes


Day 1: June 2nd, 2010

09:50-10:00   Registration
10:00-10:15   Wellcome
10:15-11:15   D. J. Wales: "Energy Landscapes for Soft Matter"
11:15-11:35   Coffe Break.
11:35-12:35   M. Palassini: "Complex free energy landscapes in disordered systems. Examples from condensed matter physics and computer science"
12:35-13:35   G. Franzese: "Free Energy Landscape of Hydration Water: Theory and Experiments"
13:35-15:30   Lunch
15:30-16:30   P. de los Rios: "Coarse-graining of configuration-space networks"
16:30-17:30   L. Bongini: "A graph theoretical analysis of the energy landscape of model proteins"
17:30-17:45   Break
17:45-18:20   B. Seoane: "Separation and fractionation of order and disorder in highly polydisperse systems."

Day 2: June 3rd, 2010

09:30-10:30  S. Krivov: "Is protein folding sub-diffusive?"
10:30-11:30   F. Noe: "Markov models of molecular kinetics: Generation, Validity and Analysis"
11:30-11:50   Coffe Break.
11:50-12:50   P. Bolhuis: "Extraction of reaction coordinates from the reweighted path ensemble. "
12:50-13:25   F. J. Cao: "How occasional backstepping can speed up a processive motor protein."
13:30-15:30   Lunch
15:30-16:30   R. Best: "Mapping protein folding dynamics to low-dimensional coordinates"
16:30-17:30   F.Rao: "Complex network analysis of protein free-energy landscapes"
17:30-17:45   Break.
17:45-18:20   A. N. Naganathan: "The Curious Case of Protein Folding Transition States"
20:30   Dinner

Day 3: June 4th, 2010

09:30-10:30   Y. Moreno: "Use and Misuse of Networks in Biology"
10:30-11:05   M. Seeber: "Exploring mutational effects on the energy landscape of a beta-hairpin"
11:05-11:25   Coffe Break.
11:25-12:25   F.Ritort: "Molecular misfolding investigated by mechanically unzipping nucleic acids"
12:25-12:40   M.Mareschal (Z-CAM Director): Concluding Remarks
12:40-13:30   Round Table

Invited Talks duration: 45' +15' discussion
Contributed Talks duration: 25'+10' discussion


D. Wales: "Energy Landscapes for Soft Matter"

Coarse-graining the potential energy surface into the basins of attraction of local minima, provides a computational framework for investigating structure, dynamics and thermodynamics in molecular science and soft matter. Steps between local minima form the basis for global optimisation via basin-hopping and for calculating thermodynamic properties using the superposition approach and basin-sampling. To treat global dynamics we must include transition states of the potential energy surface, which link local minima via steepest-descent paths. We may then apply the discrete path sampling method, which provides access to rate constants for rare events. In large systems the paths between minima with unrelated structures may involve hundreds of stationary points of the potential energy surface. New algorithms have been developed for both geometry optimisation and making connections between distant local minima, which allow us to treat such systems. Applications will be presented for a wide variety of soft matter systems, including atomic and molecular clusters, bulk glass formers, self-assembling mesoscopic systems and proteins.

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S. Krivov: "Is protein folding sub-diffusive?"

Protein folding dynamics is often described as diffusion on a free energy surface considered as a function of one or few reaction coordinates. However, a growing number of experiments and models show that, when projected onto a reaction coordinate, protein dynamics is subdiffusive. This raises the question whether the conventionally used diffusive description of the dynamics is adequate. Here we numerically construct the optimum reaction coordinate for a long equilibrium protein folding trajectory and show that the trajectory projected onto this coordinate exhibits diffusive dynamics, while dynamics of the same trajectory projected onto a sub-optimal reaction coordinate is sub-diffusive.

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R. Best: "Mapping protein folding dynamics to low-dimensional coordinates"

One of the predictions of the energy landscape theory of protein folding is that, as a consequence of the funneled nature of the energy landscape, it should be possible to project faithfully the complex, multidimensional dynamics of protein folding onto only a few collective variables. However, there has been some controversy about how the effective diffusion coefficient should vary along these coordinates, which has implications for the interpretation of experiments. I report the explicit mapping of molecular simulations onto diffusion along a one-dimensional reaction coordinate. The diffusion coefficient is shown to be strongly position-dependent for some reaction coordinates, but almost position-independent for the fraction of native contacts. Mapping the dynamics to a coordinate along which the dynamics is position-invariant allows the assumptions made in interpreting experimental kinetic data to be tested, and also reveals intermediates hidden in the original free-energy projections.

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L. Bongini: "A graph theoretical analysis of the energy landscape of model proteins"

The dynamics of systems characterized by a rough energy landscape, such as proteins and structural glasses is described in terms of diffusion on a graph. Depending on temperature changing collections of energy minima contribute in defining a web of metastable states interconnected via a subset of the existing transition states. The characterization of the topology and the spectral properties of such graphs proves an efficient mean to investigate the relaxation and folding dynamics of simple protein models where heterogeneity in the transition rates leads to a non exponential decay of fluctuations and correlations in the spatial distribution of barrier heights might lead to markedly different folding behaviors. The interplay between network topology and energy barrier heights statistic in shaping the relaxation of more general network models is further investigated by means of numerical simulations.

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G. Franzese: "Free Energy Landscape of Hydration Water: Theory and Experiments."

Recent experiments reveal the dynamic behavior of water in the hydration shell of a protein. I will show, by means of numerical simulations and theoretical calculations of a coarse-grained model for hydration water, that it is possible to clarify the mechanism of water dynamics as a consequence of structural changes in the hydrogen bond network. These changes are related to the non-trivial thermodynamics of water, that is understood thanks to the free energy analysis in terms of the relevant order parameters and their fluctuations. Free energy calculations also allow to discuss the mechanism for the dynamics of water at different pressures and temperatures. New calculations and simulations predict other structural changes in more extreme conditions, not accessible in other simulation models. These predictions have been supported by matching experimental results. If the time will allow it, I will also describe the effect of interfaces in this model and how this helps us in understanding protein folding, cold denaturation and pressure denaturation.

[1] K. Stokely, M. G. Mazza, H. E. Stanley, and G. Franzese, Effect of hydrogen bond cooperativity on the behavior of water, Proceedings of the National Academy of Sciences 107, 1301--1306 (2010)
[2] G. Franzese and F. de los Santos, Dynamically Slow Processes in Supercooled Water Confined Between Hydrophobic Plates, Journal of Physics: Condensed Matter, Journal of Physics: Condensed Matter 21, 504107 (2009)
[3] M. G. Mazza, K. Stokely, S. E. Pagnotta, F. Bruni, H. E. Stanley, and G. Franzese, Two dynamic crossovers in protein hydration water and their thermodynamic interpretation, arXiv:0907.1810v1 (2009).
[4] M. G. Mazza, K. Stokely, E. G. Strekalova, H. E. Stanley, and G. Franzese, Cluster Monte Carlo and numerical mean field analysis for the water liquid--liquid phase transition, Computer Physics Communications 180, 497-502 (2009).
[5] P. Kumar, G. Franzese, and H. E. Stanley, Dynamics and Thermodynamics of Water, Journal of Physics: Condensed Matter 20, 244114 (2008).
[6] P. Kumar, G. Franzese, and H. E. Stanley, Predictions of Dynamic Behavior under Pressure for Two Scenarios to Explain Water Anomalies, Physical Review Letters 100, 105701 (2008)

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F. Noe: "Markov models of molecular kinetics: Generation, Validity and Analysis"

There is an increasing interest in deriving simple and telling, yet precise models of the kinetics on complex molecular energy surfaces, and relating these models to experimental data. This can be achieved by Markov models on discrete conformation spaces, parametrized from simulation data. In this talk, we address the following questions:
- What is the precision and the accuracy of a Markov model in predicting long-time kinetics?
- How should a Markov model be generated in order to achieve maximum precision?
- How can we compute transition pathways from Markov models?
- How can we use Markov models to assign structural information to experimental observations?
- How can we compute rigorous uncertainty estimates from Markov models?
The methods and theories will be illustrated with applications on protein and peptide folding.

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M. Palassini: "Complex free energy landscapes in disordered systems. Examples from condensed matter physics and computer science"

Systems of many particles with heterogeneous and conflicting interactions are studied in many different fields. The frustrated interactions give rise to rugged free-energy landscapes with many local minima separated by large barriers, which translate in glassy dynamics and computational hardness. A more precise notion of a "complex" landscape can also be given in terms of the multiplicity of thermodynamic states available to the system.

I will discuss two examples of such systems currently studied in our group with various computational and analytical approaches:
a) The electron glass, a low-temperature regime in disordered insulators, in which the rugged free energy landscape gives rise to slow relaxation and characteristic transport properties [1],[2].
b) The coloring of random graphs, a prototypical NP-complete constraint satisfaction problem [3]. Here the complex landscape manifests itself in a series of phase transitions as the number of constraints per variable is varied, which directly affect the performance of solution algorithms [4].

[1] M. Goethe and M. Palassini, Phys. Rev. Lett. 103, 045702 (2009).
[2] M. Goethe and M. Palassini, Annalen der Physik, 868 (2009).
[3] M. Mézard, M. Palassini, O. Rivoire, Phys. Rev. Lett. 95, 200202 (2005)
[4] S. Mandrá and M. Palassini, 2010.

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F. Rao: "Complex network analysis of protein free-energy landscapes"

The kinetics of biomolecular isomerization processes, such as protein conformational changes, is governed by a free-energy surface of high dimensionality and complexity. As an alternative to projections into one or two dimensions, the free-energy surface can be mapped into a weighted network where nodes and links are configurations and direct transitions among them, respectively. The obtained networks result in an high-resolution representation of the underlying landscape and can be used to infer the presence of metastable states as well as the transition pathways.

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F. Ritort: "Molecular misfolding investigated by mechanically unzipping nucleic acids"

Single molecule manipulation makes possible to disrupt molecular bonds that hold native structures in nucleic acids and proteins. By exerting tiny forces in the piconewton range single molecule techniques allow scientists to monitor molecular reactions in real time (e.g. molecular folding) and characterize thermodynamics and kinetics of individual molecules (e.g. nucleic acids and proteins) with unprecedented energy accuracy within tenths of a kcal/mol.
In this talk I will show experimental results on irreversibility and dissipation in nucleic acid hairpins that are mechanically unzipped using optical tweezers. Our aim is to explore complex molecular free energy landscapes and nonequilibrium behavior in small systems . For this we have designed DNA hairpins of specific sequences that exhibit molecular misfolding to investigate the role of irreversibility and dissipation during the folding process. Our results suggest the existence of a widespread mechanism used by chaperones to assist molecular folding of RNAs and proteins.

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P. de los Rios: "Coarse-graining of configuration-space networks"

Configuration-space networks encode the sequence of biomolecular conformations explored during molecular dynamics simulations: conformations are the vertices of the network, and two vertices are connected if the are encountered in consecutive steps of the dynamics. The resulting network is a weighted network and the associated stochastic matrix represents the molecular dynamics as a Markov process over the network. Here we describe how to reduce the size of the network while preserving the main dynamical features of the Markov process. In doing so, we identify groups of nodes that play common kinetic roles, and single nodes whose identity and singularity must be preserved.

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P. Bolhuis: "Extraction of reaction coordinates from the reweighted path ensemble. "

Simulation of rare events in high-dimensional complex systems often employ biasing along a predefined reaction coordinate to compute the free energy, estimate the rate constants, and explore the mechanism. Instead, one would like to extract a reaction coordinate from the simulation itself. One option is to employ a likelihood optimization scheme that searches for the best reaction coordinate, modeled as a combination of linear candidate collective variables. We introduce an extension of this method that allows finding the best nonlinear reaction coordinate parameterized as a string of beads in collective variable space, which is able to describe the entire reaction from the initial to final state. To do so, we make use of a novel reweighted scheme to unbias the path ensembles in the transition interface sampling framework. In addition to the reaction coordinate, this reweighted path ensemble allows for the analysis of free energy landscapes and committor projections in any collective variable space. While developed for use with path sampling, this analysis method can also be applied to regular molecular dynamics trajectories.

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Y. Moreno: "Use and Misuse of Networks in Biology"

In this talk, we will discuss how network science is contributing to our understanding of several biological process at different levels of organization. The first part of the talk will address the topology of biological networks as well as different tools used by biologists and network scientists to characterize their structure. In doing this, we shall comment on the advantages and drawbacks of this approach. Finally, through several examples, we will also discuss dynamical aspects of networks in biology, the relation between the structure and the dynamics and what are the expected forthcomings in this relatively new field of research.

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A. N. Naganthan: "The Curious Case of Protein Folding Transition States."

There has been a significant amount of work in the last couple of decades to understand the nature of protein folding transition states (TS) from protein engineering experiments, i.e. f-values. In the conventional interpretation, f-values report on the degree of TS structure at the probed site with values of 0 and 1 representing mostly unfolded-like and folded-like structure, respectively. However, in reality, most f-values seem to cluster around a value of 0.3 and are further prone to large experimental errors. Naganathan and Muoz (PNAS, 2010, v. 107, p. 8611-8616) have recently provided an alternative interpretation to the fractional f-values by analyzing the data of over 800 mutants from 24 different proteins. The results include: a) folding TS have little tertiary structure and are dominated by local interactions suggesting a local first mechanism for protein folding, b) TS move monotonously with protein stability in accordance with Hammond behavior, c) transition states are ensembles with varying degrees of structure at the probed site and the conventional view of a unique expanded transition state does not hold, and d) exposed residues in a folded protein have higher f-values compared to buried residues from a simple structural argument. In light of these developments, we re-take the above issues from the point of view of a simple Ca-centric mean-field G-model. We study three proteins belonging to the three main structural classes and investigate each of the above aspects. Our results support the conclusions of the previous work further providing hitherto unexpected observations.

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B. Seoane: "Separation and fractionation of order and disorder in highly polydisperse systems."

We study by means of Monte Carlo simulation a polydisperse soft-spheres model for colloids. We focus on the high polydispersity region. We use Microcanonical Monte Carlo combined with Parallel Tempering to gently explore the energy range in between the solid and the liquid phase. Swap particle moves decrease the equilibration times by at least three orders of magnitude. We conclude that neither a crystal nor an amorphous state are thermodynamically stable. By means of finite size scaling techniques and the introduction of crystalline parameters, we infer that: a) the fluid-solid transition we find is rather a crystal-amorphous phase-separation, b) such phase-separation is preceded by the dynamic glass transition, thus slowing down the dynamics, and c) small and big particles arrange arrange themselves in the two phases following a complex pattern not predicted by any fractionation scenario.

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F.J. Cao: "How occasional backstepping can speed up a processive motor protein."

Fueled by the hydrolysis of ATP, the motor protein kinesin literally walks on two legs along the biopolymer microtubule. The number of accidental backsteps that kinesin takes appears to be much larger than what one would expect given the amount of free energy that ATP hydrolysis makes available. This indicates that backsteps are not simply the forward stepping cycle run backwards. We propose here a more appropriate model. We show how more backstepping increases the entropy of final and intermediate states, thus reducing its free energy. This free energy reduction for the final and intermediate states can speed up the catalytic cycle of the kinesin. We show how measured backstep percentages represent an optimum at which maximal net forward speed is achieved. That result suggests that, through natural selection, kinesin evolved to maximal speed.

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M. Seeber: "Exploring mutational effects on the energy landscape of a beta-hairpin."

We studied the effects of naturally occurring and artificial mutations on the energy landscape of the Ace-RYIPEGMQCSCGID-Nme beta-hairpin from the second extracellular loop (EL2) of the G protein Coupled Receptor (GPCR) rhodopsin [1]. Spontaneous mutations in the rhodopsin gene are associated with a group of retinal degenerative diseases named Retinitis Pigmentosa (RP). RP mutations that affect rhodopsin folding were found to hit 11 out of the 14 amino acids in the beta-hairpin considered in this study.

Comparative Replica Exchange Molecular Dynamics (REMD) simulations simulations, by employing the FACTS implicit solvent model [2], were carried out on the wild type and fourty spontaneous and artificial EL2 mutations known so far. For each variance, a total of 20 replicas were simulated by Langevin dynamics with a friction coefficient of 5.0 ps-1 and temperature values spanning the interval from 270 to 690 K. Analyses were run with the in-house developed program Wordom [3].

REMD simulations together with the analysis of rhodopsin structure could differentiate a minority of mutants (eight mutants over forty) likely to affect the intrinsic stability of the native beta-hairpin from those expected to perturb retinal binding or the local packing interactions of the mutation site with the surrounding receptor domains [4]. Pathogenic mutations which are likely to affect beta-hairpin stability include R1C, P4A, D14A, D14C and D14G.

[1] Fanelli F. and De Benedetti, P.G., Chem. Rev. 105:297-3351, 2005.
[2] Haberthür, U. and Caflisch A., J. Comput. Chem. 29:701-715, 2008.
[3] Seeber, M.; Cecchini, M.; Rao, F.; Settanni, G.; and Caflisch, A., Bioinformatics 23: 2625-2627, 2007.
[4] Felline A.; Seeber M.; Rao F. and Fanelli F. JCTC 5: 2472-2485, 2009.

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