A coarse-grained, ``realistic'' model for Protein Folding

A phenomenological model hamiltonian to describe the folding of a protein with any given sequence is proposed. The protein is thought of as a collection of pieces of helices; as a consequence its configuration space increases with the number of secondary structure elements rather than with the number of residues. The hamiltonian presents both local (i.e. single helix, accounting for the stiffness of the chain) and non local (interactions between hydrophobically-charged helices) terms, and is expected to provide a first tool for studying the folding of real proteins. The partition function for a simplified, but by no means trivial, version of the model is calculated almost completely in an analytical way. The latter simplified model is also applied to the study of a synthetic protein, and some preliminary results are shown.Comment: 21 pages, 5 Postscript figures, RevTex; to be published in J. Chem. Phy

Testing the helix model for protein folding on four simple proteins

We test a simplified, local version of the helix model on two synthetic and two natural proteins, to study its efficiency in predicting the native secondary structure. The results we obtain are very good for the synthetic sequences, poorer for the two natural ones. This suggests that non-local terms play a fundamental role in determining the secondary structure, even if in some cases local terms alone may be sufficient.Comment: 9 pages, 4 Postscript figures, RevTex; to be published on Mod. Phys. Lett.

Mean Field Approach for a Statistical Mechanical Model of Proteins

We study the thermodynamical properties of a topology-based model proposed by Galzitskaya and Finkelstein for the description of protein folding. We devise and test three different mean-field approaches for the model, that simplify the treatment without spoiling the description. The validity of the model and its mean-field approximations is checked by applying them to the $\beta$-hairpin fragment of the immunoglobulin-binding protein (GB1) and making a comparison with available experimental data and simulation results. Our results indicate that this model is a rather simple and reasonably good tool for interpreting folding experimental data, provided the parameters of the model are carefully chosen. The mean-field approaches substantially recover all the relevant exact results and represent reliable alternatives to the Monte Carlo simulations.Comment: RevTeX-4, 11 pages, 6 eps-figures, To Appear on J.Chem.Phy

Analysis of the Equilibrium and Kinetics of the Ankyrin Repeat Protein Myotrophin

We apply the Wako-Saito-Munoz-Eaton model to the study of Myotrophin, a small ankyrin repeat protein, whose folding equilibrium and kinetics have been recently characterized experimentally. The model, which is a native-centric with binary variables, provides a finer microscopic detail than the Ising model, that has been recently applied to some different repeat proteins, while being still amenable for an exact solution. In partial agreement with the experiments, our results reveal a weakly three-state equilibrium and a two-state-like kinetics of the wild type protein despite the presence of a non-trivial free-energy profile. These features appear to be related to a careful "design" of the free-energy landscape, so that mutations can alter this picture, stabilizing some intermediates and changing the position of the rate-limiting step. Also the experimental findings of two alternative pathways, an N-terminal and a C-terminal one, are qualitatively confirmed, even if the variations in the rates upon the experimental mutations cannot be quantitatively reproduced. Interestingly, folding and unfolding pathway appear to be different, even if closely related: a property that is not generally considered in the phenomenological interpretation of the experimental data.Comment: 27 pages, 7 figure

Estudio de las propensidades estructurales de la sequencia de alfa sinucleína

The amyloid aggregation of alpha synuclein (αS), an intrinsically disordered protein, plays a key role in the etiology of Parkinson disease. The molecular mechanisms of αS amyloid aggregation are not well described at the moment and in this research, we move the first steps towards the characterization of two different mechanisms of αS aggregation as well as the of the phase transition between them using methanol (MetOH) as an inducing agent, with an approach that will combine the experimental and theoretical perspective with a residues-based theoretical modelling.On the one hand the determination of the phase transition range between the two observed mechanisms was carried out by performing αS aggregation kinetics at different MetOH concentrations and carrying out structural analysis of the generated aggregates by Fourier Transform Infrared Spectroscopy (FTIR). On the other hand, the preference for one mechanism or the other was studied with pyrene excimer emission fluorescence spectroscopy and the coexistence of both types of aggregation mechanisms was observed in the transition phase. The results obtained from FTIR and emission fluorescence spectroscopy were in agreement and complementary to each other.From the theoretical perspective an initial approach for the αS aggregation modelling was given by computing the accuracy of online secondary structure predictors in order to determine the importance of different αS amino acid regions in the preference for the type of amyloid aggregation mechanism. <br /

Rate Determining Factors in Protein Model Structures

Previous research has shown a strong correlation of protein folding rates to the native state geometry, yet a complete explanation for this dependence is still lacking. Here we study the rate-geometry relationship with a simple statistical physics model, and focus on two classes of model geometries, representing ideal parallel and antiparallel structures. We find that the logarithm of the rate shows an almost perfect linear correlation with the "absolute contact order", but the slope depends on the particular class considered. We discuss these findings in the light of experimental results.Comment: 4 pages, 2 figure

Desarrollo de una aproximación variacional para la cinética de procesos markovianos en grafos. Aplicación a redes de regulación genética.

Gene regulatory networks come in different forms and each has its own characteristics. In this project we consider the application of a statistical physics method (the Cluster Variation Method) to the study of the kinetics of a class of gene regulatory networks, namely activity flow gene networks that also behave as Markov processes. We define the class of models we use, and for the sake of clearness we also describe in some detail a couple of examples of gene networks that we study later on: a toy model and a cell-cycle model found in the GINsim repository. We also describe briefly an approach, alternative to ours, that uses a Kinetic Monte Carlo algorithm (the Gillespie algorithm) to simulate the kinetics of gene regulatory networks: this approach, implemented in the software MaBoss, will represent a benchmark to evaluate our approach.Methodology plays an important role in our project, so we devote the second and third chapters to describe the rationale that allows to study a dynamical problem within the CVM approach from equilibrium statistical physics, and to introduce a particular approximation (the M-approximation) within the possible CVM choices. In the second chapter, we start by defining the stochastic models we aim at and proposing a standard notation. Those models must be Markovian processes and each node's state in a given time must be determined by itself and its nearest neighbours on the previous time step. Then, we lay down the thermodynamic formalism so that we can later use the statistical physics tools available. This means coming up with a variational free energy expression for the models defined, so that we can obtain the system's dynamic by minimizing that variational. In order to achieve this, we will introduce a fair amount of basic statistical physics theory in a way that it looks familiar to the reader. Finally in that same chapter, we get to derive the generalized Cluster Variational Method (CVM). In order to achieve it, we give an appropiate definition of cluster and a brief subsection about Möbius numbers. The goal of the CVM method is to approximate the variational free energy so that we can handle it when looking for its minimum. It does this by truncating the cumulant expansion of the entropy to a set of chosen maximal clusters. Applying a Möbius inversion to this expansion we get a valid entropy approximation. The choice of maximal clusters characterizes the approximation.In the third chapter, we derive an explicit algorithm for the M approximation so that it can be applied to the toy and cell cycle models. First, we derive the Möbius numbers given the choice of the M clusters as maximal, which determines the approximated entropy. Then, we lay the compatibility constraints between clusters, which are nothing but marginalization identities between probability distributions with respect to the missing variables. Finally, we derive an algorithm that obtains the system dynamic given the initial probability distribution and the transition probabilities. We do so by proving two equivalent expressions for the approximated variational free energy, which can be expressed in terms of Kullback-Leiber divergence terms. These divergence terms give a natural and direct way of obtaining the probability distribution of the maximal clusters, from which we can derive the system's evolution.In the fourth chapter, we apply the method developed to the forementioned models, along with the PQR approximation from the same CVM method and MaBoSS. This will be useful in order to check the performance of the M approximation. We discuss three possible trajectories from the toy model where we check if the new method works accordingly. Finally, we check qualitatively whether we can replicate the oscillating behaviour of the cell cycle with the M approximation or not and we draw some conclusions and perspectives.<br /

Estudio de la cromatina con modelos sencillos de física estadística

El trabajo comenta la aplicación de modelos de la física estadística aplicados a la cromatina, donde hemos utilizado algoritmos de Simulated Annealing y deterministas para optimizar los parámetros de los modelos utilizados para coincidir lo más posible con los datos experimentales, además de explicar los resultados que nos dan los modelos.<br /