Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising--like) models in equilibrium statistical mechanics, improving on the mean--field approximation and the Bethe--Peierls approximation, which can be regarded as the lowest level of the CVM. In recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models. The foundations of the CVM are briefly reviewed, and the relations with similar techniques are discussed. The main properties of the method are considered, with emphasis on its exactness for particular models and on its asymptotic properties. The problem of the minimization of the variational free energy, which arises in the CVM, is also addressed, and recent results about both provably convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure

Exactness of the cluster variation method and factorization of the equilibrium probability for the Wako-Saito-Munoz-Eaton model of protein folding

I study the properties of the equilibrium probability distribution of a protein folding model originally introduced by Wako and Saito, and later reconsidered by Munoz and Eaton. The model is a one-dimensional model with binary variables and many-body, long-range interactions, which has been solved exactly through a mapping to a two-dimensional model of binary variables with local constraints. Here I show that the equilibrium probability of this two-dimensional model factors into the product of local cluster probabilities, each raised to a suitable exponent. The clusters involved are single sites, nearest-neighbour pairs and square plaquettes, and the exponents are the coefficients of the entropy expansion of the cluster variation method. As a consequence, the cluster variation method is exact for this model.Comment: 14 pages, 1 figur

CVM ANALYSIS OF CROSSOVER IN THE SEMI-INFINITE ISING MODEL

The crossover behavior of the semi--infinite three dimensional Ising model is investigated by means of Pad\'e approximant analysis of cluster variation method results. We give estimates for ordinary critical as well as for multicritical exponents, which are in very good agreement with extensive Monte Carlo simulations.Comment: RevTeX, 7 pages + 2 uuencoded PostScript figures. To be published in J. Magn. Magn. Mat. (substituted raw with encoded PostScript

The Phase Diagram of the Gonihedric 3d Ising Model via CVM

We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in ${\bf Z}^3$. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the self-avoidance coupling $\kappa$ is varied. The value of the magnetization critical exponent $\beta = 0.062 \pm 0.003$, calculated with the cluster variation--Pad\`e approximant method, is also close to the simulation results.Comment: 8 pages text (LaTex) + 3 eps figures bundled together with uufile

NEW TOPOLOGIES IN THE PHASE DIAGRAM OF THE SEMI-INFINITE BLUME-CAPEL MODEL

The phase diagram of the Blume--Capel model on a semi--infinite simple cubic lattice with a (100) free surface is studied in the pair approximation of the cluster variation method. Six main topologies are found, of which two are new, due to the occurrence of a first order surface transition in the phase with ordered bulk, separating two phases with large and small surface order parameters. The latter is a new phase and is studied in some detail, giving the behaviour of the order parameter profiles in two typical cases. A comparison is made with the results of a low temperature expansion, where these are available, showing a great increase in accuracy with respect to the mean field approximation.Comment: RevTeX, 13 pages + 7 uuencoded PostScript figures (substituted raw with encoded PostScript

Mechanical unfolding and refolding pathways of ubiquitin

Mechanical unfolding and refolding of ubiquitin are studied by Monte Carlo simulations of a Go model with binary variables. The exponential dependence of the time constants on the force is verified, and folding and unfolding lengths are computed, with good agreement with experimental results. Furthermore, the model exhibits intermediate kinetic states, as observed in experiments. Unfolding and refolding pathways and intermediate states, obtained by tracing single secondary structure elements, are consistent with simulations of previous all-atom models and with the experimentally observed step sizes

Variational approximations for stochastic dynamics on graphs

We investigate different mean-field-like approximations for stochastic dynamics on graphs, within the framework of a cluster-variational approach. In analogy with its equilibrium counterpart, this approach allows one to give a unified view of various (previously known) approximation schemes, and suggests quite a systematic way to improve the level of accuracy. We compare the different approximations with Monte Carlo simulations on a reversible (susceptible-infected-susceptible) discrete-time epidemic-spreading model on random graphs.Comment: 29 pages, 5 figures. Minor revisions. IOP-style

Rigorous results on the local equilibrium kinetics of a protein folding model

A local equilibrium approach for the kinetics of a simplified protein folding model, whose equilibrium thermodynamics is exactly solvable, was developed in [M. Zamparo and A. Pelizzola, Phys. Rev. Lett. 97, 068106 (2006)]. Important properties of this approach are (i) the free energy decreases with time, (ii) the exact equilibrium is recovered in the infinite time limit, (iii) the equilibration rate is an upper bound of the exact one and (iv) computational complexity is polynomial in the number of variables. Moreover, (v) this method is equivalent to another approximate approach to the kinetics: the path probability method. In this paper we give detailed rigorous proofs for the above results.Comment: 25 pages, RevTeX 4, to be published in JSTA

Pathways of mechanical unfolding of FnIII10: Low force intermediates

We study the mechanical unfolding pathways of the $FnIII_{10}$ domain of fibronectin by means of an Ising--like model, using both constant force and constant velocity protocols. At high forces and high velocities our results are consistent with experiments and previous computational studies. Moreover, the simplicity of the model allows us to probe the biologically relevant low force regime, where we predict the existence of two intermediates with very close elongations. The unfolding pathway is characterized by stochastic transitions between these two intermediates

Direction dependent mechanical unfolding and Green Fluorescent Protein as a force sensor

An Ising--like model of proteins is used to investigate the mechanical unfolding of the Green Fluorescent Protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.Comment: to appear in Phys Rev