Potts Model Research Articles

Gibbs measures for the Potts model with a countable set of spin values on a Cayley tree

Рассматривается модель Поттса с конкурирующими взаимодействиями радиуса $r=2$ и счетным множеством значений спина $\Phi=\{0,1,\ldots,\}$ на дереве Кэли порядка $k=2$. Задача нахождения трансляционно-инвариантных мер Гиббса для такой модели при любой фиксированной вероятностной мере $\nu$, $\nu(i)>0$, заданной на множестве $\Phi$, сведена к решению некоторой бесконечной системы функциональных уравнений. Описан класс мер $\nu$ на $\Phi$, таких что для каждого элемента из этого класса полученная бесконечная система уравнений имеет единственное решение вида $\{a^i, i=1,2,\ldots\}$, где $a\in(0,1)$.

Neural network-derived Potts models for structure-based protein design using backbone atomic coordinates and tertiary motifs.

Designing novel proteins to perform desired functions, such as binding or catalysis, is a major goal in synthetic biology. A variety of computational approaches can aid in this task. An energy-based framework rooted in the sequence-structure statistics of tertiary motifs (TERMs) can be used for sequence design on predefined backbones. Neural network models that use backbone coordinate-derived features provide another way to design new proteins. In this work, we combine the two methods to make neural structure-based models more suitable for protein design. Specifically, we supplement backbone-coordinate features with TERM-derived data, as inputs, and we generate energy functions as outputs. We present two architectures that generate Potts models over the sequence space: TERMinator, which uses both TERM-based and coordinate-based information, and COORDinator, which uses only coordinate-based information. Using these two models, we demonstrate that TERMs can be utilized to improve native sequence recovery performance of neural models. Furthermore, we demonstrate that sequences designed by TERMinator are predicted to fold to their target structures by AlphaFold. Finally, we show that both TERMinator and COORDinator learn notions of energetics, and these methods can be fine-tuned on experimental data to improve predictions. Our results suggest that using TERM-based and coordinate-based features together may be beneficial for protein design and that structure-based neural models that produce Potts energy tables have utility for flexible applications in protein science.

Data-driven subsurface modelling using a Markov random field model

ABSTRACT This paper presents a method of subsurface modelling based on a Markov random field (MRF) model called Potts model. Potts model is an undirected graphical model and has been applied in image processing such as image denoising, restoration and inpainting. The proposed method is simple and requires only a few borehole data on soil types in both training and inference stages. Current implementations of the Potts model require substantial data for training, and they are not suitable for subsurface modelling. The proposed method was demonstrated through numerical examples for 2D and 3D virtual grounds and a real case history. In the numerical examples, the effect of the number of training datasets on the estimation results was also investigated. The proposed method can provide not only the most probable inference of subsurface model but also the spatial distribution of geological uncertainty and is compatible with reliability-based analysis in geotechnical engineering. The spatial distribution of uncertainty is informative in its own right. It directs the engineer to focus on mechanically important zones where the critical failure mechanism passes through if they coincide with the low-accuracy zones.

Phase transitions in a two-dimensional weakly diluted 4-state Potts model on a hexagonal lattice

The Monte Carlo method is used to study phase transitions in a two-dimensional weakly diluted 4-state Potts model on a hexagonal lattice. Systems with linear dimensions L=21/336 at spin concentrations p=1.00, 0.90 are considered. The use of the fourth-order binder cumulants method and histogram data analysis showed that the introduction of nonmagnetic impurities has a stabilizing role in the implementation of the second-order phase transition in the Potts model under study on a hexagonal lattice. Keywords: Potts model, Monte Carlo method, thermodynamic parameters, disorder.

Critical points in coupled Potts models and correlated percolation

We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case (). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.

Фазовые переходы в двумерной слабо разбавленной четырехкомпонентной модели Поттса на гексагональной решетке

The Monte Carlo method is used to study phase transitions in a two-dimensional weakly diluted 4-state Potts model on a hexagonal lattice. Systems with linear dimensions L=21÷336 at spin concentrations p=1.00, 0.90 are considered. The use of the fourth-order binder cumulants method and histogram data analysis showed that the introduction of non-magnetic impurities has a stabilizing role in the implementation of the second-order phase transition in the Potts model under study on a hexagonal lattice.

Potts model with invisible states on a scale-free network

Different models are proposed to understand magnetic phase transitions through the prism of competition between the energy and the entropy. One of such models is a q-state Potts model with invisible states. This model introduces r invisible states such that if a spin lies in one of them, it does not interact with the rest states. We consider such a model using the mean field approximation on an annealed scale-free network where the probability of a randomly chosen vertex having a degree k is governed by the power-law P(k) ∝ k λ. Our results confirm that q, r and λ play a role of global parameters that influence the critical behaviour of the system. Depending on their values, the phase diagram is divided into three regions with different critical behaviours. However, the topological influence, presented by the marginal value of λc(q), has proven to be dominant over the entropic influence, governed by the number of invisible states r.

Strict monotonicity, continuity, and bounds on the Kertész line for the random-cluster model on Zd

Ising and Potts models can be studied using the Fortuin–Kasteleyn representation through the Edwards–Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength h > 0. In this representation, which is also known as the random-cluster model, the Kertész line is the curve that separates two regions of the parameter space defined according to the existence of an infinite cluster in Zd. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kertész line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit h → 0 complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for h → ∞. Finally, using a cluster expansion, we investigate the continuity of the Kertész line phase transition.

Расчет относительных дисперсий намагниченности, теплоемкости и восприимчивости в двумерной слабо разбавленной трехкомпонентной модели Поттса

Using the Monte Carlo method, the relative dispersions of magnetization Rm, heat capacity Rc and susceptibility Rchi are calculated for a weakly diluted three-component Potts model on a square lattice at a spin concentration p=0.90. It is revealed that the introduction of disorder in the form of non-magnetic impurities into the two-dimensional Potts model leads to non-zero values for Rm, Rc, and Rchi at the critical point. It is found that these values decrease markedly for systems with linear dimensions L>80.

Влияние немагнитного беспорядка на фазовые переходы первого рода

Using the Monte Carlo method, weakly dilute magnetic systems described by two-dimensional five-component Potts model, which in the pure state undergoes a phase transition first order. Numerical studies have revealed that an insignificant quenched disorder, realized in the form of non-magnetic impurities, can change the order of phase transition in magnetic systems described by the two-dimensional five-component Potts model on a square lattice, for which a phase transition of the first order is observed in a homogeneous state. It is shown that such a phase change the transition is due to the fact that non-magnetic disorder prevents the coexistence of local phases characteristic of a phase transition of the first order at T = Tl .

Metastability of the three-state Potts model with general interactions

Metastability of the three-state Potts model with general interactions

Mathematical model for promotion of wound closure with ATP release.

To computationally investigate the recent experimental finding such that extracellular ATP release caused by exogeneous mechanical forces promote wound closure, we introduce a mathematical model, the Cellular Potts Model (CPM), which is a popular discretized model on a lattice, where the movement of a "cell" is determined by a Monte Carlo procedure. In the experiment, it was observed that there is mechanosensitive ATP release from the leading cells facing the wound gap and the subsequent extracellular Ca2+ influx. To model these phenomena, the Reaction-Diffusion equations for extracellular ATP and intracellular Ca2+ concentrations are adopted and combined with CPM, where we also add a polarity term because the cell migration is enhanced in the case of ATP release. From the numerical simulations using this hybrid model, we discuss effects of the collective cell migration due to the ATP release and the Ca2+ influx caused by the mechanical forces and the consequent promotion of wound closure.

Simulating 3D Cell Shape with the Cellular Potts Model.

Computer simulations have become a widely used method for the field of mechanobiology. An important question is whether one can predict the shape and forces of cells as a function of the extracellular environment. Different types of models have been described before to simulate cell and tissue shapes in structured environments. In this chapter, we give a brief overview of commonly used models and then describe the Cellular Potts Model, a lattice-based modelling framework, in more detail. We provide a hands-on guide on how to build a model that simulates the shape of a single cell on a micropattern in three dimensions in different open source software packages using the Cellular Potts framework. A simulation is set up with an initial configuration of generalized cells that change shape and position due to an energy function that incorporates cellular volume and surface area constraints as well as interaction energies between the generalized cells.

Listeria motility increases the efficiency of epithelial invasion during intestinal infection.

Listeria monocytogenes (Lm) is a food-borne pathogen that causes severe bacterial gastroenteritis, with high rates of hospitalization and mortality. Lm is ubiquitous in soil, water and livestock, and can survive and proliferate at low temperatures. Following oral ingestion of contaminated food, Lm crosses the epithelium through intestinal goblet cells in a mechanism mediated by Lm InlA binding host E-cadherin. Importantly, human infections typically occur with Lm growing at or below room temperature, which is flagellated and motile. Even though many important human bacterial pathogens are flagellated, little is known regarding the effect of Lm motility on invasion and immune evasion. Here, we used complementary imaging and computer modeling approaches to test the hypothesis that bacterial motility helps Lm locate and engage target cells permissive for invasion. Imaging explanted mouse and human intestine, we showed that Lm grown at room temperature uses motility to scan the epithelial surface and preferentially attach to target cells. Furthermore, we integrated quantitative parameters from our imaging experiments to construct a versatile "layered" cellular Potts model (L-CPM) that simulates host-pathogen dynamics. Simulated data are consistent with the hypothesis that bacterial motility enhances invasion by allowing bacteria to search the epithelial surface for their preferred invasion targets. Indeed, our model consistently predicts that motile bacteria invade twice as efficiently over the first hour of infection. We also examined how bacterial motility affected interactions with host cellular immunity. In a mouse model of persistent infection, we found that neutrophils migrated to the apical surface of the epithelium 5 hours post infection and interacted with Lm. Yet in contrast to the view that neutrophils "hunt" for bacteria, we found that these interactions were driven by motility of Lm-which moved at least ~50x faster than neutrophils. Furthermore, our L-CPM predicts that motile bacteria maintain their invasion advantage even in the presence of host phagocytes, with the balance between invasion and phagocytosis governed almost entirely by bacterial motility. In conclusion, our simulations provide insight into host pathogen interaction dynamics at the intestinal epithelial barrier early during infection.

Diversifying Design of Nucleic Acid Aptamers Using Unsupervised Machine Learning.

Inverse design of short single-stranded RNA and DNA sequences (aptamers) is the task of finding sequences that satisfy a set of desired criteria. Relevant criteria may be, for example, the presence of specific folding motifs, binding to molecular ligands, sensing properties, and so on. Most practical approaches to aptamer design identify a small set of promising candidate sequences using high-throughput experiments (e.g., SELEX) and then optimize performance by introducing only minor modifications to the empirically found candidates. Sequences that possess the desired properties but differ drastically in chemical composition will add diversity to the search space and facilitate the discovery of useful nucleic acid aptamers. Systematic diversification protocols are needed. Here we propose to use an unsupervised machine learning model known as the Potts model to discover new, useful sequences with controllable sequence diversity. We start by training a Potts model using the maximum entropy principle on a small set of empirically identified sequences unified by a common feature. To generate new candidate sequences with a controllable degree of diversity, we take advantage of the model's spectral feature: an "energy" bandgap separating sequences that are similar to the training set from those that are distinct. By controlling the Potts energy range that is sampled, we generate sequences that are distinct from the training set yet still likely to have the encoded features. To demonstrate performance, we apply our approach to design diverse pools of sequences with specified secondary structure motifs in 30-mer RNA and DNA aptamers.

Neural network evidence of a weakly first-order phase transition for the two-dimensional 5-state Potts model

A universal (supervised) neural network (NN), which is trained only once on a one-dimensional lattice of 200 sites, is employed to study the phase transition of the two-dimensional (2D) 5-state ferromagnetic Potts model on the square lattice. In particular, the NN is obtained by using two artificially made configurations as the training set. Due to the unique features of the employed NN, results associated with systems consisting of over 4,000,000 spins can be obtained with ease, and convincing NN evidence showing that the investigated phase transition is weakly first order is reached.

On the Two-Point Function of the Potts Model in the Saturation Regime

We consider the Random-Cluster model on {mathbb {Z}}^d with interactions of infinite range of the form J_x = psi (x){mathsf {scriptstyle e}}^{-rho (x)} with rho a norm on {mathbb {Z}}^d and psi a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of beta _{textrm{sat}}(s)>0 such that the inverse correlation length in the direction s is constant on [0,beta _{textrm{sat}}(s))), thus removing a regularity assumption used in our previous work (Aoun et al. in Commun Math Phys 386:433–467, 2021). Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein–Zernike form) for the two-point function in the whole saturation regime (0,beta _{textrm{sat}}(s)). We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in beta in the regime (beta _{textrm{sat}}(s), beta _{mathrm {scriptscriptstyle c}}).

Δ-wavy probability distributions and Potts model

We define the wavy probability distributions on a subset and Δ-wavy probability distributions - two generalizations of the wavy probability distributions. A classification on the Δ-waviness is given. For the Δ-wavy probability distributions having normalization constant, we give a formula for this constant, to compute this constant. We show that the Potts model is a Δ-wavy probability distribution, where Δ is a partition which will be specified. For the normalization constant of Potts model, we give formulas and bounds. As to the formulas for this constant, we give two general formulas, one of them is simple while the other is more complicated, and based on independent sets, a formula for the Potts model on connected separable graphs - closed-form expressions are then obtained in several cases -, and a formula for the Potts model on graphs with a vertex of degree 2 - a recurrence relation is then obtained for the normalization constant of Potts model on Cn; the cycle graph with n vertices; the normalization constant of Ising model on Cn is computed using this relation. As to the bounds for the normalization constant, we present two ways to obtain such bounds; we illustrate these ways giving a general lower bound, and a lower bound and an upper one when the model is the Potts model on Gn,n, the square grid graph, n = 6k, k ≥ 1 - two upper bounds for the free energy per site of this model are then obtained, one of them being in the limit. A sampling method for the Δ-wavy probability distributions is given and, as a result, a sampling method for the Potts is given. This method - that for the Potts model too - has two steps, Step 1 and Step 2, when |Δ| > 1 and one step, Step 2 only, when |Δ| = 1. For the Potts model, Step 1 is, in general, difficult. As to Step 2, for the Potts model too, using the Gibbs sampler in a generalized sense, we obtain an exact (not approximate) sampling method having p + 1 steps (p + 1 substeps of Step 2), where p = |I|; I is an independent set, best, a maximum independent set, best, a maximum independent set - for the Potts model on Gn1,n2,…,nd , the d-dimensional grid graph, d ≥ 1, n1, n2,…, nd ≥ 1, n1n2·…·nd ≥ 2; we obtain an exact sampling method for half or half+1 vertices.

Efficient sampling and counting algorithms for the Potts model on ℤd at all temperatures

AbstractFor and all we give an efficient algorithm to approximately sample from the ‐state ferromagnetic Potts and random cluster models on finite tori for any inverse temperature . This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen–Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov–Sinai theory. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures .

A universal training scheme and the resulting universality for machine learning phases

Abstract An autoencoder (AE) and a generative adversarial network (GAN) are trained only once on a one-dimensional (1D) lattice of 200 sites. Moreover, the AE contains only one hidden layer consisting of two neurons, and both the generator and the discriminator of the GAN are made up of two neurons as well. The training set employed to train both the considered unsupervised neural networks (NNs) is composed of two artificial configurations. Remarkably, despite their simple architectures, both the built AE and GAN have precisely determined the critical points of several models, including the three-dimensional classical O(3) model, the two-dimensional generalized classical XY model, the two-dimensional two-state Potts model, and the one-dimensional Bose–Hubbard model. In addition, a factor of several thousands in the speed of calculation is gained for the built AE and GAN when they are compared with the conventional unsupervised NN approaches. The results presented here, as well as those shown previously in the literature, suggest that when phase transitions are considered, an elegant universal neural network that is extremely efficient and is applicable to broad physical systems can be constructed with ease. In particular, since an NN trained with two configurations can be applied to many models, it is likely that when machine learning is concerned, the majority of phase transitions belong to a class having two elements, i.e. the Ising class.