Potts Model Research Articles

Computational modelling of cell motility modes emerging from cell-matrix adhesion dynamics.

Lymphocytes have been described to perform different motility patterns such as Brownian random walks, persistent random walks, and Lévy walks. Depending on the conditions, such as confinement or the distribution of target cells, either Brownian or Lévy walks lead to more efficient interaction with the targets. The diversity of these motility patterns may be explained by an adaptive response to the surrounding extracellular matrix (ECM). Indeed, depending on the ECM composition, lymphocytes either display a floating motility without attaching to the ECM, or sliding and stepping motility with respectively continuous or discontinuous attachment to the ECM, or pivoting behaviour with sustained attachment to the ECM. Moreover, on the long term, lymphocytes either perform a persistent random walk or a Brownian-like movement depending on the ECM composition. How the ECM affects cell motility is still incompletely understood. Here, we integrate essential mechanistic details of the lymphocyte-matrix adhesions and lymphocyte intrinsic cytoskeletal induced cell propulsion into a Cellular Potts model (CPM). We show that the combination of de novo cell-matrix adhesion formation, adhesion growth and shrinkage, adhesion rupture, and feedback of adhesions onto cell propulsion recapitulates multiple lymphocyte behaviours, for different lymphocyte subsets and various substrates. With an increasing attachment area and increased adhesion strength, the cells' speed and persistence decreases. Additionally, the model predicts random walks with short-term persistent but long-term subdiffusive properties resulting in a pivoting type of motility. For small adhesion areas, the spatial distribution of adhesions emerges as a key factor influencing cell motility. Small adhesions at the front allow for more persistent motility than larger clusters at the back, despite a similar total adhesion area. In conclusion, we present an integrated framework to simulate the effects of ECM proteins on cell-matrix adhesion dynamics. The model reveals a sufficient set of principles explaining the plasticity of lymphocyte motility.

Charge-density-wave-driven electronic nematicity in a kagome superconductor

Electronic nematicity, in which rotational symmetry is spontaneously broken by electronic degrees of freedom, has been demonstrated as a ubiquitous phenomenon in correlated quantum fluids including high-temperature superconductors and quantum Hall systems1,2. Notably, the electronic nematicity in high-temperature superconductors exhibits an intriguing entanglement with superconductivity, generating complicated superconducting pairing and intertwined electronic orders. Recently, an unusual competition between superconductivity and a charge-density-wave (CDW) order has been found in the AV3Sb5 (A = K, Rb, Cs) family with two-dimensional vanadium kagome nets3-8. Whether these phenomena involve electronic nematicity is still unknown. Here we report evidence for the existence of electronic nematicity in CsV3Sb5, using a combination of elastoresistance measurements, nuclear magnetic resonance (NMR) and scanning tunnelling microscopy/spectroscopy (STM/S). The temperature-dependent elastoresistance coefficient (m11 minus m12) and NMR spectra demonstrate that, besides a C2 structural distortion of the 2a0 × 2a0 supercell owing to out-of-plane modulation, considerable nematic fluctuations emerge immediately below the CDW transition (approximately 94 kelvin) and finally a nematic transition occurs below about 35 kelvin. The STM experiment directly visualizes the C2-structure-pinned long-range nematic order below the nematic transition temperature, suggesting a novel nematicity described by a three-state Potts model. Our findings indicate an intrinsic electronic nematicity in the normal state of CsV3Sb5, which sets a new paradigm for revealing the role of electronic nematicity on pairing mechanism in unconventional superconductors.

Commensurate-incommensurate transition in the chiral Ashkin-Teller model

We investigate the classical chiral Ashkin-Teller model on a square lattice with the corner transfer matrix renormalisation group (CTMRG) algorithm. We show that the melting of the period-4 phase in the presence of a chiral perturbation takes different forms depending on the coefficient of the four-spin term in the Ashkin-Teller model. Close to the clock limit of two decoupled Ising models, the system undergoes a two-step commensurate-incommensurate transition as soon as the chirality is introduced, with an intermediate critical floating phase bounded by a Kosterlitz-Thouless transition at high temperature and a Pokrovsky-Talapov transition at low temperature. By contrast, close to the four-states Potts model, we argue for the existence of a unique commensurate-incommensurate transition that belongs to the chiral universality class, and for the presence of a Lifshitz point where the ordered, disordered and floating phases meet. Finally, we map the whole phase diagram, which turns out to be in qualitative agreement with the 40 year old prediction by Huse and Fisher.

The complexity of approximating the complex-valued Potts model

We study the complexity of approximating the partition function ofthe q-state Potts model and the closely related Tutte polynomialfor complex values of the underlying parameters. Apart from theclassical connections with quantum computing and phase transitionsin statistical physics, recent work in approximate counting hasshown that the behaviour in the complex plane, and more preciselythe location of zeros, is strongly connected with the complexity ofthe approximation problem, even for positive real-valued parameters.Previous work in the complex plane by Goldberg and Guo focused onq = 2, which corresponds to the case of the Ising model; for q > 2,the behaviour in the complex plane is not as well understood andmost work applies only to the real-valued Tutte plane. Our mainresult is a complete classification of the complexity of theapproximation problems for all non-real values of the parameters, byestablishing #P-hardness results that apply even when restricted toplanar graphs. Our techniques apply to all q geq 2 and furthercomplement/refine previous results both for the Ising model and theTutte plane, answering in particular a question raised by Bordewich,Freedman, Lovász and Welsh in the context of quantumcomputations.

Multicellular mechanochemical hybrid cellular Potts model of tissue formation during epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT) is the transdifferentiation of epithelial cells to a mesenchymal phenotype, where cells lose epithelial-like cell-cell adhesions and gain mesenchymal-like enhanced contractility and mobility. EMT is crucial for tissue regeneration and is also implicated in pathological conditions, such as cancer metastasis. Prior work has shown that transforming growth factor-β1 (TGF-β1) is a potent inducer of EMT. In this study, we develop a computational model coupling mechanical and biochemical signaling in a multicellular tissue undergoing EMT.

Inverse Potts model improves accuracy of phylogenetic profiling.

MotivationPhylogenetic profiling is a powerful computational method for revealing the functions of function-unknown genes. Although conventional similarity metrics in phylogenetic profiling achieved high prediction accuracy, they have two estimation biases: an evolutionary bias and a spurious correlation bias. While previous studies reduced the evolutionary bias by considering a phylogenetic tree, few studies have analyzed the spurious correlation bias.ResultsTo reduce the spurious correlation bias, we developed metrics based on the inverse Potts model (IPM) for phylogenetic profiling. We also developed a metric based on both the IPM and a phylogenetic tree. In an empirical dataset analysis, we demonstrated that these IPM-based metrics improved the prediction performance of phylogenetic profiling. In addition, we found that the integration of several metrics, including the IPM-based metrics, had superior performance to a single metric.Availability and implementationThe source code is freely available at https://github.com/fukunagatsu/Ipm.Supplementary information Supplementary data are available at Bioinformatics online.

Limits to detecting epistasis in the fitness landscape of HIV.

The rapid evolution of HIV is constrained by interactions between mutations which affect viral fitness. In this work, we explore the role of epistasis in determining the mutational fitness landscape of HIV for multiple drug target proteins, including Protease, Reverse Transcriptase, and Integrase. Epistatic interactions between residues modulate the mutation patterns involved in drug resistance, with unambiguous signatures of epistasis best seen in the comparison of the Potts model predicted and experimental HIV sequence "prevalences" expressed as higher-order marginals (beyond triplets) of the sequence probability distribution. In contrast, experimental measures of fitness such as viral replicative capacities generally probe fitness effects of point mutations in a single background, providing weak evidence for epistasis in viral systems. The detectable effects of epistasis are obscured by higher evolutionary conservation at sites. While double mutant cycles in principle, provide one of the best ways to probe epistatic interactions experimentally without reference to a particular background, we show that the analysis is complicated by the small dynamic range of measurements. Overall, we show that global pairwise interaction Potts models are necessary for predicting the mutational landscape of viral proteins.

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.

Impact of a strong temperature gradient on grain growth in films

The migration of grain boundaries and, therewith, the phenomenon of grain growth depend strongly on the annealing temperature. Generally, higher temperatures are associated with higher mobilities of the boundaries and therewith faster microstructural coarsening. In the present study, the influence of a strong temperature gradient on grain growth in thin films is investigated. To that aim, a modified three-dimensional Potts model algorithm is employed, where the annealing temperature changes with the thickness of the sample taking grain boundary mobility and energy into account. The resulting drag effect has serious consequences for the temporal and spatial evolution of the grain microstructure.

Computer simulation of phase transitions in low-dimensional Potts models

The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.

Energy analysis of magnetic structures of the ground state of the Potts model with competing exchange interactions

Based on the Wang--Landau algorithm, the Monte Carlo method was used to study the magnetic structures of the ground state and the thermodynamic properties of the two-dimensional Potts model with the number of spin states q=4 on a triangular lattice, taking into account the exchange interactions of the first J1 and second J2 nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the range -2.0≤ J2≤0.0. The magnetic structures of the ground state are constructed in the interval considered. An energy analysis of the magnetic structures of the ground state has been carried out. A phase diagram of the dependence of the critical temperature on the value of J2 is constructed. It is shown that taking into account the interactions of the second nearest neighbors leads to the appearance of frustrations and violation of the magnetic ordering. Keywords: Frustrations, magnetic structures, Monte Carlo method, Potts model.

Phase transitions in the diluted 2D three-state Potts model on a square lattice

The computer simulation method was used to study phase transitions in a two-dimensional site-diluted 3-state Potts model. Systems with linear dimensions Lx L=N, L=10/160 at a spin concentration p=1.00, 0.80 are considered. The numerical data obtained indicate that in a pure 3-state Potts model on a square lattice, a phase transition of the second order is observed in accordance with the theory. The introduction of disorder in the form of non-magnetic impurities (p=0.80) in the 3-state Potts model preserves the phase transition of the second order. Keywords: Potts model, Monte Carlo method, thermodynamic parameters, disorder.

Investigation of the influence of weak magnetic fields on thermodynamic properties of the Potts model with the number of spin states q=4 on a hexagonal lattice

The replica exchange algorithm of the Monte Carlo method was used to study phase transitions and thermodynamic properties of the two-dimensional Potts model with the number of spin states q=4 on a hexagonal lattice in weak magnetic fields. The studies were carried out for the interval of the magnetic field value 0.0≤ H≤3.0 with a step of 1.0. It is found that a first-order phase transition is observed in the considered range of field values. Keywords: Frustration, Phase transitions, Monte Carlo method, Potts model.

Компьютерное моделирование фазовых переходов в низкоразмерных моделях Поттса

The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q = 3 on square and hexagonal lattices. Considered systems with linear dimensions L × L = N, L = 21 ÷ 102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values ​​of the critical points of the Potts model on different lattices.

Phase transitions in a two-dimensional antiferromagnetic Potts model on a kagome lattice

The phase transitions and thermodynamic properties of the two-dimensional antiferromagnetic Potts model with the number of spin states q=4 on the kagome lattice are studied by the Monte Carlo method, taking into account the exchange interactions of the first J1 and second J2 nearest neighbors. The studies were carried out for the value of the interaction of the second nearest neighbors in the interval -1.0≤ J_2≤0.0. An analysis of the character of phase transitions has been carried out. It is shown that in the interval -1.0≤ J_2≤-0.1, a second-order phase transition is observed, while at J_2=0.0, frustrations disturb the order in the system and lead to the disappearance of the phase transition Keywords: Frustration, phase transition, Monte Carlo method, Potts model.

Spins, percolation and height functions

To highlight certain similarities in graphical representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters. To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally coupled height function and bond percolation models, and we discuss fundamental properties of the resulting joint coupling. As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop O(n) model, the random current, double random current and XOR-Ising model.

Ab initio расчеты электронных свойств, частотная дисперсия диэлектрических коэффициентов и край оптического поглощения монокристаллов TlInS-=SUB=-2-=/SUB=-<Sn>

The method of computer simulation was used to study phase transitions in a two-dimensional site-diluted 3-state Potts model. Systems with linear dimensions L×L=N, L=10÷160 at spin concentrations p=1.00, 0.80 are considered. The obtained numerical data indicate that in the pure 3-state Potts model on a square lattice, a second-order phase transition is observed in accordance with the theory. The introduction of disorder in the form of non-magnetic impurities (p=0.80) in the Potts model with q=3 preserves the second-order phase transition.

Inference Tools for Markov Random Fields on Lattices: The R Package mrf2d

Markov random fields on two-dimensional lattices are behind many image analysis methodologies. mrf2d provides tools for statistical inference on a class of discrete stationary Markov random field models with pairwise interaction, which includes many of the popular models such as the Potts model and texture image models. The package introduces representations of dependence structures and parameters, visualization functions and efficient (C++ based) implementations of sampling algorithms, common estimation methods and other key features of the model, providing a useful framework to implement algorithms and working with the model in general. This paper presents a description and details of the package, as well as some reproducible examples of usage.

Buried points in complex dynamical systems and glassy transitions

<p indent=0mm>A new interdisciplinary research direction is addressed across the fields of statistical mechanics and complex dynamical systems together with specific research topics, such as the connection between buried points and glassy transitions. First, pertinent results on complex dynamical systems are delivered. Let<italic> f</italic> be a complex analytical mapping from the Riemann sphere (or the complex plane) to itself, i.e., <italic>f</italic> is rational or entire. The stable set <italic>F</italic>(<italic>f</italic>) of this dynamical system is called the Fatou set of <italic>f</italic>, and the unstable set <italic>J</italic>(<italic>f</italic>) the Julia set. Every component of <italic>F</italic>(<italic>f</italic>) is called a Fatou component. Obviously, the boundary of each Fatou component belongs to <italic>J</italic>(<italic>f</italic>). If a point in <italic>J</italic>(<italic>f</italic>) does not belong to the boundary of any Fatou component, then it is a buried point of <italic>f</italic>, which is often used to describe the topological complexity of the Julia set. In this quest, Makienko’s conjecture is undoubtedly one of the most important problems which is stated as follows: Let <italic>R</italic> be a rational mapping and <italic>F</italic>(<italic>R</italic>) contain infinitely many components. If every Fatou component is not completely invariant, then <italic>R</italic> has buried points. The known result is the following: If <italic>J</italic>(<italic>R</italic>) is disconnected, or connected and locally connected, then Makienko’s conjecture is true; if <italic>J</italic>(<italic>R</italic>) is disconnected and <italic>R</italic> has a buried point, then it must have buried components. Concerning entire functions, we have: If <italic>F</italic>(<italic>f</italic>) is not empty, then <italic>f</italic> has a buried point if and only if <italic>F</italic>(<italic>f</italic>) is disconnected. Following the above results, interesting topological properties on the sets of buried points are deduced. However, many open problems remain unsolved in regard to these special points. In the second part, we explained the topological complexity of the set of Yang-Lee zeros. In 1952, Yang C.N. and Lee T.D. established an analytic theory in statistical mechanics, summarized as the circle theorem: For statistical models like 2-dimensional lattice gas, the zeros of the partition function condense to the unit circle. Henceforth, the importance of the distribution of Yang-Lee zeros in the complex plane is emphasized for general models. In 1983, researchers including B. Derrida, L. De Seze and C. Itzykson found that the distribution of Yang-Lee zeros can be cast into a problem on Julia sets of renormalization transformations. In the third part, we describe the connection between buried points and glass transitions. Concerning the Potts model on a generalized diamond hierarchical lattice, we present a new type of distribution of Julia sets of renormalization transformations, which is called the real Feigenbaum-type distribution of buried points. Physically, this means that the limiting set of Yang-Lee zeros could contain an interval of the real axis, all of which are singularities of the free energy function. This important observation indicates that the exploration of renormalization transformation dynamics is essential to the understanding of the phenomenon of glassy transitions. Finally, we discussed the theoretical value and practical significance of the above research topics and directions. It is well known that <italic>Science</italic> published 125 most challenging scientific questions in 2005 to celebrate the Journal’s 125th anniversary, among which Question 47 is “What is the essence of the glassy substance?”. There are many theoretical models for glassy transitions, but up to now, a complete theory remains illusive. Perhaps, the connection between complex dynamics and glassy transition established above could shed new light on this old problem with the help of powerful tools from mathematics, as ever before.

Finitary codings for the random-cluster model and other infinite-range monotone models

A random field X=(Xv)v∈G on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as X=φ(Y) for some i.i.d. process Y=(Yv)v∈G and equivariant map φ. Such a map, also called a coding, is finitary if, for every vertex v∈G, there exists a finite (but random) set U⊂G such that Xv is determined by {Yu}u∈U. We construct a coding for the random-cluster model on G, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström–Jonasson–Lyons [18]. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model. Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications – the loop O(n) model and long-range Ising models. In the case of G= Zd, we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process Y. To do this, we extend a mixing-time result of Martinelli–Olivieri [22] to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.