Vertex Model Research Articles

Evaluation of integrals for the emptiness formation probability in the square-ice model

We study the emptiness formation probability (EFP) in the six-vertex model with domain wall boundary conditions. We present a conjecture according to which at the ice point, i.e., when all the Boltzmann weights are equal, the known multiple integral representation (MIR) for the EFP can be given as a finite-size matrix determinant of Fredholm type. Our conjecture is based on the explicit evaluation of the MIR for particular values of geometric parameters and on two kinds of identities for the boundary correlation function. The obtained representation can be further written as the Fredholm determinant of some linear integral operator. We show that as the geometric parameters of the EFP are tuned to the vicinity of the arctic curve arising in the scaling limit, the conjectured determinant turns into the GUE Tracy–Widom distribution.

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

Automated machine learning model for fundus image classification by health-care professionals with no coding experience

To assess the feasibility of code-free deep learning (CFDL) platforms in the prediction of binary outcomes from fundus images in ophthalmology, evaluating two distinct online-based platforms (Google Vertex and Amazon Rekognition), and two distinct datasets. Two publicly available datasets, Messidor-2 and BRSET, were utilized for model development. The Messidor-2 consists of fundus photographs from diabetic patients and the BRSET is a multi-label dataset. The CFDL platforms were used to create deep learning models, with no preprocessing of the images, by a single ophthalmologist without coding expertise. The performance metrics employed to evaluate the models were F1 score, area under curve (AUC), precision and recall. The performance metrics for referable diabetic retinopathy and macular edema were above 0.9 for both tasks and CFDL. The Google Vertex models demonstrated superior performance compared to the Amazon models, with the BRSET dataset achieving the highest accuracy (AUC of 0.994). Multi-classification tasks using only BRSET achieved similar overall performance between platforms, achieving AUC of 0.994 for laterality, 0.942 for age grouping, 0.779 for genetic sex identification, 0.857 for optic, and 0.837 for normality with Google Vertex. The study demonstrates the feasibility of using automated machine learning platforms for predicting binary outcomes from fundus images in ophthalmology. It highlights the high accuracy achieved by the models in some tasks and the potential of CFDL as an entry-friendly platform for ophthalmologists to familiarize themselves with machine learning concepts.

Graph topological transformations in space-filling cell aggregates.

Cell rearrangements are fundamental mechanisms driving large-scale deformations of living tissues. In three-dimensional (3D) space-filling cell aggregates, cells rearrange through local topological transitions of the network of cell-cell interfaces, which is most conveniently described by the vertex model. Since these transitions are not yet mathematically properly formulated, the 3D vertex model is generally difficult to implement. The few existing implementations rely on highly customized and complex software-engineering solutions, which cannot be transparently delineated and are thus mostly non-reproducible. To solve this outstanding problem, we propose a reformulation of the vertex model. Our approach, called Graph Vertex Model (GVM), is based on storing the topology of the cell network into a knowledge graph with a particular data structure that allows performing cell-rearrangement events by simple graph transformations. Importantly, when these same transformations are applied to a two-dimensional (2D) polygonal cell aggregate, they reduce to a well-known T1 transition, thereby generalizing cell-rearrangements in 2D and 3D space-filling packings. This result suggests that the GVM's graph data structure may be the most natural representation of cell aggregates and tissues. We also develop a Python package that implements GVM, relying on a graph-database-management framework Neo4j. We use this package to characterize an order-disorder transition in 3D cell aggregates, driven by active noise and we find aggregates undergoing efficient ordering close to the transition point. In all, our work showcases knowledge graphs as particularly suitable data models for structured storage, analysis, and manipulation of tissue data.

Incompatibility-driven growth and size control during development

Size regulation in living organisms is a major unsolved problem in developmental biology. This is due to the intrinsic complexity of biological growth, which simultaneously involves genetic, biochemical, and mechanical factors. In this article, we propose a novel theoretical framework that explores the role of incompatibility, the geometric source of residual stress in a growing body, as a possible regulator of size termination during development. We explore this paradigm both at the level of a model 2D cell, and at the level of continuous tissues. After establishing a parallel between incompatibility and the shape parameter of vertex models, we show that incompatibility-driven growth leads to size control in a model 2D cell. We then extend the same paradigm to the level of continuous bodies, where incompatibility is measured by the Ricci curvature of the growth tensor. By using the model 2D cell as a template, we now derive an evolutionary law for the growth tensor with curvature fixed at a physiological value. When the analysis is specialised to radial symmetry (discs and spheres), this model captures the salient features observed in Drosophila wing discs and multicellular spheroids: these systems have a target size and build up residual stresses that cause the tissue to open in response to a radial cut, with the cut edges curling outward. The theory proposed in this work suggests that incompatibility in a growing biological tissue is potentially controllable at the cell level, and that incompatibility-driven growth provides an effective method of controlling global information (stress, size) through local geometric controls.

Numerical assessment of the applicability of geometry-based force inference on homogeneous and heterogeneous cells.

The measurement of cellular forces, which reflect crucial biological attributes, has the potential to replace conventional cell assessment methods, such as morphology, proliferation, and molecular expression analysis, in medical cell diagnosis and cell culture studies. In medical cell evaluations, force inference techniques have gained prominence due to their non-invasiveness and lack of requirement for specialized equipment. Among those techniques, the method proposed by Ishihara et al., which estimates forces in densely packed cells based only on cell geometry, is a promising method. However, its applicability range of this method has not been fully established. In this study, we employed a two-dimensional vertex model to numerically assess the applicability of this method on homogeneous and heterogeneous cells. Our comparisons between the true values from numerical simulations and the estimated values from the inference method revealed a significant correlation between estimation accuracy and cell roundness in systems of homogeneous cell. Moreover, the method demonstrated efficient force estimations in heterogeneous-cell systems. These findings may be useful when the force inference method is employed to evaluate medical cells.

Topological balance of cell distributions in plane monolayers

Most of normal proliferative epithelia of plants and metazoans are topologically invariant and characterized by similar cell distributions according to the number of cell neighbors (DCNs). Here we study peculiarities of these distributions and explain why the DCN obtained from the location of intercellular boundaries and that based on the Voronoi tessellation with nodes located on cell nuclei may differ from each other. As we demonstrate, special microdomains where four or more intercellular boundaries converge are topologically charged. Using this fact, we deduce a new equation describing the topological balance of the DCNs. The developed theory is applied for a series of microphotographs of non-tumoral epithelial cells of the human cervix (HCerEpiC) to improve the image processing near the edges of microphotographs and reveal the topological invariance of the examined monolayers. Special contact microdomains may be present in epithelia of various natures, however, considering the well-known vertex model of epithelium, we show that such contacts are absent in the usual solid-like state of the model and appear only in the liquid-like cancer state. Also, we discuss a possible biological role of special contacts in context of proliferative epithelium dynamics and tissue morphogenesis.

Mean-field elastic moduli of a three-dimensional, cell-based vertex model

The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.

Orthogonal Polynomial Duality and Unitary Symmetries of Multi-species ASEP$$(q,\varvec{\theta })$$ and Higher-Spin Vertex Models via $$^*$$-Bialgebra Structure of Higher Rank Quantum Groups

Orthogonal Polynomial Duality and Unitary Symmetries of Multi-species ASEP$$(q,\varvec{\theta })$$ and Higher-Spin Vertex Models via $$^*$$-Bialgebra Structure of Higher Rank Quantum Groups

The Floquet Baxterisation

Quantum integrability has proven to be a useful tool to study quantum many-body systems out of equilibrium. In this paper we construct a generic framework for integrable quantum circuits through the procedure of Floquet Baxterisation. The integrability is guaranteed by establishing a connection between Floquet evolution operators and inhomogeneous transfer matrices obtained from the Yang-Baxter relations. This allows us to construct integrable Floquet evolution operators with arbitrary depths and various boundary conditions. Furthermore, we focus on the example related to the staggered 6-vertex model. In the scaling limit we establish a connection of this Floquet protocol with a non-rational conformal field theory. Employing the properties of the underlying affine Temperley-Lieb algebraic structure, we demonstrate the dynamical anti-unitary symmetry breaking in the easy-plane regime. We also give an overview of integrability-related quantum circuits, highlighting future research directions.

Emergence, Pattern, and Frequency of Spontaneous Waves in Spreading Epithelial Monolayers.

A striking phenomenon of collective cell motion is that they can exhibit a spontaneously emerging wave during epithelia expansions. However, the fundamental mechanism, governing the emergence and its crucial characteristics (e.g., the eigenfrequency and the pattern), remains an enigma. By introducing a mechanochemical feedback loop, we develop a highly efficient discrete vertex model to investigate the spatiotemporal evolution of spreading epithelia. We find both numerically and analytically that expanding cell monolayers display a power-law dependence of wave frequency on the local heterogeneities (i.e., cell density) with a scaling exponent of -1/2. Moreover, our study demonstrates the quantitative capability of the proposed model in capturing distinct X-, W-, and V-mode wave patterns. We unveil that the phase transition between these modes is governed by the distribution of active self-propulsion forces. Our work provides an avenue for rigorous quantitative investigations into the collective motion and pattern formation of cell groups.

Beyond windability: Approximability of the four-vertex model

We study the approximability of the four-vertex model, a special case of the six-vertex model. We prove that, despite being NP-hard to approximate in the worst case, the four-vertex model admits a fully polynomial randomized approximation scheme (FPRAS) when the input satisfies certain linear equation system over GF(2). The FPRAS is given by a Markov chain known as the worm process, whose state space and rapid mixing rely on the solution of the linear equation system. This is the first attempt to design an FPRAS for the six-vertex model with unwindable constraint functions. Additionally, we explore the applications of this technique on planar graphs, providing efficient sampling algorithms.

Choice of friction coefficient deeply affects tissue behaviour in stochastic epithelial vertex models.

To understand the mechanisms that coordinate the formation of biological tissues, the use of numerical implementations is necessary. The complexity of such models involves many assumptions and parameter choices that result in unpredictable consequences, obstructing the comparison with experimental data. Here, we focus on vertex models, a family of spatial models used extensively to simulate the dynamics of epithelial tissues. Usually, in the literature, the choice of the friction coefficient is not addressed using quasi-static deformation arguments that generally do not apply to realistic scenarios. In this manuscript, we discuss the role that the choice of friction coefficient has on the relaxation times and consequently in the conditions of cell cycle progression and division. We explore the effects that these changes have on the morphology, growth rate and topological transitions of the tissue dynamics. These results provide a deeper understanding of the role that an accurate mechanical description plays in the use of vertex models as inference tools. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Helical vasculogenesis driven by cell chirality.

The cellular helical structure is well known for its crucial role in development and disease. Nevertheless, the underlying mechanism governing this phenomenon remains largely unexplored, particularly in recapitulating it in well-controlled engineering systems. Leveraging advanced microfluidics, we present compelling evidence of the spontaneous emergence of helical endothelial tubes exhibiting robust right-handedness governed by inherent cell chirality. To strengthen our findings, we identify a consistent bias toward the same chirality in mouse vascular tissues. Manipulating endothelial cell chirality using small-molecule drugs produces a dose-dependent reversal of the handedness in engineered vessels, accompanied by non-monotonic changes in vascular permeability. Moreover, our three-dimensional cell vertex model provides biomechanical insights into the chiral morphogenesis process, highlighting the role of cellular torque and tissue fluidity in its regulation. Our study unravels an intriguing mechanism underlying vascular chiral morphogenesis, shedding light on the broader implications and distinctive perspectives of tubulogenesis within biological systems.

Discontinuous Shear Thickening in Biological Tissue Rheology.

During embryonic morphogenesis, tissues undergo dramatic deformations in order to form functional organs. Similarly, in adult animals, living cells and tissues are continually subjected to forces and deformations. Therefore, the success of embryonic development and the proper maintenance of physiological functions rely on the ability of cells to withstand mechanical stresses as well as their ability to flow in a collective manner. During these events, mechanical perturbations can originate from active processes at the single-cell level, competing with external stresses exerted by surrounding tissues and organs. However, the study of tissue mechanics has been somewhat limited to either the response to external forces or to intrinsic ones. In this work, we use an active vertex model of a 2D confluent tissue to study the interplay of external deformations that are applied globally to a tissue with internal active stresses that arise locally at the cellular level due to cell motility. We elucidate, in particular, the way in which this interplay between globally external and locally internal active driving determines the emergent mechanical properties of the tissue as a whole. For a tissue in the vicinity of a solid-fluid jamming or unjamming transition, we uncover a host of fascinating rheological phenomena, including yielding, shear thinning, continuous shear thickening, and discontinuous shear thickening. These model predictions provide a framework for understanding the recently observed nonlinear rheological behaviors in vivo.

Quantum solution of the relationship between the 19-vertex model and the Jones polynomial

The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.

Testing Structural Elements under Multiaxial Loading: A Numerical Model of the Bench to Understand and Predict Complex Boundary Conditions

Airworthiness certification requires proof of structure strength, which is performed generally through a building block approach. To achieve this, representative intermediate-scale experiments generated by test benches are, in general, needed, in addition to material characterization on a coupons scale and structure testing on a large scale. The VERTEX test bench can generate the combined loading of tension/compression-shear-pressure on structural elements and was modelled with Finite Elements to perform virtual testing, representative of its intermediate-scale specificity. The numerous bolted joints of the bench were modelled and their behavior was identified in previous tests, so the model could quantitatively estimate the transfer function of the bench, which is the relationship between the displacements imposed by the jacks and the resulting loads on a given sample. The VERTEX model was identified to represent load shapes and amplitudes based on a training set and was later confronted by a validation set of tests of tension and shear. A model with ideal boundary conditions was also developed for a comparison, but it failed to predict some load shape specificities and did not give any indication of the loading amplitude. Application cases of the developed model are shown to assess a range of virtual testing possibilities.

Modified rational six vertex model on the rectangular lattice

We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified Izergin determinant. The proofs are based on the quantum inverse scattering method and its representation theory together with elementary linear algebra.

Differences in boundary behavior in the 3D vertex and Voronoi models.

An important open question in the modeling of biological tissues is how to identify the right scale for coarse-graining, or equivalently, the right number of degrees of freedom. For confluent biological tissues, both vertex and Voronoi models, which differ only in their representation of the degrees of freedom, have effectively been used to predict behavior, including fluid-solid transitions and cell tissue compartmentalization, which are important for biological function. However, recent work in 2D has hinted that there may be differences between the two models in systems with heterotypic interfaces between two tissue types, and there is a burgeoning interest in 3D tissue models. Therefore, we compare the geometric structure and dynamic sorting behavior in mixtures of two cell types in both 3D vertex and Voronoi models. We find that while the cell shape indices exhibit similar trends in both models, the registration between cell centers and cell orientation at the boundary are significantly different between the two models. We demonstrate that these macroscopic differences are caused by changes to the cusp-like restoring forces introduced by the different representations of the degrees of freedom at the boundary, and that the Voronoi model is more strongly constrained by forces that are an artifact of the way the degrees of freedom are represented. This suggests that vertex models may be more appropriate for 3D simulations of tissues with heterotypic contacts.

Limit shapes from harmonicity: dominos and the five vertex model

We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and five-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of Kenyon and Prause (2022 Duke Math J. 171 3003–22).