Vertex Model Research Articles

A mechanistic model of the organization of cell shapes in epithelial tissues

The organization of cells within tissues plays a vital role in various biological processes, including development and morphogenesis. As a result, understanding how cells self-organize in tissues has been an active area of research. In our study, we explore a mechanistic model of cellular organization that represents cells as force dipoles that interact with each other via the tissue, which we model as an elastic medium. By conducting numerical simulations using this model, we are able to observe organizational features that are consistent with those obtained from vertex model simulations. This approach provides valuable insights into the underlying mechanisms that govern cellular organization within tissues, which can help us better understand the processes involved in development and disease.

Continuum theory for confluent cell monolayers: Interplay between cell growth, division, and intercalation

Mechanical forces generated by dynamic cellular activities play a crucial role in the morphogenesis and growth of biological tissues. While the influence of mechanics is clear, many questions arise regarding the way by which mechanical forces communicate with biological processes at the level of a confluent cell population. Some answers may be found in the development of mathematical models that are capable of describing the emerging behavior of a large population of active agents based on individualistic rules (single-cell response). In this perspective, the present work presents a continuum-scale model that can capture, in an average sense, the active mechanics and evolution of a confluent tissue with or without external mechanical constraints. For this, we conceptualize a confluent cell population (in a monolayer) as a deformable dynamic network, where a single cell can modify the topology of its neighborhood by swapping neighbors or dividing. With this description, we use concepts from statistical mechanics and the transient network theory to derive an equivalent active visco-elastic continuum model, which can recapitulate some of the salient features of the underlying network at the macroscale. Without loss of generality, the cell network is here assumed to follow well-known rules used in vertex model simulations, which are: (a) cell elasticity based on its bulk and cortical elasticity, (b) cell intercalation (or T1 transition), and (c) cell proliferation (expansion and division). We show, through examples and illustrations, that the model is able to characterize complex cross-talk between mechanical forces and biological processes, which are likely to drive the emergent growth and deformation of cell aggregates.

Finite temperature properties of an integrable zigzag ladder chain

We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum transfer matrix approach for the face model version of the six-vertex model. The integrable quantum Hamiltonian shares some thermodynamical properties with the Heisenberg XXZ chain, but has different ordering and critical exponents. Two gapped phases are the dimerized antiferromagnetic order and the usual antiferromagnetic (Néel) order for positive nearest neighbor Ising coupling. In between these, there is an extended critical region, which is a quantum spin-liquid with broken parity symmetry inducing an oscillatory behavior at the long distance <σ1zσℓ+1z> correlation. At finite temperatures, the numerical solution of the non-linear integral equations allows for the determination of the correlation length as well as for the momentum of the oscillation.

Correlation inequalities for the uniform eight-vertex model and the toric code model

We investigate connections between four models in statistical physics and probability theory: (1) the toric code model of Kitaev, (2) the uniform eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising model with four-body interaction. As a consequence of our analysis (and of the GKS-inequalities for the Ising model) we obtain correlation inequalities for the toric code model and the uniform eight-vertex model.

Topological properties and shape of proliferative and nonproliferative cell monolayers.

During embryonic development, structures with complex geometry can emerge from planar epithelial monolayers; studying these shape transitions is of key importance for revealing the biophysical laws involved in the morphogenesis of biological systems. Here, using the example of normal proliferative monkey kidney (COS) cell monolayers, we investigate global and local topological characteristics of this model system in dependence on its shape. The obtained distributions of cells by their valence demonstrate a difference between the spherical and planar monolayers. In addition, in both spherical and planar monolayers, the probability of observing a pair of neighboring cells with certain valences depends on the topological charge of the pair. The zero topological charge of the cell pair can increase the probability for the cells to be the nearest neighbors. We then test and confirm that analogous relationships take place in a more ordered spherical system with a larger fraction of 6-valent cells, namely, in the nonproliferative epithelium (follicular system) of ascidian species oocytes. However, unlike spherical COS cell monolayers, ascidian monolayers are prone to nonrandom agglomeration of 6-valent cells and have linear topological defects called scars and pleats. The reasons for this difference in morphology are discussed. The morphological peculiarities found are compared with predictions of the widely used vertex model of epithelium.

Colored fermionic vertex models and symmetric functions

Colored fermionic vertex models and symmetric functions

The mechanics of cephalic furrow formation in the Drosophila embryo

Cephalic furrow formation (CFF) is a major morphogenetic movement during gastrulation in Drosophila melanogaster embryos that gives rise to a deep, transitory epithelial invagination. Recent studies have identified the individual cell shape changes that drive the initiation and progression phases of CFF; however, the underlying mechanics are not yet well understood. During the progression phase, the furrow deepens as columnar cells from both the anterior and posterior directions fold inwards rotating by 90°. To analyze the mechanics of this process, we have developed an advanced two-dimensional lateral vertex model that includes multinode representation of cellular membranes and allows us to capture the membrane curvature associated with pressure variation. Our investigations reveal some key potential mechanical features of CFF, as follows. When cells begin to roll over the cephalic furrow cleft, they become wedge shaped as their apical cortices and overlying membranes expand, lateral cortices and overlying membranes release tension, internal pressures drop, and basal cortices and membranes contract. Then, cells reverse this process by shortening apical cortices and membranes, increasing lateral tension, and causing internal pressures to rise. Since the basal membranes expand, the cells recover their rotated columnar shape once in the furrow. Interestingly, our findings indicate that the basal membranes may be passively reactive throughout the progression phase. We also find that the smooth rolling of cells over the cephalic furrow cleft necessitates that internalized cells provide a solid base through high levels of membrane tension and internal pressure, which allows the transmission of tensile force that pulls new cells into the furrow. These results lead us to suggest that CFF helps to establish a baseline tension across the apical surface of the embryo to facilitate cellular coordination of other morphogenetic movements via mechanical stress feedback mechanisms.

Single and two-cells shape analysis from energy functionals for three-dimensional vertex models.

Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.

An integrated vertex model of the mesoderm invagination during the embryonic development of Drosophila

The mesoderm invagination of the Drosophila embryo is known as an archetypal morphogenic process. To explore the roles of the active cellular forces and the regulation of these forces, we developed an integrated vertex model that combines the regulation of morphogen expression with cell movements and tissue mechanics. Our results suggest that a successful furrow formation requires an apical tension gradient, decreased basal tension, and increased lateral tension, which corresponds to apical constriction, basal expansion, and apicobasal shortening respectively. Our model also considers the mechanical feedback which leads to an ectopic twist expression with external compression as observed in experiments. Our model predicts that ectopic invagination could happen if an external compressive gradient is applied.

Integrability of planar-algebraic models

The quantum inverse scattering method is a scheme for solving integrable models in dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra , at least for . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.

Construction of determinants for the six-vertex model with domain wall boundary conditions

We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions that depend on two sets of spectral parameters. In the pioneering works of Korepin and Izergin a determinant formula was proposed and proved using a recursion relation. In later works, equivalent determinant formulas were given by Kostov for the rational case and by Foda and Wheeler for the trigonometric case. Here, we develop an approach in which the recursion relation is replaced by a system of algebraic equations with respect to one of the two sets of spectral parameters. We prove that this system has a unique solution. The result can be easily given as a determinant parametrized by an arbitrary basis of polynomials. In particular, the choice of the basis of Lagrange polynomials with respect to the remaining set of spectral parameters leads to the Izergin–Korepin representation, and the choice of the monomial basis leads to the Kostov and Foda–Wheeler representations.

Complexity classification of the eight-vertex model

We prove a complexity dichotomy theorem for the eight-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit. For tractability, we find some new classes of problems computable in polynomial time. For #P-hardness, we employ Möbius transformations to prove the success of interpolations.

On the scaling behaviour of an integrable spin chain with [formula omitted] symmetry

The subject matter of this work is a 1D quantum spin-12 chain associated with the inhomogeneous six-vertex model possessing an additional Zr symmetry. The model is studied in a certain parametric domain, where it is critical. Within the ODE/IQFT approach, a class of ordinary differential equations and a quantization condition are proposed which describe the scaling limit of the system. Some remarkable features of the CFT underlying the critical behaviour are observed. Among them is an infinite degeneracy of the conformal primary states and the presence of a continuous component in the spectrum in the case of even r.

Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.

New Solutions to the Tetrahedron Equation Associated with Quantized Six-Vertex Models

We present a family of new solutions to the tetrahedron equation of the form RLLL=LLLR, where L operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the q-oscillator or q-Weyl algebras. When the three L’s are associated with the q-oscillator algebra, R coincides with the known intertwiner of the quantized coordinate ring A_q(sl_3). On the other hand, L’s based on the q-Weyl algebra lead to new R’s whose elements are either factorized or expressed as a terminating q-hypergeometric type series.

Collective migration of cells in geometric spaces: Intrinsic correlation length racing against extrinsic confinement size

Many experiments showed that cell groups can exhibit various impressive motion modes (e.g., global rotation and local swirling) in different-sized geometrical spaces, with the underlying mechanisms still unclear. However, recent experiments showed that different types of cells perform distinct motion modes (i.e., global rotation and coherent oscillation) in the same constraint conditions, which brings further puzzles on the movements of collective cells. Here, by comprehensively considering the social interactions between adjacent cells and internal mechanical processes of the cell itself, we propose an active vertex model to bring physical insights into collective cell migration in geometrical constraints. Using this model, we can reproduce all reported motion modes of cells in circular constraints observed in existing experiments. Specifically, as the confinement size decreases, local swirling and global rotation modes successively appear in cell groups with finite correlation length, while eccentric rotation and coherent oscillation modes successively emerge in cell groups with scale-free correlations. We demonstrate that these distinct modes are subtly coded by the intrinsic correlation length of cell groups and the extrinsic confinement size. Unexpectedly, we discover that, in small-size confinements, cell groups with scale-free correlations can spontaneously transition from eccentric rotation into coherent oscillation modes, because the strong social interactions reconcile the movements of all cells. In addition, we propose a velocity indicator that can directly distinguish migration modes emerging in small-scale confinements. These findings are in broad agreement with many experiments, and shed light on the spatiotemporal dynamics of active matter.

Exact results for the residual entropy of ice hexagonal monolayer.

Since the problem of the residual entropy of square ice was exactly solved, exact solutions for two-dimensional realistic ice models have been of interest. In this work, we study the exact residual entropy of ice hexagonal monolayer in two cases. In the case that the external electric field along the z-axis exists, we map the hydrogen configurations into the spin configurations of the Ising model on the kagome lattice. By taking the low temperature limit of the Ising model, we derive the exact residual entropy, which agrees with the result determined previously from the dimer model on the honeycomb lattice. In another case that the ice hexagonal monolayer is under the periodic boundary conditions in the cubic ice lattice, the residual entropy has not been studied exactly. For this case, we employ the six-vertex model on the square lattice to represent the hydrogen configurations obeying the ice rules. The exact residual entropy is obtained from the solution of the equivalent six-vertex model. Our work provides more examples of the exactly soluble two-dimensional statistical models.

Arctic curves of the 20V model on a triangle

We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Twenty Vertex model with specific domain wall boundary conditions on a triangle, in the Disordered phase, leading to a phase diagram with six types of frozen phases and one liquid one. The result relies on a relation to the Six Vertex model with domain wall boundary conditions and suitable weights, as a consequence of integrability. We also perform the exact refined enumeration of configurations.

The frustration-free fully packed loop model

We consider a quantum fully packed loop model on the square lattice with a frustration-free projector Hamiltonian and ring-exchange interactions acting on plaquettes. A boundary Hamiltonian is added to favor domain-wall boundary conditions and link ground state properties to the combinatorics and six-vertex model literature. We discuss how the boundary term fractures the Hilbert space into Krylov subspaces, and we prove that the Hamiltonian is ergodic within each subspace, leading to a series of energy-equidistant exact eigenstates in the lower end of the spectrum. Among them we systematically classify both finitely entangled eigenstates and product eigenstates. Using a recursion relation for enumerating half-plane configurations, we compute numerically the exact entanglement entropy of the ground state, confirming area law scaling. Finally, the spectrum is shown to be gapless in the thermodynamic limit with a trial state constructed by adding a twist to the ground state superposition.

Emergence of Spatial Scales and Macroscopic Tissue Dynamics in Active Epithelial Monolayers

Migrating cells in tissues are often known to exhibit collective swirling movements. In this paper, we develop an active vertex model with polarity dynamics based on contact inhibition of locomotion (CIL). We show that under this dynamics, the cells form steady-state vortices in velocity, polarity, and cell stress with length scales that depend on polarity alignment rate (ζ), self-motility (v0), and cell-cell bond tension (λ). When the ratio λ/v0 becomes larger, the tissue reaches a near jamming state because of the inability of the cells to exchange their neighbors, and the length scale associated with tissue kinematics increases. A deeper examination of this jammed state provides insights into the mechanism of sustained swirl formation under CIL rule that is governed by the feedback between cell polarities and deformations. To gain additional understanding of how active forcing governed by CIL dynamics leads to large-scale tissue dynamics, we systematically coarse-grain cell stress, polarity, and motility and show that the tissue remains polar even on larger length scales. Overall, we explore the origin of swirling patterns during collective cell migration and obtain a connection between cell-level dynamics and large-scale cellular flow patterns observed in epithelial monolayers.