Off-lattice Model Research Articles

Polar flocks with discretized directions: The active clock model approaching the Vicsek model

We consider the off-lattice two-dimensional q-state active clock model (ACM) as a natural discretization of the Vicsek model (VM) describing flocking. The ACM consists of particles able to move in the plane in a discrete set of q equidistant angular directions, as in the active Potts model (APM), with an alignment interaction inspired by the ferromagnetic equilibrium clock model. We find that for a small number of directions, the flocking transition of the ACM has the same phenomenology as the APM, including macrophase separation and reorientation transition. For a larger number of directions, the flocking transition in the ACM becomes equivalent to the one of the VM and displays microphase separation and only transverse bands, i.e., no re-orientation transition. Concomitantly also the transition of the q → ∞ limit of the ACM, the active XY model (AXYM), is in the same universality class as the VM. We also construct a coarse-grained hydrodynamic description for the ACM and AXYM akin to the VM.

Investigation of the conformational space of hydrophobic-polar heteropolymers by gyration tensor based parameters

• Different folding types such as two-state folding, folding through intermediates, as well as metastability, are presented with an effective order parameter for polymers. • Gyration tensor-based analysis does not need any reference state before the simulation as that is the need for the other order parameters which are chosen in other studies in the same content. • Our paper presents the use of very useful parameters to detect conformational phase transitions in polymer chains. In the earlier works related to this aspect presents another order parameter that needs a reference conformation. Multicanonical simulations of hydrophobic-polar heteropolymers were performed with a simple and effective coarse-grained off-lattice model to identify the folding pathways and the topology of the energy landscape with gyration tensor-based parameters. We examined six exemplified sequences of heteropolymers and illustrated three of them which present definitely different folding types such as two-state folding, folding through intermediates, as well as metastability. The major advantage of this study is our chosen ”order” parameter, namely gyration tensor-based analysis, does not require any reference state before the simulation as that is the need for the other order parameters which are chosen in other studies in the same context. Our paper presents the use of very useful parameters to detect conformational phase transitions in polymer chains. Another order parameter that requires a reference conformation is presented in earlier works related to this aspect. This extends the simulation into two steps. With our analysis, there is no need for a global minimum conformation so-called reference state and our simulation will be completed in one step. Furthermore, there is no relative evaluation according to the reference state.

Interconversion-controlled liquid-liquid phase separation in a molecular chiral model.

Liquid-liquid phase separation of fluids exhibiting interconversion between alternative states has been proposed as an underlying mechanism for fluid polyamorphism and may be of relevance to the protein function and intracellular organization. However, molecular-level insight into the interplay between competing forces that can drive or restrict phase separation in interconverting fluids remains elusive. Here, we utilize an off-lattice model of enantiomers with tunable chiral interconversion and interaction properties to elucidate the physics underlying the stabilization and tunability of phase separation in fluids with interconverting states. We show that introducing an imbalance in the intermolecular forces between two enantiomers results in nonequilibrium, arrested phase separation into microdomains. We also find that in the equilibrium case, when all interaction forces are conservative, the growth of the phase domain is restricted only by the system size. In this case, we observe phase amplification, in which one of the two alternative phases grows at the expense of the other. These findings provide novel insights on how the interplay between dynamics and thermodynamics defines the equilibrium and steady-state morphologies of phase transitions in fluids with interconverting molecular or supramolecular states.

Capacitive energy storage in single-file pores: Exactly solvable models and simulations.

Understanding charge storage in low-dimensional electrodes is crucial for developing novel ecologically friendly devices for capacitive energy storage and conversion and water desalination. Exactly solvable models allow in-depth analyses and essential physical insights into the charging mechanisms. So far, however, such analytical approaches have been mainly limited to lattice models. Herein, we develop a versatile, exactly solvable, one-dimensional off-lattice model for charging single-file pores. Unlike the lattice model, this model shows an excellent quantitative agreement with three-dimensional Monte Carlo simulations. With analytical calculations and simulations, we show that the differential capacitance can be bell-shaped (one peak), camel-shaped (two peaks), or have four peaks. Transformations between these capacitance shapes can be induced by changing pore ionophilicity, by changing cation-anion size asymmetry, or by adding solvent. We find that the camel-shaped capacitance, characteristic of dilute electrolytes, appears for strongly ionophilic pores with high ion densities, which we relate to charging mechanisms specific to narrow pores. We also derive a large-voltage asymptotic expression for the capacitance, showing that the capacitance decays to zero as the inverse square of the voltage, C ∼ u-2. This dependence follows from hard-core interactions and is not captured by the lattice model.

A rigid body framework for multicellular modeling.

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

Off-lattice and parallel implementations of the pivot algorithm

The pivot algorithm is the most efficient known method for sampling polymer configurations for self-avoiding walks and related models. Here we introduce two recent improvements to an efficient binary tree implementation of the pivot algorithm: an extension to an off-lattice model, and a parallel implementation.

Multiple metastable states in an off-lattice Potts model

The interactions between a group of components are commonly studied in several areas of science (social science, biology, material science, complex dynamical systems, among others) using the methods of thermodynamics and statistical mechanics. In this work we study the properties of the recently proposed off-lattice, two-dimensional Potts model (Davis et al., 2014), originally motivated by the dynamics of agent opinions, and which is described by a Hamiltonian obtained by a maximum entropy inference procedure. We performed microcanonical and canonical Monte Carlo simulations of the first-order phase transition in the model, revealing a caloric curve with metastable regions. Furthermore, we report a “switching” behavior between multiple metastable states. We also note that the thermodynamics of the model has striking similarities with systems having long-range interactions, even though the interactions are short-ranged.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration.

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Chemical Kinetics Simulations of Ice Chemistry on Porous Versus Non-Porous Dust Grains

The degree of porosity in interstellar dust-grain material is poorly defined, although recent work has suggested that the grains could be highly porous. Aside from influencing the optical properties of the dust, porosity has the potential to affect the chemistry occurring on dust-grain surfaces, via increased surface area, enhanced local binding energies, and the possibility of trapping of molecules within the pores as ice mantles build up on the grains. Through computational kinetics simulations, we investigate how interstellar grain-surface chemistry and ice composition are affected by the porosity of the underlying dust-grain material. Using a simple routine, idealized three-dimensional dust-grains are constructed, atom by atom, with varying degrees of porosity. Diffusive chemistry is then simulated on these surfaces using the off-lattice microscopic Monte Carlo chemical kinetics model, MIMICK, assuming physical conditions appropriate to dark interstellar clouds. On the porous grain surface, the build-up of ice mantles, mostly composed of water, leads to the covering over of the pores, leaving empty pockets. Once the pores are completely covered, the chemical and structural behavior is similar to non-porous grains of the same size. The most prominent chemical effect of the presence of grain porosity is the trapping of molecular hydrogen, formed on the grain surfaces, within the ices and voids inside the grain pores. Trapping of H2 in this way may indicate that other volatiles, such as inert gases not included in these models, could be trapped within dust-grain porous structures when ices begin to form.

Connectedness percolation in the random sequential adsorption packings of elongated particles.

Connectedness percolation phenomena in the two-dimensional packing of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption off-lattice models with preferential orientations of the particles along a given direction. The partial ordering was characterized by the order parameter S, with S=0 for completely disordered films (random orientation of particles) and S=1 for completely aligned particles along the horizontal direction x. The aspect ratio (length-to-width ratio) of the particles was varied within the range ɛ∈[1;100]. Analysis of connectivity was performed assuming a core-shell structure of the particles. The value of S affected the structure of the packings, the formation of long-range connectivity, and the behavior of the electrical conductivity. The effects can be explained by taking accounting of the competition between the particles' orientational degrees of freedom and excluded volume effects. For aligned deposition, anisotropy in the electrical conductivity was observed with the values along the alignment direction σ_{x} being larger than the values in the perpendicular direction σ_{y}. Anisotropy in the localization of the percolation threshold was also observed in finite-sized packings, but it disappeared in the limit of infinitely large systems.

Critical behavior in active lattice models of motility-induced phase separation

Lattice models allow for a computationally efficient investigation of motility-induced phase separation (MIPS) compared to off-lattice systems. Simulations are less demanding, and thus, bigger systems can be accessed with higher accuracy and better statistics. In equilibrium, lattice and off-lattice models with comparable interactions belong to the same universality class. Whether concepts of universality also hold for active particles is still a controversial and open question. Here, we examine two recently proposed active lattice systems that undergo MIPS and investigate numerically their critical behavior. In particular, we examine the claim that these systems and MIPS in general belong to the Ising universality class. We also take a more detailed look on the influence and role of rotational diffusion and active velocity in these systems.Graphic

Mobility driven coexistence of living organisms

We propose a minimal off-lattice model of living organisms where just a very few dynamical rules of growth are assumed. The stable coexistence of many clusters is detected when we replace the global restriction rule by a locally applied one. A rich variety of evolving patterns is revealed where players movement has a decisive role on the evolutionary outcome. For example, intensive individual mobility may jeopardize the survival of the population, but if we increase players movement further then it can save the population. Notably, the collective drive of population members is capable to compensate the negative consequence of intensive movement and keeps the system alive. When the drive becomes biased then the resulting unidirectional flow alters the stable pattern and produce a stripe-like state instead of the previously observed hexagonal arrangement of clusters. Interestingly, the rotation of stripes can be flipped if the individual movement exceeds a threshold value.

Machine-learned interatomic potentials for alloys and alloy phase diagrams

We introduce machine-learned potentials for Ag-Pd to describe the energy of alloy configurations over a wide range of compositions. We compare two different approaches. Moment tensor potentials (MTPs) are polynomial-like functions of interatomic distances and angles. The Gaussian approximation potential (GAP) framework uses kernel regression, and we use the smooth overlap of atomic position (SOAP) representation of atomic neighborhoods that consist of a complete set of rotational and permutational invariants provided by the power spectrum of the spherical Fourier transform of the neighbor density. Both types of potentials give excellent accuracy for a wide range of compositions, competitive with the accuracy of cluster expansion, a benchmark for this system. While both models are able to describe small deformations away from the lattice positions, SOAP-GAP excels at transferability as shown by sensible transformation paths between configurations, and MTP allows, due to its lower computational cost, the calculation of compositional phase diagrams. Given the fact that both methods perform nearly as well as cluster expansion but yield off-lattice models, we expect them to open new avenues in computational materials modeling for alloys.

Effect of deposition, detachment and aggregation processes on nanoparticle transport in porous media using Monte Carlo simulations

A novel off-lattice three-dimensional coarse-grained Monte Carlo model is developed to study engineered nanoparticle (ENP) behavior in porous media.

An off-lattice model of the Sanchez-Lacombe Eq. of state for polymers with finite flexibility

An off-lattice Sanchez-Lacombe equation of state is extended to include finite flexibility in polymer molecules using a two-energy state model that is described by two empirical parameters: the degeneracy of excited states and their energy. A protocol for determining these parameters from the glass transition temperature and heat capacity is given. The model shows that the Gibbs-DiMarzio condition for glass transition is an artifact of the lattice for Sanchez-Lacombe models. A new condition based on a fraction of maximum entropy is suggested and fixed by minimizing the degeneracy of excited states. The theory is in very good agreement with the experimental results of five pure polymers. It predicts that the glass transition temperatures behaviours of binary mixtures as predicted by a lattice-based theory from literature are incorrect at low temperatures. Instead, two new glass transition temperature behaviours are identified including retrograde vitrification behaviour in poly(methyl-methacrylate) – carbon dioxide mixture.

Kinetics of domain growth and aging in a two-dimensional off-lattice system.

We have used molecular dynamics simulations for a comprehensive study of phase separation in a two-dimensional single-component off-lattice model where particles interact through the Lennard-Jones potential. Via state-of-the-art methods we have analyzed simulation data on structure, growth, and aging for nonequilibrium evolutions in the model. These data were obtained following quenches of well-equilibrated homogeneous configurations, with density close to the critical value, to various temperatures inside the miscibility gap, having vapor-"liquid" as well as vapor-"solid" coexistence. For the vapor-liquid phase separation we observe that ℓ, the average domain length, grows with time (t) as t^{1/2}, a behavior that has connection with hydrodynamics. At low-enough temperature, a sharp crossover of this time dependence to a much slower, temperature-dependent, growth is identified within the timescale of our simulations, implying "solid"-like final state of the high-density phase. This crossover is, interestingly, accompanied by strong differences in domain morphology and other structural aspects between the two situations. For aging, we have presented results for the order-parameter autocorrelation function. This quantity exhibits data collapse with respect to ℓ/ℓ_{w},ℓ, and ℓ_{w} being the average domain lengths at times t and t_{w} (≤t), respectively, the latter being the age of a system. Corresponding scaling function follows a power-law decay: ∼(ℓ/ℓ_{w})^{-λ} for t≫t_{w}. The decay exponent λ, for the vapor-liquid case, is accurately estimated via the application of an advanced finite-size scaling method. The obtained value is observed to satisfy a bound.

Exploration of lattice Hamiltonians for functional and structural discovery via Gaussian process-based exploration–exploitation

Statistical physics models ranging from simple lattice to complex quantum Hamiltonians are one of the mainstays of modern physics that have allowed both decades of scientific discovery and provided a universal framework to understand a broad range of phenomena from alloying to frustrated and phase separated materials to quantum systems. Traditionally, exploration of the phase diagrams corresponding to multidimensional parameter spaces of Hamiltonians was performed using a combination of basic physical principles, analytical approximations, and extensive numerical modeling. However, exploration of complex multidimensional parameter spaces is subject to the classic dimensionality problem, and the behaviors of interest concentrated on low dimensional manifolds remain undiscovered. Here, we demonstrate that a combination of exploration and exploration–exploitation with Gaussian process modeling and Bayesian optimization allows effective exploration of the parameter space for lattice Hamiltonians and effectively maps the regions at which specific macroscopic functionalities or local structures are maximized. We argue that this approach is general and can be further extended well beyond the lattice Hamiltonians to effectively explore the parameter space of more complex off-lattice and dynamic models.

Mathematical modelling reveals cellular dynamics within tumour spheroids.

Tumour spheroids are widely used as an in vitro assay for characterising the dynamics and response to treatment of different cancer cell lines. Their popularity is largely due to the reproducible manner in which spheroids grow: the diffusion of nutrients and oxygen from the surrounding culture medium, and their consumption by tumour cells, causes proliferation to be localised at the spheroid boundary. As the spheroid grows, cells at the spheroid centre may become hypoxic and die, forming a necrotic core. The pressure created by the localisation of tumour cell proliferation and death generates an cellular flow of tumour cells from the spheroid rim towards its core. Experiments by Dorie et al. showed that this flow causes inert microspheres to infiltrate into tumour spheroids via advection from the spheroid surface, by adding microbeads to the surface of tumour spheroids and observing the distribution over time. We use an off-lattice hybrid agent-based model to re-assess these experiments and establish the extent to which the spatio-temporal data generated by microspheres can be used to infer kinetic parameters associated with the tumour spheroids that they infiltrate. Variation in these parameters, such as the rate of tumour cell proliferation or sensitivity to hypoxia, can produce spheroids with similar bulk growth dynamics but differing internal compositions (the proportion of the tumour which is proliferating, hypoxic/quiescent and necrotic/nutrient-deficient). We use this model to show that the types of experiment conducted by Dorie et al. could be used to infer spheroid composition and parameters associated with tumour cell lines such as their sensitivity to hypoxia or average rate of proliferation, and note that these observations cannot be conducted within previous continuum models of microbead infiltration into tumour spheroids as they rely on resolving the trajectories of individual microbeads.

Phase behavior of continuous-space systems: A supervised machine learning approach.

The phase behavior of complex fluids is a challenging problem for molecular simulations. Supervised machine learning (ML) methods have shown potential for identifying the phase boundaries of lattice models. In this work, we extend these ML methods to continuous-space systems. We propose a convolutional neural network model that utilizes grid-interpolated coordinates of molecules as input data of ML and optimizes the search for phase transitions with different filter sizes. We test the method for the phase diagram of two off-lattice models, namely, the Widom-Rowlinson model and a symmetric freely jointed polymer blend, for which results are available from standard molecular simulations techniques. The ML results show good agreement with results of previous simulation studies with the added advantage that there is no critical slowing down. We find that understanding intermediate structures near a phase transition and including them in the training set is important to obtain the phase boundary near the critical point. The method is quite general and easy to implement and could find wide application to study the phase behavior of complex fluids.

Modelling collective cell motion: are on- and off-lattice models equivalent?

Biological processes, such as embryonic development, wound repair and cancer invasion, or bacterial swarming and fruiting body formation, involve collective motion of cells as a coordinated group. Collective cell motion of eukaryotic cells often includes interactions that result in polar alignment of cell velocities, while bacterial patterns typically show features of apolar velocity alignment. For analysing the population-scale effects of these different alignment mechanisms, various on- and off-lattice agent-based models have been introduced. However, discriminating model-specific artefacts from general features of collective cell motion is challenging. In this work, we focus on equivalence criteria at the population level to compare on- and off-lattice models. In particular, we define prototypic off- and on-lattice models of polar and apolar alignment, and show how to obtain an on-lattice from an off-lattice model of velocity alignment. By characterizing the behaviour and dynamical description of collective migration models at the macroscopic level, we suggest the type of phase transitions and possible patterns in the approximative macroscopic partial differential equation descriptions as informative equivalence criteria between on- and off-lattice models. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.