Modeling cell rheology with the Subcellular Element Model

A three-dimensional computational model of collagen network mechanics

Extracellular matrix (ECM) strongly influences cellular behaviors, including cell proliferation, adhesion, and particularly migration. In cancer, the rigidity of the stromal collagen environment is thought to control tumor aggressiveness, and collagen alignment has been linked to tumor cell invasion. While the mechanical properties of collagen at both the single fiber scale and the bulk gel scale are quite well studied, how the fiber network responds to local stress or deformation, both structurally and mechanically, is poorly understood. This intermediate scale knowledge is important to understanding cell-ECM interactions and is the focus of this study. We have developed a three-dimensional elastic collagen fiber network model (bead-and-spring model) and studied fiber network behaviors for various biophysical conditions: collagen density, crosslinker strength, crosslinker density, and fiber orientation (random vs. prealigned). We found the best-fit crosslinker parameter values using shear simulation tests in a small strain region. Using this calibrated collagen model, we simulated both shear and tensile tests in a large linear strain region for different network geometry conditions. The results suggest that network geometry is a key determinant of the mechanical properties of the fiber network. We further demonstrated how the fiber network structure and mechanics evolves with a local formation, mimicking the effect of pulling by a pseudopod during cell migration. Our computational fiber network model is a step toward a full biomechanical model of cellular behaviors in various ECM conditions.

Three-dimensional multi-scale model of deformable platelets adhesion to vessel wall in blood flow

When a blood vessel ruptures or gets inflamed, the human body responds by rapidly forming a clot to restrict the loss of blood. Platelets aggregation at the injury site of the blood vessel occurring via platelet-platelet adhesion, tethering and rolling on the injured endothelium is a critical initial step in blood clot formation. A novel three-dimensional multi-scale model is introduced and used in this paper to simulate receptor-mediated adhesion of deformable platelets at the site of vascular injury under different shear rates of blood flow. The novelty of the model is based on a new approach of coupling submodels at three biological scales crucial for the early clot formation: novel hybrid cell membrane submodel to represent physiological elastic properties of a platelet, stochastic receptor-ligand binding submodel to describe cell adhesion kinetics and lattice Boltzmann submodel for simulating blood flow. The model implementation on the GPU cluster significantly improved simulation performance. Predictive model simulations revealed that platelet deformation, interactions between platelets in the vicinity of the vessel wall as well as the number of functional GPIbα platelet receptors played significant roles in platelet adhesion to the injury site. Variation of the number of functional GPIbα platelet receptors as well as changes of platelet stiffness can represent effects of specific drugs reducing or enhancing platelet activity. Therefore, predictive simulations can improve the search for new drug targets and help to make treatment of thrombosis patient-specific.

A new model for cell division and migration with spontaneous topology changes

Tissue topology, in particular proliferating epithelium topology, is remarkably similar between various species. Understanding the mechanisms that result in the observed topologies is needed for better insight into the processes governing tissue formation. We present a two-dimensional single-cell based model for cell divisions and tissue growth. The model accounts for cell mechanics and allows cell migration. Cells do not have pre-existing shapes or topologies. Shape changes and local rearrangements occur naturally as a response to the evolving cellular environment and cell-cell interactions. We show that the commonly observed tissue topologies arise spontaneously from this model. We consider different cellular rearrangements that accompany tissue growth and study their effects on tissue topology.

A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion

Cell-cell adhesion plays a key role in the collective migration of cells and in determining correlations in the relative cell positions and velocities. Recently, it was demonstrated that off-lattice individual cell based models (IBMs) can accurately capture the correlations observed experimentally in a migrating cell population. However, IBMs are often computationally expensive and difficult to analyse mathematically. Traditional continuum-based models, in contrast, are amenable to mathematical analysis and are computationally less demanding, but typically correspond to a mean-field approximation of cell migration and so ignore cell-cell correlations. In this work, we address this problem by using an off-lattice IBM to derive a continuum approximation which does take into account correlations. We furthermore show that a mean-field approximation of the off-lattice IBM leads to a single partial integro-differential equation of the same form as proposed by Sherratt and co-workers to model cell adhesion. The latter is found to be only effective at approximating the ensemble averaged cell number density when mechanical interactions between cells are weak. In contrast, the predictions of our novel continuum model for the time-evolution of the ensemble cell number density distribution and of the density-density correlation function are in close agreement with those obtained from the IBM for a wide range of mechanical interaction strengths. In particular, we observe 'front-like' propagation of cells in simulations using both our IBM and our continuum model, but not in the continuum model simulations obtained using the mean-field approximation.

Predictive modeling of post bioprinting structure formation

Cellular particle dynamics (CPD) is an effective computational method to describe the shape evolution and biomechanical relaxation processes in systems composed of micro tissues such as multicellular aggregates. Therefore, CPD is a useful tool to predict the outcome of postprinting structure formation in bioprinting. The predictive power of CPD has been demonstrated for multicellular systems composed of identical volume-conserving spherical and cylindrical bioink units. Experiments and computer simulations were related through an independently developed theoretical formalism based on continuum mechanics. Here we generalize the CPD formalism to (i) include non-identical bioink particles often used in specific bioprinting applications, (ii) describe the more realistic experimental situation in which during the post-printing structure formation via the fusion of spherical bioink units the volume of the system decreases, and (iii) directly connect CPD simulations to the corresponding experiments without the need of the intermediate continuum theory inherently based on simplifying assumptions.

The formation of tight tumor clusters affects the efficacy of cell cycle inhibitors: a hybrid model study

Cyclin-dependent kinases (CDKs) are vital in regulating cell cycle progression, and, thus, in highly proliferating tumor cells CDK inhibitors are gaining interest as potential anticancer agents. Clonogenic assay experiments are frequently used to determine drug efficacy against the survival and proliferation of cancer cells. While the anticancer mechanisms of drugs are usually described at the intracellular single-cell level, the experimental measurements are sampled from the entire cancer cell population. This approach may lead to discrepancies between the experimental observations and theoretical explanations of anticipated drug mechanisms. To determine how individual cell responses to drugs that inhibit CDKs affect the growth of cancer cell populations, we developed a spatially explicit hybrid agent-based model. In this model, each cell is equipped with internal cell cycle regulation mechanisms, but it is also able to interact physically with its neighbors. We model cell cycle progression, focusing on the G1 and G2/M cell cycle checkpoints, as well as on related essential components, such as CDK1, CDK2, cell size, and DNA damage. We present detailed studies of how the emergent properties (e.g., cluster formation) of an entire cell population depend on altered physical and physiological parameters. We analyze the effects of CDK1 and CKD2 inhibitors on population growth, time-dependent changes in cell cycle distributions, and the dynamic evolution of spatial cell patterns. We show that cell cycle inhibitors that cause cell arrest at different cell cycle phases are not necessarily synergistically super-additive. Finally, we demonstrate that the physical aspects of cell population growth, such as the formation of tight cell clusters versus dispersed colonies, alter the efficacy of cell cycle inhibitors, both in 2D and 3D simulations. This finding may have implications for interpreting the treatment efficacy results of in vitro experiments, in which treatment is applied before the cells can grow to produce clusters, especially because in vivo tumors, in contrast, form large masses before they are detected and treated.

A force based model of individual cell migration with discrete attachment sites and random switching terms

A force based model of cell migration is presented which gives new insight into the importance of the dynamics of cell binding to the substrate. The main features of the model are the focus on discrete attachment dynamics, the treatment of the cellular forces as springs, and an incorporation of the stochastic nature of the attachment sites. One goal of the model is to capture the effect of the random binding and unbinding of cell attachments on global cell motion. Simulations reveal one of the most important factor influencing cell speed is the duration of the attachment to the substrate. The model captures the correct velocity and force relationships for several cell types.

Bayesian inference of the lung alveolar spatial model for the identification of alveolar mechanics associated with acute respiratory distress syndrome

Acute respiratory distress syndrome (ARDS) is acute lung failure secondary to severe systemic inflammation, resulting in a derangement of alveolar mechanics (i.e. the dynamic change in alveolar size and shape during tidal ventilation), leading to alveolar instability that can cause further damage to the pulmonary parenchyma. Mechanical ventilation is a mainstay in the treatment of ARDS, but may induce mechano-physical stresses on unstable alveoli, which can paradoxically propagate the cellular and molecular processes exacerbating ARDS pathology. This phenomenon is called ventilator induced lung injury (VILI), and plays a significant role in morbidity and mortality associated with ARDS. In order to identify optimal ventilation strategies to limit VILI and treat ARDS, it is necessary to understand the complex interplay between biological and physical mechanisms of VILI, first at the alveolar level, and then in aggregate at the whole-lung level. Since there is no current consensus about the underlying dynamics of alveolar mechanics, as an initial step we investigate the ventilatory dynamics of an alveolar sac (AS) with the lung alveolar spatial model (LASM), a 3D spatial biomechanical representation of the AS and its interaction with airflow pressure and the surface tension effects of pulmonary surfactant. We use the LASM to identify the mechanical ramifications of alveolar dynamics associated with ARDS. Using graphical processing unit parallel algorithms, we perform Bayesian inference on the model parameters using experimental data from rat lung under control and Tween-induced ARDS conditions. Our results provide two plausible models that recapitulate two fundamental hypotheses about volume change at the alveolar level: (1) increase in alveolar size through isotropic volume change, or (2) minimal change in AS radius with primary expansion of the mouth of the AS, with the implication that the majority of change in lung volume during the respiratory cycle occurs in the alveolar ducts. These two model solutions correspond to significantly different mechanical properties of the tissue, and we discuss the implications of these different properties and the requirements for new experimental data to discriminate between the hypotheses.

Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution

Despite a growing wealth of available molecular data, the growth of tumors, invasion of tumors into healthy tissue, and response of tumors to therapies are still poorly understood. Although genetic mutations are in general the first step in the development of a cancer, for the mutated cell to persist in a tissue, it must compete against the other, healthy or diseased cells, for example by becoming more motile, adhesive, or multiplying faster. Thus, the cellular phenotype determines the success of a cancer cell in competition with its neighbors, irrespective of the genetic mutations or physiological alterations that gave rise to the altered phenotype. What phenotypes can make a cell "successful" in an environment of healthy and cancerous cells, and how? A widely used tool for getting more insight into that question is cell-based modeling. Cell-based models constitute a class of computational, agent-based models that mimic biophysical and molecular interactions between cells. One of the most widely used cell-based modeling formalisms is the cellular Potts model (CPM), a lattice-based, multi particle cell-based modeling approach. The CPM has become a popular and accessible method for modeling mechanisms of multicellular processes including cell sorting, gastrulation, or angiogenesis. The CPM accounts for biophysical cellular properties, including cell proliferation, cell motility, and cell adhesion, which play a key role in cancer. Multiscale models are constructed by extending the agents with intracellular processes including metabolism, growth, and signaling. Here we review the use of the CPM for modeling tumor growth, tumor invasion, and tumor progression. We argue that the accessibility and flexibility of the CPM, and its accurate, yet coarse-grained and computationally efficient representation of cell and tissue biophysics, make the CPM the method of choice for modeling cellular processes in tumor development.

Before the endless forms: embodied model of transition from single cells to aggregates to ecosystem engineering

The emergence of complex multicellular systems and their associated developmental programs is one of the major problems of evolutionary biology. The advantages of cooperation over individuality seem well known but it is not clear yet how such increase of complexity emerged from unicellular life forms. Current multicellular systems display a complex cell-cell communication machinery, often tied to large-scale controls of body size or tissue homeostasis. Some unicellular life forms are simpler and involve groups of cells cooperating in a tissue-like fashion, as it occurs with biofilms. However, before true gene regulatory interactions were widespread and allowed for controlled changes in cell phenotypes, simple cellular colonies displaying adhesion and interacting with their environments were in place. In this context, models often ignore the physical embedding of evolving cells, thus leaving aside a key component. The potential for evolving pre-developmental patterns is a relevant issue: how far a colony of evolving cells can go? Here we study these pre-conditions for morphogenesis by using CHIMERA, a physically embodied computational model of evolving virtual organisms in a pre-Mendelian world. Starting from a population of identical, independent cells moving in a fluid, the system undergoes a series of changes, from spatial segregation, increased adhesion and the development of generalism. Eventually, a major transition occurs where a change in the flow of nutrients is triggered by a sub-population. This ecosystem engineering phenomenon leads to a subsequent separation of the ecological network into two well defined compartments. The relevance of these results for evodevo and its potential ecological triggers is discussed.

Simulation of cellular packing in non-proliferative epithelia

The physical laws governing the morphogenesis of biological tissues remain largely misunderstood. In particular, the role of the mechanical interactions occurring in this process needs to be better understood and studied. Inner follicular cells surrounding the oocytes of Ciona intestinalis form an epithelial monolayer resulting from an accretion process (without mitosis or apoptosis). This epithelium is elementary and useful for morphogenesis studies: the cells exhibit polygon packing with a specific but non-systematically repeatable topology (i.e. the distribution of pentagons, hexagons and heptagons changes). To understand the role of mechanical forces in tissue formation, we propose an innovative "2D spherical" model based on the physics of divided media. This approach simulates the cellular mechanical behavior and epithelium structuration by allowing cells to adopt a large variety of shapes and to self-organize in response to mechanical interactions. The numerical parameters considered in the model are derived from experimental data in order to perform pertinent and realistic simulations. The results obtained are compared to biological observations using the same counting method to characterize epithelium topology. Numerical and experimental data appear close enough to validate the model. It is then used for exploratory studies dealing with "Tissue Development Speed" variation, which is not easily attainable by experimentation. We show that the formation speed of the tissue influences its topology and hence its packing organization.

The mechanics of metastasis: insights from a computational model

Although it may seem obvious that mechanical forces are required to drive metastatic cell movements, understanding of the mechanical aspects of metastasis has lagged far behind genetic and biochemical knowledge. The goal of this study is to learn about the mechanics of metastasis using a cell-based finite element model that proved useful for advancing knowledge about the forces that drive embryonic cell and tissue movements. Metastasis, the predominant cause of cancer-related deaths, involves a series of mechanical events in which one or more cells dissociate from a primary tumour, migrate through normal tissue, traverse in and out of a multi-layer circulatory system vessel and resettle. The present work focuses on the dissemination steps, from dissociation to circulation. The model shows that certain surface tension relationships must be satisfied for cancerous cells to dissociate from a primary tumour and that these equations are analogous to those that govern dissociation of embryonic cells. For a dissociated cell to then migrate by invadopodium extension and contraction and exhibit the shapes seen in experiments, the invadopodium must generate a contraction equal to approximately twice that produced by the interfacial tension associated with surrounding cells. Intravasation through the wall of a vessel is governed by relationships akin to those in the previous two steps, while release from the vessel wall is governed by equations that involve surface and interfacial tensions. The model raises a number of potential research questions. It also identifies how specific mechanical properties and the sub-cellular structural components that give rise to them might be changed so as to thwart particular metastatic steps and thereby block the spread of cancer.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.

Kinetic Monte Carlo and cellular particle dynamics simulations of multicellular systems

Computer modeling of multicellular systems has been a valuable tool for interpreting and guiding in vitro experiments relevant to embryonic morphogenesis, tumor growth, angiogenesis and, lately, structure formation following the printing of cell aggregates as bioink particles. Here we formulate two computer simulation methods: (1) a kinetic Monte Carlo (KMC) and (2) a cellular particle dynamics (CPD) method, which are capable of describing and predicting the shape evolution in time of three-dimensional multicellular systems during their biomechanical relaxation. Our work is motivated by the need of developing quantitative methods for optimizing postprinting structure formation in bioprinting-assisted tissue engineering. The KMC and CPD model parameters are determined and calibrated by using an original computational-theoretical-experimental framework applied to the fusion of two spherical cell aggregates. The two methods are used to predict the (1) formation of a toroidal structure through fusion of spherical aggregates and (2) cell sorting within an aggregate formed by two types of cells with different adhesivities.

Computational modeling of axonal microtubule bundles under tension

Microtubule bundles cross-linked by tau protein serve a variety of neurological functions including maintaining mechanical integrity of the axon, promoting axonal growth, and facilitating cargo transport. It has been observed that axonal damage in traumatic brain injury leads to bundle disorientation, loss of axonal viability, and cognitive impairment. This study investigates the initial mechanical response of axonal microtubule bundles under uniaxial tension using a discrete bead-spring representation. Mechanisms of failure due to traumatic stretch loading and their impact on the mechanical response and stability are also characterized. This study indicates that cross-linked axonal microtubule bundles in tension display stiffening behavior similar to a power-law relationship from nonaffine network deformations. Stretching of cross-links and microtubule bending were the primary deformation modes at low stresses. Microtubule stretch was negligible up to tensile stresses of ∼1 MPa. Bundle failure occurred by failure of cross-links leading to pull-out of microtubules and loss of bundle integrity. This may explain the elongation, undulation, and delayed elasticity of axons following traumatic stretch loading. More extensively cross-linked bundles withstood higher tensile stresses before failing. The bundle mechanical behavior uncovered by these computational techniques should guide future experiments on stretch-injured axons.

A 'chemotactic dipole' mechanism for large-scale vortex motion during primitive streak formation in the chick embryo

Primitive streak formation in the chick embryo involves significant coordinated cell movement lateral to the streak, in addition to the posterior-anterior movement of cells in the streak proper. Cells lateral to the streak are observed to undergo 'polonaise movements', i.e. two large counter-rotating vortices, reminiscent of eddies in a fluid. In this paper, we propose a mechanism for these movement patterns which relies on chemotactic signals emitted by a dipolar configuration of cells in the posterior region of the epiblast. The 'chemotactic dipole' consists of adjacent regions of cells emitting chemo-attractants and chemo-repellents. We motivate this idea using a mathematical analogy between chemotaxis and electrostatics, and test this idea using large-scale computer simulations. We implement active cell response to both neighboring mechanical interactions and chemotactic gradients using the Subcellular Element Model. Simulations show the emergence of large-scale vortices of cell movement. The length and time scales of vortex formation are in reasonable agreement with experimental data. We also provide quantitative estimates for the robustness of the chemotaxis dipole mechanism, which indicate that the mechanism has an error tolerance of about 10% to variation in chemotactic parameters, assuming that only 1% of the cell population is involved in emitting signals. This tolerance increases for larger populations of cells emitting signals.

Spatial organization of mesenchymal stem cells in vitro--results from a new individual cell-based model with podia

Therapeutic application of mesenchymal stem cells (MSC) requires their extensive in vitro expansion. MSC in culture typically grow to confluence within a few weeks. They show spindle-shaped fibroblastoid morphology and align to each other in characteristic spatial patterns at high cell density. We present an individual cell-based model (IBM) that is able to quantitatively describe the spatio-temporal organization of MSC in culture. Our model substantially improves on previous models by explicitly representing cell podia and their dynamics. It employs podia-generated forces for cell movement and adjusts cell behavior in response to cell density. At the same time, it is simple enough to simulate thousands of cells with reasonable computational effort. Experimental sheep MSC cultures were monitored under standard conditions. Automated image analysis was used to determine the location and orientation of individual cells. Our simulations quantitatively reproduced the observed growth dynamics and cell-cell alignment assuming cell density-dependent proliferation, migration, and morphology. In addition to cell growth on plain substrates our model captured cell alignment on micro-structured surfaces. We propose a specific surface micro-structure that according to our simulations can substantially enlarge cell culture harvest. The 'tool box' of cell migratory behavior newly introduced in this study significantly enhances the bandwidth of IBM. Our approach is capable of accommodating individual cell behavior and collective cell dynamics of a variety of cell types and tissues in computational systems biology.

Modelling platelet-blood flow interaction using the subcellular element Langevin method

In this paper, a new three-dimensional modelling approach is described for studying fluid-viscoelastic cell interaction, the subcellular element Langevin (SCEL) method, with cells modelled by subcellular elements (SCEs) and SCE cells coupled with fluid flow and substrate models by using the Langevin equation. It is demonstrated that: (i) the new method is computationally efficient, scaling as (N) for N SCEs; (ii) cell geometry, stiffness and adhesivity can be modelled by directly relating parameters to experimentally measured values; (iii) modelling the fluid-platelet interface as a surface leads to a very good correlation with experimentally observed platelet flow interactions. Using this method, the three-dimensional motion of a viscoelastic platelet in a shear blood flow was simulated and compared with experiments on tracking platelets in a blood chamber. It is shown that the complex platelet-flipping dynamics under linear shear flows can be accurately recovered with the SCEL model when compared with the experiments. All experimental details and electronic supplementary material are archived at http://biomath.math.nd.edu/scelsupplementaryinformation/.

Correlating cell behavior with tissue topology in embryonic epithelia

Measurements on embryonic epithelial tissues in a diverse range of organisms have shown that the statistics of cell neighbor numbers are universal in tissues where cell proliferation is the primary cell activity. Highly simplified non-spatial models of proliferation are claimed to accurately reproduce these statistics. Using a systematic critical analysis, we show that non-spatial models are not capable of robustly describing the universal statistics observed in proliferating epithelia, indicating strong spatial correlations between cells. Furthermore we show that spatial simulations using the Subcellular Element Model are able to robustly reproduce the universal histogram. In addition these simulations are able to unify ostensibly divergent experimental data in the literature. We also analyze cell neighbor statistics in early stages of chick embryo development in which cell behaviors other than proliferation are important. We find from experimental observation that cell neighbor statistics in the primitive streak region, where cell motility and ingression are also important, show a much broader distribution. A non-spatial Markov process model provides excellent agreement with this broader histogram indicating that cells in the primitive streak may have significantly weaker spatial correlations. These findings show that cell neighbor statistics provide a potentially useful signature of collective cell behavior.

Computational morphodynamics of plants: integrating development over space and time

The emerging field of computational morphodynamics aims to understand the changes that occur in space and time during development by combining three technical strategies: live imaging to observe development as it happens; image processing and analysis to extract quantitative information; and computational modelling to express and test time-dependent hypotheses. The strength of the field comes from the iterative and combined use of these techniques, which has provided important insights into plant development.

AFM stiffness nanotomography of normal, metaplastic and dysplastic human esophageal cells

The mechanical stiffness of individual cells is important in tissue homeostasis, cell growth, division and motility, and the epithelial-mesenchymal transition in the initiation of cancer. In this work, a normal squamous cell line (EPC2) and metaplastic (CP-A) as well as dysplastic (CP-D) Barrett's Esophagus columnar cell lines are studied as a model of pre-neoplastic progression in the human esophagus. We used the combination of an atomic force microscope (AFM) with a scanning confocal fluorescence lifetime imaging microscope to study the mechanical properties of single adherent cells. Sixty four force indentation curves were taken over the nucleus of each cell in an 8 x 8 grid pattern. Analyzing the force indentation curves, indentation depth-dependent Young's moduli were found for all cell lines. Stiffness tomograms demonstrate distinct differences between the mechanical properties of the studied cell lines. Comparing the stiffness for indentation forces of 1 nN, most probable Young's moduli were calculated to 4.7 kPa for EPC2 (n = 18 cells), 3.1 kPa for CP-A (n = 10) and 2.6 kPa for CP-D (n = 19). We also tested the influence of nuclei and nucleoli staining organic dyes on the mechanical properties of the cells. For stained EPC2 cells (n = 5), significant stiffening was found (9.9 kPa), while CP-A cells (n = 5) showed no clear trend (2.9 kPa) and a slight softening was observed (2.1 kPa) in the case of CP-D cells (n = 16). Some force-indentation curves show non-monotonic discontinuities with segments of negative slope, resembling a sawtooth pattern. We found the incidence of these 'breakthrough events' to be highest in the dysplastic CP-D cells, intermediate in the metaplastic CP-A cells and lowest in the normal EPC2 cells. This observation suggests that the microscopic explanation for the increased compliance of cancerous and pre-cancerous cells may lie in their susceptibility to 'crumble and yield' rather than their ability to 'bend and flex'.

Computational investigation of intrinsic and extrinsic mechanisms underlying the formation of carcinoma

The disruption of tissue epithelial architecture is thought to be involved in the initiation of cancer; however, little is known about the cell biology of early neoplastic lesions. Computational models can facilitate in the study of early cancer stagesby simulating different scenarios of tumour inception and by testing in silico the influence of various intrinsic and extrinsic factors on the emerging multicellular morphologies. Here, we have used a computational model to create 3D morphocharts, collections of multicellular morphologies arising from systematic modification of three model parameters defining cell sensitivity to external cues for cell growth, death and adhesion to the extracellular matrix. This systematic search revealed the ranges of parameter values for which robust epithelial structures formed and those that led to abnormal geometrical forms resembling tumour-like morphologies. We also showed how our model can be used to map morphologies of experimental multicellular systems onto their in silico counterpart via the cell sensitivity parameter space that may in turn allow us to identify genetic or epigenetic changes responsible for these morphological distortions.

Collective cell motion in endothelial monolayers

Collective cell motility is an important aspect of several developmental and pathophysiological processes. Despite its importance, the mechanisms that allow cells to be both motile and adhere to one another are poorly understood. In this study we establish statistical properties of the random streaming behavior of endothelial monolayer cultures. To understand the reported empirical findings, we expand the widely used cellular Potts model to include active cell motility. For spontaneous directed motility we assume a positive feedback between cell displacements and cell polarity. The resulting model is studied with computer simulations and is shown to exhibit behavior compatible with experimental findings. In particular, in monolayer cultures both the speed and persistence of cell motion decreases, transient cell chains move together as groups and velocity correlations extend over several cell diameters. As active cell motility is ubiquitous both in vitro and in vivo, our model is expected to be a generally applicable representation of cellular behavior.

A sub-cellular viscoelastic model for cell population mechanics

Understanding the biomechanical properties and the effect of biomechanical force on epithelial cells is key to understanding how epithelial cells form uniquely shaped structures in two or three-dimensional space. Nevertheless, with the limitations and challenges posed by biological experiments at this scale, it becomes advantageous to use mathematical and 'in silico' (computational) models as an alternate solution. This paper introduces a single-cell-based model representing the cross section of a typical tissue. Each cell in this model is an individual unit containing several sub-cellular elements, such as the elastic plasma membrane, enclosed viscoelastic elements that play the role of cytoskeleton, and the viscoelastic elements of the cell nucleus. The cell membrane is divided into segments where each segment (or point) incorporates the cell's interaction and communication with other cells and its environment. The model is capable of simulating how cells cooperate and contribute to the overall structure and function of a particular tissue; it mimics many aspects of cellular behavior such as cell growth, division, apoptosis and polarization. The model allows for investigation of the biomechanical properties of cells, cell-cell interactions, effect of environment on cellular clusters, and how individual cells work together and contribute to the structure and function of a particular tissue. To evaluate the current approach in modeling different topologies of growing tissues in distinct biochemical conditions of the surrounding media, we model several key cellular phenomena, namely monolayer cell culture, effects of adhesion intensity, growth of epithelial cell through interaction with extra-cellular matrix (ECM), effects of a gap in the ECM, tensegrity and tissue morphogenesis and formation of hollow epithelial acini. The proposed computational model enables one to isolate the effects of biomechanical properties of individual cells and the communication between cells and their microenvironment while simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue.

Particle-based model to simulate the micromechanics of biological cells

This paper is concerned with addressing how biological cells react to mechanical impulse. We propose a particle based model to numerically study the mechanical response of these cells with subcellular detail. The model focuses on a plant cell in which two important features are present: (1) the cell's interior liquidlike phase inducing hydrodynamic phenomena, and (2) the cell wall, a viscoelastic solid membrane that encloses the protoplast. In this particle modeling framework, the cell fluid is modeled by a standard smoothed particle hydrodynamics (SPH) technique. For the viscoelastic solid phase (cell wall), a discrete element method (DEM) is proposed. The cell wall hydraulic conductivity (permeability) is built in through a constitutive relation in the SPH formulation. Simulations show that the SPH-DEM model is in reasonable agreement with compression experiments on an in vitro cell and with analytical models for the basic dynamical modes of a spherical liquid filled shell. We have performed simulations to explore more complex situations such as relaxation and impact, thereby considering two cell types: a stiff plant type and a soft animal-like type. Their particular behavior (force transmission) as a function of protoplasm and cell wall viscosity is discussed. We also show that the mechanics during and after cell failure can be modeled adequately. This methodology has large flexibility and opens possibilities to quantify problems dealing with the response of biological cells to mechanical impulses, e.g., impact, and the prediction of damage on a (sub)cellular scale.

Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics

We present a micro-macro method for the simulation of large elastic deformations of plant tissue. At the microscopic level, we use a mass-spring model to describe the geometrical structure and basic properties of individual plant cells. The macroscopic domain is discretized using standard finite elements, in which the macroscopic material properties (the stress-strain relation) are not given in analytical form, but are computed using the microscopic model in small subdomains, called representative volume elements (RVEs), centered around the macroscopic quadrature points. The boundary conditions for these RVEs are derived from the macroscopic deformation gradient. The computation of the macroscopic stress tensor is based on the definition of virial stress, as defined in molecular dynamics. The anisotropic Eulerian elasticity tensor is estimated using a forward finite difference approximation for the Truesdell rate of the Cauchy stress tensor. We investigate the influence of the size of the RVE and the boundary conditions. This multi-scale method converges to the solution of the full microscopic simulation, for both globally and adaptively refined finite element meshes, and achieves a significant speedup compared to the full microscopic simulation.