A computational model for collective cellular motion in three dimensions: general framework and case study for cell pair dynamics

Bridging from single to collective cell migration: A review of models and links to experiments

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.

Matrix degradation regulates osteoblast protrusion dynamics and individual migration

Protrusions are one of the structures that cells use to sense their surrounding environment in a probing and exploratory manner as well as to communicate with other cells. In particular, osteoblasts embedded within a 3D matrix tend to originate a large number of protrusions compared to other type of cells. In this work, we study the role that mechanochemical properties of the extracellular matrix (ECM) play on the dynamics of these protrusions, namely, the regulation of the size and number of emanating structures. In addition, we also determine how the dynamics of the protrusions may lead the 3D movement of the osteoblasts. Significant differences were found in protrusion size and cell velocity, when degradation activity due to metalloproteases was blocked by means of an artificial broad-spectrum matrix metalloproteinase inhibitor, whereas stiffening of the matrix by introducing transglutaminase crosslinking, only induced slight changes in both protrusion size and cell velocity, suggesting that the ability of cells to create a path through the matrix is more critical than the matrix mechanical properties themselves. To confirm this, we developed a cell migration computational model in 3D including both the mechanical and chemical properties of the ECM as well as the protrusion mechanics, obtaining good agreement with experimental results.

To lead or to herd: optimal strategies for 3D collective migration of cell clusters

Cells migrating in clusters play a significant role in a number of biological processes such as embryogenesis, wound healing, and tumor metastasis during cancer progression. A variety of environmental and biochemical factors can influence the collective migration of cells with differing degrees of cell autonomy and inter-cellular coupling strength. For example, weakly coupled cells can move collectively under the influence of contact guidance from neighboring cells or the environment. Alternatively strongly coupled cells might follow one or more leader cells to move as a single cohesive unit. Additionally, chemical and mechanical signaling between these cells may alter the degree of coupling and determine effective cluster sizes. Being able to understand this collective cell migration process is critical in the prediction and manipulation of outcomes of key biological processes. Here we focus on understanding how various environmental and cellular factors influence small clusters of cells migrating collectively within a 3D fibrous matrix. We combine existing knowledge of single-cell migration in 2D and 3D environments, prior experimental observations of cell-cell interactions and collective migration, and a newly developed stochastic model of cell migration in 3D matrices, to simulate the migration of cell clusters in different physiologically relevant environments. Our results show that based on the extracellular environment and the strength of cell-cell mechanical coupling, two distinct optimal approaches to driving collective cell migration emerge. The ability to effectively employ these two distinct migration strategies might be critical for cells to collectively migrate through the heterogeneous tissue environments within the body.

Continuum descriptions of spatial spreading for heterogeneous cell populations: Theory and experiment

Variability in cell populations is frequently observed in both in vitro and in vivo settings. Intrinsic differences within populations of cells, such as differences in cell sizes or differences in rates of cell motility, can be present even within a population of cells from the same cell line. We refer to this variability as cell heterogeneity. Mathematical models of cell migration, for example, in the context of tumour growth and metastatic invasion, often account for both undirected (random) migration and directed migration that is mediated by cell-to-cell contacts and cell-to-cell adhesion. A key feature of standard models is that they often assume that the population is composed of identical cells with constant properties. This leads to relatively simple single-species homogeneous models that neglect the role of heterogeneity. In this work, we use a continuum modelling approach to explore the role of heterogeneity in spatial spreading of cell populations. We employ a three-species heterogeneous model of cell motility that explicitly incorporates different types of experimentally-motivated heterogeneity in cell sizes: (i) monotonically decreasing; (ii) uniform; (iii) non-monotonic; and (iv) monotonically increasing distributions of cell size. Comparing the density profiles generated by the three-species heterogeneous model with density profiles predicted by a more standard single-species homogeneous model reveals that when we are dealing with monotonically decreasing and uniform distributions a simple and computationally efficient single-species homogeneous model can be remarkably accurate in describing the evolution of a heterogeneous cell population. In contrast, we find that the simpler single-species homogeneous model performs relatively poorly when applied to non-monotonic and monotonically increasing distributions of cell sizes. Additional results for heterogeneity in parameters describing both undirected and directed cell migration are also considered, and we find that similar results apply.

Relevant biological processes for tissue development with stem cells and their mechanistic modeling: A review

A potential alternative for tissue transplants is tissue engineering, in which the interaction of cells and biomaterials can be optimized. Tissue development in vitro depends on the complex interaction of several biological processes such as extracellular matrix synthesis, vascularization and cell proliferation, adhesion, migration, death, and differentiation. The complexity of an individual phenomenon or of the combination of these processes can be studied with phenomenological modeling techniques. This work reviews the main biological phenomena in tissue development and their mathematical modeling, focusing on mesenchymal stem cell growth in three-dimensional scaffolds.

Extended logistic growth model for heterogeneous populations

Cell proliferation is the most important cellular-level mechanism responsible for regulating cell population dynamics in living tissues. Modern experimental procedures show that the proliferation rates of individual cells can vary significantly within the same cell line. However, in the mathematical biology literature, cell proliferation is typically modelled using a classical logistic equation which neglects variations in the proliferation rate. In this work, we consider a discrete mathematical model of cell migration and cell proliferation, modulated by volume exclusion (crowding) effects, with variable rates of proliferation across the total population. We refer to this variability as heterogeneity. Constructing the continuum limit of the discrete model leads to a generalisation of the classical logistic growth model. Comparing numerical solutions of the model to averaged data from discrete simulations shows that the new model captures the key features of the discrete process. Applying the extended logistic model to simulate a proliferation assay using rates from recent experimental literature shows that neglecting the role of heterogeneity can, at times, lead to misleading results.

A model of the effects of cancer cell motility and cellular adhesion properties on tumour-immune dynamics

We present a three-dimensional model simulating the dynamics of an anti-cancer T-cell response against a small, avascular, early-stage tumour. Interactions at the tumour site are accounted for using an agent-based model (ABM), while immune cell dynamics in the lymph node are modelled as a system of delay differential equations (DDEs). We combine these separate approaches into a two-compartment hybrid ABM-DDE system to capture the T-cell response against the tumour. In the ABM at the tumour site, movement of tumour cells is modelled using effective physical forces with a specific focus on cell-to-cell adhesion properties and varying levels of tumour cell motility, thus taking into account the ability of cancer cells to spread and form clusters. We consider the effectiveness of the immune response over a range of parameters pertaining to tumour cell motility, cell-to-cell adhesion strength and growth rate. We also investigate the dependence of outcomes on the distribution of tumour cells. Low tumour cell motility is generally a good indicator for successful tumour eradication before relapse, while high motility leads, almost invariably, to relapse and tumour escape. In general, the effect of cell-to-cell adhesion on prognosis is dependent on the level of tumour cell motility, with an often unpredictable cross influence between adhesion and motility, which can lead to counterintuitive effects. In terms of overall tumour shape and structure, the spatial distribution of cancer cells in clusters of various sizes has shown to be strongly related to the likelihood of extinction.

Numerical investigation of the role of intercellular interactions on collective epithelial cell migration

During collective cell migration, the intercellular forces will significantly affect the collective migratory behaviors. However, the measurement of mechanical stresses exerted at cell-cell junctions is very challenging. A recent experimental observation indicated that the intercellular adhesion sites within a migrating monolayer are subjected to both normal stress exerted perpendicular to cell-cell junction surface and shear stress exerted tangent to cell-cell junction surface. In this study, an interfacial interaction model was proposed to model the intercellular interactions for the first time. The intercellular interaction model-based study of collective epithelial migration revealed that the direction of cell migration velocity has better alignment with the orientation of local principal stress at higher maximum shear stress locations in an epithelial monolayer sheet. Parametric study of the effects of adhesion strength indicated that normal adhesion strength at the cell-cell junction surface has dominated effect on local alignment between the direction of cell velocity vector and the principal stress orientation, while the shear adhesion strength has little effect, which provides compelling evidence to help explain the force transmission via cell-cell junctions between adjacent cells in collective cell motion and provides new insights into "adhesive belt" effects at cell-cell junction.

A computational modelling framework to quantify the effects of passaging cell lines

In vitro cell culture is routinely used to grow and supply a sufficiently large number of cells for various types of cell biology experiments. Previous experimental studies report that cell characteristics evolve as the passage number increases, and various cell lines can behave differently at high passage numbers. To provide insight into the putative mechanisms that might give rise to these differences, we perform in silico experiments using a random walk model to mimic the in vitro cell culture process. Our results show that it is possible for the average proliferation rate to either increase or decrease as the passaging process takes place, and this is due to a competition between the initial heterogeneity and the degree to which passaging damages the cells. We also simulate a suite of scratch assays with cells from near–homogeneous and heterogeneous cell lines, at both high and low passage numbers. Although it is common in the literature to report experimental results without disclosing the passage number, our results show that we obtain significantly different closure rates when performing in silico scratch assays using cells with different passage numbers. Therefore, we suggest that the passage number should always be reported to ensure that the experiment is as reproducible as possible. Furthermore, our modelling also suggests some avenues for further experimental examination that could be used to validate or refine our simulation results.

Computational study of cell adhesion and rolling in flow channel by meshfree method

Tethering and rolling of circulating leukocytes on the surface of endothelium are critical steps during an inflammatory response. A soft solid cell model was proposed to study monocytes tethering and rolling behaviors on substrate surface in shear flow. The interactions between monocytes and micro-channel surface were modeled by a coarse-grained molecular adhesive potential. The computational model was implemented in a Lagrange-type meshfree Galerkin formulation to investigate the monocyte tethering and rolling process with different flow rates. From the simulation results, it was found that the flow rate has profound effects on the rolling velocity, contact area and effective stress of monocytes. As the flow rate increased, the rolling velocity would increase linearly, whereas the contact area and average effective stress in monocyte showed nonlinear increase.

Plasticity of Cell Migration In Vivo and In Silico

Cell migration results from stepwise mechanical and chemical interactions between cells and their extracellular environment. Mechanistic principles that determine single-cell and collective migration modes and their interconversions depend upon the polarization, adhesion, deformability, contractility, and proteolytic ability of cells. Cellular determinants of cell migration respond to extracellular cues, including tissue composition, topography, alignment, and tissue-associated growth factors and cytokines. Both cellular determinants and tissue determinants are interdependent; undergo reciprocal adjustment; and jointly impact cell decision making, navigation, and migration outcome in complex environments. We here review the variability, decision making, and adaptation of cell migration approached by live-cell, in vivo, and in silico strategies, with a focus on cell movements in morphogenesis, repair, immune surveillance, and cancer metastasis.

Multiscale mechanobiology: computational models for integrating molecules to multicellular systems

Mechanical signals exist throughout the biological landscape. Across all scales, these signals, in the form of force, stiffness, and deformations, are generated and processed, resulting in an active mechanobiological circuit that controls many fundamental aspects of life, from protein unfolding and cytoskeletal remodeling to collective cell motions. The multiple scales and complex feedback involved present a challenge for fully understanding the nature of this circuit, particularly in development and disease in which it has been implicated. Computational models that accurately predict and are based on experimental data enable a means to integrate basic principles and explore fine details of mechanosensing and mechanotransduction in and across all levels of biological systems. Here we review recent advances in these models along with supporting and emerging experimental findings.

Interpreting scratch assays using pair density dynamics and approximate Bayesian computation

Quantifying the impact of biochemical compounds on collective cell spreading is an essential element of drug design, with various applications including developing treatments for chronic wounds and cancer. Scratch assays are a technically simple and inexpensive method used to study collective cell spreading; however, most previous interpretations of scratch assays are qualitative and do not provide estimates of the cell diffusivity, D, or the cell proliferation rate, λ. Estimating D and λ is important for investigating the efficacy of a potential treatment and provides insight into the mechanism through which the potential treatment acts. While a few methods for estimating D and λ have been proposed, these previous methods lead to point estimates of D and λ, and provide no insight into the uncertainty in these estimates. Here, we compare various types of information that can be extracted from images of a scratch assay, and quantify D and λ using discrete computational simulations and approximate Bayesian computation. We show that it is possible to robustly recover estimates of D and λ from synthetic data, as well as a new set of experimental data. For the first time, our approach also provides a method to estimate the uncertainty in our estimates of D and λ. We anticipate that our approach can be generalized to deal with more realistic experimental scenarios in which we are interested in estimating D and λ, as well as additional relevant parameters such as the strength of cell-to-cell adhesion or the strength of cell-to-substrate adhesion.

A mathematical model of collective cell migration in a three-dimensional, heterogeneous environment

Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed in vivo, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of Drosophila oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.

Physical explanation of coupled cell-cell rotational behavior and interfacial morphology: a particle dynamics model

Previous studies have reported persistent rotational behavior between adherent cell-cell pairs cultured on micropatterned substrates, and this rotation is often accompanied by a sigmoidal deflection of the cell-cell interface. Interestingly, the cell-cell rotation runs in the opposite reference frame from what could be expected of single cell locomotion. Specifically, the rotation of the cell pair consists of each individual cell protruding from the inwardly regressive arm of the cell-cell interface, and retracting from the other outwardly protrusive arm. To this author's knowledge, the cause of this elusive behavior has not yet been clarified. Here, we propose a physical model based on particle dynamics, accounting for actomyosin forcing, viscous dissipation, and cortical tension. The results show that a correlation in actomyosin force vectors leads to both persistent rotational behavior and interfacial deflection in a simulated cell cluster. Significantly, the model, without any artificial cues, spontaneously and consistently reproduces the same rotational reference frame as experimentally observed. Further analyses show that the interfacial deflection depends predominantly on cortical tension, whereas the cluster rotation depends predominantly on actomyosin forcing. Together, these results corroborate the hypothesis that both rotational and morphological phenomena are, in fact, physically coupled by an intracellular torque of a common origin.