Bridging from single to collective cell migration: A review of models and links to experiments
aa r X i v : . [ q - b i o . CB ] N ov Bridging from single to collective cell migration: A review ofmodels and links to experiments.
Andreas Buttensch¨on
1, * , Leah Edelstein-Keshet Department of Mathematics, University of British Columbia, Vancouver, Canada* [email protected]
Abstract
Mathematical and computational models can assist in gaining an understanding of cellbehavior at many levels of organization. Here, we review models in the literature thatfocus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulatescell shape and movement, single cell motility, and collective cell behavior from a fewcells to tissues. We survey recent literature to summarize distinct computationalmethods (phase-field, polygonal, Cellular Potts, and spherical cells). We discussmodels that bridge between levels of organization, and describe levels of detail, bothbiochemical and geometric, included in the models. We also highlight links betweenmodels and experiments. We find that models that span the 3 levels are still in theminority.
Author summary
In this review paper, we summarize the literature on computational models for cellmotility, from the biochemical networks that regulate it, to the behavior of 1 andmany cells. We discuss the distinct approaches used at each level, and how models canbuild bridges between the different size scales. We find models at many different levelsof biological detail, and discuss their relative contributions to our understanding ofsingle and collective cell behavior. Finally, we indicate how models have been linked tobiological experiments in this field.
Over several decades, there has been great progress in our understanding of cellmotility. In the 1980s and 1990s, the basic machinery of eukaryotic cell motion andthe role of the actin cytoskeleton were discovered and refined. Regulation of motilityby intracellular signaling networks was then deciphered in the late 1990s and throughthe 2000s. We continue to discover links between cell signaling and cell shape andfunction, in both normal and diseased cells. Recent efforts aim to link single cellbehavior to collective behavior of many cells and emergent dynamics of tissues.Though originally descriptive, cell biology has emerged as a quantitative scienceover the same time span. Mathematical and computational modeling have becomemore universally accepted, more closely integrated with experimental research, andmore advanced in terms of methodology.Here, we survey the state of the field, emphasizing bridges that span scales: frommolecular signaling to multicellular hierarchies. We focus on the role of modeling andNovember 24, 2020 1/39 ig 1.
Mathematical models can be used to bridge from intracellular signaling (left),to single cell shape and motility (center), to cell–cell interactions (right). At the lowestscales, the goal is deciphering the interplay between stimuli to the cell (chemical,topographic, mechanical, etc.) and intracellular signaling networks that regulateF-actin (branched polymer) and the cytoskeleton (not drawn to scale). Theseinteractions lead to protrusion or retraction, cell polarization, and shape changes thatenable directed motility and chemotaxis. At a higher level, an aim is to link cellbehavior and cell–cell interactions to the outcomes of cell collisions (e.g., CIL) and tothe cohesion of tissues versus EMT, where cells break off. Interconnections existbetween all layers, only 2 of which (white arrows) are shown here. CIL, ContactInhibition of Locomotion; EMT, Epithelial Mesenchymal Transition.computational biology. Because the literature is vast and growing exponentially, welimit our review to several key themes and concentrate on 3 questions: (1)
To what extent have models provided a way to bridge between the 3 levels oforganization, from intracellular signaling, to single cell behavior, and tocollective cell/tissue behavior? (2)
What level of detail is appropriate in a computational or mathematical model?What kinds of models are suitable for a given situation? (3)
What is the relationship between models and experiments in the currentliterature on the subject?At each level, we consider these 3 questions in subsections with headings “Bridgingscales,” “Levels of detail,” and “Links with experiments.” Like any other subdivision,this is to some extent arbitrary, as literature papers often span such categories.Many excellent reviews are already available, including [1–4]. Some surveycomputational methods and others provide links to experiments. The focus on theabove set of 3 questions is, to our knowledge, unique to the current review.The paper is organized by size-scale and level of detail. As shown in Fig 1, we startwith the subcellular level of biochemical signaling (left), and move up to single cellbehavior (center). We then link to small cell groups, larger groups, and tissues (right).At each level, we revisit the 3 key themes and select a few representative contributionsNovember 24, 2020 2/39rom the literature to use as examples. A summary “mapping” of the modelingliterature into levels of detail and numbers of cells is provided in Fig 2.
Phase- (cid:1) eldVertex-based, FEM3D shapesAgents/points/disksPolygon/splines/IBMAutomataCPM
Maree 2012 Marzban 2018 Zhao 2020Merchant 2018Durney 2018Liedekerke 2020Knutsdottir 2017 Drasdo 2005Rens & Merks 2019Camley 2017 Marin-Rirra 2015George 2017Niculesku 2015 Carrillo 2018Nonomura 2012Rejniak 2007 Graner & Glazier 1992Smeets 2016Kabla 2014Zmurchok 2018 Fletcher 2014Lober 2015 Deutsch 2005Jagiella 2016
Number of Cells L e v e l o f d e t a il f o r e a c h c e ll Liedkerke 2019
Fig 2.
A mapping of computational models according to the number of cells(horizontal axis) and the level of detail for each cell (vertical axis). Citations of papersin the diagram (starting from the upper left to lower right: [5–30]).No one review paper can do justice to the entire field. Hence, we point the readerto related review articles that complement our own. In some cases, they cover similarground but with distinct emphases or points of view. In [1], the authors study theissue of cell heterogeneity, its sources at various size scales, and its role in collectivecell migration. They briefly discuss self-propelled particles (SPP), Cellular PottsModels (CPM), and vertex-based approaches that we also discuss in this review. Theyrecommend further investigation of mechanical assumptions in models, and of morerealistic tissue size. A thorough review of the biomechanics of collective cell migrationappears in [2], with a primary focus on cancer. In contrast with other reviews, thispaper also provides an excellent summary of experimental methods. Phenomenadiscussed include the “unjamming transition” where a cell collective changes from“solid-like” to “liquid-like” behavior and its connection to the epithelia-mesenchymaltransition.Force balance and energy-based models for single amoeboid and collectivemesenchymal cell migration are compared in [3]. The authors indicate the challenge ofconverting between energy-based and force-based modeling platforms, and point to thesignificance of doing so. (See [31] for this connection in the CPM.) They also describeinstances of close experiment-model integration. Finally, a recent review of multiscalemodels in [4] has a soft-matter physics perspective and describes computationalmethods (CPM, phase-field, active matter, particle systems, etc.) and their physicalbasis.November 24, 2020 3/39
From cytoskeleton and intracellular signaling tocell shape and migration
How do chemical and mechanical stimuli, together with intracellular signaling shapethe behavior of single cells? This question is central to the bridge between left andcentral panels of Fig 1.
We briefly highlight this well-established area to illustrate examples of bridging scales,diverse levels of detail, and model–experiment synergy [32].
Actin dynamics is studied at many levels: from association of actin monomers to formfilaments [33], to biophysical force production by F-actin [34], the assembly andbranching of F-actin [35], and the resultant shape and motion of cells [36, 37].An excellent review of actin dynamics models in single cell migration, [38] providesa solid bridge between molecular scale and cell scale phenomena. The authors explainthe main model formalisms and show how models can be used to understandexperimental data. A review of the link from actin and its properties to cellmechanosensing and behavior is given in [39]. Reviews of the literature on actin-basedcell migration reveal a mature field, where quantitative methods and theory haveripened in tandem. This synergy has benefited greatly from biochemical pioneers, suchas Tom Pollard (Yale University), who helped to nurture an appreciation formathematical and physical modeling in the community.
The pair of papers [33, 40] aptly illustrates the dichotomy between highly detailedcomputational models [40] and more conceptual minimal models [33]. On one hand,the highly detailed [40] synthesizes a large body of experimental data for actinassembly and branching, including many actin-related component. At the oppositeextreme, the models of [33], as well as later papers [36, 37] emphasize physicalprinciples, universal properties, and general insights. In some sense, the overallpredictions of both model types can be compared. The minimal models are moretractable for analysis, parameter sweeps, and overall insights, but are harder toconnect to detailed molecular biology experiments. The complexity of highly detailedcomputational models (matching molecular experiments) makes it harder to navigatetheir results, which are more like a “1-to-1” map. Such maps may be important forthose who “live in the neighborhood,” but are essentially baffling for newcomers.While distinct, one could argue that these approaches are complementary, so thatoverall insights can be obtained from one, and specific details from the other. Thedichotomy between the detailed and the simplified models will reappear throughoutthis review, serving as 1 hallmark of distinct views.
A review of the experiments and theory for actin dynamics in the motility ofkeratocytes is provided in [41]. A retrospective that emphasizes the importance oftheory, and case-study where theory has led the way is described in [42]. Notablecontributions linking experiments to theory for cell shape include [36, 37], whereshapes of cells are classified and related to actin branching and treadmilling.November 24, 2020 4/39 .2 Signaling networks
Fluorescence resonance energy transfer (FRET) microscopy led to breakthroughs invisualizing the activities of signaling proteins that regulate the cytoskeleton. This hasenabled live imaging of spatiotemporal activity of the Rho family GTPases [43].The roles of Rho family GTPases (Cdc42, Rac, and Rho) in regulating actinassembly and myosin contraction are well–known [44, 45]. These switch-like proteinsare central regulators of cell migration, funneling signals from the cell’s environmentto downstream components that shape its motility. Cdc42 and Rac promote F-actinassembly and cell edge protrusion, whereas Rho facilitates myosin light chain (MLC)phosphorylation, activating myosin-based cell contraction. Rho also promotes F-actinthrough formins and is a Rac antagonist. The importance of GTPase dynamics in cellmotility [46] has also attracted modelers to this arena.A review of GTPase spatio temporal models can be found in [47]. A classic paperon modeling spatio temporal dynamics of signaling in cells (not necessarily GTPases)is [48], a paper that emphasizes universal principles. We see how commonly sharedbasic motifs combine to form complex dynamics [48]. Many lessons learned from thisapproach can be applied directly to studying GTPases or other signaling networks. Ingeneral, the community stands to benefit from more expository papers of this type,where the ingredients that combine to set up specific dynamical signatures areexposed. See also [49] for a popular example of this type, based on years of experiencewith models of cell cycle regulatory networks.
Several papers link the dynamics of GTPases to cytoskeletal assembly and cell shape.Among these are experimental [50, 51] and computational [5, 52] works. A review ofsignaling that organizes the cell front and rear in Dictyostelium discoideum is givenin [53], who also survey models of these as excitable systems. Some of these papers aredescribed in fuller detail below.
GTPases act like molecular switches that cycle on and off the cell membrane andinteract via crosstalk through effector proteins. A large collection of proteinsparticipate in GTPase signaling: Guanine nucleotide exchange factors (GEFs) activateand GTPase-activating proteins (GAPs) inactivate the GTPases with varying degreesof specificity. GTPases are also sequestered in the cytosol by binding to guaninenucleotide dissociation inhibitors (GDIs). In some models, notably [54, 55], the detailsof the binding and mechanisms of activation–inactivation are lumped intophenomenological terms such as Michaelis–Menten or Hill functions. In [55], emphasison cross-talk of Cdc42, Rac, and Rho and on spatial polarization comes at the expenseof the molecular steps themselves (some of which remain unknown). In [56], moredetail on such steps is modeled and eventually reduced to 3 ordinary differentialequations (ODEs) using a quasi-steady-state approach. The authors focus on the roleof a Cdc42-GEF in oscillatory phenotype in neuronal growth cone motility. Incontrast, [57] concentrate on GDI binding of GTPases in a highly detailedcomputational model (originally crafted in BioNetGen and Virtual Cell).Crosstalk between GTPases is modeled in great detail by [58], who constructed aBoolean model for epidermal growth factor (EGF) signaling to Rac and Rho. Theirmodel has 38 intermediates and multiple reactions that activate or inhibit components.(See also [59], who combines PhysiCell with MaBoss to allow the modeling ofintracellular signaling networks as Boolean networks.) At lower level of detail areNovember 24, 2020 5/39ac-Rho mutually antagonistic models that leave out the forest of interactingnodes [60].Simpler models for single GTPase include [61] for cell polarity and [62, 63] for cellshape. There, assumptions are made to condense underlying complex interactions intosimpler “rules” of behavior. For example, “high GTPase activity leads to outwardsprotrusion force at the cell edge” (in the case of Rac) or “contraction” (in the case ofRho) [64, 65]. An advantage of this simplification is that simple models consisting offew partial differential equations (PDEs) allow for well-honed methods of appliedmathematics, including nonlinear dynamics, stability theory, asymptotic analysis,and/or bifurcation theory to explore model predictions, parameter dependence, andregimes of behavior.Serendipitously, simplified signaling models, like [61], occasionally also exposeinteresting mathematical structures to study. A case in point is “wave-pinning” [61,66]:a wave of GTPase activity initiated at 1 end of a cell stalls before reaching theopposite end, resulting in robust “cell polarization.” Simplified models also permitnumerical implementation in more complex geometries. For example, in [67], awave-pinning model for a single GTPase in a single 3D cell is implemented by bulkdiffusion methods. The cell does not deform, but the spatial localization of theGTPase is quantified in a fully 3D geometry. (See also [68], who considered neutrophilfluidization, and [69], who considered polarization in a 3D sphere.)At a gradually increasing scale are models that include not just GTPases (Rac,Rho, and Cdc42) but also other layers (phosphoinositides, actin, Arp2/3, and myosin)that interact to regulate cell polarity in response to stimuli [70], or dictate cell shapeand directed motility [5, 52]. In the latter 2 papers, a single moving “keratocyte” isrepresented using a Cellular Potts Model (CPM), an energy minimization agent-basedplatform for computing cell shapes. It is shown that GTPase signaling can account forcell polarization, reorientation [52], and resolution of conflicting cues or obstacles [5].(Compare with [71], who employ phase-field computational methods toward similargoals.)By way of comparison, in another more detailed approach, in [72], a cell is modeledas having a solid boundary that is moved using a Stefan condition. For the chemicalsignaling, Rho, Rac, 2 species of GEFs, F-actin, and G-actin are included. The modelaccounts for the basic repertoire of neutrophil motility.All in all, models of GTPase signaling have taught us several valuable lessons, somewith universal ramifications. First, modeling has provided clues to the functionalsignificance of the seemly strange cycling of GTPases between membrane and cytosol:namely, the separation of diffusion rates resulting from these distinct compartmentscould be playing a role in pattern formation processes, an ingredient enabling GTPaselocalization or patterning in cells. Second, the stripped-down models have shown thatchemical polarization in a cell need not depend on crosstalk between multiple types ofGTPases — it can be set up by a single member of the family, given sufficient positivefeedback and some depletion of its cellular pool [61]. Groups are still occasionallyrediscovering on their own, the link between cell size and cell polarization that wasimplicit in [66], suggesting the need for more expository reviews of mathematicalresults. An important concept introduced in [5], but not yet fully recognized in thecommunity, is the synergy between GTPase dynamics and its effect of cell shape andboundary curvature of the cell edge. Simply put, the “motion by curvature” of thechemical system interacts with boundary conditions to accelerate the dynamics.Altogether, the appreciation of GTPase signaling has benefited greatly from a host ofdistinct modeling and mathematical approaches.November 24, 2020 6/39 .2.3 Links with experiment
A review of the links between models of cell migration and experimental data (imageprocessing, cell tracking, and feature extraction from 1 cell to many) is given in [73].Here we focus more specifically on experiments that highlight intracellular signaling.The work by [70] provides data for the reorientation and polarity responses ofHeLa cells from various starting cell states (polar, anti-polar, or nonpolar). Theauthors showed that an internal circuit of signaling (Cdc42, Rac, Rho, andphosphoinositides) could account qualitatively for the observed responses of these cellsin microfluidic channels, with an externally controllable response to Rac. In both thisand the follow-up [74], the 1D geometry of the channels helps to reduce the geometriccomplexity of cell shape, allowing for a better match to 1D spatial modelrepresentations.How are models for GTPase crosstalk experimentally linked to cell morphology andmotility dynamics? In experimental results of [51], the authors showed that signalingcircuit of mutually antagonistic Rac and Rho could affect not only the dynamics ofF-actin, but also shapes and migration of mesenchymal breast cancer cells. Togetherwith a mathematical model for the Rac-Rho interactions, they were able tomanipulate the Rac-Rho competition, demonstrate bistable states, and show thatmanipulating the system by inhibiting PAK (a kinase that mediates inhibition of Rhoby Rac) displayed hysteresis characteristic of bistable systems. The significance of thispaper is that it demonstrates a direct link between a simple hypothetical model for theway that Rac and Rho GTPases operate in a cell and the next level up, that of overallcell morphology.Sometimes, individual papers provide only part of the story, but taken as a whole,a series of papers gives a broader view. The sequence of work in [50, 60, 75, 76] explorehow Rac-Rho mutual antagonism is linked to cell morphology. In [75], the authorsexperimentally manipulated Rac and Rho activities to show spread or contracted cells.Interestingly, they found that combining constitutively active (CA) Rac and Rhosimultaneously results in mixed morphologies in human glioma cells. The fact thathigh Rho and high Rac activity produces a mixture of possible coexisting stablesteady state cell shapes was independently predicted in a purely modeling studyby [60]. Related experiments by [50] on cell shapes were also later modeled andexplained in a follow-up paper by [76]. These papers demonstrate that relativelysimple stripped-down depictions of cellular signaling “modules” (such as Rac-Rho) canaccount for important and unexpected observations at the level of the cell as a whole.The paper [77] explores how 3D collagen microtracks and confinement affect cellmigration. The authors find that the degree of cell-extracellular matrix (ECM)interactions are key determinants of speed, morphology, and cell-generate substratestrains during motility.In recent times, a clear link has been established between GTPase activity andmechanical tension experienced by cells. A pioneering experimental paper that showedthis 2-way feedback is [78]. The effects of forces on the Rho family proteins, includingthe involvement of GEFs that activate Rac1, RhoA, or Cdc42 are reviewed in [79].Some GEFs respond to cyclic stretch, and others to tensile force or shear stress andsubstrate stiffness. Rac1 and Cdc42 are activated by stretching of adhesion bonds [80].Rho is mainly used in maintenance of focal adhesions, but it appears to play aprominent role in cell–matrix interactions.Strain and strain gradients affect cell orientation. Single cell experiments withcyclic stretch are described in [81]. The authors suggest that the Rho-ROCK pathwaythat regulates myosin light-chain activity is responsible for sensing and responding tostrain gradients.Overall, it appears that the link between mechanical stimuli and GTPase signalingNovember 24, 2020 7/39s still young, providing ample opportunities for creative modeling. So far, there is noconsensus on what are the “takeaways” from the models so far. Further, it wouldappear that this gap should be filled if we are to successfully bridge between 1 cell andmany, since the mechanics of cell collisions, as well as cell collective migration, entailsensing of both chemical and mechanical stimuli in the interacting cells.
While the meaning of “collective behavior” is intuitively clear, what is less clear is howto specify the transition between a collection of agents, acting individually, and thecollective behavior of the group (right panel, Fig 3). To some extent, the same issuearises in macroscopic models for swarming animals or interacting particles. As groupsgrow and interactions between members increase, new distinct properties emerge atthe level of the group that were absent at lower levels of organization. Characterizing,quantifying, and understanding such emergent properties remains the single mostinteresting and elusive goal in bridging between single and collective phenomena.
Fig 3.
Modeling goals can be classified into broad categories that span levels ofhierarchy. Some models attempt to span knowledge of single cell behavior plusinteractions to predict emergent multicellular behavior (bottom up, left), whereasothers start with observations of tissue dynamics and seek to infer underlying rules,feedbacks, and cell–cell (c–c) interactions that lead to those observations (top down,right).Unsurprisingly, physicists have grappled with similar questions in inanimatesystems. This goes 2 ways. If we observe the dynamics of the group, can we inferunderlying interactions? This is a top-down approach; see Fig 3. Conversely, givenrules followed by individuals, what can we predict about the structure and dynamicsof the group? For example, how do “rules” for cell–cell interactions such ascoattraction, contact inhibition of locomotion (CIL) affect cell collectives? In cellbiology, we add the question of how rules of behavior of cells are to be associated withspecific molecular/signaling pathways inside the cells.From physics, we gain various concepts and techniques such as order parameters,e.g., average movement direction, normalized separation distances, or migrationpersistence measures. Mathematicians have also contributed with useful methods,such as dynamical systems, ODEs, PDEs, and bifurcation methods. More recently, theNovember 24, 2020 8/39se of topological data analysis (TDA) has also entered the mix, to help address someof these questions. The examples provided by [82–85] show how TDA can be appliedeven more effectively than the traditional physics-based order parameters to comparedata and model output. Such novel methods could help to begin addressing questionswhere model hypotheses are to be compared with observed behavior.
A number of illuminating papers suggest that to understand the dynamics of tissues,we should start by first understanding 1, 2, or a few interacting cells in smallgroups [25, 86]. (See top right panel, Fig 3.) Then, the behavior of single cells and theeffect of cell–cell interaction can be explored in detail. Collisions between cell pairs or“cell trains,” contact inhibition of locomotion, and similar responses fall into thiscategory, as do “cell swarms.” The migration of neural crest cells is 1 example of aloose cell swarm, as reviewed in [87]. Adherent cells in tissues (and their cohesion inepithelia) will be discussed in another section.A number of key questions posed by papers in this area include the following: (1)
What mechanism, chemical and/or mechanical, can account for CIL ininteracting cell pairs [7]? (2)
How do cells reconcile conflicting cues [74, 87]? (3)
Under what conditions would a cell reverse its direction [74]?The study of small numbers of interacting cells provides a useful paradigm forconnecting molecular mechanisms to group behavior, an area that has only recentlyreceived the attention it deserves [86].
Phase-field methods have been used to model cell shape and motility in 2 spatialdimensions (2D) [71]. The outline of a cell is represented by a level curve of somefunction φ ( x, y, t ). For example, in [88], a minimal model for Rac signaling was used todescribe the shape and motility dynamics of a single phase-field cell. The model isthen extended to pairwise and cell–train interactions in a narrow 2D strip.Interestingly, the authors had to assume an intracellular signaling “inhibitor” that isactivated at cell–cell contacts to obtain appropriate behavior. This is one of theearliest models spanning the three levels of organization, from subcellular, to singleand multiple cells. The model exploits a reasonably simple level of detail at each stageto reproduce several behaviors, such as reversals, cells walking-past one another, cellsticking, and forming a “cell–train.” We feel that this paper provides good prototypeto be emulated by community members: a clear and well-studied intracellular system,linked to evolving cell shape, with feedback from cell–cell interactions. The paperleads to natural follow-up questions, amenable to experimental investigation: what cellcomponents play the role of the putative “inhibitor”?Models for neural crest cell (NCC) swarms in vivo and in vitro were proposedin [87, 89, 90], focusing on distinct behaviors of leader and follower cells. (See “Linkswith experiments” for a summary.) For the same NCC system, the paper [7] developeda multiscale computational model linking signaling to cell group migration. ARac-Rho GTPase circuit was represented by differential equations (ODEs) at nodes onthe perimeter of a deforming polygonal cell. Co-attraction was represented as a nonlocal tendency of cells to cluster, short range contact inhibition by up-regulating RhoNovember 24, 2020 9/39t nodes in contact with other cells. Stochastic noise in Rac activity generatedcharacteristic run lengths and reorientation seen in typical NCCs.In [7], it was shown that these underlying biochemical systems could account forCIL. This study can be compared with the GTPase and inhibitor system describedabove, [91]. Coherent migration of a group of n cells, each of diameter d , was possiblealong a confinement corridor of width on the order of d √ n . This paper, [7], has awider corridor and larger numbers of cells than [88], but confinement serves a similarfunction of enhancing directed cell movement. In real embryos, such “corridors” couldbe defined by permissive and inhibitory regions for NCC migration, as shownexperimentally in developing embryos [92]. As a comparison to the models of [7, 88],in [93] a “rule-based” approach was applied to the same problem of NCC groupmigration, without bridging to the biochemical circuit.In somewhat related work, [8], 2D cells are described by viscoelastic triangularfinite elements. Cells can adhere to one another using linear springs. At each cellvertex, intracellular signalling controls cell protrusion and cell–matrix adhesion. Thesignaling networks consists of Rac and several other components, including thedownstream kinase PAK, the focal adhesion protein Paxillin, the cell adhesionmolecule cadherin, and Merlin, a protein implicated in contact inhibition of cells in afeedback loop with Rac1 and PAK. Key findings of this study were (1) cells elongateon stiffer substrates; and (2) stiffer substrates allow for better directional guidanceand collective cell migration. Interestingly, the authors state that the principal role ofintracellular signaling is to coordinate cell movement direction over distances of ≈ µm . There are many “simple” SPP models in the literature on collective cell behavior,reviewed, for example, in [94, 95]. It is relatively straightforward to create rules thatresult in collective behaviors, but harder to do so when more details are included.In 1 SPP cell model in a 1D spatial domain, [25], CIL and force-induced cellrepolarization result in properties of collective cell behavior. Rules governing pairwiseinteractions and probabilities for CIL are derived from experiments by [96] describedin the next section. It turns out that there is an optimal number of cells in a group formigration persistence. This paper joins [70, 74, 96] and others in proposing and usingsimplified 1D geometry to examine the basics of cell–cell interactions. The synergybetween papers [25, 96] is very helpful, and points to the need for more pairedexperiment-model studies.SPP models are easily extended to higher dimensions, and/or more realisticpairwise forces. For example, [29] considers a cell swarming model in 2D that is closelyrelated to spherical cell models in [27, 97]. While [29] neglects cell–cell friction, thepaper couples the cells’ compressibility, which plays the role of a local force ofrepulsion, to adhesion due to long protrusions such as filopodia that leads tononlocality.The highlight of [29] is the inclusion of these non local cell–cell attraction forces. Inthis way, the paper makes a seminal link between extended cell–cell interactions and abody of theoretical work on non local interactions in swarms or physical particles. Thesame paper also features the concept of H-stability (preservation of bounded densityas the number of particles increases). This idea is used to ascertain correct cell spacingin swarming models, facilitating links to experimental observations. This paper henceimports a panoply of techniques, methods of analysis, and results, from theoreticalwork on non local models dating back to the 1990s [98], and inspires new ways ofinterpreting experiments.November 24, 2020 10/39arallels with macroscopic swarms and flocks are useful, so long as we rememberthat these operate in vastly different friction regimes. (Cells experience an overdampedregime, where inertial forces are negligible.) In yet another example of a cell swarmingmodel, the paper [99] accounts for the development of stripes in zebrafish pigmentationpatterns. The model comprises 2 cell types, with cell differentiation and death, andsuccessfully reproduces a wide array of observed wild-type and mutant patterns.In a different approach at a lower level of detail, [100] describes a group of bordercells in Drosophila as a single sphere to determine the role of cluster size inchemotactic migration speed. The authors assume a spherical cluster moving byStokes’ law in a viscous environment. The migration force is proportional to cellsurface receptor occupancy in an exponential gradient of attractant. The results showgood fits to both normal cells and cells that have a deficiency of receptors. The samebiological problem is treated in [101], using a force-based approach for individualspherical cells (rather than cluster as a whole as in [100]) in 2D, followed by 3D (forsmall clusters of up to 8 cells). The computation incorporates cellular adhesion andrepulsion, with some stochastic effects.At the next level of detail, the shape of interacting cells is explicitly represented.The papers [91, 102] apply phase-field methods to model cell shapes in 2D withintracellular signaling. Interestingly, the signaling system is itself “minimal:” Theauthors assume basic “wave-pinning” GTPase dynamics, as in [61], with additionalstochastic elements. The levels of internal GTPase give rise to either protruding orretracting forces on the cell edge. Then, [91] models the rotational cell motion when 2cells interact (compare with [31] who showed a similar behavior in CPM cells). Forcoordinated movement, the directions of these gradients need to be linked. A keyfinding of [91] was that the emergence of rotational motion is strongly dependent onthe type of cell–cell coordination mechanism (which includes the previous “inhibitor”field and other phenomenological cell polarization mechanisms).We see gradual ascent in the numbers of cells considered and level of detail. Wenow find groups addressing the question “What can cells do well together that they dopoorly on their own?” For example, collective chemotaxis was shown to be better ingroups than in single cells [18, 103, 104]. (See also the related work by [22] for largercell groups, reviewed in the next section.)Overall, our sampling of the literature reveals a paucity of models at the small-cellgroup level, relative to subcellular and tissue-scale modeling. We believe that paperssuch as [29] open opportunities for swarm-centered modelers to make a contribution tothe cellular biology realm, by translating numerous results of physics andmathematics, suitably reinterpreted, to cells. We also contend that understanding theways that cell interact in small groups is an important and relatively tractable step toformulating more informed tissue-scale models.
Pairwise interactions of MTLn3 rat breast adenocarcinoma cells inside microfluidicchannels with chemotactic cues were quantified in [74]. These experiments provideimportant information for the spatio temporal evolution of GTPase activity duringcell collisions. When cells collided in [74], they either stalled, adhered and movedtogether, or exhibited CIL. More recently, [105] studied how mechanics, and cellpolarity drive collective cell migration in larger groups of cells. More experiments ofthis type would be very important in laying the groundwork for understandingpairwise cell interactions and in directly informing the development of models. Suchmodels could then be used to interpret and deepen experimental observations.In a similar vein is [96], a study of CIL experiments on micropatterned surfaces,where data were collected for the probability of cell repolarization after contact withNovember 24, 2020 11/39nother cell. The authors also explored persistence of cell trains, bridging from pairsto larger groups. An agent-based model was then used to predict cell–train outcomesbased on the observed repolarization probabilities.A model that replicates the experimental setup of [106] appears in [107]. Flatcylindrical model cells are assumed to align ECM fibers. Primarily a single-cell model,it is also extended to 2 interacting cells where, interestingly, leader-follower behavior isseen: The leader creates a “track” of aligned ECM fibers that the second cell canfollow.A number of works shed light on the molecular basis of cell–cell interactions. Forinstance, Wnt signaling is a ubiquitous pathway that regulates polarity, migration,and many other cellular processes. Wnt signaling at cell–cell interfaces that locallyupregulates Rho is studied in [108]. Ephrins form cell–cell ligand–receptor bonds thatfunnel signals to GTPases to control cell repulsion or attraction. Involvement ofEphrin receptors in CIL [109], effect of tension on CIL [110, 111], and on the activationof RhoA [112] have been similarly studied.Experiments on migration of primordial progenitor cells in the developingzebrafish [113] established the importance of guidance cues such as repulsion due toephrins. Physical barriers also guide cell migration and organ placement. Actin wasvisualized for cells colliding with barriers expressing ephrin receptors. A particle-basedmodel with an attraction–repulsion Lennard-Jones potential was then used to test theeffects of reflecting and nonreflecting boundaries. (The Lennard-Jones potential iscomposed of power functions of the form r − and r − for local repulsion andlong-range attraction.)NCCs chemotax toward some cell types in a chase-and-run behavior [114]. Forexample, NCCs chase the placode cells, namely cells that are destined to form feathersor teeth or hair follicles. Before contact, each cell has high Rac at its front. Uponcontact, the cell-adhesion protein N-cadherin, at the cell–cell interface, inhibits Rac inboth cells leading to separation and escape of the placodes. Once the cells separate,chemotaxis is reestablished.The group of Danuser [115] used high-resolution traction-force microscopy (TFM)to measure traction forces for a single cell and small groups of 2 to 6 epithelial cells.They then used a thin-plate finite element model (assuming homogeneous elasticity)to reconstruct forces along the interfaces between cells in a cluster. It was shown thatthe forces were correlated with E-cadherin localization. The authors reduce the cellgroup to a network representation, with vectors connecting cell centroids.A recent paper explores the relationship between initial tissue size and thedynamics of tissue growth [116].Experiments on the migration of chick neural crest cells in vivo and in vitro aredescribed by several groups, including that of Kulesa. His experiments have beendirectly linked to models previously mentioned [87, 89, 90]. The group has investigatedhow cells reconcile mixed chemotactic signals, as in breast cancer cells in [74], and howinformation is shared between cells. The differences between leaders and followers is ofrecent interest in the group.While papers such as [25, 117] argue eloquently for the need to carry outsmall-group cell experiments, we find this, too, to be a missed opportunity. Possibly,the in vitro work that this entails is viewed as less compelling than in vivoexperiments, and so, less persuasive from the grantsmanship perspective. Hence, thereare still relatively few experiments to map out the links between cell signaling andcell–cell collisions, between events inside the cell and the small-scale swarms that form.We recommend this area as a promising one for experiment-model synergy.November 24, 2020 12/39 .2 A related example: Spacing of nuclei While not directly related to cell migration, we nevertheless decided to briefly highlightthe following example as it illustrates close synergy between model and experimentusing modern methods. In [118], models complement experiments to investigatepositioning of nuclei in Drosophila larvae muscle cells. The authors used data formany hundreds of nuclei and proposed a diverse set of particle-based models. Modelswere formulated specifically to test distinct sets of hypotheses for how nuclei interactwith microtubules and molecular motors. Interestingly, machine learning was used forcomputational filtering, comparing large numbers of simulation outcomes to simplesummary statistics such as nuclei positions. The filters are applied in stages, eachstage rejecting inappropriate or inaccurate models that are inconsistent with the data.Ideally, this process requires high-throughput screening, mandating both clear-cutsummary statistics, and a large dataset. As noted by [118], the availability of dataimposes a limitation on possible model complexity that could be adequately fit.Eventually, the best-fit model predicts that nuclei push against one another andagainst cell boundaries by sending out growing microtubules.This example suggests a number of important questions for futuremodel-experiment investigations. (1) How can one ascertain the optimal level of modelcomplexity in the first place? (2) How do we extract clear identifiable, meaningfulsummary statistics from a large dataset? (positions of hundreds of cells) (3) What areoptimal methods for massive filtering to obtain the best model? This example alsoillustrates a creative use of machine learning as a tool in understanding and filteringmany possible mechanistic explanations. Clearly, this approach goes beyond thecommon machine learning application as a “black box” to sort or classify or learn rotefeatures of cells or images.
Even before details of cell–cell signaling were known, theorists were formulatingcell-based computational models for tissue dynamics. Garret Odell and GeorgeOster [119] considered cell–cell pulling forces, coupled to a bistable stretch-sensingsignaling module. They showed that this mechanism could account for localcontractions that folds a tissue in a developmental process such as gastrulation. Theirseminal idea still resonates today [16], now that signaling components are familiar tous. Reviews of the recent biological literature on collective cell migrationinclude [120–122]. (See also brief highlights in Table 1.) A thought-provoking reviewof the biology, [86], focuses on the role of adhesion mechanics, bridging between forceregulation in single cells and in collective cell migration. The paper gives detailedsummary of the underlying molecular players, synthesizing a vast literature on thesubject. The mechanical and mechanotransduction players at the molecular level arethen linked to consequent cell-level properties. The paper proposes a rational programof experiments from 1 to 2 to many cells (see Fig 4 in [86]) to bridge the scales fromsingle cells to tissues.Several key questions appear in papers at this level. (Q1)
How do internal dynamics of cells influence the emergence of collectivebehavior? How do cells share and transfer information between one another?How and to what extent are mechanical forces transmitted overdistance [136]? (Q2)
What is the role of the leading front in guiding collective migration? [132]November 24, 2020 13/39 ain-Target Main-Finding Ref.
Rac opto-activation Cell migration by photo-activation (HeLa cells) [123]Cdc42 opto-activation Cdc42 activates Rac at the front, and Rho at theback of the cell (Immune cells) [124]Rho opto-activation Rho at cell’s rear can control directional migration.Rho activity regulates switch between amoeboidand mesenchymal migration (macrophages) [125]Rho-ROCK pathway This pathway senses and responds to strain gradi-ents; cyclic stretching (single cells) [81]Chemotactic response If gradient switches too rapidly, cells get stuck(Dictyostelium) [126]Cell polarization Comparison of three RDEs, parameter fitting todata [127]Merlin Merlin is a negative regulator of Rac and may alsobe regulated by the Rho pathway [128]Merlin Key role in cell polarity, and leadership. Spatio-temporal data (cell monolayers) [129]Signaling at cell-cell interfaces. Non-canonical Wnt signaling at cell-cell contactscauses an upregulation of Rho [108]Forces and GTPase activity Cell-cell forces affect Rho activation [112]Topographic cues Single cells have different organizations on groovedvs flat substrata (fibroblasts) [130]NCCs and placode cells Intricate interplay between chemotaxis, integrinsand their effect on internal GTPase activity [114]Ephrins Ephrins in repolarization of cells (zebrafish devel-opment) [113]CIL and Ephrin receptors Ephrin receptors affect CIL [109]CIL Biphasic relationship between probability of CILand collective migration [96]CIL Tension built up during CIL [110,111]CIL Spatio-temporal GTPase patterns during cell-cellcollisions and CIL events (microfluidic channels) [70,74]Collective strand formation Interesting difference between polarization ofleader and followers [131]Small groups of epithelial cells Forces are correlated with E-cadherin localization [115]Effect of forces on Rho and Rac GEFs Some GEFs respond to cyclic stretch, others to ten-sile force or shear stress and substrate stiffness [79]Mechanical GTPase activation Rac1, Cdc42 activated by stretching adhesionbonds. Rho maintains focal adhesions [80]Epithelia “Push-pull” or “caterpillar” collective motion innarrow grooves, more complicated movement inwide channels [132]Epithelia Sustained oscillations in epithelial sheets [133]Epithelia Cells crawl in the direction of maximal principalstress. Leaders impose mechanical cues on follow-ers [134]GTPases and GEFs Coordinated waves of motion, cells at front reactfirst (scratch-wound assay in human bronchial ep-ithelial cells) [135]
Table 1.
Experimental summary: CIL, contact inhibition of locomotion; GEF,guanine nucleotide exchange factor; NCCs, neural crest cells; RDEs, reaction diffusionequations. (Q3)
Do actin cables or ECM fibers transmit long-range stresses [2, 131]? Aresuch mechanisms analogous to the non local sensing mechanisms that play adominant role in models of animal swarms? (Q4)
What cellular and tissue-based mechanisms (long- and short-range cues,barriers, cell–cell adhesion, etc.) are essential for proper formation of organsduring development [113, 137]?Parallels between the polarization and directed migration of single cells and of acell collective is highlighted in many recent works. See, for example, the recentperspective paper, [117], demonstrating parallels in the GTPase distribution,chemotaxis, and mechanosensing between the single cell level and tissue level.November 24, 2020 14/39s shown in Fig 5, many computational platforms are currently used to simulatemulticellular migration. These are reviewed in [4, 95, 138–140]. Rigid or deformablespheres or ellipsoids (in 3D or 2D, Fig 5A1 and 5B1, respectively), polygonal“vertex-based,” or CPM cell representations are common. Software platforms such asCHASTE (Oxford University, [141–143], and Fig 5A2), CompuCell3D (UIndiana, [144], and Fig 5A3 and 5B2), or Morpheus (TU Dresden, [145]) have made itincreasingly easier to simulate complex tissue dynamics without having to reinventcomputational algorithms and graphics.In our humble opinion, there is a vital need for support and sharing ofstandardized open-source software packages, with suitable plug-ins donated by groupsutilizing those resources. First, such tools would save time, person-power, expertise,and expense of individual “kludges.” It would reduce duplication of effort — howmany of us want to reinvent our own finite-element computations? Even morecompellingly, such standardization can bring about much easier communication andappraisal of published models, and in-depth scrutiny of exactly what those modelsinclude. Having learned the basic steps of Morpheus, one of us (LEK) has become anenthusiastic user. See, for example, [146], where figures are linked directly toexecutable Morpheus .xml model files that produced them.
Here we concentrate on bridging between models for biochemical signaling andmulticellular behavior. In the next section, we will focus on bridges between 1 or a fewcells to larger populations and from simple to more detailed geometry (points, spheres,and cell shapes) and dimensionality (1D, 2D, and 3D).The modeling paper by [13] is a vertex-based computation of epithelial dynamics,where force balance of cell vertices includes an active contractile force on the cellperimeter due to myosin. ODEs track Rho activation by cell perimeter stretching anddownstream myosin activation (by phosphorylation) at a given vertex. Vertex motionis a result of force balance. Friction and passive forces are derived from energypenalties for area constraint and from the energy of adhesion to neighbor edges. Thisaspect of the paper resembles the energy-based approach in CPM computations, e.g.,in [16]. The authors quantified the effects of cell density on correlation lengths, andregimes of behavior such as streaming, contractile waves, group movement, rotation ina circular domain, presence of vortices, and non uniformity of myosin distribution.Some similarities are shared by [13, 16]. Both are tissue-based computations, withODEs for GTPase biochemistry, an assumption that GTPase leads to cell contractionor expansion, and feedback from cell tension back to the activation of the GTPase.Both papers find fluctuations in cell shape or size from the full feedback betweensignaling and contractility (Fig 2C in [13], and the comparable Fig 3B in [16]). Thesignaling biochemistry is assigned to vertices in [13] versus the cell interior in [16].In [16], a single cell is first linked to a 1D chain of cells, and then to 2D CPM cells,where waves of cell contraction are observed. The levels of detail in these 2 papers arecomparable, though [13] also includes predictions for myosin, unlike [16]. See also [147]for a similar approach restricted to 1D cell chains.In the same general class, the work of [10] links cell contraction (in a vertex-basedhexagonal cell with 6 spokes) to biochemical details for a circuit of proteins known toregulate myosin contractility in Drosophila dorsal closure. This model is more detailedon the specific known interactions of protease-activated receptor (PAR) proteins thatform a negative feedback loop with actomyosin (Baz, Par-6, aPKC), including a set of9 ODEs for components linked to myosin dynamics along cell edges and spokes. Theauthors account for several phases in the developmental process, including oscillationsobserved in the tissue (using time delays) and the eventual contraction of the tissue.November 24, 2020 15/39y way of comparison, [148] has a greater level of detail of the cell shape, includingmore viscoelastic spokes and edges, but essentially no biochemical detail. An exampleof 3D vertex-based simulations with cell rearrangement and out-of-plane bending canbe found in [149].
The gene regulatory network involved in Drosophila ventral furrow formation ismodeled at a fine level of molecular detail in [150] using a Boolean modeling approach.The model encapsulates interactions of transcription factors to explain 3 phenotypicattractor states, and identify missing and alternative pathways. Spatial distribution ofcells and cell shape dynamics is not considered.At the opposite extreme, internal cell signaling is omitted in favor of positions andspatiotemporal dynamics of cells. SPP “agent-based” models with forces of attraction,adhesion, and repulsion include [29], as reviewed in a previous section. There, nonlocalforces are considered in groups of hundreds of cells.Modified agent-based models in [151] can be compared to the cell-pair study in [91].Additional assumptions are included in [151], notably Vicsek-type alignment, cellmembrane curvature, elasticity, actomyosin cable force, and a tendency to move in theoutwards normal direction. The authors show the dependence of tissue wound-healingvelocity on the width of a confining stripe. On a similar topic, CPM computations areused in [152] to investigate the effect of adhesive micro patterns on the motion ofsingle cells and collective cell migration. Similarly, modified Viscek-type models areconsidered by [25] endowing each cell with a polarization direction. Various types ofcell–cell polarity coordination mechanisms are considered.Yet another modification of particle-based models is [153], where a cell isrepresented by a spring linking 2 discs, to depict the “cell body” and the “pseudopod.”The model demonstrates various outcomes of binary collisions in a 2D domain, as wellas order–disorder transitions and velocity waves in large 2D cell groups with andwithout CIL. This paper explicitly bridges from 1 to many cells.At the next level of complexity, we find representations of cells as spheroids orellipsoids (Fig 5A1, [27, 154–156]). Drasdo computed a 2D growing monolayer [154],and later extended it to 3D [27]. Palsson built 3D simulations of deformable off-latticeellipsoids, with cell division, chemotaxis, cell–cell adhesion, and volume exclusion. Theplatform has been used to model the dynamics of Dictyostelium aggregation and cyclicAMP (cAMP) signaling [155], as well as the posterior lateral line primordium, acluster of about 100 migrating cells in zebrafish embryos that generate sensorystructures on the surface of the fish [14].Other recent papers add variations on these themes. For example, [157] provides acomputational model for 3D off-lattice spheroid cells, but with greater level of detailfor cell–cell and cell–matrix adhesion. The model is coarse grained from the biologicallevels of 10 down to 100 sites per cell. After quantifying the stable distance betweencell pairs (for a given set of forces, intrinsic cell properties, and adhesion sites), theauthors go on to show the behavior of a larger group of cells.Recently, these approaches have been extended to more finely resolve cell shapes in3D [9]. Here, each cell is a triangulated 3D object, composed of discrete viscoelasticelements. Simulations track up to a thousand liver cells. While computationally veryexpensive, these approaches can provide valuable insights. For example, it is shownthat highly deformable cells move more easily to heal a lesion than stiffer or more rigidcells [9]. They are also useful for specific systems where it is crucial to capture detailsaccurately, as, for example, in drug design in silico for liver disease.In [158, 159], a confluent epithelium in confinement is modeled by vertex-basedpolygonal cells, and the type of collective migration (directed flow, vortex chain, orNovember 24, 2020 16/39urbulent) is related to a dimensionless quantity (“cell motility number”) thatcombines cell motility, cell density, and size of the confinement pattern.In the phase-field category, the paper by [22] scales up from small numbers of cellsin [91] to larger populations. Cell collisions are modeled by an energetic penalty foroverlapping phase fields, and cell adhesion is described by a reaction–diffusionequation in [22]. Their model cells exhibit bistable shapes, (either symmetric orkeratocyte-like), and both elastic and adherent collisions are predicted. Strongadhesion results in bands of dense closely packed cells that move as a collective. Theauthors compare their work with that of [160] (see Links with experiments) where aCPM model is used to describe large confluent cell rotations in a confined region. Asnoted by [22], the level of detail in their phase-field description is better suited fornon-confluent cell models, but likely too detailed for confluent layers of cells whereindividual cell details average out. In [161], similar methods are extended to a 3Dphase field computation for a single cell interacting with curved or grooved surface.This sequence illustrates the trade-off between geometric complexity (1D versus 2Dversus 3D) and collective complexity (1 cell versus few versus many).Overall, it emerges from many papers that the vertex-based models are good atdescribing epithelia, where fragmentation or EMT is absent or not important.Vertex-based models are not well suited to track cell death or fragmenting clusters,since edges and nodes are shared by neighboring cells. Breakage of a cell requires thatshared edges and nodes be duplicated or reassigned an inconvenience. To track tissuefragmentation or loss of cells, center-based models or CPM platforms have anadvantage, since these represent each cell by an individual geometric object or by a setof pixels.Continuum models are also commonly used at the level of tissue dynamics. Here,the cell identity is omitted altogether, in favor of cell densities, local flows, andmaterial properties. Analogies are made with fluids, viscoelastic material, elasticsheets, or foams. Advantages include the ability to harness traditional methods ofphysics, mathematics, and fluid or engineering computational software. Examples ofthis approach are numerous, and we mention only a few.One example of this class, [162], is a 3D physical description of an epithelial sheet.The authors show bistability in cell aspect ratio, bending and buckling instabilities ofepithelia, and transitions that lead to formation of epithelial tubes and spheres.Experimentally testable scaling laws are given for such morphogenetic transitions.While there is no subcellular molecular detail, the connection between local cell shapeand epithelial behavior is an important contribution.In [163], the tissue is modeled as an active continuous medium, with a Maxwellviscoelastic constitutive law. A reaction–diffusion-transport equation accounts foractomyosin, whose contractility is coupled to cell motion and polarity. The authorsexplore predicted motion of tissue in confined regions, closure of wound gaps, and therelationship between traction forces and sizes of cohesive cell clusters. Mechanicalwaves, as well as tissue rigidity cycles, are observed, where fast fluidization is followedby a gradual period of stiffening.The continuum model in [164] treats a tissue as a compressible fluid and includesboth cell division and death in a type of Stefan free boundary problem. The authorsshow that behavior of the tissue in wound closure and in colony expansion depends on3 parameters: 2 physical constants and the proliferation rate. They argue that allthese can be estimated from limited experimental data, with examples calculated forIEC-6, a rat cancer cell line and MDCK cells, a canine kidney cell line. As the authorspoint out, continuum models are appropriate, provided the size of the tissue or thecharacteristic length of the wound is sufficiently large relative to the size of individualcells.November 24, 2020 17/39n some cases, an approach that combines features of both continuum and discretecell identity is implemented. One example is [165]. Here, a 1D cell monolayer consistsof a long contractile element with myosin creating a strain rate resulting in lengthchanges. This element is flanked by cells in front and rear. Binding and unbinding ofadhesion sites are also included. The model tissue exhibits durotaxis, movementdirected up a stiffness gradient, under appropriate conditions on the myosin andadhesion parameters. The authors use this model to conclude that a monolayer ismore effective at durotaxis than single cells.Other examples of hybrid treatments of particle-based and continuum approachesinclude [166, 167]. These papers tackle the important question of how to deriveappropriate continuum models from underlying SPP models. This kind of work and,in particular, expository papers that summarize the conclusions in ways that biologicalmodelers can understand, could help link work in the literature that is currentlyunderappreciated or not understood.A recent work, [168], describes fingering at the front of an epithelial sheet usingseveral of the above approaches. Cells are represented by pairs of points, as in [153],and also by Voronoi polygons for computations corresponding to experiments in [131].The authors develop an active fluid model for the epithelium. Using these combinedapproaches, they demonstrate that stable fingering of the tissue edge requires leadercells. They characterize the wavenumber of the fingering instabilities (distancebetween stable fingers) using stability analysis of the continuum PDE model.In summary, while the literature on tissue-scale collective migration is rapidlygrowing, it is still in stages of infancy as far as coherence, coordination, and cleardirection are concerned. We see many exploratory steps in many individualisticdirections. The list of core questions and the discovery of unifying principles is beyondthe horizon, providing ample challenges for the community. We can draw an analogybetween the current state of the art and the behavior of a growing but uncoordinatedpopulation of cells. There is no emerging unified front. Each is exploringindependently and occasionally aggregating with a few others, but the globalorganizing principles are yet to emerge. The links between molecular players such as Rho family GTPases and epithelialmorphogenesis have been known for some time. (Experimental literature reviewedin [169].) The role of Rho GTPases in the leader-follower identities and in front–reartissue polarity is reviewed in [170].Over the years, investigators have been asking how external constraints, geometry,strain fields, gradients, and other factors affect collective behavior. How doescollective cell migration emerge from transfer of mechanical information between cells?For a cell to be a “leader,” should it be more sensitive to stimulation than other cells?Should it have elevated or more responsive GTPase activity, for example?These and similar questions are posed in the experimental work of [134]. Here, theauthors investigate the roles of the GTPases RhoA and RhoC and their GEFs incollective cell migration. It is found that cells tend to crawl in the direction ofmaximal principal stress, a process called plithotaxis. In their scratch-wound assay ofMDCK and human bronchial epithelial cells, wounding leads to coordinated waves ofmotion, with cells at the front edge reacting first, followed by those successivelyfurther back. The authors speculate that mechanical cues are induced by leader cellson followers behind then by normal strain, and alongside them by shear strain tocoordinate motion.In [134], the relationship of tissue speed to distance from the leading edge isquantified using particle image velocimetry. The authors employ shRNA to knockNovember 24, 2020 18/39own GTPase Cdc42 or Rac1 and many of their GEFs (screening some 81 GEFs intotal). They find that the speed and directionality of the cells drop everywhere. WhenRhoA is depleted, there is a reduction in the spatial gradient of cell speed. Theysuggest that the molecular mechanism may include signaling downstream of cadherin,as well as Merlin-Rac1 signaling.The experiments in [129] elucidate the effect of pulling stress on Rac and RhoGTPases. The authors investigated which properties of underlying molecularmachinery allow for coupling between mechanical forces and correlated cell motion.The correlation length scale of collective force transmission was determinedexperimentally in [171]. Observed behavior was then modeled using a thin elasticsheet of height h with some elasticity and isotropic contraction stress. The authorsinvestigate the emergence of leader cells and found a typical length scale of about 170 µ m.A number of experimental studies have explored how epithelia behave ingrooves [130, 132], various topographic surfaces, or confined settings [105], includingarrays of posts [172], arrays of grooves [173], or adhesive [160] surfaces. In [132], thewidth of grooves and adhesive strips is varied to investigate how these affect collectivemigration of an epithelial sheet of MDCK cells. The authors map out the force andvelocity fields in each case. Their narrowest grooves are 20 µ m wide, so cells are in asingle file, and the authors observe “push-pull” or “caterpillar” type motion,resembling the relaxation–contraction cycles in the model by [16]. Larger tissue widthsexhibit vortices of cell motion. The authors deduce that the constraints of thegeometry influence cell rearrangement, as well as junctional forces between cells.(Compare with [133], who observed sustained oscillations in epithelial sheets.)Micropillar arrays form the playing field in [172] to observe collective migrationand EMT in mammary epithelia and breast cancer cell lines. Dispersal of single,highly motile mesenchymal cells from a spreading epithelial front was shown to agreequantitatively with a minimal physical model of binary mixture solidification. Whilesuch a model lacks cell detail, it has several advantages, including simplicity, basicsummary statistics for comparison with experiments, and availability of an analyticsolution. Methods from physics can be used to inform the link between the materialproperties and the overall macroscopic behavior.Both single fibroblasts and epithelial monolayers were studied in [130], showingthat cells tend to be more organized on grooved versus flat substrata. Mechanicalexclusion interactions, rather than strength of junctions between cells, affect thedistance over which the topographic guidance signal propagates between cells. Thesame paper proposes a CPM computational model to describe the observations. In theCPM model, following the the style of [15], each cell is assigned a phenomenological“polarity” vector that both guides and is affected by cell displacement. Aside from thecustomary Hamiltonian area constraint and cell–cell adhesion energy, there is also aterm for a phenomenological motile force, implemented as a “migration energy” (dotproduct of the polarity vector and the cell centroid position). The authors investigatehow this motile force affects spatial correlation of the velocity. They observeemergence of streaming patterns. The authors also describe the influence of “leadercells,” whose polarity vector is set to an external cue. Leaders are embedded in thetissue interior, and they coordinating neighbors over some “interaction distance.”In a similar style of experiment-model study, MDCK cells are seeded on a circularadhesive domain in [160]. Once a critical density is attained, the confluent culturecollectively rotates. The same behavior is then captured in a CPM computationalmodel that includes a motile force and a polarity term that is reinforced by celldisplacement, with some persistence time. The authors show that rotation takes placewhen the size of the circular domain is on the order of the correlation length of cells.November 24, 2020 19/39hat correlation length, in turn, depends on the cell—cell adhesion energy, the motileforce magnitude, and the polarity persistence time. Note the comparison with thepurely computational paper by [22] using phase-field methods, where cell collisions,rather than adhesive confluent cells, are the subject of focus. It is also interesting tocompare the CPM model in [160] with the vertex-based treatment of a similarsituation in [158].A recent paper, [105], demonstrates the fact that single cell properties affect muchof the collective behavior of a cell population. In this ground breaking work, theauthors link single cell Rac1 polarity to the emergent rotations of confluent cells (fromfew to many) confined to a ring or closed curve. A simple mechanical model capturesthe balance of directed motility and contact forces to demonstrate the principles atwork. This paper spans subcellular to multicellular scales and provides a greatexample of elegant experiments to be emulated by others. At present, there are still relatively few models that bridge from detailed underlyingmolecular mechanisms through individual cell motility, all the way to tissue dynamicsand collective cell migration, though the number of such papers is growing. There are2 major issues that hamper such efforts. First, it is unclear how to deal with theproblem of combinatorial complexity in trying to understand numerous players andinteractions at each level. Decisions made at 1 stage affect other stages, and exploringa multiplicity of assumptions is a challenge (but see [118]). Second, complexity of theresulting models makes it challenging to determine the range of possible behavior, letalone make sense of overall principles and key components.Few papers specifically address the signaling modules that get recruited incollective migration, beyond those that serve single cells migrating on their own. Here,we refer to signaling that is triggered by cell contact or junctions, and that specificallyaffects the way that cells then interact. Downstream responses might include changesin cell adhesion, migratory potential, permissive or inhibitory control of cell division orapoptosis, or relative sizes, polarity, or other aspects of cells. Input from theenvironment in the form of mechanical tension or topography can influence thesesignaling networks, promoting or inhibiting EMT. Examples of this sort include someof the following.The role of Merlin is highlighted in [129]. Merlin, a negative regulator of Rac [128],is a member of the Ezrin-radixin-Moesin (ERM) family. When Merlin is bound totight junctions between cells, it inhibits Rac1, but when it is in the cytoplasm, itreleases that inhibition. At the same time, low Rac1 activity leads Merlin to becomestabilized at tight junctions [129]. Hence, Merlin and Rac1 compete in a mutuallyinhibitory circuit, and a balance between mechanical and chemical factors controlsMerlin activity and localization. The balance, in turn, regulates cell states that favor(high Rac, low Merlin) or inhibit (high Merlin, low Rac) protrusion of cells at the front.Spatial separation of Rac and Merlin activities can result in front–rear polarization incollectively motile cells.Mechanical cues and ECM topography appear to influence the adhesion andmigration of cells in an epithelial sheet. The signaling of YAP-Merlin-Trio (YAP,Yes-associated protein) in regulation of Rac and the expression of E-cadherin wereinvestigated in [173] on nanostructure ridge arrays that mimic ECM. A minimal modelfor 2 signaling modules was proposed to account for the transition of YAP activitywith distance from the front edge of the sheet. These examples of how cell–cellNovember 24, 2020 20/39 ridgingscales Levels of detail
Refs.
Experimental linksS Boolean signaling network [150] Motivated by expts.
SCB 3D phase-field model [161] Motivated by expts.
SCB 3D triangulated cells [9] Links to liver regeneration expts.
CCB SPP , pairwise forces, cell types [29] Pure theory, applied to expt’l design.
CCB SPP , Spherical cells, 2D, 3D; adhesion, repulsion,random forces [101] Compared to live images.
CCB SPP , 2D swarm, cell types, differentiation [99] Compared to fish skin patterns.
CCB vertex-based , 3D [149] Integrated expt.-model
CCB polygonal cells , 2D, 3D, viscoelastic elements [148] Motivated by expts.
CCB
Cells as pairs of spheres [153, 168] Motivated by [131].
CCB CPM with cell polarity [15, 130] Motivated by expts.
CCB 3D vertex based model , epithelia [162] Proposes new expts.
CCB Spherical cluster ; Langevin eqn. [100] Integrated expt.-model
CCB Continuum compressible fluid tissue [164] Reproduces expts.
SCB , CCB Spherical cells , detailed adhesion dynamics [157] Motivated by expts.
SCB , CCB SPP , disk shaped cells, ECM, Cell–ECM interaction [107] Reproduces expts. of [106]. S , CCB SPP with alignment rules [151] Reproduces expts. S , CCB Spherical cells , internal details [27, 154] Reproduces expts. S , CCB Ellipsoidal cells [14, 155, 156] Motivated by expts. S , CCB Vertex based [158, 159] Motivated by expts. S , CCB Vertex-based ; sub-cellular components [13] Motivated by expts. S , CCB Vertex-based , intracellular signaling [10] Motivated by expts. S , CCB FEM , 1D approximation of 3D [165] Motivated by expts. S , CCB Cell agents , repolarize after collisions [96] Integrated expt.-model. S , CCB SPP , leaders, followers, filopodia, chemical gradients [87, 89, 90] Integrated expt.-model. S , CCB SPP cells, polarity vectors, c-c coordination [18, 103] – S , CCB FEM cells, signaling at cell edge, adhesion [8] Wound-healing, compared to expts. S , CCB Active media , RDEs, actomyosin, cell motion [163] – S , SCB , CCB 1D or 2D cells , ODE signaling networks [16, 147] Theoretical study. S , SCB , CCB SPP , 1D, rules for polarity coordination [25] Motivated by expts. S , SCB , CCB deforming polygon cells, signaling at nodes [7] Motivated by expts. S , SCB , CCB CPM , minimal intracellular signaling [31] Real cell traction forces [174]. S , SCB , CCB Phase-field method, minimal intracellular signaling [22,71,88,91,102] Collision assays of [160].
Table 2.
Summary of the modeling papers. Papers classified by 3 levels of organization: S, SCB, and CCB ortissue behavior. SPP models represent the cell shape statistically; common cell shape choices are spherical,ellipsoidal, or cylindrical cells. Models are categorized into: (1) CPM; (2) phase-field models; (3) vertex models; (4)particle models; (5) continuum models. CPM and phase-field models resolve cell shape well. c-c, cell-cell; CCB,collective cell behavior; CPM, Cellular Potts Models; expts., experiments; FEM, finite element methods; ODE,ordinary differential equations; RDE, reaction diffusion equation; S, signaling; SCB, single cell behavior; SPP,self-propelled particles.nteractions depend on and influence both intracellular Rac gradients and adhesionmotivate future modeling efforts at the multiscale level.An intriguing stepping stone on the route to the challenging eventual targets areengineered tissues studied in synthetic biology. Since these are designed with knowncomponents, they allow for more direct development of models based on underlyingmechanisms. We mention and example of this sort in what follows.An interesting question is whether and how lessons from 1 level can be simplifiedinto rules for components at the next level and how to best include the additionalsignaling pathways that get recruited at distinct levels of organization.Here, we mention a few representative examples but recognize that others mayexist of which we are as yet unaware.
In many respects, we are currently seeing first steps in models that bridge severallayers of organization. Examples of this type can be found in the work of Marzbanand colleagues [6, 175]. Their model combines several modules: cell polarization asin [5], a viscoelastic cytoskeleton, stress fiber structure, cell motility as in [176], andcell–substrate interaction. The authors first model the polarization, motility, anddurotaxis in single cells, and then combine these with a cell–cell interaction module tosimulate the rotation of tens of cells in a confining 2D annulus. The paperdemonstrates a creative combination of a number of known cell representations tobridge from subcellular properties to those of the collective.In [12], the authors study tumor spheroid growth under high mechanicalcompression. Their model represents cells as 3D spheres. Commonly used cell contactmodels (e.g., the Hertz model) do not take such large volume compression intoaccount. The required corrections to the contact models were calibrated using ahigh-resolution mechanical models of cells [9]. We expect that similar approaches willin the future improve the accuracy of coarse-grained tissue level simulations.The intersection of cellular and developmental biology has provided additionalimportant examples of computational models that span multiple scales. Someexamples predate recent efforts by 2 decades, notably computational studies of cellularslime mold,
D. discoideum morphogenesis, and signaling in [177, 178]. Other examplessuch as [179] are purely theoretical, aimed at exploring how random gene circuitslinked to cell adhesion, cell division, and some overall measure of “fitness” evolve intoa zoological garden of multicellular structures. See also [180] for a recent work along asimilar “EvoDevo” evolutionary developmental biology line.
By using very simple representations of subcellular events, [26] bridge from 1 cell tohundred(s) using CPM computations. The authors assumed an elementary processthat mimics, but does not explicitly depict, the effect of actin on persistence of motion.Local protrusion is self-reinforcing, and decays on some time scale. First showing how2 simple computational parameters tune the cell shape from keratocyte-like toamoeboid, they then simulate the collective migration of a monolayer. Theabbreviated level of intracellular detail permits efficient computations. Note thecomparison with [6], where more costly FEM computations and greater detail makesfor greater computational cost and smaller number of cells that can be readilysimulated. Software packages as in [181] may eventually make it more realistic toincorporate intracellular detail into multiscale models.November 24, 2020 22/39 .3 Links with experiments
In creating synthetic gene networks that regulate the adhesion protein E-cadherin inreal cells, Toda and colleagues [182] succeeded to design multicellular clusters thatself-organize into distinct layers. The expression of E-cadherin genes was placeddownstream of cell-surface notch receptors. Notch ligands on some cells activatednotch receptors on neighbors, and in this way, cell–cell interactions both influenced,and were influenced by intracellular signaling. The experiments were later linked tocomputational models in [146, 183]. When details of the relatively “simple” geneticcircuits are known, as in such synthetic biology experiments, modelers can bootstrapsignaling models to learn how spatial influences and cell–cell interactions shape theemergent tissue structures.In [184], we find a link between intracellular signaling and tissue morphogenesis.The authors show that the mutual antagonism of Rac and Rho can affect theinvagination and bending of epithelial that form a lens pit in the eye development in amouse. Rho apparently controls the apical constriction of cells via myosin, and Racthe elongation of those cells via F-actin assembly, hence accounting for the conicalangle formed by each cell and the overall curvature of the tissue.Overall, more experimental papers that probe the circuits that get recruited in thecollective cell migration are needed. The papers [129, 173] on the Merlin-Rac loop andon the link to YAP and E-cadherin [173] should be followed up with more detailedcomputational modeling and future rounds of experiments.
In this paper, we described a small selection of modeling works on cell migration thatbridge from intracellular, to cellular and multicellular scales, as shown in Fig 3. Manyother excellent papers have been omitted due to space limitations. That said, evenfrom the fraction surveyed, we find that a variety of computational and analyticmethods are used at various scales. Table 2 organizes modeling papers by their subjectand methodological approaches, Table 1 classifies a few experimental papers by theirbiological targets, and Fig 4 summarizes the ranges of relevance of both computationaland experimental methods. Our review has focused on the topic of single andcollective cell migration and its regulation. Likely motivated by development ofdisease therapies and NIH funding, or drug targets and support from pharmaceuticalcompanies, the more medically oriented subjects such as cancer, liver toxicity, or lungmorphogenesis, have fostered many generations of computational models. Bycomparison, the level of basic scientific computational research on multiscale cellbiology modeling is still emerging.We have seen that some computational techniques that work well at the single-celllevel, become too costly or excessive at the tissue level. (See also review ofcomputational models in [4, 38, 95, 140, 143] and [185] for cancer.) We also encounteredtopics where diverse computational techniques lead to similar predictions. (See Fig 5,and [143] for comparisons.) We still find instances where 1 or another group claimsthat their computational method of choice outperforms others or has fewer unrealisticfeatures. In many cases, such claims are, at best, unfortunate and overlook essentialshared attributes. In other cases, they skip over relative advantages versusdisadvantages of the distinct schemes. We believe that the field needs more rationalcomparisons of how custom-build computations perform against a host of “benchmark”test problems, as for example, shown in [143], and/or deeper comparative analysis ofcell-surface mechanics models that demonstrate equivalence of distinct approaches, asin [186].November 24, 2020 23/39 ig 4.
A summary of common modeling (top, red box) and experimental (bottom,green box) methods used to study cell behaviors from single cells to cell groups and upto tissues (left to right in increasing number of cells and increasing cell–cell adhesion).While more journals now require that simulation codes be made publicly available,in practice, this is only a half measure. Many researchers, and most biologists do nothave the software packages or expertise to read, execute, and run codes in differentformats. We recommend that, moving forward, the community should invest moredirectly in standardizing open-source software, with several specific aims: (1) ease ofoperation, user friendliness, and rapid learning curve; (2) basic built-in code for mostmajor computations, including reaction–diffusion solvers, particle position andcollision solvers, cell shape and cell–cell adhesion, intra and intercellular signaling, andso on; (3) ease of sharing “code,” as, for example, in the simple small Morpheus xml files — these preserve exact details of each simulation run; and (4) ease ofdevelopment of new plug-ins to allow the capabilities to expand with the needs of thecommunity. We recommend that groups invest resources in helping to develop suchshared platforms and that funding bodies (NIH, NSF, and NSERC) make it a priorityto support such developments. Certainly, at initial stages, this development requiresteams of computational experts who can establish solid, robust platforms, and createtraining manuals or instructional videos to recruit new users. Morpheus, a softwaresystem developed at the Technische Universit¨at Dresden, is 1 example along theselines, but others are needed.Returning to questions posed in the Introduction, we can draw a few generalconclusions.
Building from the bottom up, models that include detailed molecular interaction orgene networks (such as the Boolean models in [58, 150]) encode many details andoccasionally reveal attractors or identify missing components [150]. They are harder tounderstand on their own. Detailed models such as [40] benefit from insights ofpreceding more basic models [33]. At lower levels of detail but still arguably “bottomup” are studies that show how certain properties of molecular components andinteractions result in specific cell behavior such as polarization [5, 52, 54, 55]. OtherNovember 24, 2020 24/39 ig 5.
Cell sorting (left) and wound-healing (right) captured by several distinctcomputational methods. (A1) Cells represented by deformable ellipsoids in 3D.Simulations by Hildur Knutsdottir based on code originally created by Eirikur Palsson(A2) Vertex-based simulations using CHASTE open source platform, run byDhananjay Bhaskar. (A3) CompuCell3D cell-sorting simulations run by DhananjayBhaskar. In (A1–A3), there are 2 cell types with differing adhesion strengths to selfand other cell type. (B1) and (B2) show deformable ellipsoids and CPM scratchwound models with 2 cell types. Cells with weaker adhesion (red) tend to segregate tothe edge of the monolayer, acting as leader cells, and causing fingering of the front.(Compare with [168]). Papers that compare distinct computational platformsinclude [143].work starts from observed cell dynamics to infer the likely signaling pathways atplay [64, 70, 74] or likely underlying mechanisms [36, 118], that we could denote “topdown.” The latter, [118], also demonstrates the idea that it is preferable to weedthrough and discuss many models, particularly some that fail, with reasons for thatfailure, not merely aiming for a single optimal model. This kind of process helps tobuild a greater intuitive understanding of the specific key mechanisms required toexplain observed phenotypes.We recommend that modelers avoid publishing large-scale models as a “faitaccompli,” so as to “be the first to accurately simulate” some entire comprehensivebehavior of choice. Instead, we suggest that modelers should provide due descriptionof the steps taken in developing such models, with all informative failures andsuccesses. This would help the community to assimilate the insights that resulted froman entire program of research.In a related, but separate vein, the field is in need of rational consensus andstandard practices for stepping between size scales and levels of organization. To drawa common analogy, our ability to easily use and appreciate global geography as well aslocal structure has been transformed by the way that Google maps seamlessly allowsus to zoom down to street-level views and back out to the globe as a whole. Thealgorithms that reveal or blur over specific details as we step up or down were logicallyconstructed for easy navigation and optimal viewing of many levels of complexity.Nothing like this currently exists in the realm of cell biology. The fact that structuresand interactions change on a rapid timescale make this a tough issue, to be sure, butNovember 24, 2020 25/39ne deserving more attention.Associating rule of behavior with a specific hierarchy is possible once we havesufficient familiarity with the biology and predictions of basic models. This can helpto bridge hierarchies and avoid the fog of complexity. Mathematical methods such asdynamical systems, PDEs, and bifurcation analysis can help to find and account foremergent properties and universal principles in such basic models [48]. This is one ofthe strengths of the mathematical tools. A weakness is that these methods currentlywork well for small systems of differential equations, but not for large and complexsystems.Once we understand the repertoire of a single cell, we can move up to 2, 3, or manycells. Experimental observations of small cell groups provide good opportunities forunderstanding how to bridge from single to collective behavior. Modeling can thenalso explore the advantages of group migration, chemotaxis [18, 103], durotaxis [165],etc., in larger groups. For a large enough number of cells, condensing the details intosimpler rules becomes expedient. For example, while polarity is represented by PDEsand patterns inside a single cell, it can then be simplified, depicted by a directionvector [15, 25] in place of a full internal gradient of Rho or Rac for multiple cells.
The current computing power at our disposal allows increasingly detailed models to beconstructed, with tens or even hundreds of components, and many more parameters.The temptation to create such a model and to present it as a mechanisticrepresentation of real cells is hard to resist. Yet, before going down this very intricateroute, important questions should be considered: What do we expect to learn? Mightwe have left out something important? Have we included superfluous detail thatobscures the basic structure? Are we certain that our parameter regime faithfullycorresponds with true rates and quantities? As previously noted, even though we canconstruct highly complex models, our comprehension of that detail is limited.In general, best practices seem to invoke back and forth attempts to includeimportant aspects, simplify to understand, throw out secondary factors, and go back toincluding detail. Simplifications lead to insights and rigorous analytic results but leavelarge gaps to real biology. However, returning to detailed versions of a model once themajor insights are at hand (or vice versa) is helpful. While [52] explains polarizationin a motile cell, for example, [61] extracts some of the key properties and interactionsthat guarantee that it could work in a mathematically tractable mini-model.Furthermore, “toy models” consisting of small sets of ODEs or PDEs can help toformulate universal principles that work across many biological examples and manyscales. Here, we can mention the concepts that mutual inhibition or positive feedback,which result in bistability and hysteresis. Simple “modules” consisting of specific typesof interacting components, whether molecules, cells, or animals have prototypicalbehaviors as switches, oscillators, or other dynamics, and, with molecular diffusion orrandom motion, engender patterns or waves [48, 49].
Biology is an experimental science at its core, and models that illuminate its mysteriesmust eventually meet and concur with biological evidence. Modern methods such asmachine learning can help match a vast dataset with the best candidate model(s) asshown elegantly by [118]. At the same time, modelers, physicists, and biophysicistscan make real contributions by using their craft and theories to unlock important factsthat are not at all apparent otherwise [42].November 24, 2020 26/39everal of our examples demonstrate that some experimental papers have explicitlyaddressed signaling pathways that mediate cell–cell communication in collective cellbehavior. As pointed out eloquently by a reviewer of this paper, “experiments canonly tell [us ... what are] upstream and downstream regulators. We needmathematical models to incorporate this information with the intracellular networksto form a signaling network on a multicellular level to study how a group of cellsprocesses signals collaboratively.”
Acknowledgements:
We are grateful to the Pacific Institute for MathematicalSciences for providing space and resources for AB’s postdoctoral research. We aregrateful to former and current members of the Keshet-Feng Research Group forhelpful discussions over many years that solidified our appreciation of this area ofresearch. We thank Hildur Knutsdottir and Dhananjay Bhaskar for permission to useresults shown in Fig 5 that they produced while they were UBC students.
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