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

Andreas Buttenschön1, *, Leah Edelstein-Keshet1

1 Department of Mathematics, University of British Columbia, Vancouver, Canada

*<EMAIL_ADDRESS>
## Abstract

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.

## Author summary

In this review paper, we summarize the literature on computational models for
cell motility, from the biochemical networks that regulate it, to the behavior
of 1 and many cells. We discuss the distinct approaches used at each level,
and how models can build bridges between the different size scales. We find
models at many different levels of biological detail, and discuss their
relative contributions to our understanding of single and collective cell
behavior. Finally, we indicate how models have been linked to biological
experiments in this field.

##  1 Introduction

Over several decades, there has been great progress in our understanding of
cell motility. In the 1980s and 1990s, the basic machinery of eukaryotic cell
motion and the role of the actin cytoskeleton were discovered and refined.
Regulation of motility by intracellular signaling networks was then deciphered
in the late 1990s and through the 2000s. We continue to discover links between
cell signaling and cell shape and function, in both normal and diseased cells.
Recent efforts aim to link single cell behavior to collective behavior of many
cells and emergent dynamics of tissues.

Though originally descriptive, cell biology has emerged as a quantitative
science over the same time span. Mathematical and computational modeling have
become more universally accepted, more closely integrated with experimental
research, and more advanced in terms of methodology.

Here, we survey the state of the field, emphasizing bridges that span scales:
from molecular signaling to multicellular hierarchies. We focus on the role of
modeling and computational biology. Because the literature is vast and growing
exponentially, we limit our review to several key themes and concentrate on 3
questions:

  1. (1)

To what extent have models provided a way to bridge between the 3 levels of
organization, from intracellular signaling, to single cell behavior, and to
collective cell/tissue behavior?

  2. (2)

What level of detail is appropriate in a computational or mathematical model?
What kinds of models are suitable for a given situation?

  3. (3)

What is the relationship between models and experiments in the current
literature on the subject?

At each level, we consider these 3 questions in subsections with headings
“Bridging scales,” “Levels of detail,” and “Links with experiments.” Like any
other subdivision, this is to some extent arbitrary, as literature papers
often span such categories.

Fig 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 lowest scales, the goal is deciphering the interplay between
stimuli to the cell (chemical, topographic, mechanical, etc.) and
intracellular signaling networks that regulate F-actin (branched polymer) and
the cytoskeleton (not drawn to scale). These interactions lead to protrusion
or retraction, cell polarization, and shape changes that enable directed
motility and chemotaxis. At a higher level, an aim is to link cell behavior
and cell–cell interactions to the outcomes of cell collisions (e.g., CIL) and
to the cohesion of tissues versus EMT, where cells break off. Interconnections
exist between all layers, only 2 of which (white arrows) are shown here. CIL,
Contact Inhibition of Locomotion; EMT, Epithelial Mesenchymal Transition.

Many excellent reviews are already available, including [1, 2, 3, 4]. Some
survey computational methods and others provide links to experiments. The
focus on the above 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 start with the subcellular level of biochemical signaling (left), and move
up to single cell behavior (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 contributions from the literature to use as
examples. A summary “mapping” of the modeling literature into levels of detail
and numbers of cells is provided in Fig 2.

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 papers in the diagram (starting from the upper left to lower
right: [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30]).

No one review paper can do justice to the entire field. Hence, we point the
reader to related review articles that complement our own. In some cases, they
cover similar ground but with distinct emphases or points of view. In [1], the
authors study the issue of cell heterogeneity, its sources at various size
scales, and its role in collective cell migration. They briefly discuss self-
propelled particles (SPP), Cellular Potts Models (CPM), and vertex-based
approaches that we also discuss in this review. They recommend further
investigation of mechanical assumptions in models, and of more realistic
tissue size. A thorough review of the biomechanics of collective cell
migration appears in [2], with a primary focus on cancer. In contrast with
other reviews, this paper also provides an excellent summary of experimental
methods. Phenomena discussed include the “unjamming transition” where a cell
collective changes from “solid-like” to “liquid-like” behavior and its
connection to the epithelia-mesenchymal transition.

Force balance and energy-based models for single amoeboid and collective
mesenchymal cell migration are compared in [3]. The authors indicate the
challenge of converting between energy-based and force-based modeling
platforms, and point to the significance of doing so. (See [31] for this
connection in the CPM.) They also describe instances of close experiment-model
integration. Finally, a recent review of multiscale models in [4] has a soft-
matter physics perspective and describes computational methods (CPM, phase-
field, active matter, particle systems, etc.) and their physical basis.

##  2 From cytoskeleton and intracellular signaling to cell shape and
migration

How do chemical and mechanical stimuli, together with intracellular signaling
shape the behavior of single cells? This question is central to the bridge
between left and central panels of Fig 1.

###  2.1 Actin dynamics

We briefly highlight this well-established area to illustrate examples of
bridging scales, diverse levels of detail, and model–experiment synergy [32].

####  2.1.1 Bridging scales

Actin dynamics is studied at many levels: from association of actin monomers
to form filaments [33], to biophysical force production by F-actin [34], the
assembly and branching 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]
provides a solid bridge between molecular scale and cell scale phenomena. The
authors explain the main model formalisms and show how models can be used to
understand experimental data. A review of the link from actin and its
properties to cell mechanosensing and behavior is given in [39]. Reviews of
the literature on actin-based cell migration reveal a mature field, where
quantitative methods and theory have ripened in tandem. This synergy has
benefited greatly from biochemical pioneers, such as Tom Pollard (Yale
University), who helped to nurture an appreciation for mathematical and
physical modeling in the community.

####  2.1.2 Levels of detail

The pair of papers [40, 33] aptly illustrates the dichotomy between highly
detailed computational models [40] and more conceptual minimal models [33]. On
one hand, the highly detailed [40] synthesizes a large body of experimental
data for actin assembly and branching, including many actin-related component.
At the opposite extreme, the models of [33], as well as later papers [37, 36]
emphasize physical principles, universal properties, and general insights. In
some sense, the overall predictions of both model types can be compared. The
minimal models are more tractable for analysis, parameter sweeps, and overall
insights, but are harder to connect to detailed molecular biology experiments.
The complexity of highly detailed computational models (matching molecular
experiments) makes it harder to navigate their results, which are more like a
“1-to-1” map. Such maps may be important for those who “live in the
neighborhood,” but are essentially baffling for newcomers. While distinct, one
could argue that these approaches are complementary, so that overall insights
can be obtained from one, and specific details from the other. The dichotomy
between the detailed and the simplified models will reappear throughout this
review, serving as 1 hallmark of distinct views.

####  2.1.3 Links with experiments

A review of the experiments and theory for actin dynamics in the motility of
keratocytes is provided in [41]. A retrospective that emphasizes the
importance of theory, and case-study where theory has led the way is described
in [42]. Notable contributions linking experiments to theory for cell shape
include [36, 37], where shapes of cells are classified and related to actin
branching and treadmilling.

###  2.2 Signaling networks

Fluorescence resonance energy transfer (FRET) microscopy led to breakthroughs
in visualizing the activities of signaling proteins that regulate the
cytoskeleton. This has enabled live imaging of spatiotemporal activity of the
Rho family GTPases [43].

The roles of Rho family GTPases (Cdc42, Rac, and Rho) in regulating actin
assembly and myosin contraction are well–known [44, 45]. These switch-like
proteins are central regulators of cell migration, funneling signals from the
cell’s environment to downstream components that shape its motility. Cdc42 and
Rac promote F-actin assembly and cell edge protrusion, whereas Rho facilitates
myosin light chain (MLC) phosphorylation, activating myosin-based cell
contraction. Rho also promotes F-actin through formins and is a Rac
antagonist. The importance of GTPase dynamics in cell motility [46] has also
attracted modelers to this arena.

A review of GTPase spatio temporal models can be found in [47]. A classic
paper on modeling spatio temporal dynamics of signaling in cells (not
necessarily GTPases) is [48], a paper that emphasizes universal principles. We
see how commonly shared basic motifs combine to form complex dynamics [48].
Many lessons learned from this approach can be applied directly to studying
GTPases or other signaling networks. In general, the community stands to
benefit from more expository papers of this type, where the ingredients that
combine to set up specific dynamical signatures are exposed. See also [49] for
a popular example of this type, based on years of experience with models of
cell cycle regulatory networks.

####  2.2.1 Bridging scales

Several papers link the dynamics of GTPases to cytoskeletal assembly and cell
shape. Among these are experimental [50, 51] and computational [52, 5] works.
A review of signaling that organizes the cell front and rear in Dictyostelium
discoideum is given in [53], who also survey models of these as excitable
systems. Some of these papers are described in fuller detail below.

####  2.2.2 Levels of detail

GTPases act like molecular switches that cycle on and off the cell membrane
and interact via crosstalk through effector proteins. A large collection of
proteins participate in GTPase signaling: Guanine nucleotide exchange factors
(GEFs) activate and GTPase-activating proteins (GAPs) inactivate the GTPases
with varying degrees of specificity. GTPases are also sequestered in the
cytosol by binding to guanine nucleotide dissociation inhibitors (GDIs). In
some models, notably [54, 55], the details of the binding and mechanisms of
activation–inactivation are lumped into phenomenological terms such as
Michaelis–Menten or Hill functions. In [55], emphasis on cross-talk of Cdc42,
Rac, and Rho and on spatial polarization comes at the expense of the molecular
steps themselves (some of which remain unknown). In [56], more detail on such
steps is modeled and eventually reduced to 3 ordinary differential equations
(ODEs) using a quasi-steady-state approach. The authors focus on the role of a
Cdc42-GEF in oscillatory phenotype in neuronal growth cone motility. In
contrast, [57] concentrate on GDI binding of GTPases in a highly detailed
computational model (originally crafted in BioNetGen and Virtual Cell).

Crosstalk between GTPases is modeled in great detail by [58], who constructed
a Boolean model for epidermal growth factor (EGF) signaling to Rac and Rho.
Their model has 38 intermediates and multiple reactions that activate or
inhibit components. (See also [59], who combines PhysiCell with MaBoss to
allow the modeling of intracellular signaling networks as Boolean networks.)
At lower level of detail are Rac-Rho mutually antagonistic models that leave
out the forest of interacting nodes [60].

Simpler models for single GTPase include [61] for cell polarity and [62, 63]
for cell shape. There, assumptions are made to condense underlying complex
interactions into simpler “rules” of behavior. For example, “high GTPase
activity leads to outwards protrusion force at the cell edge” (in the case of
Rac) or “contraction” (in the case of Rho) [64, 65]. An advantage of this
simplification is that simple models consisting of few partial differential
equations (PDEs) allow for well-honed methods of applied mathematics,
including nonlinear dynamics, stability theory, asymptotic analysis, and/or
bifurcation theory to explore model predictions, parameter dependence, and
regimes of behavior.

Serendipitously, simplified signaling models, like [61], occasionally also
expose interesting 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 the opposite end, resulting in robust “cell
polarization.” Simplified models also permit numerical implementation in more
complex geometries. For example, in [67], a wave-pinning model for a single
GTPase in a single 3D cell is implemented by bulk diffusion methods. The cell
does not deform, but the spatial localization of the GTPase is quantified in a
fully 3D geometry. (See also [68], who considered neutrophil fluidization, 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 shape and directed motility [52, 5]. In the latter 2 papers, a
single moving “keratocyte” is represented using a Cellular Potts Model (CPM),
an energy minimization agent-based platform for computing cell shapes. It is
shown that GTPase signaling can account for cell polarization, reorientation
[52], and resolution of conflicting cues or obstacles [5]. (Compare with [71],
who employ phase-field computational methods toward similar goals.)

By way of comparison, in another more detailed approach, in [72], a cell is
modeled as having a solid boundary that is moved using a Stefan condition. For
the chemical signaling, Rho, Rac, 2 species of GEFs, F-actin, and G-actin are
included. The model accounts for the basic repertoire of neutrophil motility.

All in all, models of GTPase signaling have taught us several valuable
lessons, some with universal ramifications. First, modeling has provided clues
to the functional significance of the seemly strange cycling of GTPases
between membrane and cytosol: namely, the separation of diffusion rates
resulting from these distinct compartments could be playing a role in pattern
formation processes, an ingredient enabling GTPase localization or patterning
in cells. Second, the stripped-down models have shown that chemical
polarization in a cell need not depend on crosstalk between multiple types of
GTPases — it can be set up by a single member of the family, given sufficient
positive feedback and some depletion of its cellular pool [61]. Groups are
still occasionally rediscovering on their own, the link between cell size and
cell polarization that was implicit in [66], suggesting the need for more
expository reviews of mathematical results. An important concept introduced in
[5], but not yet fully recognized in the community, is the synergy between
GTPase dynamics and its effect of cell shape and boundary curvature of the
cell edge. Simply put, the “motion by curvature” of the chemical system
interacts with boundary conditions to accelerate the dynamics. Altogether, the
appreciation of GTPase signaling has benefited greatly from a host of distinct
modeling and mathematical approaches.

####  2.2.3 Links with experiment

A review of the links between models of cell migration and experimental data
(image processing, 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 of
HeLa cells from various starting cell states (polar, anti-polar, or nonpolar).
The authors showed that an internal circuit of signaling (Cdc42, Rac, Rho, and
phosphoinositides) could account qualitatively for the observed responses of
these cells in microfluidic channels, with an externally controllable response
to Rac. In both this and the follow-up [74], the 1D geometry of the channels
helps to reduce the geometric complexity of cell shape, allowing for a better
match to 1D spatial model representations.

How are models for GTPase crosstalk experimentally linked to cell morphology
and motility dynamics? In experimental results of [51], the authors showed
that signaling circuit of mutually antagonistic Rac and Rho could affect not
only the dynamics of F-actin, but also shapes and migration of mesenchymal
breast cancer cells. Together with a mathematical model for the Rac-Rho
interactions, they were able to manipulate the Rac-Rho competition,
demonstrate bistable states, and show that manipulating the system by
inhibiting PAK (a kinase that mediates inhibition of Rho by Rac) displayed
hysteresis characteristic of bistable systems. The significance of this paper
is that it demonstrates a direct link between a simple hypothetical model for
the way that Rac and Rho GTPases operate in a cell and the next level up, that
of overall cell 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 [75,
50, 60, 76] explore how Rac-Rho mutual antagonism is linked to cell
morphology. In [75], the authors experimentally manipulated Rac and Rho
activities to show spread or contracted cells. Interestingly, they found that
combining constitutively active (CA) Rac and Rho simultaneously results in
mixed morphologies in human glioma cells. The fact that high Rho and high Rac
activity produces a mixture of possible coexisting stable steady state cell
shapes was independently predicted in a purely modeling study by [60]. Related
experiments by [50] on cell shapes were also later modeled and explained in a
follow-up paper by [76]. These papers demonstrate that relatively simple
stripped-down depictions of cellular signaling “modules” (such as Rac-Rho) can
account 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
cell migration. The authors find that the degree of cell-extracellular matrix
(ECM) interactions are key determinants of speed, morphology, and cell-
generate substrate strains during motility.

In recent times, a clear link has been established between GTPase activity and
mechanical tension experienced by cells. A pioneering experimental paper that
showed this 2-way feedback is [78]. The effects of forces on the Rho family
proteins, including the 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 and substrate 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 a prominent role in
cell–matrix interactions.

Strain and strain gradients affect cell orientation. Single cell experiments
with cyclic stretch are described in [81]. The authors suggest that the Rho-
ROCK pathway that regulates myosin light-chain activity is responsible for
sensing and responding to strain gradients.

Overall, it appears that the link between mechanical stimuli and GTPase
signaling is still young, providing ample opportunities for creative modeling.
So far, there is no consensus on what are the “takeaways” from the models so
far. Further, it would appear that this gap should be filled if we are to
successfully bridge between 1 cell and many, since the mechanics of cell
collisions, as well as cell collective migration, entail sensing of both
chemical and mechanical stimuli in the interacting cells.

##  3 From single to collective cell behavior

While the meaning of “collective behavior” is intuitively clear, what is less
clear is how to specify the transition between a collection of agents, acting
individually, and the collective behavior of the group (right panel, Fig 3).
To some extent, the same issue arises in macroscopic models for swarming
animals or interacting particles. As groups grow and interactions between
members increase, new distinct properties emerge at the level of the group
that were absent at lower levels of organization. Characterizing, quantifying,
and understanding such emergent properties remains the single most interesting
and elusive goal in bridging between single and collective phenomena.

Fig 3: Modeling goals can be classified into broad categories that span levels
of hierarchy. Some models attempt to span knowledge of single cell behavior
plus interactions to predict emergent multicellular behavior (bottom up,
left), whereas others 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 inanimate
systems. This goes 2 ways. If we observe the dynamics of the group, can we
infer underlying interactions? This is a top-down approach; see Fig 3.
Conversely, given rules followed by individuals, what can we predict about the
structure and dynamics of the group? For example, how do “rules” for cell–cell
interactions such as coattraction, contact inhibition of locomotion (CIL)
affect cell collectives? In cell biology, we add the question of how rules of
behavior of cells are to be associated with specific 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 migration persistence measures. Mathematicians have also contributed with
useful methods, such as dynamical systems, ODEs, PDEs, and bifurcation
methods. More recently, the use of topological data analysis (TDA) has also
entered the mix, to help address some of these questions. The examples
provided by [82, 83, 84, 85] show how TDA can be applied even more effectively
than the traditional physics-based order parameters to compare data and model
output. Such novel methods could help to begin addressing questions where
model hypotheses are to be compared with observed behavior.

###  3.1 Small cell groups

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 small groups [25, 86]. (See top right panel, Fig 3.) Then, the
behavior of single cells and the effect 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 this category, as do
“cell swarms.” The migration of neural crest cells is 1 example of a loose
cell swarm, as reviewed in [87]. Adherent cells in tissues (and their cohesion
in epithelia) will be discussed in another section.

A number of key questions posed by papers in this area include the following:

  1. (1)

What mechanism, chemical and/or mechanical, can account for CIL in interacting
cell pairs [7]?

  2. (2)

How do cells reconcile conflicting cues [74, 87]?

  3. (3)

Under what conditions would a cell reverse its direction [74]?

The study of small numbers of interacting cells provides a useful paradigm for
connecting molecular mechanisms to group behavior, an area that has only
recently received the attention it deserves [86].

####  3.1.1 Bridging scales

Phase-field methods have been used to model cell shape and motility in 2
spatial dimensions (2D) [71]. The outline of a cell is represented by a level
curve of some function $\phi(x,y,t)$. For example, in [88], a minimal model
for Rac signaling was used to describe the shape and motility dynamics of a
single phase-field cell. The model is then extended to pairwise and cell–train
interactions in a narrow 2D strip. Interestingly, the authors had to assume an
intracellular signaling “inhibitor” that is activated at cell–cell contacts to
obtain appropriate behavior. This is one of the earliest models spanning the
three levels of organization, from subcellular, to single and multiple cells.
The model exploits a reasonably simple level of detail at each stage to
reproduce several behaviors, such as reversals, cells walking-past one
another, cell sticking, and forming a “cell–train.” We feel that this paper
provides good prototype to be emulated by community members: a clear and well-
studied intracellular system, linked to evolving cell shape, with feedback
from cell–cell interactions. The paper leads to natural follow-up questions,
amenable to experimental investigation: what cell components play the role of
the putative “inhibitor”?

Models for neural crest cell (NCC) swarms in vivo and in vitro were proposed
in [89, 90, 87], focusing on distinct behaviors of leader and follower cells.
(See “Links with experiments” for a summary.) For the same NCC system, the
paper [7] developed a multiscale computational model linking signaling to cell
group migration. A Rac-Rho GTPase circuit was represented by differential
equations (ODEs) at nodes on the perimeter of a deforming polygonal cell. Co-
attraction was represented as a non local tendency of cells to cluster, short
range contact inhibition by up-regulating Rho at nodes in contact with other
cells. Stochastic noise in Rac activity generated characteristic run lengths
and reorientation seen in typical NCCs.

In [7], it was shown that these underlying biochemical systems could account
for CIL. This study can be compared with the GTPase and inhibitor system
described above, [91]. Coherent migration of a group of $n$ cells, each of
diameter $d$, was possible along a confinement corridor of width on the order
of $d\sqrt{n}$. This paper, [7], has a wider corridor and larger numbers of
cells than [88], but confinement serves a similar function of enhancing
directed cell movement. In real embryos, such “corridors” could be defined by
permissive and inhibitory regions for NCC migration, as shown experimentally
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 group
migration, without bridging to the biochemical circuit.

In somewhat related work, [8], 2D cells are described by viscoelastic
triangular finite elements. Cells can adhere to one another using linear
springs. At each cell vertex, intracellular signalling controls cell
protrusion and cell–matrix adhesion. The signaling networks consists of Rac
and several other components, including the downstream kinase PAK, the focal
adhesion protein Paxillin, the cell adhesion molecule cadherin, and Merlin, a
protein implicated in contact inhibition of cells in a feedback loop with Rac1
and PAK. Key findings of this study were (1) cells elongate on stiffer
substrates; and (2) stiffer substrates allow for better directional guidance
and collective cell migration. Interestingly, the authors state that the
principal role of intracellular signaling is to coordinate cell movement
direction over distances of $\approx 500\mu m$.

####  3.1.2 Levels of detail

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 that result 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 cell
repolarization result in properties of collective cell behavior. Rules
governing pairwise interactions and probabilities for CIL are derived from
experiments by [96] described in the next section. It turns out that there is
an optimal number of cells in a group for migration persistence. This paper
joins [96, 70, 74] and others in proposing and using simplified 1D geometry to
examine the basics of cell–cell interactions. The synergy between papers [25,
96] is very helpful, and points to the need for more paired experiment-model
studies.

SPP models are easily extended to higher dimensions, and/or more realistic
pairwise forces. For example, [29] considers a cell swarming model in 2D that
is closely related to spherical cell models in [27, 97]. While [29] neglects
cell–cell friction, the paper couples the cells’ compressibility, which plays
the role of a local force of repulsion, to adhesion due to long protrusions
such as filopodia that leads to nonlocality.

The highlight of [29] is the inclusion of these non local cell–cell attraction
forces. In this way, the paper makes a seminal link between extended cell–cell
interactions and a body of theoretical work on non local interactions in
swarms or physical particles. The same paper also features the concept of
H-stability (preservation of bounded density as the number of particles
increases). This idea is used to ascertain correct cell spacing in swarming
models, facilitating links to experimental observations. This paper hence
imports a panoply of techniques, methods of analysis, and results, from
theoretical work on non local models dating back to the 1990s [98], and
inspires new ways of interpreting experiments.

Parallels with macroscopic swarms and flocks are useful, so long as we
remember that these operate in vastly different friction regimes. (Cells
experience an overdamped regime, where inertial forces are negligible.) In yet
another example of a cell swarming model, the paper [99] accounts for the
development of stripes in zebrafish pigmentation patterns. The model comprises
2 cell types, with cell differentiation and death, and successfully 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
border cells in Drosophila as a single sphere to determine the role of cluster
size in chemotactic migration speed. The authors assume a spherical cluster
moving by Stokes’ law in a viscous environment. The migration force is
proportional to cell surface receptor occupancy in an exponential gradient of
attractant. The results show good fits to both normal cells and cells that
have a deficiency of receptors. The same biological problem is treated in
[101], using a force-based approach for individual spherical cells (rather
than cluster as a whole as in [100]) in 2D, followed by 3D (for small clusters
of up to 8 cells). The computation incorporates cellular adhesion and
repulsion, with some stochastic effects.

At the next level of detail, the shape of interacting cells is explicitly
represented. The papers [102, 91] apply phase-field methods to model cell
shapes in 2D with intracellular signaling. Interestingly, the signaling system
is itself “minimal:” The authors assume basic “wave-pinning” GTPase dynamics,
as in [61], with additional stochastic elements. The levels of internal GTPase
give rise to either protruding or retracting forces on the cell edge. Then,
[91] models the rotational cell motion when 2 cells interact (compare with
[31] who showed a similar behavior in CPM cells). For coordinated movement,
the directions of these gradients need to be linked. A key finding of [91] was
that the emergence of rotational motion is strongly dependent on the 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.
We now find groups addressing the question “What can cells do well together
that they do poorly on their own?” For example, collective chemotaxis was
shown to be better in groups than in single cells [18, 103, 104]. (See also
the related work by [22] for larger cell groups, reviewed in the next
section.)

Overall, our sampling of the literature reveals a paucity of models at the
small-cell group level, relative to subcellular and tissue-scale modeling. We
believe that papers such as [29] open opportunities for swarm-centered
modelers to make a contribution to the cellular biology realm, by translating
numerous results of physics and mathematics, suitably reinterpreted, to cells.
We also contend that understanding the ways that cell interact in small groups
is an important and relatively tractable step to formulating more informed
tissue-scale models.

####  3.1.3 Links with experiments

Pairwise interactions of MTLn3 rat breast adenocarcinoma cells inside
microfluidic channels with chemotactic cues were quantified in [74]. These
experiments provide important information for the spatio temporal evolution of
GTPase activity during cell collisions. When cells collided in [74], they
either stalled, adhered and moved together, or exhibited CIL. More recently,
[105] studied how mechanics, and cell polarity drive collective cell migration
in larger groups of cells. More experiments of this type would be very
important in laying the groundwork for understanding pairwise cell
interactions and in directly informing the development of models. Such models
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 with another cell. The authors also explored persistence of cell
trains, bridging from pairs to larger groups. An agent-based model was then
used to predict cell–train outcomes based on the observed repolarization
probabilities.

A model that replicates the experimental setup of [106] appears in [107]. Flat
cylindrical 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 is seen: The leader creates a “track” of aligned ECM
fibers that the second cell can follow.

A number of works shed light on the molecular basis of cell–cell interactions.
For instance, Wnt signaling is a ubiquitous pathway that regulates polarity,
migration, and many other cellular processes. Wnt signaling at cell–cell
interfaces that locally upregulates Rho is studied in [108]. Ephrins form
cell–cell ligand–receptor bonds that funnel signals to GTPases to control cell
repulsion or attraction. Involvement of Ephrin receptors in CIL [109], effect
of tension on CIL [110, 111], and on the activation of RhoA [112] have been
similarly studied.

Experiments on migration of primordial progenitor cells in the developing
zebrafish [113] established the importance of guidance cues such as repulsion
due to ephrins. Physical barriers also guide cell migration and organ
placement. Actin was visualized for cells colliding with barriers expressing
ephrin receptors. A particle-based model with an attraction–repulsion Lennard-
Jones potential was then used to test the effects of reflecting and
nonreflecting boundaries. (The Lennard-Jones potential is composed of power
functions of the form $r^{-12}$ and $r^{-6}$ for local repulsion and long-
range attraction.)

NCCs chemotax toward some cell types in a chase-and-run behavior [114]. For
example, NCCs chase the placode cells, namely cells that are destined to form
feathers or teeth or hair follicles. Before contact, each cell has high Rac at
its front. Upon contact, the cell-adhesion protein N-cadherin, at the
cell–cell interface, inhibits Rac in both 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 that the forces were correlated with
E-cadherin localization. The authors reduce the cell group to a network
representation, with vectors connecting cell centroids.

A recent paper explores the relationship between initial tissue size and the
dynamics of tissue growth [116].

Experiments on the migration of chick neural crest cells in vivo and in vitro
are described by several groups, including that of Kulesa. His experiments
have been directly linked to models previously mentioned [89, 90, 87]. The
group has investigated how cells reconcile mixed chemotactic signals, as in
breast cancer cells in [74], and how information is shared between cells. The
differences between leaders and followers is of recent interest in the group.

While papers such as [117, 25] argue eloquently for the need to carry out
small-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 vivo experiments, and so, less persuasive from the grantsmanship
perspective. Hence, there are still relatively few experiments to map out the
links between cell signaling and cell–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.

###  3.2 A related example: Spacing of nuclei

While not directly related to cell migration, we nevertheless decided to
briefly highlight the following example as it illustrates close synergy
between model and experiment using modern methods. In [118], models complement
experiments to investigate positioning of nuclei in Drosophila larvae muscle
cells. The authors used data for many hundreds of nuclei and proposed a
diverse set of particle-based models. Models were formulated specifically to
test distinct sets of hypotheses for how nuclei interact with microtubules and
molecular motors. Interestingly, machine learning was used for computational
filtering, comparing large numbers of simulation outcomes to simple summary
statistics such as nuclei positions. The filters are applied in stages, each
stage rejecting inappropriate or inaccurate models that are inconsistent with
the data.

Ideally, this process requires high-throughput screening, mandating both
clear-cut summary statistics, and a large dataset. As noted by [118], the
availability of data imposes a limitation on possible model complexity that
could be adequately fit. Eventually, the best-fit model predicts that nuclei
push against one another and against cell boundaries by sending out growing
microtubules.

This example suggests a number of important questions for future model-
experiment investigations. (1) How can one ascertain the optimal level of
model complexity in the first place? (2) How do we extract clear identifiable,
meaningful summary statistics from a large dataset? (positions of hundreds of
cells) (3) What are optimal methods for massive filtering to obtain the best
model? This example also illustrates a creative use of machine learning as a
tool in understanding and filtering many possible mechanistic explanations.
Clearly, this approach goes beyond the common machine learning application as
a “black box” to sort or classify or learn rote features of cells or images.

###  3.3 Collective cell behavior and tissue dynamics

Even before details of cell–cell signaling were known, theorists were
formulating cell-based computational models for tissue dynamics. Garret Odell
and George Oster [119] considered cell–cell pulling forces, coupled to a
bistable stretch-sensing signaling module. They showed that this mechanism
could account for local contractions that folds a tissue in a developmental
process such as gastrulation. Their seminal idea still resonates today [16],
now that signaling components are familiar to us.

Reviews of the recent biological literature on collective cell migration
include [120, 121, 122]. (See also brief highlights in Table 1.) A thought-
provoking review of the biology, [86], focuses on the role of adhesion
mechanics, bridging between force regulation in single cells and in collective
cell migration. The paper gives detailed summary of the underlying molecular
players, synthesizing a vast literature on the subject. The mechanical and
mechanotransduction players at the molecular level are then linked to
consequent cell-level properties. The paper proposes a rational program of
experiments from 1 to 2 to many cells (see Fig 4 in [86]) to bridge the scales
from single cells to tissues.

Main-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 the back of the cell (Immune cells) |  [124]
Rho opto-activation |  Rho at cell’s rear can control directional migration. Rho activity regulates switch between amoeboid and mesenchymal migration (macrophages) |  [125]
Rho-ROCK pathway |  This pathway senses and responds to strain gradients; 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 to data |  [127]
Merlin |  Merlin is a negative regulator of Rac and may also be 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 contacts causes an upregulation of Rho |  [108]
Forces and GTPase activity |  Cell-cell forces affect Rho activation |  [112]
Topographic cues |  Single cells have different organizations on grooved vs flat substrata (fibroblasts) |  [130]
NCCs and placode cells |  Intricate interplay between chemotaxis, integrins and their effect on internal GTPase activity |  [114]
Ephrins |  Ephrins in repolarization of cells (zebrafish development) |  [113]
CIL and Ephrin receptors |  Ephrin receptors affect CIL |  [109]
CIL |  Biphasic relationship between probability of CIL and collective migration |  [96]
CIL |  Tension built up during CIL |  [110, 111]
CIL |  Spatio-temporal GTPase patterns during cell-cell collisions and CIL events (microfluidic channels) |  [70, 74]
Collective strand formation |  Interesting difference between polarization of leader 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 tensile force or shear stress and substrate stiffness |  [79]
Mechanical GTPase activation |  Rac1, Cdc42 activated by stretching adhesion bonds. Rho maintains focal adhesions |  [80]
Epithelia |  “Push-pull” or “caterpillar” collective motion in narrow grooves, more complicated movement in wide channels |  [132]
Epithelia |  Sustained oscillations in epithelial sheets |  [133]
Epithelia |  Cells crawl in the direction of maximal principal stress. Leaders impose mechanical cues on followers |  [134]
GTPases and GEFs |  Coordinated waves of motion, cells at front react first (scratch-wound assay in human bronchial epithelial cells) |  [135]
Table 1: Experimental summary: CIL, contact inhibition of locomotion; GEF,
guanine nucleotide exchange factor; NCCs, neural crest cells; RDEs, reaction
diffusion equations.

Several key questions appear in papers at this level.

  1. (Q1)

How do internal dynamics of cells influence the emergence of collective
behavior? How do cells share and transfer information between one another? How
and to what extent are mechanical forces transmitted over distance [136]?

  2. (Q2)

What is the role of the leading front in guiding collective migration? [132]

  3. (Q3)

Do actin cables or ECM fibers transmit long-range stresses [131, 2]? Are such
mechanisms analogous to the non local sensing mechanisms that play a dominant
role in models of animal swarms?

  4. (Q4)

What cellular and tissue-based mechanisms (long- and short-range cues,
barriers, cell–cell adhesion, etc.) are essential for proper formation of
organs during development [137, 113]?

Parallels between the polarization and directed migration of single cells and
of a cell collective is highlighted in many recent works. See, for example,
the recent perspective paper, [117], demonstrating parallels in the GTPase
distribution, chemotaxis, and mechanosensing between the single cell level and
tissue level.

As shown in Fig 5, many computational platforms are currently used to simulate
multicellular migration. These are reviewed in [138, 95, 139, 140, 4]. Rigid
or deformable spheres or ellipsoids (in 3D or 2D, Fig 5A1 and 5B1,
respectively), polygonal “vertex-based,” or CPM cell representations are
common. Software platforms such as CHASTE (Oxford University, [141, 142, 143],
and Fig 5A2), CompuCell3D (U Indiana, [144], and Fig 5A3 and 5B2), or Morpheus
(TU Dresden, [145]) have made it increasingly easier to simulate complex
tissue dynamics without having to reinvent computational algorithms and
graphics.

In our humble opinion, there is a vital need for support and sharing of
standardized open-source software packages, with suitable plug-ins donated by
groups utilizing those resources. First, such tools would save time, person-
power, expertise, and expense of individual “kludges.” It would reduce
duplication of effort — how many of us want to reinvent our own finite-element
computations? Even more compellingly, such standardization can bring about
much easier communication and appraisal of published models, and in-depth
scrutiny of exactly what those models include. Having learned the basic steps
of Morpheus, one of us (LEK) has become an enthusiastic user. See, for
example, [146], where figures are linked directly to executable Morpheus .xml
model files that produced them.

####  3.3.1 Bridging scales

Here we concentrate on bridging between models for biochemical signaling and
multicellular behavior. In the next section, we will focus on bridges between
1 or a few cells 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 cell perimeter due to myosin. ODEs track Rho activation by cell
perimeter stretching and downstream myosin activation (by phosphorylation) at
a given vertex. Vertex motion is a result of force balance. Friction and
passive forces are derived from energy penalties for area constraint and from
the energy of adhesion to neighbor edges. This aspect 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, and regimes of
behavior such as streaming, contractile waves, group movement, rotation in a
circular domain, presence of vortices, and non uniformity of myosin
distribution.

Some similarities are shared by [13, 16]. Both are tissue-based computations,
with ODEs for GTPase biochemistry, an assumption that GTPase leads to cell
contraction or 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 between signaling and contractility (Fig 2C in [13],
and the comparable Fig 3B in [16]). The signaling 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 are
comparable, 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-based hexagonal cell with 6 spokes) to biochemical details for a
circuit of proteins known to regulate myosin contractility in Drosophila
dorsal closure. This model is more detailed on the specific known interactions
of protease-activated receptor (PAR) proteins that form a negative feedback
loop with actomyosin (Baz, Par-6, aPKC), including a set of 9 ODEs for
components linked to myosin dynamics along cell edges and spokes. The authors
account for several phases in the developmental process, including
oscillations observed in the tissue (using time delays) and the eventual
contraction of the tissue. By way of comparison, [148] has a greater level of
detail of the cell shape, including more viscoelastic spokes and edges, but
essentially no biochemical detail. An example of 3D vertex-based simulations
with cell rearrangement and out-of-plane bending can be found in [149].

####  3.3.2 Levels of detail

The gene regulatory network involved in Drosophila ventral furrow formation is
modeled at a fine level of molecular detail in [150] using a Boolean modeling
approach. The model encapsulates interactions of transcription factors to
explain 3 phenotypic attractor states, and identify missing and alternative
pathways. Spatial distribution of cells and cell shape dynamics is not
considered.

At the opposite extreme, internal cell signaling is omitted in favor of
positions and spatiotemporal dynamics of cells. SPP “agent-based” models with
forces of attraction, adhesion, and repulsion include [29], as reviewed in a
previous section. There, nonlocal forces 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, cell membrane curvature, elasticity, actomyosin cable force, and a
tendency to move in the outwards normal direction. The authors show the
dependence of tissue wound-healing velocity on the width of a confining
stripe. On a similar topic, CPM computations are used in [152] to investigate
the effect of adhesive micro patterns on the motion of single cells and
collective cell migration. Similarly, modified Viscek-type models are
considered by [25] endowing each cell with a polarization direction. Various
types of cell–cell polarity coordination mechanisms are considered.

Yet another modification of particle-based models is [153], where a cell is
represented 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 well as order–disorder transitions and velocity waves in large
2D cell groups with and without CIL. This paper explicitly bridges from 1 to
many cells.

At the next level of complexity, we find representations of cells as spheroids
or ellipsoids (Fig 5A1, [154, 155, 27, 156]). Drasdo computed a 2D growing
monolayer [154], and later extended it to 3D [27]. Palsson built 3D
simulations of deformable off-lattice ellipsoids, with cell division,
chemotaxis, cell–cell adhesion, and volume exclusion. The platform has been
used to model the dynamics of Dictyostelium aggregation and cyclic AMP (cAMP)
signaling [155], as well as the posterior lateral line primordium, a cluster
of about 100 migrating cells in zebrafish embryos that generate sensory
structures on the surface of the fish [14].

Other recent papers add variations on these themes. For example, [157]
provides a computational model for 3D off-lattice spheroid cells, but with
greater level of detail for cell–cell and cell–matrix adhesion. The model is
coarse grained from the biological levels of $10^{6}$ down to 100 sites per
cell. After quantifying the stable distance between cell pairs (for a given
set of forces, intrinsic cell properties, and adhesion sites), the authors go
on to show the behavior of a larger group of cells.

Recently, these approaches have been extended to more finely resolve cell
shapes in 3D [9]. Here, each cell is a triangulated 3D object, composed of
discrete viscoelastic elements. Simulations track up to a thousand liver
cells. While computationally very expensive, these approaches can provide
valuable insights. For example, it is shown that highly deformable cells move
more easily to heal a lesion than stiffer or more rigid cells [9]. They are
also useful for specific systems where it is crucial to capture details
accurately, as, for example, in drug design in silico for liver disease.

In [158, 159], a confluent epithelium in confinement is modeled by vertex-
based polygonal cells, and the type of collective migration (directed flow,
vortex chain, or turbulent) is related to a dimensionless quantity (“cell
motility number”) that combines 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
cells in [91] to larger populations. Cell collisions are modeled by an
energetic penalty for overlapping phase fields, and cell adhesion is described
by a reaction–diffusion equation in [22]. Their model cells exhibit bistable
shapes, (either symmetric or keratocyte-like), and both elastic and adherent
collisions are predicted. Strong adhesion results in bands of dense closely
packed cells that move as a collective. The authors compare their work with
that of [160] (see Links with experiments) where a CPM model is used to
describe large confluent cell rotations in a confined region. As noted by
[22], the level of detail in their phase-field description is better suited
for non-confluent cell models, but likely too detailed for confluent layers of
cells where individual cell details average out. In [161], similar methods are
extended to a 3D phase field computation for a single cell interacting with
curved or grooved surface. This sequence illustrates the trade-off between
geometric complexity (1D versus 2D versus 3D) and collective complexity (1
cell versus few versus many).

Overall, it emerges from many papers that the vertex-based models are good at
describing 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 that shared edges and nodes be duplicated or reassigned an
inconvenience. To track tissue fragmentation or loss of cells, center-based
models or CPM platforms have an advantage, since these represent each cell by
an individual geometric object or by a set of 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, and material properties. Analogies are made with fluids, viscoelastic
material, elastic sheets, or foams. Advantages include the ability to harness
traditional methods of physics, mathematics, and fluid or engineering
computational software. Examples of this 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 of epithelia, 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 shape and
epithelial behavior is an important contribution.

In [163], the tissue is modeled as an active continuous medium, with a Maxwell
viscoelastic constitutive law. A reaction–diffusion-transport equation
accounts for actomyosin, whose contractility is coupled to cell motion and
polarity. The authors explore predicted motion of tissue in confined regions,
closure of wound gaps, and the relationship between traction forces and sizes
of cohesive cell clusters. Mechanical waves, as well as tissue rigidity
cycles, are observed, where fast fluidization is followed by a gradual period
of stiffening.

The continuum model in [164] treats a tissue as a compressible fluid and
includes both cell division and death in a type of Stefan free boundary
problem. The authors show that behavior of the tissue in wound closure and in
colony expansion depends on 3 parameters: 2 physical constants and the
proliferation rate. They argue that all these can be estimated from limited
experimental data, with examples calculated for IEC-6, a rat cancer cell line
and MDCK cells, a canine kidney cell line. As the authors point out, continuum
models are appropriate, provided the size of the tissue or the characteristic
length of the wound is sufficiently large relative to the size of individual
cells.

In some cases, an approach that combines features of both continuum and
discrete cell identity is implemented. One example is [165]. Here, a 1D cell
monolayer consists of a long contractile element with myosin creating a strain
rate resulting in length changes. This element is flanked by cells in front
and rear. Binding and unbinding of adhesion sites are also included. The model
tissue exhibits durotaxis, movement directed up a stiffness gradient, under
appropriate conditions on the myosin and adhesion parameters. The authors use
this model to conclude that a monolayer is more effective at durotaxis than
single cells.

Other examples of hybrid treatments of particle-based and continuum approaches
include [166, 167]. These papers tackle the important question of how to
derive appropriate continuum models from underlying SPP models. This kind of
work and, in particular, expository papers that summarize the conclusions in
ways that biological modelers can understand, could help link work in the
literature that is currently underappreciated or not understood.

A recent work, [168], describes fingering at the front of an epithelial sheet
using several 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 combined approaches, they demonstrate
that stable fingering of the tissue edge requires leader cells. They
characterize the wavenumber of the fingering instabilities (distance between
stable fingers) using stability analysis of the continuum PDE model.

In summary, while the literature on tissue-scale collective migration is
rapidly growing, it is still in stages of infancy as far as coherence,
coordination, and clear direction are concerned. We see many exploratory steps
in many individualistic directions. The list of core questions and the
discovery of unifying principles is beyond the horizon, providing ample
challenges for the community. We can draw an analogy between the current state
of the art and the behavior of a growing but uncoordinated population of
cells. There is no emerging unified front. Each is exploring independently and
occasionally aggregating with a few others, but the global organizing
principles are yet to emerge.

####  3.3.3 Links with experiments

The links between molecular players such as Rho family GTPases and epithelial
morphogenesis have been known for some time. (Experimental literature reviewed
in [169].) The role of Rho GTPases in the leader-follower identities and in
front–rear tissue 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 does collective 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,
the authors investigate the roles of the GTPases RhoA and RhoC and their GEFs
in collective cell migration. It is found that cells tend to crawl in the
direction of maximal principal stress, a process called plithotaxis. In their
scratch-wound assay of MDCK and human bronchial epithelial cells, wounding
leads to coordinated waves of motion, with cells at the front edge reacting
first, followed by those successively further back. The authors speculate that
mechanical cues are induced by leader cells on followers behind then by normal
strain, and alongside them by shear strain to coordinate motion.

In [134], the relationship of tissue speed to distance from the leading edge
is quantified using particle image velocimetry. The authors employ shRNA to
knock down GTPase Cdc42 or Rac1 and many of their GEFs (screening some 81 GEFs
in total). They find that the speed and directionality of the cells drop
everywhere. When RhoA is depleted, there is a reduction in the spatial
gradient of cell speed. They suggest 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 Rho
GTPases. The authors investigated which properties of underlying molecular
machinery allow for coupling between mechanical forces and correlated cell
motion. The correlation length scale of collective force transmission was
determined experimentally in [171]. Observed behavior was then modeled using a
thin elastic sheet of height $h$ with some elasticity and isotropic
contraction stress. The authors investigate the emergence of leader cells and
found a typical length scale of about 170 $\mu$m.

A number of experimental studies have explored how epithelia behave in grooves
[132, 130], various topographic surfaces, or confined settings [105],
including arrays of posts [172], arrays of grooves [173], or adhesive [160]
surfaces. In [132], the width of grooves and adhesive strips is varied to
investigate how these affect collective migration of an epithelial sheet of
MDCK cells. The authors map out the force and velocity fields in each case.
Their narrowest grooves are 20 $\mu$m wide, so cells are in a single file, and
the authors observe “push-pull” or “caterpillar” type motion, resembling the
relaxation–contraction cycles in the model by [16]. Larger tissue widths
exhibit vortices of cell motion. The authors deduce that the constraints of
the geometry 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
migration and EMT in mammary epithelia and breast cancer cell lines. Dispersal
of single, highly motile mesenchymal cells from a spreading epithelial front
was shown to agree quantitatively with a minimal physical model of binary
mixture solidification. While such a model lacks cell detail, it has several
advantages, including simplicity, basic summary statistics for comparison with
experiments, and availability of an analytic solution. Methods from physics
can be used to inform the link between the material properties and the overall
macroscopic behavior.

Both single fibroblasts and epithelial monolayers were studied in [130],
showing that cells tend to be more organized on grooved versus flat substrata.
Mechanical exclusion interactions, rather than strength of junctions between
cells, affect the distance over which the topographic guidance signal
propagates between cells. The same paper proposes a CPM computational model to
describe the observations. In the CPM 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 the customary Hamiltonian
area constraint and cell–cell adhesion energy, there is also a term for a
phenomenological motile force, implemented as a “migration energy” (dot
product of the polarity vector and the cell centroid position). The authors
investigate how this motile force affects spatial correlation of the velocity.
They observe emergence of streaming patterns. The authors also describe the
influence of “leader cells,” whose polarity vector is set to an external cue.
Leaders are embedded in the tissue interior, and they coordinating neighbors
over some “interaction distance.”

In a similar style of experiment-model study, MDCK cells are seeded on a
circular adhesive domain in [160]. Once a critical density is attained, the
confluent culture collectively rotates. The same behavior is then captured in
a CPM computational model that includes a motile force and a polarity term
that is reinforced by cell displacement, with some persistence time. The
authors show that rotation takes place when the size of the circular domain is
on the order of the correlation length of cells. That correlation length, in
turn, depends on the cell–-cell adhesion energy, the motile force magnitude,
and the polarity persistence time. Note the comparison with the purely
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 to compare the CPM model in [160] with the vertex-based treatment
of a similar situation in [158].

A recent paper, [105], demonstrates the fact that single cell properties
affect much of the collective behavior of a cell population. In this ground
breaking work, the authors link single cell Rac1 polarity to the emergent
rotations of confluent cells (from few to many) confined to a ring or closed
curve. A simple mechanical model captures the balance of directed motility and
contact forces to demonstrate the principles at work. This paper spans
subcellular to multicellular scales and provides a great example of elegant
experiments to be emulated by others.

##  4 Modeling recruited signaling networks and multiscale behavior in
collective cell systems

At present, there are still relatively few models that bridge from detailed
underlying molecular mechanisms through individual cell motility, all the way
to tissue dynamics and collective cell migration, though the number of such
papers is growing. There are 2 major issues that hamper such efforts. First,
it is unclear how to deal with the problem of combinatorial complexity in
trying to understand numerous players and interactions at each level.
Decisions made at 1 stage affect other stages, and exploring a multiplicity of
assumptions is a challenge (but see [118]). Second, complexity of the
resulting models makes it challenging to determine the range of possible
behavior, let alone make sense of overall principles and key components.

Few papers specifically address the signaling modules that get recruited in
collective 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 specifically affects the way that cells then interact.
Downstream responses might include changes in cell adhesion, migratory
potential, permissive or inhibitory control of cell division or apoptosis, or
relative sizes, polarity, or other aspects of cells. Input from the
environment in the form of mechanical tension or topography can influence
these signaling networks, promoting or inhibiting EMT. Examples of this sort
include some of 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 to tight junctions between cells, it inhibits Rac1, but when it is in
the cytoplasm, it releases that inhibition. At the same time, low Rac1
activity leads Merlin to become stabilized at tight junctions [129]. Hence,
Merlin and Rac1 compete in a mutually inhibitory circuit, and a balance
between mechanical and chemical factors controls Merlin 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 in collectively motile cells.

Mechanical cues and ECM topography appear to influence the adhesion and
migration 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 were investigated in [173] on nanostructure ridge arrays that mimic
ECM. A minimal model for 2 signaling modules was proposed to account for the
transition of YAP activity with distance from the front edge of the sheet.
These examples of how cell–cell interactions depend on and influence both
intracellular Rac gradients and adhesion motivate future modeling efforts at
the multiscale level.

An intriguing stepping stone on the route to the challenging eventual targets
are engineered tissues studied in synthetic biology. Since these are designed
with known components, they allow for more direct development of models based
on underlying mechanisms. We mention and example of this sort in what follows.

An interesting question is whether and how lessons from 1 level can be
simplified into rules for components at the next level and how to best include
the additional signaling pathways that get recruited at distinct levels of
organization.

Here, we mention a few representative examples but recognize that others may
exist of which we are as yet unaware.

###  4.1 Bridging scales

In many respects, we are currently seeing first steps in models that bridge
several layers of organization. Examples of this type can be found in the work
of Marzban and colleagues [6, 175]. Their model combines several modules: cell
polarization as in [5], a viscoelastic cytoskeleton, stress fiber structure,
cell motility as in [176], and cell–substrate interaction. The authors first
model the polarization, motility, and durotaxis in single cells, and then
combine these with a cell–cell interaction module to simulate the rotation of
tens of cells in a confining 2D annulus. The paper demonstrates a creative
combination of a number of known cell representations to bridge from
subcellular properties to those of the collective.

In [12], the authors study tumor spheroid growth under high mechanical
compression. Their model represents cells as 3D spheres. Commonly used cell
contact models (e.g., the Hertz model) do not take such large volume
compression into account. The required corrections to the contact models were
calibrated using a high-resolution mechanical models of cells [9]. We expect
that similar approaches will in the future improve the accuracy of coarse-
grained tissue level simulations.

The intersection of cellular and developmental biology has provided additional
important examples of computational models that span multiple scales. Some
examples predate recent efforts by 2 decades, notably computational studies of
cellular slime mold, D. discoideum morphogenesis, and signaling in [177, 178].
Other examples such as [179] are purely theoretical, aimed at exploring how
random gene circuits linked to cell adhesion, cell division, and some overall
measure of “fitness” evolve into a zoological garden of multicellular
structures. See also [180] for a recent work along a similar “EvoDevo”
evolutionary developmental biology line.

###  4.2 Levels of detail

By using very simple representations of subcellular events, [26] bridge from 1
cell to hundred(s) using CPM computations. The authors assumed an elementary
process that 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 how 2 simple computational parameters tune the
cell shape from keratocyte-like to amoeboid, they then simulate the collective
migration of a monolayer. The abbreviated level of intracellular detail
permits efficient computations. Note the comparison with [6], where more
costly FEM computations and greater detail makes for greater computational
cost and smaller number of cells that can be readily simulated. Software
packages as in [181] may eventually make it more realistic to incorporate
intracellular detail into multiscale models.

###  4.3 Links with experiments

In creating synthetic gene networks that regulate the adhesion protein
E-cadherin in real cells, Toda and colleagues [182] succeeded to design
multicellular clusters that self-organize into distinct layers. The expression
of E-cadherin genes was placed downstream of cell-surface notch receptors.
Notch ligands on some cells activated notch receptors on neighbors, and in
this way, cell–cell interactions both influenced, and were influenced by
intracellular signaling. The experiments were later linked to computational
models in [183, 146]. When details of the relatively “simple” genetic circuits
are known, as in such synthetic biology experiments, modelers can bootstrap
signaling models to learn how spatial influences and cell–cell interactions
shape the emergent 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 the invagination and bending of epithelial that form a lens pit in the
eye development in a mouse. Rho apparently controls the apical constriction of
cells via myosin, and Rac the elongation of those cells via F-actin assembly,
hence accounting for the conical angle formed by each cell and the overall
curvature of the tissue.

Overall, more experimental papers that probe the circuits that get recruited
in the collective cell migration are needed. The papers [129, 173] on the
Merlin-Rac loop and on the link to YAP and E-cadherin [173] should be followed
up with more detailed computational modeling and future rounds of experiments.

##  5 Discussion

In this paper, we described a small selection of modeling works on cell
migration that bridge from intracellular, to cellular and multicellular
scales, as shown in Fig 3. Many other excellent papers have been omitted due
to space limitations. That said, even from the fraction surveyed, we find that
a variety of computational and analytic methods are used at various scales.
Table LABEL:Table:ModelSummaryCombined organizes modeling papers by their
subject and methodological approaches, Table 1 classifies a few experimental
papers by their biological targets, and Fig 4 summarizes the ranges of
relevance of both computational and experimental methods. Our review has
focused on the topic of single and collective cell migration and its
regulation. Likely motivated by development of disease therapies and NIH
funding, or drug targets and support from pharmaceutical companies, the more
medically oriented subjects such as cancer, liver toxicity, or lung
morphogenesis, have fostered many generations of computational models. By
comparison, the level of basic scientific computational research on multiscale
cell biology modeling is still emerging.

Fig 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 up to tissues (left to right in increasing number of cells and
increasing cell–cell adhesion).

We have seen that some computational techniques that work well at the single-
cell level, become too costly or excessive at the tissue level. (See also
review of computational models in [38, 140, 95, 4, 143] and [185] for cancer.)
We also encountered topics where diverse computational techniques lead to
similar predictions. (See Fig 5, and [143] for comparisons.) We still find
instances where 1 or another group claims that their computational method of
choice outperforms others or has fewer unrealistic features. In many cases,
such claims are, at best, unfortunate and overlook essential shared
attributes. In other cases, they skip over relative advantages versus
disadvantages of the distinct schemes. We believe that the field needs more
rational comparisons of how custom-build computations perform against a host
of “benchmark” test problems, as for example, shown in [143], and/or deeper
comparative analysis of cell-surface mechanics models that demonstrate
equivalence of distinct approaches, as in [186].

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 not have the software packages or expertise to read,
execute, and run codes in different formats. We recommend that, moving
forward, the community should invest more directly in standardizing open-
source software, with several specific aims: (1) ease of operation, user
friendliness, and rapid learning curve; (2) basic built-in code for most major
computations, including reaction–diffusion solvers, particle position and
collision solvers, cell shape and cell–cell adhesion, intra and intercellular
signaling, and so 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 of development of new plug-ins to allow the
capabilities to expand with the needs of the community. We recommend that
groups invest resources in helping to develop such shared platforms and that
funding bodies (NIH, NSF, and NSERC) make it a priority to support such
developments. Certainly, at initial stages, this development requires teams of
computational experts who can establish solid, robust platforms, and create
training manuals or instructional videos to recruit new users. Morpheus, a
software system developed at the Technische Universität Dresden, is 1 example
along these lines, but others are needed.

Returning to questions posed in the Introduction, we can draw a few general
conclusions.

Fig 5: Cell sorting (left) and wound-healing (right) captured by several
distinct computational 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 by Dhananjay Bhaskar. (A3) CompuCell3D cell-sorting
simulations run by Dhananjay Bhaskar. In (A1–A3), there are 2 cell types with
differing adhesion strengths to self and other cell type. (B1) and (B2) show
deformable ellipsoids and CPM scratch wound models with 2 cell types. Cells
with weaker adhesion (red) tend to segregate to the edge of the monolayer,
acting as leader cells, and causing fingering of the front. (Compare with
[168]). Papers that compare distinct computational platforms include [143].

###  5.1 Bridging scales

Building from the bottom up, models that include detailed molecular
interaction or gene networks (such as the Boolean models in [58, 150]) encode
many details and occasionally reveal attractors or identify missing components
[150]. They are harder to understand on their own. Detailed models such as
[40] benefit from insights of preceding more basic models [33]. At lower
levels of detail but still arguably “bottom up” are studies that show how
certain properties of molecular components and interactions result in specific
cell behavior such as polarization [54, 55, 52, 5]. Other work starts from
observed cell dynamics to infer the likely signaling pathways at play [70, 74,
64] or likely underlying mechanisms [36, 118], that we could denote “top
down.” The latter, [118], also demonstrates the idea that it is preferable to
weed through and discuss many models, particularly some that fail, with
reasons for that failure, not merely aiming for a single optimal model. This
kind of process helps to build a greater intuitive understanding of the
specific key mechanisms required to explain observed phenotypes.

We recommend that modelers avoid publishing large-scale models as a “fait
accompli,” so as to “be the first to accurately simulate” some entire
comprehensive behavior of choice. Instead, we suggest that modelers should
provide due description of the steps taken in developing such models, with all
informative failures and successes. This would help the community to
assimilate the insights that resulted from an entire program of research.

In a related, but separate vein, the field is in need of rational consensus
and standard practices for stepping between size scales and levels of
organization. To draw a common analogy, our ability to easily use and
appreciate global geography as well as local structure has been transformed by
the way that Google maps seamlessly allows us to zoom down to street-level
views and back out to the globe as a whole. The algorithms that reveal or blur
over specific details as we step up or down were logically constructed 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 structures
and interactions change on a rapid timescale make this a tough issue, to be
sure, but one deserving more attention.

Associating rule of behavior with a specific hierarchy is possible once we
have sufficient familiarity with the biology and predictions of basic models.
This can help to bridge hierarchies and avoid the fog of complexity.
Mathematical methods such as dynamical systems, PDEs, and bifurcation analysis
can help to find and account for emergent properties and universal principles
in such basic models [48]. This is one of the strengths of the mathematical
tools. A weakness is that these methods currently work well for small systems
of differential equations, but not for large and complex systems.

Once we understand the repertoire of a single cell, we can move up to 2, 3, or
many cells. Experimental observations of small cell groups provide good
opportunities for understanding how to bridge from single to collective
behavior. Modeling can then also 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 into simpler rules becomes
expedient. For example, while polarity is represented by PDEs and patterns
inside a single cell, it can then be simplified, depicted by a direction
vector [25, 15] in place of a full internal gradient of Rho or Rac for
multiple cells.

###  5.2 Level of detail

The current computing power at our disposal allows increasingly detailed
models to be constructed, with tens or even hundreds of components, and many
more parameters. The temptation to create such a model and to present it as a
mechanistic representation of real cells is hard to resist. Yet, before going
down this very intricate route, important questions should be considered: What
do we expect to learn? Might we have left out something important? Have we
included superfluous detail that obscures the basic structure? Are we certain
that our parameter regime faithfully corresponds with true rates and
quantities? As previously noted, even though we can construct highly complex
models, our comprehension of that detail is limited.

In general, best practices seem to invoke back and forth attempts to include
important aspects, simplify to understand, throw out secondary factors, and go
back to including detail. Simplifications lead to insights and rigorous
analytic results but leave large gaps to real biology. However, returning to
detailed versions of a model once the major insights are at hand (or vice
versa) is helpful. While [52] explains polarization in a motile cell, for
example, [61] extracts some of the key properties and interactions that
guarantee that it could work in a mathematically tractable mini-model.

Furthermore, “toy models” consisting of small sets of ODEs or PDEs can help to
formulate universal principles that work across many biological examples and
many scales. Here, we can mention the concepts that mutual inhibition or
positive feedback, which result in bistability and hysteresis. Simple
“modules” consisting of specific types of interacting components, whether
molecules, cells, or animals have prototypical behaviors as switches,
oscillators, or other dynamics, and, with molecular diffusion or random
motion, engender patterns or waves [48, 49].

###  5.3 Model-experiment links

Biology is an experimental science at its core, and models that illuminate its
mysteries must eventually meet and concur with biological evidence. Modern
methods such as machine learning can help match a vast dataset with the best
candidate model(s) as shown elegantly by [118]. At the same time, modelers,
physicists, and biophysicists can make real contributions by using their craft
and theories to unlock important facts that are not at all apparent otherwise
[42].

Several of our examples demonstrate that some experimental papers have
explicitly addressed signaling pathways that mediate cell–cell communication
in collective cell behavior. As pointed out eloquently by a reviewer of this
paper, “experiments can only tell [us … what are] upstream and downstream
regulators. We need mathematical models to incorporate this information with
the intracellular networks to form a signaling network on a multicellular
level to study how a group of cells processes signals collaboratively.”

Acknowledgements:  We are grateful to the Pacific Institute for Mathematical
Sciences for providing space and resources for AB’s postdoctoral research. We
are grateful to former and current members of the Keshet-Feng Research Group
for helpful discussions over many years that solidified our appreciation of
this area of research. We thank Hildur Knutsdottir and Dhananjay Bhaskar for
permission to use results shown in Fig 5 that they produced while they were
UBC students.

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