Embryo as an active granular fluid: stress-coordinated cellular constriction chains

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by  Guo-Jie Jason Gao

Embryo as an active granular fluid: stress-coordinated cellular constriction chains
Guo-Jie Jason Gao
Department of Mechanical Engineering, National Taiwan University
Michael C. Holcomb
Department of Physics, Texas Tech University
Jeffrey H. Thomas
Department of Cell Biology and Biochemistry, Texas Tech University Health Sciences Center
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1 Jerzy Blawzdziewicz
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Department of Mechanical Engineering, Texas Tech University
2

r Mechanical stress plays an intricate role in gene expression in individual cells and sculpting of
a developing tissues. However, systematic methods of studying how mechanical stress and feedback
M
help to harmonize cellular activities within a tissue have yet to be developed. Motivated by our
observation of the cellular constriction chains (CCCs) during the initial phase of ventral furrow
4 formationintheDrosophilamelanogaster embryo,weproposeanactivegranularfluid(AGF)model

that provides valuable insights into cellular coordination in the apical constriction process. In our
]
model, cells are treated as circular particles connected by a predefined force network, and they
t
f undergoarandomconstrictionprocessinwhichtheparticleconstrictionprobabilityP isafunction
o
of the stress exerted on the particle by its neighbors. We find that when P favors tensile stress,
s
. constricted particles tend to form chain-like structures. In contrast, constricted particles tend to
t
a form compact clusters when P favors compression. A remarkable similarity of constricted-particle
m chainsandCCCsobservedinvivoprovidesindirectevidencethattensile-stressfeedbackcoordinates
theapicalconstrictionactivity. Ourparticle-basedAGFmodelwillbeusefulinanalyzingmechanical
-
d feedback effects in a wide variety of morphogenesis and organogenesis phenomena.
n
o
c I. INTRODUCTION process known as gastrulation).
[ Gastrulationoccursinmostanimals. Theactivegran-

2 Multicellular organismsutilize mechanicalstressfields ular fluid (AGF) model proposed in this study is moti-
v as a means of guiding tissue growth, triggering genetic vated by specific features of gastrulation in the common
2 expression and cell division, and enhancing the robust- fruit fly (Drosophila melanogaster). The first morpho-
9 genetic movement of gastrulation—i.e., ventral furrow
ness of morphogenetic processes [1–8]. This emerging
6 evidenceoftheroleofmechanicalfeedbackinorchestrat- formation—is initiated by the constriction of the outer
2
(apical) faces of the cells on what will become the un-
ingcellular-levelactivityhasgivenanimpetustoanalyze
0
. mechanicalprocessesinvolvedinbiologicaldevelopment. derside (ventral side) of the fruit fly (see the schematic
1 representationinFig.1). Theconstrictionsproduceneg-
While the existing evidence of the crucial role of me-
0 ative spontaneous curvature of the active region of the
chanical triggering in sculpting tissues and coordinating
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cell monolayer, eventually leading to its invagination.
1 cell behavior is indisputable, quantitative understand-
In the modeling effort presented in this study, we are
: ingofhowcellcommunicationvialong-rangemechanical
v concernedwiththe initialphaseofventralfurrowforma-
stressfieldsharmonizescellactivityisfragmentedandin-
i
X complete. We proposethatlocalmechanicalinteractions tion, before the actual tissue invagination occurs. Dur-
ing this initial constriction phase, approximately 40% of
r andglobalstressfieldsintissuescanbequalitativelyrep-
a
resentedandanalyzedbymodelingthetissueasanactive
granular medium.
Tissues are a conglomeration of deformable, discrete
objects (cells) that mechanically interact through direct
contactandadhesion,andtheyarelargeenoughforther-
mal motion fluctuations to be neglected. Cells, however,
are not merely passive, deformable objects. They are in
factsubjecttogeneticallyprescribedactivedeformations
thatcangiveriseto large-scalecellularflowsresultingin FIG. 1: Schematic of a Drosophila embryo with the meso-
tissue-wide structural changes. One particularly strik- dermprimordium(ventralfurrow)regionmarkedbytheblack
ing example of such active cellular flows is the collection stripe. (a) Lateral (side) view; (b) ventral (bottom) view.
of regional cellular motions (morphogenetic movements) Cells in the marked region are active and in the unmarked
region are passive during the initial phase of the apical con-
by which an embryo changes from a single layer of cells
striction process.
around a yolk center into a triple-layered structure (the
2
tures of constricted cells.
Wenotethatitwasrecentlyreportedthatacorrelation
between the rachetted contractile pulses of constricting
cells had been observed [12]. However, a robust descrip-
tion of mechanical interactions and, possibly, coordina-
tion between the apically constricting cells has yet to be
formulated.
Our approach draws on ideas developed for granular
matter. Below we introduce an AGF model to describe
collective cell behavior during the initial phase of apical
constrictions in the ventral furrow region. We present
ourproof-of-conceptcalculationsalongwithaqualitative
comparison with in vivo observations.
II. EPITHELIAL TISSUE AS AN ACTIVE
GRANULAR FLUID
A. A Brief review of relevant biology
GastrulationinDrosophila beginsaround3hoursafter
fertilization and is completed through multiple morpho-
genetic movements which are driven by region-specific
cell activities [13]. These regions are established as a
result of a cascading pattern formation caused by sym-
metry breaking events which occur during the creation
of the egg (oogenesis) [14–17]. The region of cells that
FIG. 2: Time-lapse images of the ventral side of a wild-type undergoes the apical constriction of interest is known as
Drosophila embryo (the region indicated by the black stripe the mesoderm primordium and is actually internalized
in Fig. 1) during the slow phase of the apical constriction by ventral furrow formation.
process. Markedcells(inbrown)illustratethepropagationof The mesoderm primordium is composed of a band of
cellular constriction chains (CCCs). Scale bar=20µm.
cells on the ventral side of the embryo which take up
approximately 80% of its length and 20% of its circum-
ference [13], as schematically depicted in Fig. 1. Meso-
the cells gradually constrict in a seemingly random or- derm primordial cells are capable of mechanical activ-
der. Furrow formation is subsequently completed with ity, due to expressionof regulatory genes twist and snail
a rapid, coordinatedconstrictionof the remaining active [6, 18–22] established during the preceding phase of em-
cells (fast phase) [9]. bryopatterning. Cellsoutsidethemesodermprimordium
undergopassivedeformationsunderappliedstresses,but
While apical constrictions during the initial slower
otherwiseremainmechanicallyinactiveduringtheinitial
phase of ventralfurrowformationare generally accepted
slower stage of ventral furrow formation [9].
tobeanuncorrelatedstochasticprocess[9],weshowhere
that this phase is not completely random. A close in-
spection of the distribution of constricted cells (see Fig.
2anddiscussioninSec.II)revealsthe presenceofchain- B. Force chains vs. cellular-constriction chains
like arrangements. We call these arrangements cellular
constriction chains (CCCs), because they are remark- Duringtheslowerphaseofventralfurrowformation,a
ably reminiscent of force chains which are observed in growing number of mesoderm primordium cells undergo
granular media [10, 11]. apical constriction. Initially the constrictions occur at
We propose that CCCs in the mesoderm primordium randomlocations,but as time progressesthe constricted
oftheDrosophila embryoformasaresultofcoordination cells tend to form CCCs, correlated chain-like patterns
ofcellactivitythroughmechanicalstresses. Namely,con- (see the highlighted cells in time-lapse images in Fig.
strictions of cells (which are bonded to the surrounding 2; the experimental details are described in Appendix
cells) produce tensile stresses that propagate along ten- A). Similar CCC-like patterns can also be discerned in
sile force chains (analogous to compressive force chains classical images available in the literature (see Fig. 4 of
in granular matter). The presumed coupling of these Ref. [9]), but to our knowledge such structures have not
strongly correlated stresses to the constriction probabil- been explicitly reported, and their significance has not
ityofindividualcellscausesformationofchain-likestruc- yet been analyzed.
3
We argue that CCCs occur as a result of coordina- strictions by stress.
tion of apical constrictions via mechanical feedback. As
discussed in Sec. I, our mechanical feedback conjecture
is based on the close resemblance between CCCs and B. System geometry
chains of interparticle forces that occur in granular me-
dia [10, 11]. In compressed or strained granular matter, Our system begins as a mechanically stable packing
individual force chains consist of a sequence of pairwise of N discs interacting via finite-range repulsive forces
compressiveforces between interactingparticles that are (which represent elastic cell interactions). The system,
jammed together; the chains act as a path along which preparedusingthe algorithmdescribedinAppendix B1,
the stressinthe materialis propagating. Similar tensile- occupies a square simulation box of size L. For a given
force chains occur in systems of bonded particles [23]. particle number N and disk diameters d , the box size
i
Force-chainrelatedstructures havebeen observedin a is determined from the condition that the configuration
variety of systems including emulsions, foams, and col- is closely packed (area packing fraction φ ≈ 0.84) and
loidalglasses [24]. Force chains,resulting fromcollective mechanically stable.
interactionsbetweenindividualconstituentparticles,are After the initial packing is prepared, we generate a
a prevalent phenomenon in condensed particular matter list N of interacting neighbors. We then add attractive
I
and, therefore,we propose that they also occur in active forces(representingcelladhesion)betweenthe neighbor-
cell packings constituting a developing tissue. ing particles. Particles i and j are assumed to be the
Epithelial cells in a Drosophila embryo are bonded to interacting neighbors, (i,j)∈N , if the condition
I
their immediate neighbors through specific protein for-
r0 ≤1.1d (1)
mations (adherens junctions). Thus, both tensile and ij ij
compressive stresses can be transmitted in cellular sys- is satisfied in the initial closely packed state, where r0
tems. The distribution of stress through force chains ef- ij
is the initial distance between the particles i and j, and
fectively shields the rest of the material, creating low d = 1(d +d ) is their average diameter.
stress regions. Assuming that the constriction probabil- ij 2 i j
Intheinitialstate(i.e.,beforethediskconstrictionsoc-
ity for a given cell is affected by the forces exerted on it
cur),the systemis a disordered50%mixture ofparticles
by the neighboring cells, such mechanical coupling may
with the diameter ratio r = 1.1. We use the bidisperse
result in a non-random microstructured distribution of
disk system to mimic polydispersity of Drosophila cells
the constricted cells. In the following sections we exam-
andtopreventformationofhexagonalorderedstructures
ine this possibility using our AGF model.
intheinitialcloselypackedstate(suchstructuresarenot
observedforcells). Subsequently,thesystemundergoesa
sequenceofparticleconstrictions,d →f d ,accordingto
i c i
III. ACTIVE GRANULAR FLUID MODEL
the algorithm described in Sec. IIIF. In our simulations
we use the constrictionfactor f =0.6, correspondingto
c
A. A simplified representation of an active cell the size of constricted cells [9].
monolayer
The cellular constrictions that motivate our AGF C. Active, inactive, and constricted particles
model occur onthe outer surface of the embryonic cellu-
lar monolayer. The interior (basal) cellular ends remain The disks (see Fig. 3) are divided into three distinct
relatively inactive throughout the initial slower phase of categories: active A (blue particles), inactive I (gray),
apical constrictions. Thus, to investigate coordination and particles already constricted C (brown). Inactive
of the constrictions via stress distribution we can use a discs cannot undergo constriction and will remain the
simplified description in which only the relevant ventral same size throughout the simulation. Each active disk
portion of the outer surface of the embryo is explicitly can instantaneously constrict, and will do so by follow-
represented. ing the triggering conditions described in Sec. IIIF (the
We approximate this area of interest (i.e., the meso- constricted disks cease to be active).
derm primordium and its immediate surroundings) as Particles that in the initial state are in the domain
a two-dimensional plane; apical cell ends are modeled 0.25L < y < 0.75L are active, and the remaining parti-
as interacting active discs that constrict in a stress- cles (in the regions 0 < y < 0.25L and 0.75L < y < L)
sensitive stochastic process. In the past, complex sys- are inactive. Our numerical simulations have been per-
tems of strongly coupled particles (e.g., emulsions and formed for N =512 particles, so the initial stripe of the
foams) were successfully modeled using interacting disks active particles is approximately 11 particles wide, simi-
(orspheres)[25–28]. We thus expectthatusing aclosely lar to the width of the ventral region of active cells in a
packed system of active disks to approximate the me- Drosophila embryo [9].
chanically excitable cell layer will reproduce the key fea- Our calculations are performed with periodic bound-
tures of the stress-driven constriction process, and will ary conditions in the horizontal (anteroposterior) direc-
yield valuable insights into coordination of cellular con- tion x and a free boundary in the vertical(dorsoventral)
4
FIG.3: Comparison oftheprogression ofcellularconstrictionsfordifferentvaluesofsensitivitystressparameterβ (aslabeled)
for Nc =25, 50, and 100. Particle color key: active (A) blue,inactive (I) gray, and constricted (C) brown.
5
direction y. The use of the free boundary condition is F. Particle constriction protocol
motivatedbytherelationshipbetweenthemesodermpri-
mordium(the blackstripe inFig. 1)andthe cells onthe Particle constrictions are performed by iteratively re-
sides (the unmarked lateral region). Cells of the lateral peating the following procedure:
regionsarebelievedtopassivelydeformandprovidevery
little resistance to the apical constrictions that occur in a. the system is fully equilibrated;
the mesoderm primordium [9, 29]; this condition is ap-
b. particlestressσ(i)isevaluatedaccordingtoEq.(4)
proximated by the free boundary condition.
for each particle i;
c. constrictionprobability(peronestep)P(i)isevalu-
D. Interparticle potentials atedforeachactive(unconstricted)particle,i∈A,
according to Eq. (7) provided below;
All particles interact via the finite-range, pairwise ad- d. diametersofthe activeparticlesaredecreasedby a
ditive, purely repulsive spring potential factor f with the probability P(i).
c
ǫ Theaboveparticleconstrictionstepisrepeateduntilthe
V (r )= (1−r /d )2Θ(d /r −1), (2)
r ij 2 ij ij ij ij number of constricted particles Nc reaches a prescribed
limit.
where ǫ is the characteristic energy scale, rij is the sep- Constriction probabilities TheprobabilityP(i,N )of
c
aration between particles i and j, and Θ(x) is the Heav-
constriction of particle i in a system with N particles
c
iside step function. In addition to the repulsion (2), the already constricted is evaluated in step c of the above
neighboringparticlesi,j ∈N interactviathe attractive
I protocol from the expression
spring potential
α{1+β[σ(i,N )/σmax]p}
Vanp(rij)= 2ǫ(1−rij/dij)2Θ(rij/dij −1) (3) P(i,Nc)= (1+|β|)cNa A , (7)
where σ(i,N ) is the current value of the particle stress
c
that mimics the adhesion of neighboring cells. (4),βisthestresssensitivityparameter,N isthecurrent
a
In the initial state and after each particle constriction number of active particles, and
step, the system is fully equilibrated (see Appendix B2
for details of the equilibration algorithm). Thus, the se- σAmax =max[sign(β)σ(i,Nc)] (8)
i∈A
quence of particle constriction steps is quasistatic.
is the maximal tensile (β > 0) or maximal compressive
(β < 0) stress acting on these particles. The parameter
E. Evaluation of particle stress α > 0 sets the average number of particles constricting
in a single constriction step, and the parameter p > 0
(odd integer) determines the sensitivity profile for the
To characterize the overall stress exerted on particle
dependence of the constriction process on the particle
i by the surrounding particles, we choose the following
stress. We use α = 1 and p = 3 in all our simulations.
expression,
The chosennon-unity value of p gives a higher weightto
the largest (tensile or compressive) stresses.
σ(i)=− (d /ǫ)f , (4)
ij ij The parameter β determines the sign and magnitude
Xj6=i of the overall sensitivity of particle constrictions to the
particle stress σ(i). We distinguish three fundamental
where
cases:
fij =−dVij/drij, (5) 1. uncorrelated random constrictions: β =0;
2. constrictions promoted by tensile stresses: β >0;
with
3. constrictions promoted by compressive stresses:
V (r )+Vnp(r ), (i,j)∈N
V (r )= r ij a ij I (6) β <0.
ij ij (cid:26)Vr(rij), (i,j)6∈NI
Incase1,thesystemdoesnothaveanysensitivitytopar-
are the interparticle central forces (which can be tensile, ticle stresses. This purely random constriction process
f <0,orcompressive,f >0). Accordingly,forσ(i)> provides a reference system for qualitatively describing
ij ij
0 the dimensionless particle stress (4) is predominantly how mechanical sensitivity influences the propagationof
tensile,andforσ(i)<0itispredominantlycompressive. constrictions through our medium. The other two trig-
Theparticlestress(4)isalwaysevaluatedforasystemin geringconditionsintroducethesensitivityoftheconstric-
full mechanicalequilibrium, in which there is no particle tionprobabilitytotensilestresses(case2)orcompressive
motion, and all interparticle forces balance. stress (case 3).
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A. Microstructural evolution
The results of our numerical simulations of the mi-
crostructural evolution are summarized in Figs. 3 and 4
[also see Movies 1(a)–1(e) in Supplementary Data]. The
presented images of particle configurations reveal that
thespatialarrangementoftheconstrictedparticlesshows
a striking dependence on the sign and magnitude of the
constriction-triggeringstress.
In systems in which particle constrictions are induced
by tensile stress (the top two rows of images in Fig. 3),
constricted particles form strongly correlated chain-like
structures, which closely resemble CCCs that we have
identified in the Drosophila embryo (see Fig. 2). In con-
trast, the microstructure of the system in which active
particles are sensitive to compressive stresses (the bot-
tom two rows of images in Fig. 3) is dominated by com-
pact constricted-particle clusters.
The chains that form for β > 0 are partially aligned
with the x-direction (i.e., the directionof the system pe-
riodicity). The chains are initially disconnected (see the
imagesforN =25and50),butatlatertimes(N =100
c c
for β =104) they grow into a percolating network span-
FIG. 4: Same as Fig. 3, but for a larger value of Nc. ningthe systeminthe x-direction. This behaviorclosely
resemblesthe constricted-celldynamicsinthe mesoderm
primordium shown in Fig. 2. Our results thus provide
powerful (though indirect) evidence of the tensile-stress
The simulations for the stress-dependent cases were
feedback involved in cellular constrictions in the early
performed at a variety of β values. Representative re-
phase of ventral furrow formation in the Drosophila em-
sults, for β = ±10 (moderate stress sensitivity) and
β = ±104 (strong stress sensitivity), are discussed be- bryo.
low. Thechainingis mostpronouncedforlargemagnitudes
of the stress sensitivity parameter β. We find that for a
moderate value β = 10, the chains are fragmented and
less aligned with the x-axis than for β = 104; and per-
IV. RESULTS AND DISCUSSION colation occurs later, at N ≈ 130 [see Movie 1(b) in
c
Supplementary Data].
Here we describe the results of our numerical simu- Insystemswithparticleconstrictionsinducedbycom-
lations of correlated particle constrictions in the AGF pressive stresses, β < 0, we observe the formation of
modelintroducedinSec.III.Wediscusstheevolvingmi- compactclusters,withsignificantsizepolydispersityand
crostructure of constricted-particle regions and analyze inhomogeneousspatialdistribution. The clusters remain
the stressdistributionfor differentsigns andmagnitudes disconnected and do not percolate, until much later in
of the stress sensitivity parameter β in the constriction the process.
probability function (7), to determine mechanisms that For a moderate sensitivity-parameter value β = −10,
control the constriction patterns. the clusters are much smaller than for the system with
We focus on the initial stage of the system evolution, strong stress sensitivity β = −104. In fact, during the
until approximately 40% of particles have constricted, evolutionstagedepictedinFig.3,theclusterdistribution
because this stage is most relevant to the initial phase in systems with β = −10 and β = 0 (no stress sensitiv-
of ventral furrow formation in the Drosophila embryo. ity) looks similar; however, at later stages the difference
However, since our results may be relevant also to other betweenthefullyrandomandweak-stress-sensitivitysys-
systemsofmechanicallyactivecells,weexamineavariety tems becomes much larger (see Fig. 4).
ofconstrictiontriggeringconditionsandprovidealimited We note that even in a fully random case, β = 0, a
set of results for longer times. significantnumberofclustersformalreadyatarelatively
All simulationsareperformedfor the same initialcon- early stage of evolution (see Fig. 3 for N = 50). This
c
dition with N0 =258 initially active particles. Thus, for is because an isolated constricted particle typically has
a
N = 25 approximately 10% particles have constricted, several active neighbors, which increases the probability
c
for N = 50 approximately 20%, and for N = 100 ap- of formation of small groups of constricted particles in
c c
proximately 40%. an uncorrelated random process.
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circles in Fig. 6(a) and the tensile stress coloring in Fig.
6(c)). Thus the chain increases in length. This mech-
anism not only promotes chain growth, but also results
in increased chain connectivity, stimulating the expan-
sion of a constricted-particle-chain network, and leading
to its eventual percolation.
In a system sensitive to compressive stresses (see Fig.
6(b)),theabovemechanismofgrowthofaconstrictedre-
gion is inverted: the compressed particles alongside the
chain (i.e., in the regions indicated by the dashed blue
ovals) are now likely to constrict, which results in re-
structuringofthechainintoacompactcluster. Asimilar
mechanism governs growth of already formed clusters.
C. Evolution of the particle stress distribution
A comparison of the stress distribution for a tensile-
stress-sensitive system (top panel of Fig. 5) and for a
compressive-stress-sensitivesystem(bottompanelofFig.
FIG.5: Avisualrepresentationofthedistributionofstressσ 5)showsthatthestressesaredistributedverydifferently.
throughtheactiveregion ofagranularfluidatNc =100and For tension-sensitive triggering, the tensile stress propa-
β as labeled. Shades of blue and red show a net compressive gatesalonganetworkofconstricted-particlechains,leav-
and tensile stress, respectively. (Thecolor scale is relative to
ing pockets of compressive stress between the chains. In
the maximal compressive and tensile stresses in the system
contrast, for compression-sensitive triggering (i.e., when
σmax and σmax.)
c t a network of constricted particles does not form), the
tensile stress distribution is much more uniform, and,
moreover,the tensile stresses are predominant, and only
B. The mechanism of chain and cluster formation
small pockets of compressive stress remain.
To gain further insights into stress feedback mecha-
To elucidate the mechanisms of chain and cluster for-
nisms that may be important in ventral furrow forma-
mation, we examine the stress distribution in the evolv-
tion(andmoregenerallyinbehaviorofactiveparticulate
ing AGF undergoing particle constrictions. A color map
systems), we examine the evolution of particle stress for
of the particle stress, Eq. (4), is presented in Fig. 5 for
different particle populations. The distribution of parti-
a system with constrictions triggered by tensile stresses
cle stress for the populations of active particles A, con-
(top panel, β = 104) and compressive stresses (bottom
stricted particles C, and inactive particles I is shown in
panel, β = −104). Both panels show the same stage of Fig.7foratensile-sensitivesystemwithβ =104. Figure
the evolution, N = 100. The images were obtained by
c 8 shows the corresponding results for the particle pres-
recoloring the corresponding panels in Fig. 3, to visual-
sure
ize the distribution of stress in the active particle band.
Movies 2(a) and 2(b) in Supplementary Data show the σ¯ =−σ (9)
evolution of the stress distribution during the constric-
tion process. in a compression-sensitive system with β = −104. (We
A close examination of the top panel of Fig. 5 reveals call quantity (9) the particle pressure, in line with the
the following key features of the stress distribution that standard convention in fluid mechanics, in which the
are essential for understanding the microstructural evo- pressure tensor is the negative of the stress tensor.)
lution (see the schematics in Fig. 6): In both cases the results are sorted from the smallest
to largest value of the stress σ (pressure σ¯) and shown
1. chains of constricted particles are predominantly vs. particle index i . Accordingly, the largest triggering
p
subject to tensile stress; stress (pressure) corresponds to the upper-right end of
thecurvesrepresentingthestress(pressure)distribution.
2. unconstricted particles that are positioned along-
The results reveal several important features com-
sidethechainarepredominantlycompressed,while
mon to all particle populations and both constriction-
thosenearthe chainends arepredominantlyunder
triggering conditions. First, the slope of the curves
tension.
sharply increases near the ends of the curves σ(i ) and
p
Inasystemsensitivetotensile stresses(Fig. 6(a)), ac- σ¯(i ). This behavior indicates that there exist small
p
tiveparticlesaremostlikelytoconstrictneartheendsof particle subsets for which the stress σ (or pressure σ¯)
analreadyformedchain,becausethetensilestressispre- significantly differs from the stress (pressure) for typi-
dominantintheseregions(asindicatedbythedashedred cal particles. Second, the stress distribution is initially
8
FIG.6: Mechanism of formation of (a)chainsand(b)clustersofconstricted particles (browncircles); and(c) thedistribution
of particle stress (color scale as in Fig. 5) near a chain of four constricted particles. Areas of tensile stress (TS, red dashed
circles)andareasofcompressivestress(CS,bluedashedovals)denoteregionsoftensile-stress-andcompressive-stress-triggered
constrictions, respectively.
FIG. 7: Stress distribution σ = σ(i,Nc) for (a) active, (b) constricted, and (c) inactive particles, in a system sensitive to
tensile stress (β =104), with Nc =0 (blue dotted line), Nc =25 (green dashed), Nc =50 (purple dash–dot), Nc =75 (orange
dash–dot–dot), andNc =100(redsolid) constricted particles. Foragiven Nc,theresults aresortedbytheincreasing valueof
stress and shown vs. particle index ip.
FIG.8: SameasFig.7,butfor asystem sensitivetocompressive stress(β=−104),andfor thevaluesoftheparticlepressure
σ¯ =−σ.
9
The non-monotonic behavior of the tensile stress in the
population A is a consequence of the formationof a net-
workof connected constricted-particlechains whichsup-
port most of the tensile stress (see Fig. 3 and the top
panel of Fig. 5).
Because the constriction probability (7) is normalized
by the maximal stress (8) in the current configuration,
oursimulationscannotbe continuedbeyondthe pointat
whichthemaximaltensilestressintheactivepopulation
Avanishes. Inamodifiedmodelwithnormalizationbya
fixed characteristic stress, the constriction process could
be continued, but would significantly slow down at the
point σmax ≈ 0. (In wild type Drosophila the slowdown
A
would not occur because of the earlier transition to the
rapid phase of apical constrictions, which is controlled
by different mechanisms; the slowdown, however, could
perhaps be observed in Drosophila mutants.)
The growth of the stress-supporting network of con-
stricted particles is reflected in a steadily increasing
stress in the particle population C (see Fig. 7(b) and
the dashed red line in 9(a)). We hypothesize that the
stress-supporting network of CCCs in the Drosophila
embryo (analogous to the interconnected constricted-
particle chains in our AGF model) plays a biologically
useful role. First, since the chains distribute stresses
non-locallyintheentireactiveregion,theymaymitigate
the effect of decreasedcell contractility in some domains
(such domains may result from random fluctuations or
FIG. 9: (a) Evolution of themaximal stress σ =σmax (thick genetic defects). Second, a tightly stretched band of in-
lines) and minimal stress σ = σmin (thin lines) in a system terconnected CCCs may help to organize a coherent tis-
with stress sensitivity parameter β =104 for the three types
sue motion at the onset of the second phase of ventral
of particles: active (solid lines), constricted (dashed), and
furrowformation. Inbothcases,CCCs wouldcontribute
inactive (dotted). (b) The same as (a), except that for a
system with β =−104 and for the valuesof particle pressure to robustness of the invagination process. However, the
σ¯ =−σ. Inbothpanels,theredlinescorrespondtothetensile role of CCCs requires further investigations.
endofthestressrange,andblueonestothecompressiveend. Compressive-stress-sensitive triggering As discussed
at the beginning of Sec. IVC, compression-sensitive sys-
tems develop large continuous areas of tensile stress.
narrow, and becomes significantly wider when particles These areas include compact domains of active and con-
start to constrict creating local inhomogeneities within stricted particles. As a consequence of this morphology,
the medium, and generating large positive and negative thetensilestress(i.e.,thenegativepressureσ¯ intheplots
values of particle stress. The latter feature of the stress shown in Figs. 8 and 9(b)) is similarly distributed in the
distribution is also visible in Fig. 9, where the minimal populations A and C. As seen in Fig. 9(b), the tensile
and maximal particle stress stress is somewhat larger for constricted particles (red
dashedline) thanforactiveparticles(redsolidline), but
σmin(N )=minσ(i,N ), σmax(N )=maxσ(i,N ), thedifferenceismuchsmallerthanthecorrespondingdif-
c c c c
i∈P i∈P ference for tension-sensitive triggering (see dashed and
(10a)
solid red lines in Fig. 9(a)).
and particle pressure
Thecompressivestress(i.e.,the positivepressureσ¯)is
σ¯min =−σmax, σ¯max =−σmin (10b) less evenly distributed. In the population of active par-
ticles A, the maximal value σ¯max is relatively small and
inthepopulationP =A,C,I areplottedversusthenum- decreases to zero at long times. Only a few constricted
berofconstrictedparticlesfortension-andcompression- particles are under compression according to the stress
sensitive systems. map shown in the bottom panel of Fig. 5, and the maxi-
Tensile-stress-sensitive triggering The results de- malpressureisnegativeorclosetozeroduringtheentire
picted in Figs. 7(a) and 9(a) show that the maximal evolution.
tensile stress in the population of active particles A in We note that the behavior of the stress distribution
a tension-sensitive medium initially increases, but sub- within the population of constricted particles C is qual-
sequently gradually decreases, and eventually vanishes. itatively different in the compression-sensitive medium
10
(see Fig. 8(b)) and the tension-sensitive medium (see of constricted-particle chains. It follows that mechanical
Fig. 7(b)). In the former case, the slope of the curve feedback can be used to control the system morphology.
σ¯ = σ¯(i ) decreases with the increasing N , and in the
p c The cell dynamics during the initial apical constric-
lattercasetheslopeofthecorrespondingcurveσ =σ(i )
p tion phase of ventral furrow formation (considered here)
remains constant (only the middle, nearly linear, part of
can be modeled using a 2D AGF approach, because
the curve varies in length). This universal slope is likely
cells at this stage are mechanically active only in a nar-
to be a signature of scaling properties of the developing
row, nearly planar region. However, subsequent mor-
constricted-particle network.
phogenetic movements (e.g., the later phase of ventral
Behavior of the inactive region As depicted in Figs.
furrow formation, cephalic furrow formation, and germ
7(c) and 8(c), the stress distribution in the inactive-
bandextension)involvelarge-scalecollectivecellmotions
particleregionI isrelativelyfeatureless. Mostofthepar-
in different mechanically coupled domains. Understand-
ticles experience small positive or negative stresses, and,
ing the role of mechanical feedback in coordinating cell
according to stress maps (not shown) only a small num-
activity will thus require development of comprehensive
ber of particles on the border between the active andin-
full-embryo 3D models in which motion of all cells (ap-
active regions are affected by particle constrictions. The
proximately 6000), arranged in an epithelial monolayer
maximal tensile stress grows with N , and the compres-
c surrounding the yolk sac, will be explicitly followed; we
sive stress saturates (see Fig. 9).
are working on such models.
With the free boundary condition in the y-direction,
the inactive particles do not affect the active region in Weexpectthatavarietyofmorphogeneticandorgano-
a significant way. However, for more resistive boundary genetic processes can be studied by modeling a devel-
conditions (e.g., periodic boundary conditions in both oping tissue as an active granular fluid. For example,
x and y directions; results not shown), the interaction mechanical stresses have been shown to influence mul-
between the passive and active regions is much stronger. tiple aspects of heart development in Zebrafish. Stress
In our future studies, this effect will be investigated in exerted upon cells by fluid flow influences the number
the context of interactions between active cells in the of chambers which are developed [30], valve growth [31],
mesodermprimordiumandthesurroundingcellsinmore and the establishment of pacemaker cells. It is possible
lateral regions. that these mechanically sensitive aspects of cardiogene-
sis can be evaluated by considering an appropriate AGF
model.
V. CONCLUSIONS Statistical mechanics methods and simulation tech-
niques that were initially developed for investigations of
Mechanical stress fields are now believed to play a complex fluids have significantly contributed to the un-
pivotal role in many biological developmental processes. derstanding of molecular-level mechanisms in biological
Therefore,itiscrucialtoestablishmethodstoinvestigate systems. In particular, fundamental studies of protein
local cell–cell mechanical interactions and global stress folding [32–34] (also advanced by George Stell’s group
distributionsacrosstissuesandevaluatetheeffectofsuch [35–37])haveledtothedevelopmentofdesignerproteins
local and global phenomena on mechanical cell activity. [38–40]. Other examples of cross-pollination between
We haveshownthatmodelinga tissueas anactivegran- fluid-state physics and biology include rapid progress in
ular medium can offer a means of analyzing mechanical areassuchascytoskeletondynamics[41–43]andbehavior
feedback involved in tissue development. of cell membranes [44–46].
Our AGF model of apical constrictions in the Weanticipatethatstudiesofthecollectivephenomena
Drosophila embryo during the early stage of ventral fur- associated with mechanical cellular activity and inter-
row formation has demonstrated constriction patterns cellular interactions, including results of multicellular
that are qualitatively similar to those observed in vivo. modeling of mechanical feedback during tissue forma-
The key new element of the model is the quantification tion,willbeofsimilarimportanceforunderstandingmor-
of the mechanical sensitivity of cells to tensile stresses, phogenesis and organogenesis. Further, the knowledge
which are responsible for an increase in the cell con- gained will also lead to applications in tissue engineer-
striction probability. The agreement between the model ing.
predictions and constriction patterns observed in the
Drosophila mesoderm primordium provides evidence of
theroleofmechanicalfeedbackintheearlystageofmor-
phogenesis examined here.
We have considered a wide range of constriction trig-
ACKNOWLEDGMENTS
gering conditions and analyzed the associated stress dis-
tribution, which evolves as the cellular constriction pro-
cess progresses. We have also shown that in systems in GJG gratefully acknowledges financial support from
which cells are sensitive to compressive stresses, grow- NTU startup funding 104R7417. Imaging experiments
ing clusters of constricted particles are formed instead were supported by funds from TTUHSC to JHT.