Journal of the Mechanics and Physics of Solids 112 (2018) 650–666
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An oscillating dynamic model of collective cells in a monolayer ShaoZhen Lin, ShiLei Xue, Bo Li∗, XiQiao Feng∗ Institute of Biomechanics and Medical Engineering, AML and CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 24 July 2017 Revised 9 September 2017 Accepted 23 September 2017 Available online 29 September 2017 Keywords: Cytodynamics Cellular vertex model Oscillation Chemomechanical coupling Hopf bifurcation
a b s t r a c t Periodic oscillations of collective cells occur in the morphogenesis and organogenesis of various tissues and organs. In this paper, an oscillating cytodynamic model is presented by integrating the chemomechanical interplay between the RhoA effector signaling pathway and cell deformation. We show that both an isolated cell and a cell aggregate can undergo spontaneous oscillations as a result of Hopf bifurcation, upon which the system evolves into a limit cycle of chemomechanical oscillations. The dynamic characteristics are tailored by the mechanical properties of cells (e.g., elasticity, contractility, and intercellular tension) and the chemical reactions involved in the RhoA effector signaling pathway. External forces are found to modulate the oscillation intensity of collective cells in the monolayer and to polarize their oscillations along the direction of external tension. The proposed cytodynamic model can recapitulate the prominent features of cell oscillations observed in a variety of experiments, including both isolated cells (e.g., spreading mouse embryonic ﬁbroblasts, migrating amoeboid cells, and suspending 3T3 ﬁbroblasts) and multicellular systems (e.g., Drosophila embryogenesis and oogenesis). © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Oscillations occur in a wealth of biological processes such as circadian rhythms (Dunlap, 1999; Reppert and Weaver, 2002), calcium sparks (Cheng and Lederer, 2008; Jaggar et al., 20 0 0), DNA synthesis (Zhang et al., 20 07), cytokinesis (Grill et al., 2005; Sedzinski et al., 2011), embryogenesis (Blanchard et al., 2010; Solon et al., 2009), and collective cell migration (Cox and Snead, 2016; SerraPicamal et al., 2012), with a period ranging from a few seconds to dozens of hours. These dynamic behaviors may stem from either an active programmed regulation for certain biological functions such as circadian rhythms (Dunlap, 1999; Reppert and Weaver, 2002), or a passive response to the circumstance such as the oscillation of cyclic adenosine monophosphate in starved Dictyostelium cells (Kamino et al., 2011). Biological oscillations are usually triggered by Hopf bifurcation, beyond which the system loses its stability at equilibrium and enters into a limit cycle (Goldbeter, 2002). Thus, they can be viewed as a nonequilibrated selforganization of the dissipative biological systems (Nicolis and Prigogine, 1977). As an important form of biological oscillations, cell shape or area oscillations play a pivotal role in morphogenesis including tissue elongation (Rauzi et al., 2010), embryo invagination (Martin et al., 2009), and tissue sealing (Blanchard et al., 2010). For example, during Drosophila embryogenesis, cell shape oscillations occur in ventral furrow cells at the onset of ∗
Corresponding authors. Email addresses: libo
[email protected] (B. Li),
[email protected] (X.Q. Feng).
https://doi.org/10.1016/j.jmps.2017.09.013 00225096/© 2017 Elsevier Ltd. All rights reserved.
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mesoderm invagination (Martin et al., 2009) and in amnioserosa cells during dorsal closure (Solon et al., 2009). Shape oscillations are also found in the follicle cells during stage 9 in the development of Drosophila egg chambers (He et al., 2010; Koride et al., 2014). Besides, in Xenopus embryogenesis, ectodermal and mesodermal cells can exhibit ﬂuctuating contractions during convergence and extension (Kim and Davidson, 2011). In addition to the aforementioned collective oscillations, isolated cells may also display oscillatory features, as observed in some biological systems, e.g., spreading mouse embryonic ﬁbroblast cells (Giannone et al., 2004), locomotive amoeboid cells (Satoh et al., 1985), suspended 3T3 ﬁbroblast cells (Salbreux et al., 2007), and cytokinesis of Hela cells and L929 ﬁbroblasts (Sedzinski et al., 2011). Among them, an isolated amoeboid cell exhibits distinct shape oscillation with a period of 2–10 min when migrating on an adherent glass surface (Satoh et al., 1985); suspended 3T3 ﬁbroblasts can undergo spontaneous shape oscillations with the period ranging from 1 to 5 min, depending on the culture condition (Salbreux et al., 2007). It has been recognized that the dynamics of cytoskeletal systems is of vital importance for both physiology of individual cells (Jülicher and Prost, 1997; Kruse and Jülicher, 2005; ViatchenkoKarpinski et al., 1999) and morphogenesis of multicellular epithelia (He et al., 2015), organs or organisms (Angelini et al., 2010; Gorﬁnkiel and Blanchard, 2011; Paluch and Heisenberg, 2009; Vedula et al., 2014). Cell oscillations involve biological, chemical, and mechanical mechanisms that couple at the molecular, subcellular, and supracellular scales. Along with shape oscillations, myosin activity within the oscillating cells has been revealed to ﬂuctuate in nearly an antiphase behavior (Blanchard et al., 2010; Dierkes et al., 2014; He et al., 2010; Koride et al., 2014; Sedzinski et al., 2011). Besides, experiments also evidenced that cell area oscillations disappear when actomyosin contraction is inhibited (He et al., 2010; Sedzinski et al., 2011), suggesting that the oscillations are intimately associated with cell contractility generated by the actomyosin cytoskeleton network. The actomyosin cytoskeleton network is a thin layer consisting of crosslinked actin ﬁlaments beneath the plasma membrane and links the neighboring cells through adherens junctions (Lecuit and Lenne, 2007; Lecuit et al., 2011; Murrell et al., 2015). The contractility of the actomyosin network is attributed to the activity of myosin II, which is an ATPdependent motor protein and “walks” along the actin ﬁlaments to generate contractile forces (Murrell et al., 2015). Recently, considerable attentions have been paid to the dynamics of cell oscillations. It was found that amnioserosa cells undergo remarkable collective area oscillations with a period of ∼4 min after germband retraction and these oscillations can last for over an hour until the midstage of dorsal closure during Drosophila embryogenesis (Blanchard et al., 2010; Solon et al., 2009). A theoretical model involving sinusoidal contractions or arearelated contraction forces has been proposed to reproduce the amnioserosa oscillations (Solon et al., 2009). Representing the amnioserosa by 81 connected regular hexagons, Wang et al. (2012) investigated the biomechanical mechanism behind cell oscillations. The contractility of cells was described by springs connecting neighboring vertices and an interactive regulation between the myosin activity and the contraction force was involved. Dierkes et al. (2014) proposed a onedimensional contractile rod model which involved myosin turnover to capture the characteristics of cell oscillations. Recently, Lin et al. (2017b) established a timedelayed biomechanical model to address collective oscillations in Drosophila amnioserosa. These studies provide mechanistic insights into the collective cell oscillations. To date, however, there is still a lack of a generic model that rationalizes real biochemical signaling pathways and is capable of dissecting cell oscillations in systems with wide varied cell populations ranging from single to multiple cells. In fact, actomyosinregulated cell contractility may be mediated by such complex signaling pathways as the RhoA effector signaling pathway and calcium pathway (Ahmadzadeh et al., 2017; Shenoy et al., 2016). Insights into the coupling between detailed biochemical signaling pathway and mechanical cues would deepen our understanding of cell shape oscillations. Taking RhoA effector signaling pathway as a representative one, we here develop a chemomechanical model that integrates a clear interactive regulation between myosin activity and cell deformation to address cell oscillations. We aim to providing a generic theoretical framework to delineate oscillatory dynamics observed in vastly diverse cellular systems ranging from isolated cells to cell aggregations. The paper is organized as follows. In Section 2, the chemomechanical cytodynamic model is provided by considering the mechanical and biochemical coupling mechanisms. In Section 3, we investigate the dynamic stability for both isolated cells and collective cells in an epitheliallike monolayer. In Section 4, we examine the characteristics of collective cell oscillations via spectral analysis. In Section 5, we explore the role of mechanical stretches in modulating collective cell oscillations. Finally, the main conclusions are drawn in Section 6.
2. Model 2.1. Collective cells in a conﬂuent monolayer We apply the dynamic vertex model to investigate the collective cell oscillations in a conﬂuent cell monolayer. The vertex model has been extensively used to study epithelial morphodynamics, e.g., wound closure (Nagai and Honda, 2009), tumor invasion (Lin et al., 2017c), and embryo morphogenesis (Osterﬁeld et al., 2013). In the cellular vertex model, cells are approximated by connected polygons with intercellular boundaries linking to neighboring cells (Farhadifar et al., 2007; Lin et al., 2017a), as shown in Fig. 1(a) and (b). Different from the traditional vertex model, we will involve chemomechanical feedback. Within the vertex description, the dynamics of an epitheliallike monolayer is determined by the evolution of the cell vertices, at which three neighboring cells joint together. The total potential energy in the cell monolayer can be expressed
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Fig. 1. Cellular vertex model. (a) Phase contrast image of a conﬂuent Madin–Darby canine kidney (MDCK) cell monolayer, in which cells manifest polygonal shape (see the marked cells). (b) Geometric description of epithelial cell monolayer. Refer to the coordinate system (x, y), where the x axis is along the horizontal direction and the y axis is along the vertical direction. (c) Forces acting at a vertex in a cell monolayer. Fci , Fai , Fsi , and Fri denote forces arising from cell contractility, cell area elasticity, intercellular interfacial tension, and friction, respectively. (d) Schematic diagram of the interactive regulation between RhoA effector signaling pathway and cell deformation.
as (Farhadifar et al., 2007; Fletcher et al., 2014; Nagai and Honda, 2001; Xu et al., 2016b)
U=
1 1 2 (J ) 2 km fMLC LJ + Ka AJ − A0 + li j , 2 2 J
J
(1)
i, j
where the ﬁrst term on the righthand side accounts for cell contraction due to active contractility of the actomyosin cortex layer (Farhadifar et al., 2007; Salbreux et al., 2012); the second term corresponds to cell area elasticity resulting from the resistance to monolayer’s height ﬂuctuations and cell apical area changes (Zehnder et al., 2015); the third term refers to the interfacial tension set by a competition between the cortical tension and the intercellular adhesion (Manning et al., 2010). LJ and AJ represent the perimeter and area of the Jth cell, respectively, and A0 is the preferred area. Ka denotes the area stiffness of a cell. represents the interfacial tension acting on the intercellular interface, and can be either positive (when the cortical tension dominates over the intercellular adhesive energy) or negative (when the intercellular adhesive energy (J ) dominates); lij is the length of intercellular interface ij connecting vertices i and j. f MLC refers to fraction of activated myosin (J )
light chain (MLC), i.e. the phosphorylation level of MLC (pMLC) which dictates the activity of myosin II in cell J. km fMLC
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characterizes the contractility of the actin cortex of the Jth cell, where we have assumed a linear relationship between cell contractility and pMLC. Consequently, km will be referred to as the cell contractility constant hereafter. It is noted that the intercellular adhesion may also be related to myosin II activity through the complex cytoskeleton network. For example, the intercellular cohesion of epithelium is attributed to the adhesive contacts constituted mainly by Ecadherin molecules, which concentrate in the adherens junctions and form an adhesive belt, stitching cells together (Guillot and Lecuit, 2013). The adhesive belt is mechanically coupled to the cortical actomyosin network via a complex long chain, which includes α catenin, β catenin, vinculin, and so on (Lecuit et al., 2011). In this paper, we take as a constant for simplicity. Since cellular motion is mainly damped by the energy dissipation due to viscosity, the inertia of cells is ignored in the dynamic vertex model (Fletcher et al., 2014). Therefore, the mechanical equilibrium of each vertex requires the balance p between the dissipative frictional force Fri = −ηdri /dt and the potential force Fi = −∂ U /∂ ri arising from cell contraction, cell elasticity, and interfacial tension (Fig. 1(c)), with ri the position of vertex i and η the effective coeﬃcient of friction. Accordingly, the temporal evolution of vertex i is determined by
η
dri ∂U =− . dt ∂ ri
(2)
In the present model, we do not consider the detailed cell–substrate interactions, which may play a key role in cell differentiation, migration, adhesion, polarity, and reorientation (Chen et al., 2015; Engler et al., 2006; He et al., 2014; Qian et al., 2013; Sunyer et al., 2016; Xu et al., 2016a). The friction between the cells and their substrate is taken into account by the dissipative frictional force Fri . Substituting Eq. (1) into (2) leads to
dr (J ) η i = − km fMLC LJ dt
J∈Ci
1 ri − r j + − , Ka AJ − A0 k × r j2 − r j1 − ri − r j1 ri − r j2 r i − r j 2 J∈Ci j∈Vi ri − r j1
ri − r j2
where k denotes a unit vector normal to the cell monolayer; the summation
vertex i, while the summation
J∈Ci
(3)
computes over all neighboring cells Ci of
is made over all neighboring vertices Vi of vertex i; j1 and j2 are the neighboring vertices
j∈Vi
of vertex i in cell J, as shown in Fig. 1(c). Besides, we here focus on the dynamics of cell shape oscillations, and thus do not consider topological transitions in the polygon network for brevity. To reveal the biochemical mechanism of cell oscillations, we invoke the RhoA effector signaling pathway, which is responsible for cell contractility (Ahmadzadeh et al., 2017; Lessey et al., 2012; Miller and Davidson, 2013; Shenoy et al., 2016). RhoA is a small GTPase protein of Rho family, which is the key regulator of cytoskeleton and thus crucial for morphogenesis (Lecuit and Lenne, 2007; Lecuit et al., 2011). It has been illustrated that RhoA can be activated by tensile forces (Smith et al., 2003; Zhao et al., 2007). The active RhoA (GTPbinding form) acts upon Rho kinase (ROCK) to promote pMLC, which facilitates the assembly of myosin II and thus enhances its ATPase activity. Consequently, the contractile force of actin ﬁlaments generated by myosin II is enhanced. Taken together, there exists a feedback mechanism between cell deformation and myosin II activity through the RhoA effector signaling pathway, as shown in Fig. 1(d). Different upstream regulators for myosin II activity may also exist, e.g., Rho1 (Munjal et al., 2015) and Ca2+ (Ahmadzadeh et al., 2017; Shenoy et al., 2016). However, the corresponding regulating processes and outcomes of these upstream regulators for myosin II activity are similar (Ahmadzadeh et al., 2017; Munjal et al., 2015; Shenoy et al., 2016). Therefore, replacing the RhoA effector signaling pathway with another similar signaling pathway (e.g., Rho1 and Ca2+ ) would not interfere with our main conclusions. We employ the Hill function (Ferrell et al., 2011; Mackey and Glass, 1977) to describe the chemomechanical coupling of myosin activation and cell deformation. The kinetics of Rho, ROCK, and MLC activation and deactivation in the Jth cell is quantiﬁed as () d fRho
J
dt
= αRho
λL(J )
KRho +
() d fROCK = αROCK dt K J
nRho
λL
() fRho J
ROCK
() d fMLC = αMLC dt K J
MLC
(J ) (J ) 1 − fRho − βRho fRho , n Rho (J )
+
() fROCK J
nROCK
(J ) nROCK fRho
(J ) (J ) 1 − fROCK − βROCK fROCK ,
(4)
(5)
nMLC
(J )
+ fROCK
(J ) (J ) nMLC 1 − fMLC − βMLC fMLC ,
(6)
where f represents the fraction of activated biochemical factor (i.e., activity), α the activation coeﬃcient, β the deactivation rate, K the apparent dissociation constant, and n the Hill coeﬃcient. The subscripts Rho, ROCK, and MLC denote the quantities pertaining to Rho, ROCK, and MLC, respectively. λL = L/Ls characterizes the cell deformation, where Ls is the perimeter of the cell at rest. Because the cells in a monolayer statistically prefer a hexagonal shape (Nagai and Honda, 2001), Ls can be obtained by theoretically solving the equilibrium state of a regular hexagonal cell, which will be shown in Section 2.2. We
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notice the chemical communication among cells is not a necessity for an oscillatory behavior in some systems (FernandezGonzalez and Zallen, 2011; Jayasinghe et al., 2013). For simplicity, therefore, we ignore the transmembrane transport of myosin II between cells. So far, we have developed a chemomechanical model that integrates a feedback mechanism between myosin activity and cell deformation to address cellular dynamics in a monolayer. This model will be used to investigate the cellular morphody namics of both isolated cells and collective interacting cells. We initially normalize Eqs. (3)–(6) by using A0 as the length scale and τ = η/(Ka A0 ) as the time scale. The normalized equations can be written as
d ri (J ) ˜ =− k˜ m fMLC LJ dt˜
J∈Ci
1 ri − r j + − , A˜ J − 1 k × r j2 − r j1 − ri − r j1 ri − r j2 ri − r j 2 J∈Ci j∈Vi ri − r j1
ri − r j2
n (J ) L˜J Rho d fRho (J ) (J ) = α˜ Rho 1 − fRho − β˜Rho fRho , n n dt˜ KRho L˜s Rho + L˜ Rho
(7)
(8)
J
() d fROCK = α˜ ROCK dt˜ K J
() fRho J
ROCK
() d fMLC = α˜ MLC dt˜ K J
nROCK
+ fRho
() fROCK J
MLC
(J ) nROCK
(J ) (J ) 1 − fROCK − β˜ROCK fROCK ,
(9)
nMLC
(J )
+ fROCK
(J ) (J ) nMLC 1 − fMLC − β˜MLC fMLC ,
(10)
where the dimensionless parameters are deﬁned as
˜ = 3/2 , L˜J = √LJ , r˜i = √ri , t˜ = τt , k˜ m = KkamA0 , Ka A0 A0 A0 A L˜s = √Ls , A˜ J = J , α˜ p = αp τ , β˜p = βp τ , A0
(11)
A0
with the subscript p stands for Rho, ROCK, and MLC. Eqs. (7)–(10) establish a dimensionless system controlling the dynamic evolution of cells, and they will be numerically solved by the ﬁnite difference method. 2.2. Isolated cell The above model can analyze cell oscillations in a monolayer system consisting of an arbitrary number of cells. To clearly reveal the physical mechanisms for the occurrence of cell oscillation, we ﬁrst consider an isolated cell, simpliﬁed as a regular hexagonal unit. In this case, Eqs. (3)–(6) reduce to
η
√ 1 dR 3 = −6km fMLC R − Ka (A − A0 )R − , dt 2 2
(12)
λL Rho d fRho = αRho (1 − fRho ) − βRho fRho , n dt KRho + λL Rho
(13)
nROCK fRho d fROCK = αROCK nROCK (1 − f ROCK ) − βROCK f ROCK , dt KROCK + fRho
(14)
nMLC fROCK d fMLC = αMLC nMLC (1 − f MLC ) − βMLC f MLC , dt KMLC + fROCK
(15)
n
where R is the radius (i.e. edge length) of the cell. Note that /2 in the interfacial tension term in Eq. (12) refers to the surface tension between the cell and environment for the isolated cell, but rather intercellular surface tension for collective √ interacting cells. Using R0 = 2A0 /(3 3 ) (i.e., the cell radius corresponding to the preferred area A0 ) as the length scale and τ = η/(Ka A0 ) as the time scale, we have the following normalized equations:
√ 1 ¯ dR¯ 3 2 = −6k¯ m fMLC R¯ − , R¯ − 1 R¯ − 2 2 dt¯
(16)
d fRho R¯ nRho = α¯ Rho (1 − fRho ) − β¯ Rho fRho , n dt¯ KRho R¯ s Rho + R¯ nRho
(17)
nROCK fRho d fROCK = α¯ ROCK (1 − fROCK ) − β¯ ROCK fROCK , KROCK + f nROCK dt¯
(18)
Rho
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nMLC fROCK d fMLC ¯ = α¯ MLC nMLC (1 − f MLC ) − βMLC f MLC , KMLC + fROCK dt¯
(19)
where the dimensionless parameters are deﬁned as
√
¯ = 3 3 3/2 = t¯ = τt , R¯ = RR0 , k¯ m = KkamA0 = k˜ m , 2 Ka A 0 R R¯ s = R0s , α¯ p = αp τ = α˜ p , β¯ p = βp τ = β˜p .
√ 3 3 2
˜,
(20)
The equilibrium state of the isolated cell satisﬁes dR¯ /dt¯ = 0 and d fp /dt¯ = 0, which give
√ √ 3 3 3 1 ¯ (s ) − = 0, R¯ s + 6k¯ m fMLC R¯ s + 2 2 2 ( ) fRho = s
(21)
α¯ Rho , ¯ α¯ Rho + βRho (KRho + 1 )
(22)
(s) nROCK α¯ ROCK fRho fROCK = (s) nROCK (s) nROCK , α¯ ROCK fRho + β¯ ROCK KROCK + fRho
(23)
(s) nMLC α¯ MLC fROCK fMLC = (s) nMLC (s) nMLC . α¯ MLC fROCK + β¯ MLC KMLC + fROCK
(24)
(s )
(s )
√
Substituting Eqs. (22)–(24) into Eq. (21), we can obtain the cell radius R¯ s at rest, or equivalently, R˜s = Rs / A0 = √ 4 2R¯ s / 27. Thus, the cell perimeter at equilibrium is Ls = 6Rs = 6Rs R0 = 6R˜s A0 , with the dimensionless form L˜s = 6R˜s .
2.3. Estimation of the dimensionless parameters According to experimental measurements, the viscosity is in the order of 104 − 105 N s m−2 for embryonic tissues (Forgacs et al., 1998); and the cell contraction force is in the range of 1 − 10 nN for epithelial cell sheets (Hannezo et al., 2014). Hence, we can estimate the friction coeﬃcient η ∼ 0.01 − 0.1 N s m−1 and the cell contractility constant km ∼ 10−4 − 10−3 N m−1 . Taking the cell areal stiffness Ka ∼ 105 − 107 N m−3 for epithelial tissues (Girard et al., 2007; Solon et al., 2009), and the preferred area A0 = 400 μm2 , we have the scale of the dimensionless cell contractility k˜ m ∼ 0.1, the length scale = A0 = 20 μm and the time scale τ = η/(Ka A0 ) ∼ 100 s. The activation coeﬃcients and deactivation rates of Rho, ROCK, and MLC have the scale ∼ 0.01 s−1 (Koride et al., 2014). Therefore, the corresponding dimensionless activation coeﬃcients and deactivation rates are α˜ p ∼ 1 and β˜p ∼ 1, respectively. Furthermore, we take the apparent dissociation constants Kp ∼ 1. ˜ = 0, α˜ Rho = 2.0, Speciﬁcally, the parameter values used in our calculation are set, unless stated otherwise, as: k˜ m = 0.2, β˜Rho = 0.3, α˜ ROCK = 1.0, β˜ROCK = 0.1, α˜ MLC = 100, β˜MLC = 1.0, KRho = 1.0, nRho = 8, KROCK = 0.8, nROCK = 8, KMLC = 0.8, and nMLC = 8. It should be noted that here we have set a large value of α˜ MLC (= 100) for clear illustration of cell oscillation. In fact, only if α˜ MLC and β˜MLC are in a proper range, can cell oscillations take place, as we will show in Section 3. 3. Hopf bifurcation Experiments have evidenced that the cell shape oscillations that occur during Drosophila germband extension and dorsal closure are autonomous for individual cells, which do not require stable contact with neighboring cells (FernandezGonzalez and Zallen, 2011; Jayasinghe et al., 2013). In this section, we examine the cellular dynamics for both isolated cells and collective interacting cells dictated by chemomechanical coupling. To gain insights into the dynamic mechanisms of cell oscillations, we will ﬁrst derive the analytical solution for the oscillation of a single cell and then analyze the dynamic behavior of collective cells in a monolayer. 3.1. Oscillation of an isolated cell Introducing a small perturbation to the equilibrium state of an isolated cell, the incremental equations of the dynamic system Eqs. (16)–(19) become
d
δ R¯ dt¯
(s ) = − 6k¯ m fMLC +
√ 3 2 ¯ 3Rs − 1 δ R¯ − 6k¯ m R¯ s δ fMLC , 2
d(δ fRho ) n K α¯ (s ) = Rho Rho Rho 1 − fRho δ R¯ − 2 ¯ dt¯ K + 1 R ( Rho ) s
α¯ Rho KRho + 1
+ β¯ Rho
(25)
δ fRho ,
(26)
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nROCK −1
( ) nROCK KROCK fRho d(δ fROCK ) (s ) = α¯ ROCK 1 − fROCK δ fRho − 2 ¯ dt (s ) nROCK KROCK + fRho
s
nMLC −1
( ) nMLC KMLC fROCK d(δ fMLC ) (s ) = α¯ MLC 1 − fMLC δ fROCK − 2 ¯ nMLC dt (s ) KMLC + fROCK s
α¯ ROCK
α¯ MLC
( ) fRho s
( ) KROCK + fRho
( ) fROCK s
nROCK s
nROCK + β¯ ROCK δ fROCK ,
nMLC
( ) KMLC + fROCK s
nMLC
(27)
+ β¯ MLC
δ fMLC ,
(28)
where δ R¯ , δ fRho , δ fROCK , and δ fMLC are the increments of cell radius, Rho activity, ROCK activity, and MLC activity from the equilibrium state, respectively. From the above perturbed Eqs. (25)–(28), the Jacobian (or characteristic) matrix for the system at the equilibrium state is
⎛
⎞
−bS 0 0 − aS ⎜ cS − dS 0 0 ⎟ J=⎝ , 0 eS − f S 0 ⎠ 0 0 gS − hS where
(29)
√ 3 2 (s ) aS = 6k¯ m R¯ s , bS = 6k¯ m fMLC + 3R¯ s − 1 , 2 n K α¯ α¯ Rho (s ) cS = Rho Rho Rho 1 − f , dS = + β¯ Rho , Rho 2 ¯ K Rho + 1 (KRho + 1 ) Rs
eS = α¯ ROCK
( ) nROCK KROCK fRho
nROCK −1
(s ) 2 1 − fROCK , n (s ) ROCK s
KROCK + fRho
fS = α¯ ROCK
( ) fRho s
nROCK
nROCK + β¯ ROCK , nMLC −1 (s ) nMLC KMLC fROCK (s ) gS = α¯ MLC (s) nMLC 2 1 − fMLC ,
(30)
( ) KROCK + fRho s
KMLC + fROCK
hS = α¯ MLC
( ) fROCK s
nMLC
( ) KMLC + fROCK s
nMLC + β¯ MLC .
The dynamics of the isolated cell is characterized by the eigenvalue of the corresponding Jacobian matrix, ζ , which satisﬁes
det (ζ I − J ) = 0,
(31)
where I is the identity matrix. Substituting Eq. (29) into (31), we obtain the characteristic equation for the chemomechanical coupling dynamics of an isolated cell
( ζ + b S ) ( ζ + dS ) ( ζ + f S ) ( ζ + hS ) + aS cS e S gS = 0 .
(32)
Setting ζ = r + iω, Eq. (32) becomes
( r + bS ) ( r + dS ) ( r + f S ) ( r + hS ) − [ ( r + bS ) ( r + dS ) + ( r + bS ) ( r + f S ) + ( r + bS ) ( r + hS ) + (r + dS )(r + fS ) + (r + dS )(r + hS ) + (r + fS )(r + hS )]ω2 + ω4 + aS cS eS gS = 0,
(33)
[ ( r + bS ) ( r + dS ) ( r + f S ) + ( r + bS ) ( r + dS ) ( r + hS ) + ( r + bS ) ( r + f S ) ( r + hS ) + (r + dS )(r + fS )(r + hS )]ω − (4r + bS + dS + fS + hS )ω3 = 0,
(34)
where r and ω are real. Solving Eqs. (33) and (34) numerically, we can obtain the eigenvalue ζ that determines the stability of the isolated cell. Depending on ζ , three states can be distinguished: (i) If Re(ζ ) < 0, the equilibrium of the cell is stable and cannot be broken up by perturbations, which will be referred to as a stable state. (ii) If Re(ζ ) > 0 and Im(ζ ) = 0, the equilibrium of the cell is unstable. In this case, the equilibrium state may be broken due to inﬁnitesimal perturbations, and then the system will evolve into a limit cycle of oscillation around the equilibrium state in the phase space. Such a state will be referred to as an oscillatory one hereafter. (iii) If Re(ζ ) > 0 and Im(ζ ) = 0, the cell will collapse, referred to as the collapsed state.
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Fig. 2. Oscillation of an isolated cell. (a) Temporal evolution of cell radius R, Rho activity fRho , ROCK activity fROCK and MLC activity fMLC in an oscillating cell. (b) Projection on the R − f MLC plane of trajectory of an oscillating cell, with the red arrows indicating the moving direction. The green dash line corresponds to the intersection of the nullclines of fRho , fROCK , and fMLC ; whereas the blue dash line refers to the intersection of the nullclines of R, fRho , and fROCK . The black arrows indicate the direction of motion of trajectories as they cross the nullclines. The gray spot denotes the unstable equilibrium state. Here, the nullcline is the locus of points where the corresponding rate is zero. For instance, the nullcline of R is controlled by the equation dR/dt = 0. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
There exists a critical state, upon which a Hopf bifurcation happens and the system could undergo a transition from the stable state to the oscillatory state. In particular, a Hopf bifurcation occurs when the eigenvalue ζ becomes pure imaginary, i.e. Re(ζ ) = 0 and Im(ζ ) = 0. Thus, the critical parameters for such a stability transition satisfy
ω4 + [bS dS + bS fS + bS hS + dS fS + dS hS + fS hS ]ω2 + (aS cS eS gS + bS dS fS hS ) = 0,
(35)
(bS + dS + fS + hS )ω2 − (bS dS fS + bS dS hS + bS fS hS + dS fS hS ) = 0,
(36)
where ω > 0 is the dimensionless angular frequency at the occurrence of Hopf bifurcation in the dynamic system. From Eq. (36), the corresponding angular frequency ωH is given as
ωH =
bS dS f S + bS dS hS + bS f S hS + dS f S hS . bS + dS + f S + hS
(37)
Thus, the corresponding oscillation period is
TH =
2π
ωH
= 2π
bS + dS + f S + hS . bS dS f S + bS dS hS + bS f S hS + dS f S hS
(38)
It is noted that the critical angular frequency ωH in Eq. (37) is deduced from Eq. (36), and thus the parameters bS , dS , fS , and hS should satisfy relations in Eqs. (35) and (36). We also perform direct simulations of Eqs. (16)–(19) to investigate the dynamic behavior of an isolated cell. The numerical results demonstrate that the proposed chemomechanical model can produce spontaneous cell oscillation. Fig. 2(a) gives the temporal evolution of the cell radius R, Rho activity fRho , ROCK activity fROCK and MLC activity fMLC of the cell, displaying apparent oscillations. The trajectory of (R, fRho , fROCK , fMLC ) is a limit cycle, which is a closed curve in a fourdimensional space. The projection of the trajectory on the R − fMLC plane is shown in Fig. 2(b). Thus, the cell oscillation reported here is autonomous, in agreement with experimental observations on, for example, germband extension (FernandezGonzalez and Zallen, 2011), dorsal closure (Jayasinghe et al., 2013), and follicle cell oscillation (He et al., 2010) in Drosophila embryogenesis or oogenesis. Note that this projection moves clockwise (indicated by red arrows). The encapsulated area by the projection corresponds to the energy dissipation per cycle, replenished by the chemical reactions of RhoA pathway. Besides, the three different states (i–iii) can also be identiﬁed through numerical simulations, as shown in Fig. 3. Here, the collapsed state also includes the case where there exists no equilibrium state for the isolated cell, i.e., the equilibrium Eq. (21) has no positive solution of R¯ s . Now we perform a parametric study to capture the predominant features of cell oscillation. The characteristic equation (32) is solved to obtain the eigenvalue ζ . Comparing the theoretical and numerical results, we ﬁnd that the above linear stability analysis well delineates the dynamic features of an isolated cell. In what follows, we will discuss the inﬂuence of cellular mechanical properties and the chemical reactions of the RhoA pathway on the cell dynamics. Speciﬁcally, we concentrate on the roles of cell contraction, surface tension, and the activation coeﬃcient and deactivation rate of MLC. Firstly, we scrutinize how cell contraction and surface tension dictate the dynamic behavior of a single cell. A phase diagram is constructed by using both the linear stability analysis and numerical simulations, as shown in Fig. 4(a). It is seen (cr ) that when the cell contractility constant k˜ m is small (i.e., k˜ m < k˜ m ), the eigenvalue given by the characteristic equation
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Fig. 3. Dynamics of an isolated cell. (a) Evolution of cell radius R corresponding to three different states, i.e. stable, oscillatory, and collapsed. (b) Projection on the R − f MLC plane of trajectories of the three states shown in (a), with the black arrows indicating the moving directions.
Fig. 4. Dynamics of an isolated cell regulated by cell’s mechanical properties. (a) The phase diagram regulated by the cell contractility constant k˜ m and ˜ . In the oscillatory region, the heat map corresponds to the oscillatory amplitude of cell radius RD , obtained from spectral analysis. The interfacial tension white solid line indicates the critical parameters for a Hopf bifurcation, whereas the white dashed line corresponds to a transition from oscillatory state to collapsed state. (b) The eigenvalue ζ of an isolated cell system versus the cell contractility constant k˜ m . Here, the four eigenvalues are distinguished by different colors, and Re(ζ ) and Im(ζ ) are plotted as solid and dashed lines, respectively. The magenta point marks the Hopf bifurcation.
(32) has negative real parts (Fig. 4(b)) and thus the cell is arrested in its stable equilibrium state (Fig. 4(a)). When the (cr ) cell contractility constant k˜ m is large enough (i.e., k˜ m > k˜ m ), the eigenvalue of the dynamic system could have positive real parts (Fig. 4(b)) and spontaneous oscillation may occur (Fig. 4(a)). It suggests that suﬃciently strong cell contractility is necessitated to trigger sustained oscillation. However, when the cell contractility constant k˜ m is too large, the cell will ˜ is below a critical value collapse. Besides, Fig. 4(a) reveals that surface tension inﬂuences the cellular dynamics. When ˜ exceeds (the white solid line in Fig. 4(a)), the originally oscillatory cell may transit to a stable state. In addition, when a second critical value (the white dashed line in Fig. 4(a)), the cell will collapse because of strong contraction. Therefore, ˜ ) tends to stabilize the cell. enhancing cell–substrate adhesion (i.e., decreasing We next examine how the chemical reactions of the RhoA effector signaling pathway regulate the cell dynamics by identifying the roles of the activation coeﬃcient α˜ MLC and the deactivation rate β˜MLC of MLC. Fig. 5(a) provides a phase diagram of (α˜ MLC , β˜MLC ). The coeﬃcient of MLC activation, α˜ MLC , inﬂuences the cellular dynamics in the following manner. (cr ) When α˜ MLC is too small (α˜ MLC < α˜ MLC ), the eigenvalue of the chemomechanical coupling system has negative real parts, (cr ) and thus the cell can stay in a stable equilibrium state (Fig. 5(b)). When α˜ MLC is large enough (α˜ MLC > α˜ MLC ), the real parts of the eigenvalue could become positive, and the cell will undergo spontaneous oscillation (Fig. 5(a)). When α˜ MLC is too large, however, the cell will collapse. Moreover, Fig. 5(a) also reveals that an enhanced MLC deactivation rate β˜MLC tends to stabilize the cell dynamics, since cell contractility is reduced due to downregulated MLC activity. A small β˜MLC favors spontaneous oscillations, whereas the cell will collapse when β˜MLC is too small (Fig. 5(a)). In addition, we ﬁnd that the physical parameters involved in the chemomechanical coupling can also tailor the oscillatory period within the range from 6τ to 30τ (Fig. 6), i.e. 10 min −50 min, comparable to experimental observations in a rich variety of biological systems (Giannone et al., 2004; Salbreux et al., 2007; Satoh et al., 1985; Sedzinski et al., 2011).
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Fig. 5. Dynamics of an isolated cell regulated by the chemical reaction of MLC. (a) The phase diagram regulated by the MLC activation coeﬃcient α˜ MLC and MLC deactivation rate β˜MLC . (b) The eigenvalues ζ of an isolated cell system versus the MLC activation coeﬃcient α˜ MLC .
˜ ) and (b) biochemical reactions (α˜ MLC , β˜MLC ). In the oscilFig. 6. Oscillation period of an isolated cell. Regulations of (a) cell mechanical properties (k˜ m , latory region, the heat map corresponds to the oscillatory period T, obtained from spectral analysis. The white solid line indicates the critical parameters for a Hopf bifurcation, whereas the white dashed line corresponds to a transition from oscillatory state to collapsed state.
3.2. Collective cell oscillations in a monolayer Now we examine the collective dynamics of a cell monolayer through numerical computations of Eqs. (7)–(10). Consider a cell monolayer consisting of plenty of cells (∼10,0 0 0). Periodic boundary conditions are adopted to eliminate the inﬂuence of boundary constraints. In order to eliminate the inﬂuence of stresses or stretches, we perform simulations based on the stressfree conﬁguration of the cell monolayer. We now analyze how cell contraction and intercellular surface tension inﬂuence the collective dynamics of a cell mono˜ ) is established in Fig. 7(a). Similar to the dynamics of an isolated cell, there exist three diflayer. A phase diagram of (k˜ m , ferent dynamic states for collective cells, i.e., the stable state, the oscillatory state, and the collapsed state. For the collapsed state of a cell monolayer, there exists no stressfree conﬁguration due to strong cell contraction or intercellular tension. ˜ are within a proper range. Collective cell oscillations occur when the cell contractility constant k˜ m and surface tension Speciﬁcally, a small k˜ m renders a stable state, whereas suﬃciently large k˜ m may trigger chemomechanical Hopf bifurcation, as shown in Fig. 7(b). Above the Hopf bifurcation, the cell monolayer undergoes sustained collective shape oscillations. When k˜ m is too large, however, the cells in the monolayer may collapse. Besides, the intercellular adhesion is found to stabilize the ˜ < 0, i.e., cell–cell adhesion dominates over cell cortex tension, strong enough intercellular adhesion cell dynamics: when ˜ > 0, a may bring collective cell oscillations to a stable state that all cells in the monolayer are arrested; whereas when large intercellular surface tension may result in collapse (Fig. 7(a)). The roles of the activation coeﬃcient α˜ MLC and the deactivation rate β˜MLC of MLC in the collective cell dynamics are illustrated in Fig. 8(a). It demonstrates that small α˜ MLC corresponds to a stable state, and only when α˜ MLC is large enough that collective cell oscillations could take place, as a consequence of Hopf bifurcation (Fig. 8(b)). When α˜ MLC is too large, however, cell elasticity cannot resist so high cell contraction and the cells will collapse.
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Fig. 7. Collective dynamics of a cell monolayer regulated by cell’s mechanical properties. (a) A phase diagram regulated by the cell contractility constant ˜ . AD = A(J ) J is the averaged areal oscillatory amplitude of the cell monolayer, with A(J ) being the areal oscillatory amplitude k˜ m and interfacial tension D D of the Jth cell, obtained from spectral analysis. (b) The bifurcation diagram of the cell dynamics in a monolayer dictated by the cell contractility constant ˜km . Here, Amax = maxt [AJ (t )]J and Amin = mint [AJ (t )]J .
Fig. 8. Collective dynamics of a cell monolayer regulated by the chemical reaction of MLC. (a) The phase diagram regulated by the MLC activation coeﬃcient
α˜ MLC and MLC deactivation rate β˜MLC . (b) The bifurcation diagram of the cell dynamics in a monolayer dictated by the MLC activation coeﬃcient α˜ MLC .
4. Dynamic features of collective cell oscillations 4.1. Collective oscillatory morphodynamics In this section, we investigate the collective oscillatory dynamics in a cell monolayer. Fig. 9(a) illustrates the morphological evolution of an arbitrarily selected cell (denoted by C0) and its six neighbors (denoted by C1–C6). It shows the shape changes of the connected cells with time. The temporal evolution of the area, RhoA activity, ROCK activity and MLC activity of the selected cell C0 are plotted in Fig. 9(b), displaying apparent oscillations. During this dynamic process, oscillations are transmitted and synchronized through physical linkages between cells. Fig. 9(c) shows the collective areal oscillations of a small ensemble consisting of those seven cells (C0–C6). Spectral analysis is performed to examine the collective dynamic behavior of the whole monolayer. It can be seen from Fig. 10(a) that almost all cells oscillate with the same frequency of f ∼ 0.12τ −1 . Thus, the oscillation period is T = 8τ ∼ 13 min, comparable to experimental measurements (Blanchard et al., 2010; Sokolow et al., 2012; Solon et al., 2009). We calculate the oscillation phases of cell area ϕ A , RhoA activity ϕ Rho , ROCK activity ϕ ROCK , and MLC activity ϕ MLC for all cells by using the discrete Fourier transform method. Fig. 10(b) shows that almost all values of ϕ A and ϕ MLC are close to the two lines ϕA − ϕMLC = ±180o , indicating that the dynamics of cell area and pMLC (i.e., myosin activity) exhibits approximately an antiphase feature in the whole domain. The distribution of ϕA − ϕMLC shown in Fig. 10(c) further veriﬁes the antiphase feature between ϕ A and ϕ MLC in the monolayer. As stated in the oscillation of an isolated cell, the antiphase feature arises (J ) from the negative feedback between myosin activity and cell deformation (Fig. 1(d)): cell expansion (λL > 1) promotes myosin activation, while elevated MLC activity (i.e., myosin activity) hinders cell expansion, as described in Eqs. (3)–(6). Such antiphase oscillatory dynamic behaviors of cell area and myosin activity have been observed in experiments (Blanchard et al., 2010; Dierkes et al., 2014; Martin et al., 2009; Solon et al., 2009). In addition, we also examine the oscillatory phases of RhoA activity ϕ Rho and ROCK activity ϕ ROCK . It demonstrates that the dynamics of cell area and RhoA activity are nearly
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Fig. 9. Oscillating dynamics of collective cells. (a) Temporal evolution of cellular morphology: a local perspective, focusing on a typical cell (C0) and its neighbors (C1–C6). (b) Evolution of the area A, Rho activity fRho , ROCK activity fROCK , and MLC activity fMLC in an typical oscillating cell, i.e., the cell marked as C0 in (a). (c) Evolution of cell area of a typical oscillating cell C0 and its neighbors C1–C6, as marked in (a).
Fig. 10. Spectral analysis of collective cell oscillations. (a) Spectrograms of area and pMLC of the cells. (b) Scatter diagram of the oscillation phases ϕ MLC and ϕ A of the cells. (c) Distribution of phase differences ϕA − ϕRho , ϕA − ϕROCK , and ϕA − ϕMLC .
inphase due to the direct activation of RhoA from cell deformation, whereas ϕ A and ϕ ROCK have an phase difference of about 100o (Fig. 10(c)). 4.2. Supracellular oscillatory pattern To elucidate the spatial features of collective cell oscillations, we make a heat map of oscillation phases of cell area
ϕ A (Fig. 11(a)). It shows that cells with parallel contraction and expansion phases are orchestrated into random distributed
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Fig. 11. Supracellular pattern of collective cell oscillations. (a) Spatial distribution of the areal oscillation phases ϕ A of the cells. (b) Histogram of the (J ) (J ) . (c) Spatial correlation of the oscillatory pattern shown in (c), with dc being the mean and the oscillatory orientation θaniso alignment orientation θalign distance between neighboring cells.
supracellular strings. Similar features are also observed in the spatial distributions of ϕ Rho , ϕ ROCK , and ϕ MLC . The oscillatory pattern appears to be labyrinthine, reminiscent of the patterns generated in growing soft tissues (Li et al., 2011). To examine (J ) the cell alignment during this phase pattern formation, we deﬁne the alignment orientation θalign of cell J as
rJ − rI J · ex θalign = arccos , rJ − rIJ (J )
(39)
where rJ is the geometric center of cell J; rIJ is the center of the neighboring cell I whose oscillation phase is the closest (J )
to cell J. Fig. 11(b) shows that the alignment orientation θalign distributes uniformly within the range (0o , 90o ), which is indicative of random oscillations. We further deﬁne the oscillatory orientation of cell J as
(J )
θaniso = arctan (J )
(J )
y(J ) x(J )
,
(40)
where x and y are the projective components of the area oscillation amplitude of cell J on the x and y axes, respectively. It is found that the distribution of oscillatory orientations is peaked and symmetric around 45o , suggesting that the oscillations in the whole tissue are isotropic (Fig. 11(b)). The supracellular oscillatory pattern shown in Fig. 11(a) manifests an intrinsic spatial correlation of collective cell oscillations. To identify the spatial correlation (i.e., wavelength) in the supracellular stringlike patterns, we deﬁne the spatial
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Fig. 12. Mechanical stretches mediate the collective cellular dynamics in an oscillating cell monolayer. (a) The averaged areal oscillation amplitude AD versus mechanical stretches. Here, four kinds of stretches are examined, including uniaxial stretch (λy = 1), equiaxial stretch (λx = λy ), isometric stretch (λx λy = 1), and nonequiaxial stretch (λy = (λx − 1 )/2 + 1). (b) Dynamic evolution of area, Rho activity, ROCK activity and MLC activity of an arbitrary cell (J ) for different stretches. (d) Comparison in the monolayer sustained an increasing stretch. (c) Comparison of the distribution of oscillatory orientation θaniso (J ) of the distribution of oscillatory alignment θalign for different stretches. (e) The mean oscillatory orientation θ aniso and the mean oscillatory alignment θ align (J ) (J ) J and θalign = θalign J . versus stretch λx , with λy = 1/λx . Here, θaniso = θaniso
correlation function of oscillatory phases as
Cϕ (r ) = cos
ϕA(I ) − ϕA(J )


r −r < rI −rJ