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Theoretical model for morphogenesis and cell sorting in Dictyostelium discoideum
Physica D 126 (1999) 189–200
Theoretical model for morphogenesis and cell sorting in Dictyostelium discoideum T. Umeda a,∗ , K. Inouye b a
b
Kobe University of Mercantile Marine, Higashinada, Kobe 658-0022, Japan Department of Botany, Division of Biological Science, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Received 10 November 1997; received in revised form 2 October 1998; accepted 7 October 1998 Communicated by Y. Kuramoto
Abstract The morphogenetic movement and cell sorting in cell aggregates from the mound stage to the migrating slug stage of the cellular slime mold Dictyostelium discoideum were studied using a mathematical model. The model postulates that the motive force generated by the cells is in equilibrium with the internal pressure and mechanical resistance. The moving boundary problem derived from the force balance equation and the continuity equation has stationary solutions in which the aggregate takes the shape of a spheroid (or an ellipse in two-dimensional space) with the pacemaker at one of its foci, moving at a constant speed. Numerical calculations in two-dimensional space showed that an irregularly shaped aggregate changes its shape to become an ellipse as it moves. Cell aggregates consisting of two cell types differing in motive force exhibit cell sorting and become elongated, suggesting the importance of prestalk/prespore differentiation in the morphogenesis of Dictyostelium. c 1999 Elsevier Science B.V. All rights reserved.
PACS: 87.10.+e; 87.22.-q; 87.45.k Keywords: Cellular slime mold; Slug; Movement; Model
1. Introduction Organized cell movements in cell masses is one of the most important processes in morphogenesis and pattern formation of multicellular organisms. With its very simple structures, the aggregate of the cellular slime mold Dictyostelium discoideum, which undergoes a series of highly organized movements, is an ideal model system for theoretical as well as experimental studies of the mechanism of morphogenesis. ∗
Corresponding author.; e-mail: [email protected]
Cells of D. discoideum typically exist as independent soil-inhabiting amoebae that multiply by binary fission. Upon exhaustion of food supply, the amoebae switch from the growth phase to developmental phase in which they aggregate to form a multicellular structure. During aggregation, a cell or a small group of cells at the center gives off cyclic AMP pulses. The other cells relay the cyclic AMP signal outwards while moving towards the signal source. These processes result in aggregation of amoebae from a wide area to form a mound of cells, which then elongates vertically by forming on its apex a small protrusion which grows upwards so that the entire mass becomes like a
c 0167-2789/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 8 ) 0 0 2 7 3 - 5
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finger. In moist conditions, this cell mass usually lies down on the substratum and crawls in a manner resembling the movement of a garden slug (thus called a “migrating slug”). There is ample evidence indicating that the movement of individual cells within the slug are guided by chemotactic signals, possibly of cAMP, propagating from the tip of the slug towards its rear by the signal relay mechanism, a situation similar to aggregation [1–4] (but [5], see Section 5). The migrating slug consists of two major cell types; approximately a quarter are prestalk cells, and the front part of the slug is occupied by these cells. The rest are prespore cells which constitute the majority in the main body of the slug except the anterior prestalk region. The slug eventually forms a fruiting body consisting of a mass of spores and a supporting stalk. The migrating movement of the slug has been a subject of experimental and theoretical studies over decades (reviewed in [6]). While some hypotheses emphasize a role of the extracellular matrix in slug migration [7–9], others consider active movement of the cells to be the main source of the driving force [10– 15]. Measurements of the motive force of migrating slugs have revealed that the motive force of the entire slug is proportional to its volume, suggesting that all cells, not only those in the outermost layer, of the slug contribute to its movement [12,16]. It was also shown that the anterior prestalk cells exert much greater motive force than posterior prespore cells [12]. Based on these observations, we have proposed theoretical models for slug movement in which the sum of the motive force exerted by individual cells are in equilibrium with the counteracting resistance due to continuous expansion of the surface sheath and viscoelastic properties of the cells [17,18]. With these models, we showed that some of the basic properties of the migrating slug, such as its shape, velocity, and prestalk-prespore pattern, can be explained on a mechanical basis. It remained to be clarified, however, how the elongated shape of a slug and its anteroposterior pattern of cell differentiation are generated from a hemispherical mound of cells without distinct pattern. In the present study, we focused on the dynamical aspect of this process, namely the changes in the shape of a cell aggregate and cell sorting during the formation of an elongated slug. The cell mass was regarded as a continuum of which each volume element is capable of exerting motive force against the intrin-
sic resistance of cells. Propagation of the chemotactic signals from the “organizing center” was incorporated in a simplified way. The movement of the aggregate was then expressed as a moving boundary problem represented by a set of the force balance equation and the continuity equation of the cell mass. Analysis and numerical calculations of the moving boundary problem showed that the essential features of the morphogenetic movements from the mound through slug migration are expressed within a unified framework. Furthermore, differentiation of prestalk cells was shown to be essential in the formation of a tip and subsequent elongation leading to slug migration.
2. Formulation of the model 2.1. Basic model Our model is based on the hypothesis, first proposed by Shaffer [11], that the cell moves by extending its front ahead and retracting its back with its relatively solid lateral side remaining stationary with respect to the ground (Fig. 1(a)). This hypothesis is based on the behavior of cell surface markers during cell locomotion [11], and is also consistent with the current theory for the mechanism of amoeboid movement, namely, the interface between the rigid surface layer of the cytoplasm (which is in the “gel” state, and called the “cortex” of the cell) and the inner fluid (which is in the “sol” state and called the “cytosol”) is the site of force generation, and gel–sol interconversion takes place at the front and rear of the cell so that the solid lateral side of the cell is continuously formed in its leading edge while it is absorbed from its trailing end [19]. The solid lateral surface of a cell in a threedimensional cell mass can then serve as a substratum for the neighboring cells (Fig. 1(b)). For example, while the cells constituting the first layer of the cell mass move actively by getting traction from the surface sheath, their lateral surface remains stationary with respect to the surface sheath. The cells of the second layer can then obtain traction equally efficiently from the surface of the cells in the first layer, because the latter’s solid and stationary lateral surface serves as the substratum for their movement. Likewise, the cells of the subsequent layers can also move actively by getting traction from the lateral surface of the cells
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Based on these considerations, the following mathematical formulation can be derived (see also [17,18]). We consider a cell aggregate as a continuum and denote by v(x, t) its velocity at position x and time t. Each volume element of the continuum actively moves with the force generated by individual cells contained in the volume. We assume that the propulsion force per unit volume of the continuum can be written as F = f − av,
(1)
where f is the “motive force” of cell mass per unit volume. The second term of the right-hand side of Eq. (1) represents the intrinsic resistance due to cytoplasmic streaming which takes place inside the cell and the renewing of the lateral sides, where the drag coefficient a is constant. Since F is in equilibrium with the force exerted by surrounding volume elements, the following force balance equation holds for each unit volume of the continuum: f − av − ∇p = 0, Fig. 1. (a) Schematic diagram for the movement of a cell. The lateral side of the cell, which is the cytoplasm in the “gel” state, remains stationary relative to the ground while the cell moves. The inner cytoplasm which is in the “sol” state flows forward by getting traction against the solid gel. Gel–sol interconversion takes place at the front and rear of the cell. (b) The movement of cells within a three-dimensional cell mass. The lateral surfaces of all the cells are stationary relative to the ground so that they can serve as a substratum for the neighboring cells to move on.
in the previous layers. In this way, all the cells can exert motive force equally efficiently irrespective of whether they are in the surface layer of the aggregate or inside. In other words, the lateral sides (the cell surface and the cortex) of all the cells of the cell mass form a kind of meshwork connected to the surface sheath which in turn is firmly attached to the external substratum. The cytosol of individual cells moves forward obtaining traction from this meshwork while renewing the meshwork at its front end. The meshwork on the other hand receives all the forces exerted by the cytosol and transmits them to the external substratum. Consequently, the motive force of a cell mass equals to the sum of the motive force generated by all the cells. This is supported by the observation that the motive force generated by a migrating slug is proportional to its volume, but not to its cross-section or surface area [12,16].
(2)
where p(x, t) is the hydrostatic pressure. Because of the small size and speed involved in the phenomena under consideration, the effects of gravity and inertial force can be neglected. Note that unlike standard equations for viscous fluid, Eq (2) does not contain terms that are dependent on the gradient of the velocity. This is because the lateral sides of two adjacent cells always remain attached and there is no velocity difference irrespective of the velocities of the cells (i.e. the velocities of the cytosols). Therefore the difference in the cell velocity between adjacent cells causes no tangential stress. The second equation of the model is the conservation equation for the cell mass. Since the cell mass is incompressible, the velocity v must satisfy ∇ · v = 0.
(3)
Eqs. (2) and (3) are the set of basic equations that govern the movement of cell masses. 2.2. Free boundary problem We consider an idealized cell aggregate consisting of a single cell type and formulate its movement as a free boundary problem. Suppose a freely moving cell aggregate which occupies a domain Ω(t) at time t. Eqs. (2) and (3) then
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hold in Ω(t). Since the boundary ∂Ω of the aggregate moves with the cell mass, its normal velocity u satisfies the following equation: u=v·ν
on ∂Ω(t),
(4)
where ν is the unit outward normal to ∂Ω. For the boundary value of p, we assume free movement of the boundary without considering the surface tension and the effect of the surface sheath surrounding the aggregate for simplicity, and adopt the following boundary condition: p = p0
on ∂Ω(t),
(5)
where p0 is the atmospheric pressure. We next specify the motive force f(x, t) in Eq. (2). We assume that a chemotactic signal is propagated within the aggregate and that the amoebae exert motive force in the direction the signal comes from. In aggregation of D. discoideum, propagation of cyclic AMP signals over the aggregation field can be described as wave propagation on an excitable medium [20,21], and accumulating evidence shows that the same signal controls the multicellular development under normal conditions [1–4] (see Section 5 for morphogenesis in mutants lacking cAMP). Here we adopt a simplified model based on the following assumptions: First, though both concentric and spiral wavefronts are possible on an excitable media, we consider only the case that the waves are concentrically propagated from the pacemaker. Consequently, rotating movement of cells which are often observed in mounds and slugs, are not represented in the model. Secondly, we assume that the speed of the signal propagation is constant and sufficiently fast compared with the speed of cell movement. The validity of these assumptions is discussed later. In its simplified form, the presence of a single pacemaker cell (or a pacemaker region) within the aggregate is postulated and its position at time t is denoted by xp (t). The chemotactic signal emanating from xp is propagated throughout the aggregate. If the speed of the signal propagation is constant and sufficiently fast, the wavefronts form part of spheres in Ω with its center at xp . Accordingly, the motive force at each position x is directional to xp . On the other hand, the magnitude of the motive force at each position will vary periodically if the signal is periodical. However, if averaged over time sufficiently longer than the sig-
nal period, we can regard it as constant. Therefore, the motive force can be written in the following form: f(x, t) = f
xp (t) − x , |xp (t) − x|
(6)
where f is constant. Note that f is not defined at xp in Eq. (6) so that Eq. (2) does not hold at this position. Nevertheless, the velocity v can be uniquely defined by assuming continuity at x = xp . Finally, we assume that the pacemaker moves at the same speed as the surrounding cells, i.e. the position of the pacemaker is determined by the following equation: dxp = v(xp , t). dt
(7)
3. Movement of cell aggregate If we introduce a variable p f φ = − |xp − x| − , a a
(8)
Eqs. (2) and (3) can be rewritten using (6) as v = ∇φ
in Ω
(9)
and ∇ 2φ = 0
in Ω,
(10)
respectively. The boundary condition (5) is rewritten as p0 f φ = − |xp − x| − a a
on ∂Ω.
(11)
Therefore, the velocity v at any point in the aggregate is obtained by solving Laplace equation. Eqs. (4) and (7) then determine the movement of the boundary and the pacemaker, respectively. We first look for the stationary solution, in which the boundary of the aggregate does not change its shape and moves at a constant speed U. The normal velocity of the boundary is then given by u = U · ν. Substituting this equation and Eq. (9) into (4), we have the following boundary condition for φ: ∂φ = U · ν. ∂ν
(12)
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Putting φ(x, t) = φ(ˆx), where xˆ = x − Ut, we obtain the solution of Eq. (10) with the condition (12) as φ(ˆx) = U · xˆ + c,
(13)
where c is a constant. With this, Eq. (9) yields v=U
in Ω.
(14)
Therefore, all unit volumes in Ω move with equal velocity and no convection flow occurs within the aggregate. Substituting (13) for the left-hand side of (11), we obtain the following equation which determines the boundary shape of the moving aggregate: p0 f |ˆxp − xˆ | + U · xˆ + + c = 0. a a
(15)
The shape given by (15) is the revolving surface of a conic section whose eccentricity is η=
a|U| , f
(16)
and one of its foci coincides the position of the pacemaker. Since the aggregate has a finite volume, the boundary shape is a spheroid where η satisfies 0 ≤ η < 1. If the movement of the aggregate is restricted in two-dimensional space, the boundary shape is an ellipse with the pacemaker at one of its foci. Eq. (16) indicates that the velocity does not depend on the volume of Ω but depends on the relative position of the pacemaker in the aggregate. Fig. 2 illustrates some examples of cell aggregate at steady state, showing the relationship between the position of the pacemaker, the shape of the aggregate, and its speed of movement. Arrows represent the velocity U of the aggregate. As the pacemaker approaches the boundary (i.e. η becoming larger), the aggregate becomes more elongated and moves faster. If η approximates 1, the aggregate elongates infinitely and |U| approaches to f/a. Conversely, if the pacemaker is located at the center of the aggregate (i.e. η = 0), the boundary of the aggregate becomes a sphere and the aggregate does not move. To examine whether the aggregate starting with an arbitrary shape becomes the above stationary shape, we numerically solved the moving boundary problem represented by Eqs. (4),(7), and (9)–(11) (see Appendix A for the detail of calculation). To avoid the computational difficulty, we considered the model in
Fig. 2. Shapes and the corresponding velocities of moving cell aggregates. The solid circles indicate the location of the pacemaker and the arrows indicate the velocity of the aggregate. The shape of the aggregate is a three-dimensional spheroid with the pacemaker at one of its foci. The eccentricities of the spheroids shown in figure are 0.3, 0.7 and 0.9. Aggregates with their pacemaker close to the edge are more elongated and move faster.
Fig. 3. Numerical simulation of the morphogenetic movement of cell aggregate in two-dimensional space. The initial shape of the aggregate is irregular with its pacemaker positioned near the boundary. The aggregate changes its shape as it moves and then finally becomes an ellipse moving at a constant speed. Contour lines (t = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2) are obtained by solving Eqs. (4),(7), and (9)–(11), where parameter f/a = 1.
two-dimensional space, which corresponds to the situation that cell aggregate is confined in a region between flat plates. Fig. 3 shows a result, in which the initial boundary shape is irregular with the pacemaker positioned near the boundary. The aggregate changes its shape as it moves, and finally becomes an ellipse
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moving with a constant velocity. Similar results were observed in other numerical calculations with various initial shapes, though the eccentricity of the final shape varies depending on the initial boundary shape and the initial position of the pacemaker.
4. Cell sorting and slug movement So far we have assumed that the aggregate consists of a single cell type. However, there are at least two types of cells differing in their motive force in Dictyostelium cell aggregates. In this section, we consider the effect of the heterogeneity of cell types on the movement of aggregates. Let us consider a cell aggregate consisting of two types of cells, prestalk and prespore cells. Since cell type conversion is scarce under normal conditions [22], the cell number of each cell type is considered invariant. We assume that the two cell types differ in the magnitude of their motive force but both motive forces are directed to the pacemaker, i.e. the motive forces generated by prestalk and prespore cells, f1 and f2 , are given by fi = fi
xp − x |xp − x|
(i = 1, 2),
(17)
∇ ·v =0
in Ω,
∂n + v · ∇n = −∇ · J ∂t
in Ω,
(18) (19)
in Ω
(20)
and dxp = v(xp , t). dt
(21)
Eq. (20) is the conservation equation for the prestalk fraction, where the flux J is given by J=
f1 − f2 (1 − n)n − D∇n. a
J·ν =0
(22)
on ∂Ω,
(23)
which represents the impermeability of prestalk and prespore cells at the boundary. Now the movement of the mixed aggregate can be obtained by solving Eqs. (18)–(21) with the conditions (4),(5) and (23). However, we here calculate a mathematically simpler system instead of solving the full moving boundary problem. We first consider a cell aggregate with fixed boundary, in order to examine how the cell distribution changes in time within the aggregate. The boundary conditions (4) and (5) are then replaced by v·ν =0
where fi ’s are constant. Since the motive force of prestalk cells is greater than that of prespore cells, f1 > f2 . Let v(x, t) be the mass-averaged velocity of the two types of cells and n(x, t) the fraction of prestalk cells at position x and time t. Then the movement of the mixed cell mass is described by the following equations [18]: f1 n + f2 (1 − n) − av − ∇p = 0
The first term of the right-hand side of (22) represents the differential movement due to the difference in motive force between the two cell types. The second term is the diffusion term that comes from the random movement of cells and the coefficient D is constant. In addition to the differential motive force, Umeda [23] considered the effect of differential cell adhesion on the change of n. However, we do not consider the differential adhesion in this study to simplify the model. The boundary conditions (4) and (5) are unchanged in this case, but the following condition should be added for Eq. (20):
on ∂Ω
(24)
and ∂p = [f1 n + f2 (1 − n)] · ν ∂ν
on ∂Ω,
(25)
respectively. We numerically solved the set of equations (18)– (21) with boundary conditions (23)–(25) in twodimensional space (see Appendix B for the method of calculation). The boundary shape was assumed to be a square for simplicity. Fig. 4 shows a result, where the shading expresses the fraction of prestalk cells n and the open circle represents the location of the pacemaker xp . By the time t = 0.6, initially uniformly distributed prestalk cells sort out to cluster around the pacemaker, forming a well-defined prestalk region. Then the prestalk region, along with the pacemaker, gradually moves towards the periphery of the aggregate. This movement is caused by the deviation of the initial pacemaker position from the center; initially the position of the pacemaker is off the center to its right so that more prestalk cells would gather
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Fig. 4. Cell sorting within the fixed square of size 1. The graded shading shows the fraction of prestalk cells present in each position (a gray scale of 10% intervals) and the open circle represents the position of the pacemaker. During t = 0 to t = 0.6, prestalk cells sort out to form a prestalk region around the pacemaker. After the prestalk region is formed, it gradually moves to the edge of the aggregate, along with the pacemaker. The figures are obtained by solving Eqs. (18)–(21) with boundary conditions (23) and (24). Parameters are f1 /a = 1.5, f2 /a = 0.5 and D = 0.01. Initial distribution of prestalk fraction is uniformly 0.2 and initial position of the pacemaker is (0.6,0.55).
to the left of the pacemaker when the prestalk region is formed. This asymmetry of cell distribution brings about rightward movements of the pacemaker and the prestalk region. We next consider the case that cells are completely sorted out within an aggregate before the aggregate moves. Though experimental observations show that cell sorting and the movement of aggregate simultaneously proceed, we use this assumption for mathematical simplicity to solve the moving boundary problem. Then the flux of prestalk fraction vanishes everywhere in the moving aggregate, i.e. J=0
in Ω.
(26)
Substituting (17) and (22) into (26) and integrating it with respect to |xp − x|, we have the following distribution of prestalk cells: n(x, t) =
1 . 1+exp[((f1 −f2 )/aD)(|xp −x|−r0 (t))]
(27)
The function r0 (t) in (27), which represents the distance from the pacemaker to the position where n is 0.5, is determined by the conservation of the total number of prestalk cells. Fig 5 depicts typical prestalk distributions given by (27). Prestalk cells sort out toward the pacemaker. The boundary of the two types of cells becomes more clear as (f1 − f2 )/aD increases.
Fig. 5. Stationary distribution of the prestalk cells in the aggregate. The horizontal axis represents the distance from the pacemaker. Prestalk cells sort out toward the pacemaker, where the boundary of the two types of cells become more clear as (f1 − f2 )/aD increases. The parameter (f1 − f2 )/aD is: (a) 10.0; (b) 20.0; (c) 100.0.
If we substitute (27) into (18) and couple it with (19) and (21), we can use the same method as in Section 3 to solve the moving boundary problem for the mixed cell mass (see Appendix A). Fig. 6 shows a typical result of the numerical calculation, where the solid circle indicates the position of the pacemaker and the solid curve within the aggregate represents the curve where n = 0.5. Initial shape of the aggregate is a circle with prestalk region being located at the edge of the aggregate. As time advances, the prestalk region is projected from the aggregate. Then the
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Fig. 6. Movement of a cell aggregate which comprises prestalk and prespore regions. The prestalk region projects from the round aggregate while the entire aggregate moves forward. As the aggregate moves, it is gradually elongated. These figures are obtained by solving Eqs. (18),(19),(21) and (27). Parameters are f1 /a = 1.5, f2 /a = 0.5 and D = 0.01. Initial shape of the aggregate is a circle of diameter 1 and the initial position of the pacemaker is 0.37 from the center. The area of the prestalk region is 20% of the aggregate.
aggregate gradually narrows and lengthens its shape, and finally moves at a constant speed keeping a spatial pattern of two cell types. We can see the model reproduces the basic feature of the real slug formation and slug movement, except that the final shape of the aggregate has thicker prestalk region than real slugs. 5. Discussion We first summarize the main assumptions of the present model: (i) the movements of cell aggregates, and cell sorting therein, occur as a result of active cell movement; (ii) the movement of individual cells is controlled by chemotactic signals propagated from a single “pacemaker”; (iii) the boundary of the aggregate has no effect on the movement of cells. In assumption (i), we considered that cells in the aggregate move as a “stationary surface model”, in which the motive force of the whole aggregate is simply given
as the accumulation of the motive forces generated by individual cells. Besides the present model, several models have been proposed for the movement of an aggregate. Williams et al. [9] proposed a “squeeze–pull model”, which involves circumferential cells squeezing forward a cellular core, followed by pulling up of the rear. Odell and Bonner [13] proposed a hydrodynamical model, in which cells turn the “membrane tractor” at a speed that depends on the cell position, and the difference in the membrane velocity among adjacent cells is responsible for generating thrust of cells. However, the fact that cells that have become non-motile within a slug are stationary with respect to the substratum [24] seems to support our model, because Williams’s model predicts forward movement of non-motile cells and the model of Odell–Bonners predicts backward movement of them. The fact that the motive force of the entire slug is proportional to its volume [12,16] also supports our model. Assumption (ii) is known to hold during cell aggregation [25], and there is evidence showing that cell movements within cell aggregate are also under the control of propagating signals [1–4,26]. During cell aggregation and in mound stage, both concentric and spiral wavefronts are observed [27,28]. However, we did not consider the spiral wave propagation for the following reasons. First, not all slugs exhibit rotating cell movements while undergoing morphogenetic movements indistinguishable from those accompanying rotating cell movements, suggesting that the rotating cell movements are not an essential part of the formation and migration of a slug. Secondly, the direction of the motive force generated by the cells would be virtually the same as that driven by concentric waves throughout the aggregate except the vicinity of the center of spiral signals. Even in the vicinity of the center where the diagonal components of the motive force prevail, they would not affect significantly the shape of the cell aggregate and cell sorting pattern observed over timescales sufficiently larger than the rotation period of spiral waves. Thus, most of the conclusion of the present study will be unaffected whether the cell aggregate is controlled by concentric or spiral waves. For the rotating movement of cells, work is under way to incorporate spiral wave propagation in our model. Instead of considering the evolution equation of the cyclic AMP concentration, we adopted a simplified version of the model, i.e. we assumed instantaneous
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transmission of signals from the pacemaker and generation of constant motive force by cells in the direction of the signal source. These approximations dramatically facilitate numerical calculation and enables one to conduct simulations involving cell sorting and shape changes. As regards the quickness of the signal propagation, the speed of wave propagation is over 10-fold larger than the speed of cell movements during aggregation [29] and most likely within migrating slugs as well [2]. Furthermore, changes in the relative position of the pacemaker is very slow compared with the speed of cell movements seen during cell sorting, as shown in the numerical simulations, so any errors due to this assumption must be small. Although the individual cells show periodical motion during aggregation [30] and within slugs [1], the assumption that motive force is constant is good approximation for describing phenomena occurring over the developmental timescale. On the other hand, we ignored the effect of unevenness of wave propagation. If the speed of signal varies with cell types or is changed due to boundary effects, the wave would be refracted so that the direction of the motive force might be affected. The magnitude of the motive force may also vary due to the effects of the local factors such as oxygen tension and ammonia concentration. In order to take these effects into account, it will be necessary to formulate the evolution equation of the cyclic AMP concentration and to assume chemotactic movements of cells against cyclic AMP gradient. Some recent models deal with the Dictyostelium morphogenesis using such approaches [14,15,31]. The recent results by Wang and Kuspa [5] indicate that mutants lacking the enzyme adenylate cyclase which produces cAMP can complete its morphogenesis if cAMP-dependent protein kinase (PKA) is made constitutively active. Although this observation suggests that cAMP is dispensable for the Dictyostelium morphogenesis, a body of evidence points to the presence of propagating signals, most likely of cAMP, that organize the morphogenetic movements of cells in later development, at least in normal situations [4,32]. Furthermore, mutants lacking the cell surface cAMP receptor subtype CAR2 which is expressed during the multicellular stages, as well as strains overexpressing extracellular phosphodiesterase during the multicellular stages (resulting in diminished levels of extracellular cAMP), are deficient in tip formation and sub-
197
sequent morphogenesis [33,34], indicating the importance of extracellular cAMP during this period. This, however, does not exclude the possibility of the presence of chemotactic agents other than cAMP that act during the later development, and our model is open to that possibility. Assumption (iii) was adopted solely for the sake of simplicity of the model. In reality, a slug is covered with a mucous film called surface sheath, which without doubt constrains the movement of the slug, and its omission is likely to be the cause of the unrealistic results in some of the simulations (see below). Surface tension is another factor that would affect the movement and shape of the aggregate [18]. The results of the model based on these assumptions showed several properties of the cell aggregate that agree with experimental observations, i.e. the elongated shape of a migrating slug, positive correlation between the length and speed for slugs with the same volume, and sorting-out of prestalk cells towards the anterior of the slug. These agreements support the view that the morphogenetic movement and cell sorting in aggregates are caused by active cell movement under the control of propagating signals. Although numerical calculations were performed in twodimensional space, the essence of these results will be unchanged in three-dimensional space. Furthermore, the results shown in Figs. 4 and 6 can be seen as simulating the process of translocation of the pacemaker from the central part of the aggregate to the anterior tip while the shape of the entire aggregate becomes elongated. These results demonstrate the essential role of cell differentiation in the formation of the tip and subsequent elongation of the aggregate; if the aggregate consisted of single cell type, the pacemaker would stand still at the initial position after cell aggregation and the entire aggregate would keep its rounded shape. Although the present study deals with only two cell types, the model can easily be extended to cell aggregates containing three or more cell types. Prestalk cells are now known to consist of at least three subtypes [35]. They differ from each other in the ability of chemotactic movement [36,37], which seems to suggest their motive force within a slug might also be different. In cell aggregates consisting of more than two cell types differing in the motive force, the cells would be sorted out according to their motive force.
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Then the translocation of the pacemaker as seen in Fig. 4 will be accelerated, because each cell type region would move to a peripheral position of the surrounding cell type region, and a more elongated shape than the ones obtained with two cell types would result. Recent studies revealed that changes in spatial distribution of the prestalk sub-types occur before slug formation [37]. Such changes in spatial distribution of the prestalk sub-types may be caused by changes in the relative magnitudes of the motive force among the prestalk sub-types and prespore cells. On the other hand, some of the results do not agree with the known facts; the speed of slugs with the same length:width ratio is independent of their volumes (in reality bigger ones migrate faster), and the anterior prestalk region of a slug becomes thicker than real. The former disagreement is likely due to the negligence of the effects of the surface sheath. Taking into account the resistance of the surface sheath at the front of the slug, the force balance would favor among slugs with the same length:width ratio bigger ones for faster migration, because the total motive force of the slug is proportional to its volume while the resistance of the surface is proportional to the surface area of the front part of the slug (see [17]). The latter disagreement may also be due to the negligence of the surface effects. The rigidity and thickness of the surface sheath decrease towards the tip of a slug, which may be the cause of the tapering shape of the slug [38]. To analyze these effects, more realistic model considering the mechanical properties of the surface sheath will be necessary. In the present study, we focused on the morphogenesis and the early stage of slug migration. However, with regard to the movement of a slug, there remain many unsolved problems, e.g. the mechanism of its turning behavior in response to external stimuli such as light and temperature gradient. The mechanism of fruiting body formation is another interesting problem. Approaches from mechanical viewpoints like the present study will be necessary for thorough understanding of these interesting phenomena.
Acknowledgements T.U. thanks K. Maruo and N. Shigesada for helpful discussions and comments.
Fig. A.1. Parametrization of the boundary shape of a two-dimensional aggregate. vn (s) and vs (s), respectively, are the normal and the tangential component of the velocity v on the boundary.
Appendix A We consider Eqs. (4),(7) and (9)–(11) in twodimensional space. For a fixed time t, we express the velocity in the domain Ω(t) as a complex function f (z) = vx − ivy ,
(A.1)
where z = x + iy.
(A.2)
Let z(s) be a parametric representation of the boundary ∂Ω, where s is the arc length of the boundary, and let vn (s) and vs (s) be the normal and the tangential component of the velocity v on ∂Ω, respectively (Fig. A.1). Then, for z ∈ Ω, f (z) can be expressed as I vn (s) − ivs (s) 1 ds, (A.3) f (z) = 2π ∂Ω z(s) − z since Eqs. (9) and (10) indicate that f (z) is an analytic function [39]. If z is on ∂Ω, f (z) = (vn − ivs )e−iθ I 1 vn (s) − ivs (s) = ds, π ∂Ω z(s) − z
(A.4)
where θ is the angle between the x-axis and the normal to ∂Ω, and the integral means the Cauchy principal value. For the tangential component of the velocity, Eqs. (9) and (11) yield vs (s) =
f sin (θ (s) − ϕ(s)), a
(A.5)
T. Umeda, K. Inouye / Physica D 126 (1999) 189–200
where ϕ is the angle between the x-axis and the line connecting z(s) and xp (see Fig. A.1). The normal component vn is then obtained by solving the boundary integral equation (A.4). For each iteration time step tk , we divide the boundary into finite elements and calculate the velocities of the boundary points using (A.4) and (A.5), and the velocity of the pacemaker using (A.3). Then the boundary points and the pacemaker are displaced by using the following equation: x(k+1) = x(k) + v(k) 1t.
(A.6)
If the aggregate consists of two types of cells, f1 n + f2 (1 − n) sin (θ(s) − ϕ(s)) (A.7) a is used instead of (A.5), where n is given by (27). Function r0 (t) in (27) is determined by the constraint that the area of the prestalk region (n > 0.5) is invariant.
vs (s) =
Appendix B We consider Eqs. (18)–(21) with boundary conditions (23)–(25) in two-dimensional space. Eq. (19) implicates that there exists a function 9 which satisfies vx =
∂9 , ∂y
vy = −
∂9 . ∂x
(B.1)
By using this, Eqs. (18) and (20) are transformed as ∇ 2 9 + ∇n ×
f1 − f2 = 0, a
(B.2)
and f1 −f2 ∂n +∇n×∇9=∇ · − (1−n)n+D∇n . (B.3) ∂t a Boundary condition (24) becomes 9 = constant.
(B.4)
For each time step tk , equations were solved on 49 × 49 grid as follows: (i) 9 (k) is calculated from (B.2) and boundary condition (B.4). (ii) n(k+1) is calculated from (B.3) and boundary condition (23) by using an explicit method. (iii) The location of the pacemaker is calculated using xp(k+1) = xp(k) +
∂9 (k) 1t, ∂y
yp(k+1) = yp(k) −
∂9 (k) 1t. ∂x
199
(B.5)
References [1] F. Siegert, C.J. Weijer, Physica D 49 (1991) 224. [2] F. Siegert, C.J. Weijer, Proc. Natl. Acad. Sci. U.S.A. 89 (1992) 6433. [3] T. Bretschneider, F. Siegert, C.J. Weijer, Proc. Natl. Acad. Sci. U.S.A. 92 (1995) 4387. [4] F. Siegert, C.J. Weijer, in: Y. Maeda, K. Inouye, I. Takeuchi (Eds.), Dictyostelium A Model System for Cell and Developmental Biology, Universal Academy Press, Tokyo, Japan, 1997, p. 425. [5] B. Wang, A. Kuspa, Science 277 (1997) 251. [6] J.T. Bonner, FEMS Microbiol. Lett. 120 (1994) 1. [7] D.R. Garrod, J. Cell Sci. 4 (1969) 781. [8] W.F. Loomis, Nature New Biol. 240 (1972) 6. [9] K.L. Williams, P.H. Vardy, L.A. Segel, Bioessays 6 (1986) 148. [10] J.T. Bonner, P.G. Koontz Jr., D. Paton, Mycologia 45 (1953) 235. [11] B.M. Shaffer, Adv. Morphogenesis 2 (1962) 109. [12] K. Inouye, I. Takeuchi, J. Cell. Sci. 41 (1980) 53. [13] G.M. Odell, J.T. Bonner, Phil. Trans. Roy. Soc. 312 (1986) 487. [14] N.J. Savill, P. Hogeweg, J. Theoret. Biol. 184 (1997) 229. [15] B. Vasiev, F. Siegert, C.J. Weijer, J. Theoret. Biol. 184 (1997) 441. [16] K. Inouye, Protoplasma 121 (1984) 171. [17] K. Inouye, I. Takeuchi, Protoplasma 99 (1979) 289. [18] T. Umeda, Bull. Math. Biol. 51 (1989) 485. [19] R.L. Debiasio, G.M. Larocca, P.L. Post, D.L. Taylor, Mol. Biol. Cell 7 (1996) 1259. [20] A.J. Durston, J. Theoret. Biol. 42 (1973) 483. [21] J.J. Tyson, K.A. Alexander, V.S. Manoranjan, J.D. Murray, Physica D 34 (1989) 193. [22] Y. Akiyama, K. Inouye, Differentiation 35 (1987) 83. [23] T. Umeda, Bull. Math. Biol. 55 (1994) 451. [24] J. Sternfeld, Roux Arch. Dev. Biol. 201 (1992) 354. [25] P. Devreotes, Science 245 (1989) 1054. [26] J. Rubin, A. Robertson, J. Embryol. Exp. Morphol. 33 (1975) 227. [27] J.D. Gross, M.J. Peacey, D.J. Trevan, J. Cell Sci. 22 (1976) 645. [28] F. Siegert, C.J. Weijer, Current Biol. 5 (1995) 937. [29] F. Siegert, C.J. Weijer, J. Cell Sci. 93 (1989) 325. [30] F. Alcantara, M. Monk, J. Gen. Microbiol. 85 (1974) 321. [31] H. Levine, L. Tsimring, D. Kessler, Physica D 106 (1997) 375. [32] I.V. Wijk, P. Schaap, in: Y. Maeda, K. Inouye, I. Takeuchi (Eds.), Dictyostelium A Model System for Cell and Developmental Biology, Universal Academy Press, Tokyo, Japan, 1997, p. 145. [33] C.L. Saxe III, G.T. Ginsburg, J.M. Louis, R. Johnson, P.N. Devreotes, A.R. Kimmel, Genes and Dev. 7 (1993) 262.
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[34] D. Trayor, R.H. Kessin, J.G. Williams, Proc. Natl. Acad. Sci. U.S.A. 89 (1992) 8303. [35] J. Williams, in: Y. Maeda, K. Inouye, I. Takeuchi (Eds.), Dictyostelium A Model System for Cell and Developmental Biology, Universal Academy Press, Tokyo, Japan, 1997, p. 293.
[36] T. Abe, A. Early, F. Siegert, C. Weijer, J. Williams, Cell 77 (1994) 687. [37] A. Early, T. Abe, J. Williams, Cell 83 (1995) 91. [38] P.A. Farnsworth, W.F. Loomis, Dev. Biol. 46 (1975) 349. [39] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, translated by J.B. Sykes, W.H. Reid, Pergamon Press, Oxford, 1959.
Theoretical model for morphogenesis and cell sorting in Dictyostelium discoideum T. Umeda a,∗ , K. Inouye b a
b
Kobe University of Mercantile Marine, Higashinada, Kobe 658-0022, Japan Department of Botany, Division of Biological Science, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Received 10 November 1997; received in revised form 2 October 1998; accepted 7 October 1998 Communicated by Y. Kuramoto
Abstract The morphogenetic movement and cell sorting in cell aggregates from the mound stage to the migrating slug stage of the cellular slime mold Dictyostelium discoideum were studied using a mathematical model. The model postulates that the motive force generated by the cells is in equilibrium with the internal pressure and mechanical resistance. The moving boundary problem derived from the force balance equation and the continuity equation has stationary solutions in which the aggregate takes the shape of a spheroid (or an ellipse in two-dimensional space) with the pacemaker at one of its foci, moving at a constant speed. Numerical calculations in two-dimensional space showed that an irregularly shaped aggregate changes its shape to become an ellipse as it moves. Cell aggregates consisting of two cell types differing in motive force exhibit cell sorting and become elongated, suggesting the importance of prestalk/prespore differentiation in the morphogenesis of Dictyostelium. c 1999 Elsevier Science B.V. All rights reserved.
PACS: 87.10.+e; 87.22.-q; 87.45.k Keywords: Cellular slime mold; Slug; Movement; Model
1. Introduction Organized cell movements in cell masses is one of the most important processes in morphogenesis and pattern formation of multicellular organisms. With its very simple structures, the aggregate of the cellular slime mold Dictyostelium discoideum, which undergoes a series of highly organized movements, is an ideal model system for theoretical as well as experimental studies of the mechanism of morphogenesis. ∗
Corresponding author.; e-mail: [email protected]
Cells of D. discoideum typically exist as independent soil-inhabiting amoebae that multiply by binary fission. Upon exhaustion of food supply, the amoebae switch from the growth phase to developmental phase in which they aggregate to form a multicellular structure. During aggregation, a cell or a small group of cells at the center gives off cyclic AMP pulses. The other cells relay the cyclic AMP signal outwards while moving towards the signal source. These processes result in aggregation of amoebae from a wide area to form a mound of cells, which then elongates vertically by forming on its apex a small protrusion which grows upwards so that the entire mass becomes like a
c 0167-2789/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 8 ) 0 0 2 7 3 - 5
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finger. In moist conditions, this cell mass usually lies down on the substratum and crawls in a manner resembling the movement of a garden slug (thus called a “migrating slug”). There is ample evidence indicating that the movement of individual cells within the slug are guided by chemotactic signals, possibly of cAMP, propagating from the tip of the slug towards its rear by the signal relay mechanism, a situation similar to aggregation [1–4] (but [5], see Section 5). The migrating slug consists of two major cell types; approximately a quarter are prestalk cells, and the front part of the slug is occupied by these cells. The rest are prespore cells which constitute the majority in the main body of the slug except the anterior prestalk region. The slug eventually forms a fruiting body consisting of a mass of spores and a supporting stalk. The migrating movement of the slug has been a subject of experimental and theoretical studies over decades (reviewed in [6]). While some hypotheses emphasize a role of the extracellular matrix in slug migration [7–9], others consider active movement of the cells to be the main source of the driving force [10– 15]. Measurements of the motive force of migrating slugs have revealed that the motive force of the entire slug is proportional to its volume, suggesting that all cells, not only those in the outermost layer, of the slug contribute to its movement [12,16]. It was also shown that the anterior prestalk cells exert much greater motive force than posterior prespore cells [12]. Based on these observations, we have proposed theoretical models for slug movement in which the sum of the motive force exerted by individual cells are in equilibrium with the counteracting resistance due to continuous expansion of the surface sheath and viscoelastic properties of the cells [17,18]. With these models, we showed that some of the basic properties of the migrating slug, such as its shape, velocity, and prestalk-prespore pattern, can be explained on a mechanical basis. It remained to be clarified, however, how the elongated shape of a slug and its anteroposterior pattern of cell differentiation are generated from a hemispherical mound of cells without distinct pattern. In the present study, we focused on the dynamical aspect of this process, namely the changes in the shape of a cell aggregate and cell sorting during the formation of an elongated slug. The cell mass was regarded as a continuum of which each volume element is capable of exerting motive force against the intrin-
sic resistance of cells. Propagation of the chemotactic signals from the “organizing center” was incorporated in a simplified way. The movement of the aggregate was then expressed as a moving boundary problem represented by a set of the force balance equation and the continuity equation of the cell mass. Analysis and numerical calculations of the moving boundary problem showed that the essential features of the morphogenetic movements from the mound through slug migration are expressed within a unified framework. Furthermore, differentiation of prestalk cells was shown to be essential in the formation of a tip and subsequent elongation leading to slug migration.
2. Formulation of the model 2.1. Basic model Our model is based on the hypothesis, first proposed by Shaffer [11], that the cell moves by extending its front ahead and retracting its back with its relatively solid lateral side remaining stationary with respect to the ground (Fig. 1(a)). This hypothesis is based on the behavior of cell surface markers during cell locomotion [11], and is also consistent with the current theory for the mechanism of amoeboid movement, namely, the interface between the rigid surface layer of the cytoplasm (which is in the “gel” state, and called the “cortex” of the cell) and the inner fluid (which is in the “sol” state and called the “cytosol”) is the site of force generation, and gel–sol interconversion takes place at the front and rear of the cell so that the solid lateral side of the cell is continuously formed in its leading edge while it is absorbed from its trailing end [19]. The solid lateral surface of a cell in a threedimensional cell mass can then serve as a substratum for the neighboring cells (Fig. 1(b)). For example, while the cells constituting the first layer of the cell mass move actively by getting traction from the surface sheath, their lateral surface remains stationary with respect to the surface sheath. The cells of the second layer can then obtain traction equally efficiently from the surface of the cells in the first layer, because the latter’s solid and stationary lateral surface serves as the substratum for their movement. Likewise, the cells of the subsequent layers can also move actively by getting traction from the lateral surface of the cells
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191
Based on these considerations, the following mathematical formulation can be derived (see also [17,18]). We consider a cell aggregate as a continuum and denote by v(x, t) its velocity at position x and time t. Each volume element of the continuum actively moves with the force generated by individual cells contained in the volume. We assume that the propulsion force per unit volume of the continuum can be written as F = f − av,
(1)
where f is the “motive force” of cell mass per unit volume. The second term of the right-hand side of Eq. (1) represents the intrinsic resistance due to cytoplasmic streaming which takes place inside the cell and the renewing of the lateral sides, where the drag coefficient a is constant. Since F is in equilibrium with the force exerted by surrounding volume elements, the following force balance equation holds for each unit volume of the continuum: f − av − ∇p = 0, Fig. 1. (a) Schematic diagram for the movement of a cell. The lateral side of the cell, which is the cytoplasm in the “gel” state, remains stationary relative to the ground while the cell moves. The inner cytoplasm which is in the “sol” state flows forward by getting traction against the solid gel. Gel–sol interconversion takes place at the front and rear of the cell. (b) The movement of cells within a three-dimensional cell mass. The lateral surfaces of all the cells are stationary relative to the ground so that they can serve as a substratum for the neighboring cells to move on.
in the previous layers. In this way, all the cells can exert motive force equally efficiently irrespective of whether they are in the surface layer of the aggregate or inside. In other words, the lateral sides (the cell surface and the cortex) of all the cells of the cell mass form a kind of meshwork connected to the surface sheath which in turn is firmly attached to the external substratum. The cytosol of individual cells moves forward obtaining traction from this meshwork while renewing the meshwork at its front end. The meshwork on the other hand receives all the forces exerted by the cytosol and transmits them to the external substratum. Consequently, the motive force of a cell mass equals to the sum of the motive force generated by all the cells. This is supported by the observation that the motive force generated by a migrating slug is proportional to its volume, but not to its cross-section or surface area [12,16].
(2)
where p(x, t) is the hydrostatic pressure. Because of the small size and speed involved in the phenomena under consideration, the effects of gravity and inertial force can be neglected. Note that unlike standard equations for viscous fluid, Eq (2) does not contain terms that are dependent on the gradient of the velocity. This is because the lateral sides of two adjacent cells always remain attached and there is no velocity difference irrespective of the velocities of the cells (i.e. the velocities of the cytosols). Therefore the difference in the cell velocity between adjacent cells causes no tangential stress. The second equation of the model is the conservation equation for the cell mass. Since the cell mass is incompressible, the velocity v must satisfy ∇ · v = 0.
(3)
Eqs. (2) and (3) are the set of basic equations that govern the movement of cell masses. 2.2. Free boundary problem We consider an idealized cell aggregate consisting of a single cell type and formulate its movement as a free boundary problem. Suppose a freely moving cell aggregate which occupies a domain Ω(t) at time t. Eqs. (2) and (3) then
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hold in Ω(t). Since the boundary ∂Ω of the aggregate moves with the cell mass, its normal velocity u satisfies the following equation: u=v·ν
on ∂Ω(t),
(4)
where ν is the unit outward normal to ∂Ω. For the boundary value of p, we assume free movement of the boundary without considering the surface tension and the effect of the surface sheath surrounding the aggregate for simplicity, and adopt the following boundary condition: p = p0
on ∂Ω(t),
(5)
where p0 is the atmospheric pressure. We next specify the motive force f(x, t) in Eq. (2). We assume that a chemotactic signal is propagated within the aggregate and that the amoebae exert motive force in the direction the signal comes from. In aggregation of D. discoideum, propagation of cyclic AMP signals over the aggregation field can be described as wave propagation on an excitable medium [20,21], and accumulating evidence shows that the same signal controls the multicellular development under normal conditions [1–4] (see Section 5 for morphogenesis in mutants lacking cAMP). Here we adopt a simplified model based on the following assumptions: First, though both concentric and spiral wavefronts are possible on an excitable media, we consider only the case that the waves are concentrically propagated from the pacemaker. Consequently, rotating movement of cells which are often observed in mounds and slugs, are not represented in the model. Secondly, we assume that the speed of the signal propagation is constant and sufficiently fast compared with the speed of cell movement. The validity of these assumptions is discussed later. In its simplified form, the presence of a single pacemaker cell (or a pacemaker region) within the aggregate is postulated and its position at time t is denoted by xp (t). The chemotactic signal emanating from xp is propagated throughout the aggregate. If the speed of the signal propagation is constant and sufficiently fast, the wavefronts form part of spheres in Ω with its center at xp . Accordingly, the motive force at each position x is directional to xp . On the other hand, the magnitude of the motive force at each position will vary periodically if the signal is periodical. However, if averaged over time sufficiently longer than the sig-
nal period, we can regard it as constant. Therefore, the motive force can be written in the following form: f(x, t) = f
xp (t) − x , |xp (t) − x|
(6)
where f is constant. Note that f is not defined at xp in Eq. (6) so that Eq. (2) does not hold at this position. Nevertheless, the velocity v can be uniquely defined by assuming continuity at x = xp . Finally, we assume that the pacemaker moves at the same speed as the surrounding cells, i.e. the position of the pacemaker is determined by the following equation: dxp = v(xp , t). dt
(7)
3. Movement of cell aggregate If we introduce a variable p f φ = − |xp − x| − , a a
(8)
Eqs. (2) and (3) can be rewritten using (6) as v = ∇φ
in Ω
(9)
and ∇ 2φ = 0
in Ω,
(10)
respectively. The boundary condition (5) is rewritten as p0 f φ = − |xp − x| − a a
on ∂Ω.
(11)
Therefore, the velocity v at any point in the aggregate is obtained by solving Laplace equation. Eqs. (4) and (7) then determine the movement of the boundary and the pacemaker, respectively. We first look for the stationary solution, in which the boundary of the aggregate does not change its shape and moves at a constant speed U. The normal velocity of the boundary is then given by u = U · ν. Substituting this equation and Eq. (9) into (4), we have the following boundary condition for φ: ∂φ = U · ν. ∂ν
(12)
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193
Putting φ(x, t) = φ(ˆx), where xˆ = x − Ut, we obtain the solution of Eq. (10) with the condition (12) as φ(ˆx) = U · xˆ + c,
(13)
where c is a constant. With this, Eq. (9) yields v=U
in Ω.
(14)
Therefore, all unit volumes in Ω move with equal velocity and no convection flow occurs within the aggregate. Substituting (13) for the left-hand side of (11), we obtain the following equation which determines the boundary shape of the moving aggregate: p0 f |ˆxp − xˆ | + U · xˆ + + c = 0. a a
(15)
The shape given by (15) is the revolving surface of a conic section whose eccentricity is η=
a|U| , f
(16)
and one of its foci coincides the position of the pacemaker. Since the aggregate has a finite volume, the boundary shape is a spheroid where η satisfies 0 ≤ η < 1. If the movement of the aggregate is restricted in two-dimensional space, the boundary shape is an ellipse with the pacemaker at one of its foci. Eq. (16) indicates that the velocity does not depend on the volume of Ω but depends on the relative position of the pacemaker in the aggregate. Fig. 2 illustrates some examples of cell aggregate at steady state, showing the relationship between the position of the pacemaker, the shape of the aggregate, and its speed of movement. Arrows represent the velocity U of the aggregate. As the pacemaker approaches the boundary (i.e. η becoming larger), the aggregate becomes more elongated and moves faster. If η approximates 1, the aggregate elongates infinitely and |U| approaches to f/a. Conversely, if the pacemaker is located at the center of the aggregate (i.e. η = 0), the boundary of the aggregate becomes a sphere and the aggregate does not move. To examine whether the aggregate starting with an arbitrary shape becomes the above stationary shape, we numerically solved the moving boundary problem represented by Eqs. (4),(7), and (9)–(11) (see Appendix A for the detail of calculation). To avoid the computational difficulty, we considered the model in
Fig. 2. Shapes and the corresponding velocities of moving cell aggregates. The solid circles indicate the location of the pacemaker and the arrows indicate the velocity of the aggregate. The shape of the aggregate is a three-dimensional spheroid with the pacemaker at one of its foci. The eccentricities of the spheroids shown in figure are 0.3, 0.7 and 0.9. Aggregates with their pacemaker close to the edge are more elongated and move faster.
Fig. 3. Numerical simulation of the morphogenetic movement of cell aggregate in two-dimensional space. The initial shape of the aggregate is irregular with its pacemaker positioned near the boundary. The aggregate changes its shape as it moves and then finally becomes an ellipse moving at a constant speed. Contour lines (t = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2) are obtained by solving Eqs. (4),(7), and (9)–(11), where parameter f/a = 1.
two-dimensional space, which corresponds to the situation that cell aggregate is confined in a region between flat plates. Fig. 3 shows a result, in which the initial boundary shape is irregular with the pacemaker positioned near the boundary. The aggregate changes its shape as it moves, and finally becomes an ellipse
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moving with a constant velocity. Similar results were observed in other numerical calculations with various initial shapes, though the eccentricity of the final shape varies depending on the initial boundary shape and the initial position of the pacemaker.
4. Cell sorting and slug movement So far we have assumed that the aggregate consists of a single cell type. However, there are at least two types of cells differing in their motive force in Dictyostelium cell aggregates. In this section, we consider the effect of the heterogeneity of cell types on the movement of aggregates. Let us consider a cell aggregate consisting of two types of cells, prestalk and prespore cells. Since cell type conversion is scarce under normal conditions [22], the cell number of each cell type is considered invariant. We assume that the two cell types differ in the magnitude of their motive force but both motive forces are directed to the pacemaker, i.e. the motive forces generated by prestalk and prespore cells, f1 and f2 , are given by fi = fi
xp − x |xp − x|
(i = 1, 2),
(17)
∇ ·v =0
in Ω,
∂n + v · ∇n = −∇ · J ∂t
in Ω,
(18) (19)
in Ω
(20)
and dxp = v(xp , t). dt
(21)
Eq. (20) is the conservation equation for the prestalk fraction, where the flux J is given by J=
f1 − f2 (1 − n)n − D∇n. a
J·ν =0
(22)
on ∂Ω,
(23)
which represents the impermeability of prestalk and prespore cells at the boundary. Now the movement of the mixed aggregate can be obtained by solving Eqs. (18)–(21) with the conditions (4),(5) and (23). However, we here calculate a mathematically simpler system instead of solving the full moving boundary problem. We first consider a cell aggregate with fixed boundary, in order to examine how the cell distribution changes in time within the aggregate. The boundary conditions (4) and (5) are then replaced by v·ν =0
where fi ’s are constant. Since the motive force of prestalk cells is greater than that of prespore cells, f1 > f2 . Let v(x, t) be the mass-averaged velocity of the two types of cells and n(x, t) the fraction of prestalk cells at position x and time t. Then the movement of the mixed cell mass is described by the following equations [18]: f1 n + f2 (1 − n) − av − ∇p = 0
The first term of the right-hand side of (22) represents the differential movement due to the difference in motive force between the two cell types. The second term is the diffusion term that comes from the random movement of cells and the coefficient D is constant. In addition to the differential motive force, Umeda [23] considered the effect of differential cell adhesion on the change of n. However, we do not consider the differential adhesion in this study to simplify the model. The boundary conditions (4) and (5) are unchanged in this case, but the following condition should be added for Eq. (20):
on ∂Ω
(24)
and ∂p = [f1 n + f2 (1 − n)] · ν ∂ν
on ∂Ω,
(25)
respectively. We numerically solved the set of equations (18)– (21) with boundary conditions (23)–(25) in twodimensional space (see Appendix B for the method of calculation). The boundary shape was assumed to be a square for simplicity. Fig. 4 shows a result, where the shading expresses the fraction of prestalk cells n and the open circle represents the location of the pacemaker xp . By the time t = 0.6, initially uniformly distributed prestalk cells sort out to cluster around the pacemaker, forming a well-defined prestalk region. Then the prestalk region, along with the pacemaker, gradually moves towards the periphery of the aggregate. This movement is caused by the deviation of the initial pacemaker position from the center; initially the position of the pacemaker is off the center to its right so that more prestalk cells would gather
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Fig. 4. Cell sorting within the fixed square of size 1. The graded shading shows the fraction of prestalk cells present in each position (a gray scale of 10% intervals) and the open circle represents the position of the pacemaker. During t = 0 to t = 0.6, prestalk cells sort out to form a prestalk region around the pacemaker. After the prestalk region is formed, it gradually moves to the edge of the aggregate, along with the pacemaker. The figures are obtained by solving Eqs. (18)–(21) with boundary conditions (23) and (24). Parameters are f1 /a = 1.5, f2 /a = 0.5 and D = 0.01. Initial distribution of prestalk fraction is uniformly 0.2 and initial position of the pacemaker is (0.6,0.55).
to the left of the pacemaker when the prestalk region is formed. This asymmetry of cell distribution brings about rightward movements of the pacemaker and the prestalk region. We next consider the case that cells are completely sorted out within an aggregate before the aggregate moves. Though experimental observations show that cell sorting and the movement of aggregate simultaneously proceed, we use this assumption for mathematical simplicity to solve the moving boundary problem. Then the flux of prestalk fraction vanishes everywhere in the moving aggregate, i.e. J=0
in Ω.
(26)
Substituting (17) and (22) into (26) and integrating it with respect to |xp − x|, we have the following distribution of prestalk cells: n(x, t) =
1 . 1+exp[((f1 −f2 )/aD)(|xp −x|−r0 (t))]
(27)
The function r0 (t) in (27), which represents the distance from the pacemaker to the position where n is 0.5, is determined by the conservation of the total number of prestalk cells. Fig 5 depicts typical prestalk distributions given by (27). Prestalk cells sort out toward the pacemaker. The boundary of the two types of cells becomes more clear as (f1 − f2 )/aD increases.
Fig. 5. Stationary distribution of the prestalk cells in the aggregate. The horizontal axis represents the distance from the pacemaker. Prestalk cells sort out toward the pacemaker, where the boundary of the two types of cells become more clear as (f1 − f2 )/aD increases. The parameter (f1 − f2 )/aD is: (a) 10.0; (b) 20.0; (c) 100.0.
If we substitute (27) into (18) and couple it with (19) and (21), we can use the same method as in Section 3 to solve the moving boundary problem for the mixed cell mass (see Appendix A). Fig. 6 shows a typical result of the numerical calculation, where the solid circle indicates the position of the pacemaker and the solid curve within the aggregate represents the curve where n = 0.5. Initial shape of the aggregate is a circle with prestalk region being located at the edge of the aggregate. As time advances, the prestalk region is projected from the aggregate. Then the
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Fig. 6. Movement of a cell aggregate which comprises prestalk and prespore regions. The prestalk region projects from the round aggregate while the entire aggregate moves forward. As the aggregate moves, it is gradually elongated. These figures are obtained by solving Eqs. (18),(19),(21) and (27). Parameters are f1 /a = 1.5, f2 /a = 0.5 and D = 0.01. Initial shape of the aggregate is a circle of diameter 1 and the initial position of the pacemaker is 0.37 from the center. The area of the prestalk region is 20% of the aggregate.
aggregate gradually narrows and lengthens its shape, and finally moves at a constant speed keeping a spatial pattern of two cell types. We can see the model reproduces the basic feature of the real slug formation and slug movement, except that the final shape of the aggregate has thicker prestalk region than real slugs. 5. Discussion We first summarize the main assumptions of the present model: (i) the movements of cell aggregates, and cell sorting therein, occur as a result of active cell movement; (ii) the movement of individual cells is controlled by chemotactic signals propagated from a single “pacemaker”; (iii) the boundary of the aggregate has no effect on the movement of cells. In assumption (i), we considered that cells in the aggregate move as a “stationary surface model”, in which the motive force of the whole aggregate is simply given
as the accumulation of the motive forces generated by individual cells. Besides the present model, several models have been proposed for the movement of an aggregate. Williams et al. [9] proposed a “squeeze–pull model”, which involves circumferential cells squeezing forward a cellular core, followed by pulling up of the rear. Odell and Bonner [13] proposed a hydrodynamical model, in which cells turn the “membrane tractor” at a speed that depends on the cell position, and the difference in the membrane velocity among adjacent cells is responsible for generating thrust of cells. However, the fact that cells that have become non-motile within a slug are stationary with respect to the substratum [24] seems to support our model, because Williams’s model predicts forward movement of non-motile cells and the model of Odell–Bonners predicts backward movement of them. The fact that the motive force of the entire slug is proportional to its volume [12,16] also supports our model. Assumption (ii) is known to hold during cell aggregation [25], and there is evidence showing that cell movements within cell aggregate are also under the control of propagating signals [1–4,26]. During cell aggregation and in mound stage, both concentric and spiral wavefronts are observed [27,28]. However, we did not consider the spiral wave propagation for the following reasons. First, not all slugs exhibit rotating cell movements while undergoing morphogenetic movements indistinguishable from those accompanying rotating cell movements, suggesting that the rotating cell movements are not an essential part of the formation and migration of a slug. Secondly, the direction of the motive force generated by the cells would be virtually the same as that driven by concentric waves throughout the aggregate except the vicinity of the center of spiral signals. Even in the vicinity of the center where the diagonal components of the motive force prevail, they would not affect significantly the shape of the cell aggregate and cell sorting pattern observed over timescales sufficiently larger than the rotation period of spiral waves. Thus, most of the conclusion of the present study will be unaffected whether the cell aggregate is controlled by concentric or spiral waves. For the rotating movement of cells, work is under way to incorporate spiral wave propagation in our model. Instead of considering the evolution equation of the cyclic AMP concentration, we adopted a simplified version of the model, i.e. we assumed instantaneous
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transmission of signals from the pacemaker and generation of constant motive force by cells in the direction of the signal source. These approximations dramatically facilitate numerical calculation and enables one to conduct simulations involving cell sorting and shape changes. As regards the quickness of the signal propagation, the speed of wave propagation is over 10-fold larger than the speed of cell movements during aggregation [29] and most likely within migrating slugs as well [2]. Furthermore, changes in the relative position of the pacemaker is very slow compared with the speed of cell movements seen during cell sorting, as shown in the numerical simulations, so any errors due to this assumption must be small. Although the individual cells show periodical motion during aggregation [30] and within slugs [1], the assumption that motive force is constant is good approximation for describing phenomena occurring over the developmental timescale. On the other hand, we ignored the effect of unevenness of wave propagation. If the speed of signal varies with cell types or is changed due to boundary effects, the wave would be refracted so that the direction of the motive force might be affected. The magnitude of the motive force may also vary due to the effects of the local factors such as oxygen tension and ammonia concentration. In order to take these effects into account, it will be necessary to formulate the evolution equation of the cyclic AMP concentration and to assume chemotactic movements of cells against cyclic AMP gradient. Some recent models deal with the Dictyostelium morphogenesis using such approaches [14,15,31]. The recent results by Wang and Kuspa [5] indicate that mutants lacking the enzyme adenylate cyclase which produces cAMP can complete its morphogenesis if cAMP-dependent protein kinase (PKA) is made constitutively active. Although this observation suggests that cAMP is dispensable for the Dictyostelium morphogenesis, a body of evidence points to the presence of propagating signals, most likely of cAMP, that organize the morphogenetic movements of cells in later development, at least in normal situations [4,32]. Furthermore, mutants lacking the cell surface cAMP receptor subtype CAR2 which is expressed during the multicellular stages, as well as strains overexpressing extracellular phosphodiesterase during the multicellular stages (resulting in diminished levels of extracellular cAMP), are deficient in tip formation and sub-
197
sequent morphogenesis [33,34], indicating the importance of extracellular cAMP during this period. This, however, does not exclude the possibility of the presence of chemotactic agents other than cAMP that act during the later development, and our model is open to that possibility. Assumption (iii) was adopted solely for the sake of simplicity of the model. In reality, a slug is covered with a mucous film called surface sheath, which without doubt constrains the movement of the slug, and its omission is likely to be the cause of the unrealistic results in some of the simulations (see below). Surface tension is another factor that would affect the movement and shape of the aggregate [18]. The results of the model based on these assumptions showed several properties of the cell aggregate that agree with experimental observations, i.e. the elongated shape of a migrating slug, positive correlation between the length and speed for slugs with the same volume, and sorting-out of prestalk cells towards the anterior of the slug. These agreements support the view that the morphogenetic movement and cell sorting in aggregates are caused by active cell movement under the control of propagating signals. Although numerical calculations were performed in twodimensional space, the essence of these results will be unchanged in three-dimensional space. Furthermore, the results shown in Figs. 4 and 6 can be seen as simulating the process of translocation of the pacemaker from the central part of the aggregate to the anterior tip while the shape of the entire aggregate becomes elongated. These results demonstrate the essential role of cell differentiation in the formation of the tip and subsequent elongation of the aggregate; if the aggregate consisted of single cell type, the pacemaker would stand still at the initial position after cell aggregation and the entire aggregate would keep its rounded shape. Although the present study deals with only two cell types, the model can easily be extended to cell aggregates containing three or more cell types. Prestalk cells are now known to consist of at least three subtypes [35]. They differ from each other in the ability of chemotactic movement [36,37], which seems to suggest their motive force within a slug might also be different. In cell aggregates consisting of more than two cell types differing in the motive force, the cells would be sorted out according to their motive force.
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Then the translocation of the pacemaker as seen in Fig. 4 will be accelerated, because each cell type region would move to a peripheral position of the surrounding cell type region, and a more elongated shape than the ones obtained with two cell types would result. Recent studies revealed that changes in spatial distribution of the prestalk sub-types occur before slug formation [37]. Such changes in spatial distribution of the prestalk sub-types may be caused by changes in the relative magnitudes of the motive force among the prestalk sub-types and prespore cells. On the other hand, some of the results do not agree with the known facts; the speed of slugs with the same length:width ratio is independent of their volumes (in reality bigger ones migrate faster), and the anterior prestalk region of a slug becomes thicker than real. The former disagreement is likely due to the negligence of the effects of the surface sheath. Taking into account the resistance of the surface sheath at the front of the slug, the force balance would favor among slugs with the same length:width ratio bigger ones for faster migration, because the total motive force of the slug is proportional to its volume while the resistance of the surface is proportional to the surface area of the front part of the slug (see [17]). The latter disagreement may also be due to the negligence of the surface effects. The rigidity and thickness of the surface sheath decrease towards the tip of a slug, which may be the cause of the tapering shape of the slug [38]. To analyze these effects, more realistic model considering the mechanical properties of the surface sheath will be necessary. In the present study, we focused on the morphogenesis and the early stage of slug migration. However, with regard to the movement of a slug, there remain many unsolved problems, e.g. the mechanism of its turning behavior in response to external stimuli such as light and temperature gradient. The mechanism of fruiting body formation is another interesting problem. Approaches from mechanical viewpoints like the present study will be necessary for thorough understanding of these interesting phenomena.
Acknowledgements T.U. thanks K. Maruo and N. Shigesada for helpful discussions and comments.
Fig. A.1. Parametrization of the boundary shape of a two-dimensional aggregate. vn (s) and vs (s), respectively, are the normal and the tangential component of the velocity v on the boundary.
Appendix A We consider Eqs. (4),(7) and (9)–(11) in twodimensional space. For a fixed time t, we express the velocity in the domain Ω(t) as a complex function f (z) = vx − ivy ,
(A.1)
where z = x + iy.
(A.2)
Let z(s) be a parametric representation of the boundary ∂Ω, where s is the arc length of the boundary, and let vn (s) and vs (s) be the normal and the tangential component of the velocity v on ∂Ω, respectively (Fig. A.1). Then, for z ∈ Ω, f (z) can be expressed as I vn (s) − ivs (s) 1 ds, (A.3) f (z) = 2π ∂Ω z(s) − z since Eqs. (9) and (10) indicate that f (z) is an analytic function [39]. If z is on ∂Ω, f (z) = (vn − ivs )e−iθ I 1 vn (s) − ivs (s) = ds, π ∂Ω z(s) − z
(A.4)
where θ is the angle between the x-axis and the normal to ∂Ω, and the integral means the Cauchy principal value. For the tangential component of the velocity, Eqs. (9) and (11) yield vs (s) =
f sin (θ (s) − ϕ(s)), a
(A.5)
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where ϕ is the angle between the x-axis and the line connecting z(s) and xp (see Fig. A.1). The normal component vn is then obtained by solving the boundary integral equation (A.4). For each iteration time step tk , we divide the boundary into finite elements and calculate the velocities of the boundary points using (A.4) and (A.5), and the velocity of the pacemaker using (A.3). Then the boundary points and the pacemaker are displaced by using the following equation: x(k+1) = x(k) + v(k) 1t.
(A.6)
If the aggregate consists of two types of cells, f1 n + f2 (1 − n) sin (θ(s) − ϕ(s)) (A.7) a is used instead of (A.5), where n is given by (27). Function r0 (t) in (27) is determined by the constraint that the area of the prestalk region (n > 0.5) is invariant.
vs (s) =
Appendix B We consider Eqs. (18)–(21) with boundary conditions (23)–(25) in two-dimensional space. Eq. (19) implicates that there exists a function 9 which satisfies vx =
∂9 , ∂y
vy = −
∂9 . ∂x
(B.1)
By using this, Eqs. (18) and (20) are transformed as ∇ 2 9 + ∇n ×
f1 − f2 = 0, a
(B.2)
and f1 −f2 ∂n +∇n×∇9=∇ · − (1−n)n+D∇n . (B.3) ∂t a Boundary condition (24) becomes 9 = constant.
(B.4)
For each time step tk , equations were solved on 49 × 49 grid as follows: (i) 9 (k) is calculated from (B.2) and boundary condition (B.4). (ii) n(k+1) is calculated from (B.3) and boundary condition (23) by using an explicit method. (iii) The location of the pacemaker is calculated using xp(k+1) = xp(k) +
∂9 (k) 1t, ∂y
yp(k+1) = yp(k) −
∂9 (k) 1t. ∂x
199
(B.5)
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