Bulletin of Mathematical Biology Vol. 51, No. 5, pp, 549-578, 1989. Printed in Great Britain

0092-8240/8953.00 + 0.00 Pergamon Press plc Society for Mathematical Biology

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS KEMBLE YATES*

Department of Mathematics, Southern Oregon State College, Ashland, OR 97520, U.S.A. m

EDWARD PATE

Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164, U.S.A.

The mesodermal tissue of some amphibian gastrula develops into a dorsal-to-ventral sequence of notochord, somite, pronephros, and lateral plate cell types. The cellular proportions regulate with respect to embryo size. The dorsal blastoporal lip appears to function as an organizer for the embryo. The transplantation of a donor lip to the ventral side of a host causes a second, opposed embryo to form and the system commits similar total proportions of cells as do normally developing embryos. Transplantation of donor somite to the ventral side of a host causes a reduction in the proportion of host somite developed. A modified reaction~liffusion system governing embryo development is proposed. Developmental simulations consistent with experimental observations are presented and analyzed. The results suggest that the degree of somite inhibition is positively correlated with the size of the somite transplant. Further predictions are that sufficiently large somite transplants would induce ectopic, ventral pronephros to form and ventral pronephros transplants would inhibit host pronephros development.

1. Introduction. Amphibian embryos provide a challenging system for developmental models. Many features of their development from the pregastrula stage to the late tailbud stage have been previously discussed (reviewed by Spemann, 1938 ; Slack et al., 1984 ; Nieuwkoop, 1985). The morphogenesis of the mesodermal tissue is of particular interest. With the onset of gastrulation, the presumptive mesoderm and endoderm regions of the blastula are enveloped by the ectoderm. As development proceeds towards the late tailbud stage, a dorsal-to-ventral, systematic sequence of four distinct cell types emerges in the mesoderm : notochord, followed circumferentially by bilateral, This paper has been reproduced directly from disc using a LA-TEX system. * Author to whom correspondence should be addressed. 549

550

K. YATES AND E. PATE

paired regions of somite and pronephros, and finally lateral plate. The proportions of these cell types regulate with respect to overall embryo size. If ventral cells are removed from pregastrula embryos, the relative proportions are preserved while the resulting specimen may have as little as 20% of the mesodermal tissue found in normal-sized embryos (Cooke, 1981, 1982). Often present in embryonic systems is one or more specialized group of cells capable of inducing a subsequent developmental pattern. The blastoporal lip, located dorsally in the mesoderm, appears to serve as such an "organizer" region of cells for amphibian embryos. It initiates the differentiation of the medullary plate and provides a reference point for a developmental axis in the mesoderm. As the classic experiments of Spemann and Mangold showed (Spemann, 1938 ; see also Cooke, 1972a), the normal differentiation and development of the gastrula stage embryo, including the size-proportioning in the mesoderm, is dependent on an intact future dorsal lip region. The transplantation of a donor blastoporal lip to the ventral side of a late blastula stage host results in a second developmental axis forming on the ventral side. When assayed at the late tailbud stage, an embryo with a bipolar, essentially mirrorsymmetric pattern of the four cell types has formed, and each cell type appears in nearly normal proportion. In another grafting experiment, transplantation of dorsolateral mesoderm (tissue destined to be somite) from early neurula stage donors to the mid-ventral region of pregastrula hosts results in the transplanted tissue continuing development to ectopic somite while the host embryo emerges with a pattern having less somite tissue and more lateral plate tissue than normally developed embryos. This experiment has been interpreted to suggest the existence of feedback mechanisms between the emerging ceil types which affect subsequent developmental events (Cooke, 1981, 1982 ; Dale and Slack, 1987). A satisfactory explanation of how a spatial pattern of distinct cell types becomes established is a fundamental goal in developmental biology. How do immature cells know into what cell types they should ultimately differentiate? A theoretically appealing idea is that some intercellular mechanism exists which informs individual cells of their position in the group. Presumably each cell has a genetic blueprint of the prospective organism, and development can proceed once the cell knows its location. This idea of "positional information" was first formally considered by Wolpert (1969) and Crick (1970). Many developmental models which have been proposed rest on the hypothesis that positional information in the organism is provided by a monotonic concentration profile of one or more diffusing chemical "morphogens". With specific concentration ranges of morphogens corresponding to activation thresholds for the various cell types, individual cells could monitor the concentration of the appropriate morphogen(s) and thus choose the correct developmental pathways (Turing, 1952 ; Wolpert, 1969). For these models, a

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

55 !

biologically plausible scheme is then needed to set up the concentration profile of the crucial chemical species. Two general types of models have been advanced which utilize chemical morphogens for positional information. Source-sink models suppose that specialized organizer cells on each end of a developmental field maintain fixed, unequal concentrations of a morphogen. With diffusion present in the field, a monotonic profile arises. A source-sink scheme can establish a linear morphogen distribution in a 1 m m system such as a pregastrula, amphibian embryo in a time scale consistent with the observed differentiation (Wolpert, 1969; Crick, 1970; Munro and Crick, 1971). Furthermore, owing to the linearity of the profile and the fixed boundaries, it can easily be shown that this model proportions an n-element pattern in a size-invariant manner. Unfortunately, biological considerations would appear to preclude the application of source-sink models to amphibian embryo development. A fundamental requirement of source-sink models is two sets of specialized organizer cells, one located dorsally and the other ventrally. However, there is evidence for only one group of organizer cells in amphibian embryos, namely the dorsal blastoporal lip. Also, the experiment involving the transplant of an extra blastoporal lip to the ventral side of a host would result in a uniform profile of the morphogen according to a source-sink scheme. Hence, it could not possibly explain the appearance of a bipolar pattern. Reaction-diffusion systems were first proposed by Turing (1952). In this class of models, two or more chemicals diffuse through a system and interact kinetically. In the simplest case, two morphogens could be interacting in a one (spatial) dimensional developmental field of length 50. The governing partial differential equations of the morphogens (representing balance of mass relations) then take the form:

(Ui), = D~(U~)x~+R,(U),

i = 1,2;

0 < x < 5 ~,

t>O.

(1)

U = [Ul(x, t), U2(x, t)] is a chemical concentration vector at point x at time t, while the x and t subscripts denote partial differentiation. The Di are constant diffusion rates and the Ri(U, V) are (typically nonlinear) reaction fnnctions of the two concentrations. No-flux boundary conditions are natural for chemicals in a closed field: (E)x(o,

t) = ( u 3 x ( 5 0 ,

t) = o,

i = 1,2.

(2)

Natural initial conditions are constant, steady-state profiles : U,(x, 0) = G,0,

with

R,(U0)=0,

for

i = 1,2.

(3)

The system can be nondimensionalized using the following dinaensionless

552

K. YATES A N D E. PATE

quantities :

= t/g, z = x/so, u(z, r) = U(x, t)/C, u0 = Uo/C, and ri(u) = gRi(U)/C.

(4) Here g is a time scale, 5 ~ is the length scale, and C is a characteristic concentration scale. Equations (1)-(3) then become :

Dig

(ui)~=~(u,)~z+ri(u),

i = 1,2;

0
1,

~>0.

(5)

The boundary and initial conditions convert to : (u~)~(O,'r) = (Ui)z(1,z) = O,

i = 1,2

(6)

for

(7)

and

ui(z,O)=ui.o,

with

re(U0)=0,

i = 1,2.

When the reaction kinetics have an appropriate structure and occur at comparable rates to diffusion, stable monotonic gradients can be formed from the initially homogeneous concentration profiles. This conclusion can be reached by doing a standard linear stability analysis of equations (5)-(7) (Turing, 1952). A primary problem with applying these models to amphibian embryos is that they have very limited size-regulatory capability. The concentration profile depends on the length of the system because, with fixed, constant values for kinetic and diffusion parameters in the reaction-diffusion equations governing the evolution of the morphogens, changes in the length of the system have dissimilar effects on kinetics and diffusion. As can be seen in the nondimensionalized equations in (5), the length of the system, 5 ~ appears only in the diffusion terms. Chemical reactions depend on chemical concentration values and thus not directly on the length of the system. Chemical diffusion, on the other hand, does depend directly on the length of the system in the sense that the relative speed of a diffusive process slows down as the length is increased. The morphogen profiles thus vary with the size of the developmental axis. In particular, for fixed diffusion and kinetic coefficients, a monotonic profile can only be generated for a fixed range of lengths (Othmer and Pate, 1980). Turing-like models have been demonstrated to size-regulate over a length factor of about five (Gierer and Meinhardt, 1972), enough perhaps for the observed morphallaxis in amphibian embryos. However, these "activatorinhibitor" models have difficulty regulating more than two pattern elements. A similar scheme (Lacalli and Harrison, 1978) also size-regulates two pattern elements over a length range compatible with the regulation exhibited by amphibian embryos, but again the extension of this type of model to three or

A C A S C A D I N G D E V E L O P M E N T M O D E L F O R A M P H I B I A N EMBRYOS

553

more cell types would be difficult. Both models divide the developmental field by holding the node of a wave-shaped, steady-state concentration profile in the same relative position as the field length changes. The amplitudes of the profiles change significantly as length is varied, however, making any additional threshold assignment(s) difficult to regulate. Many strategies to modify the structure of reaction~tiffusion schemes in order to guarantee size-regulating pattern formation have been suggested. By altering the assumption that kinetic rates (Harrison and Lacalli, 1978) or diffusivities (Babloyantz and Hiernaux, 1975 ; Othmer and Pate, 1980 ; Gierer, 1981) are constant and making them vary in a size-dependent fashion, pattern regulation with respect to developmental field size results. For a one-dimensional system of length LP, the nondimensionalized reaction-diffusion equations (5) make clear the observations that diffusion coefficients proportional to ~Ct92o r kinetics proportional to 5~-2 guarantee length-independent morphogen concentrations. Othmer and Pate (1980) have analyzed one mechanism by which this change to the morphogen diffusivities could be implemented. By assuming that the diffusion rates of the morphogens are functions of the length-dependent concentration of a special chemical "control species", the concentration profile of the morphogens can be made perfectly size-invariant. In order to make the control species evolve to a steady-state which is proportional to 5r 2, the control species is assumed to be produced at a constant rate by all cells, diffuse at a constant rate through the system, and be destroyed by cells at one end of the system while obeying a no-flux boundary condition on the other. This in turn guarantees the length-invariant evolution of all other morphogens (Othmer and Pate, 1980). Pate (1984) applied this type of model to amphibian embryos and suggested that the sink boundary condition could be found at the blastoporal lip, correlating the special mathematical condition imposed with the special biological behavior observed in the organizing tissue. Furthermore, by making the equivalent change to the boundary condition on the other end to simulate the blastoporal transplant, the correctly proportioned, bipolar pattern could be reproduced. A linear stability analysis of the two systems (Pate, 1984) confirms that the same kinetic and diffusion parameters for which monotonic profiles of the morphogens emerge in the "one organizer" case yield symmetric bipolar profiles of the morphogens in the "two organizer" case. The deficiency of this model, and of all models previously discussed, is the inability to easily explain the somite inhibition observed by Cooke (1983). Cooke has demonstrated the possibility that feedback mechanisms exist between developing parts in the mesoderm. This would contrast sharply with the strict positional information postulate that a prepattern is established along the length of an organism to provide a blueprint for development.

554

K. YATES A N D E. PATE

Cooke (1983) observed that donor, presumptive, somite tissue grafted on the ventral side of amphibian embryos results in a significant deficit of somite developing in the host (as compared to normally developing embryos). Cooke theorized that the organizer establishes dorso-ventral polarity in the embryo. Then, as each cell type emerges, it releases an inhibitor of itself. After enough inhibitor is released, development of the corresponding cell type would be arrested and the next (ventral) cell type would commence. He thus suggested the possibility that determination in amphibian embryo development proceeds as a wave originating in the organizer and propagating ventrally down the developmental axis. In particular, somite might form and then eventually inhibit its own development, thus contributing dynamically to the establishment of the pattern. The explanation of the somite grafting experiment would then be that the additional somite combines with the emerging host somite to cause a relative reduction in the final host somite region. The general idea that a developmental pattern could be dynamically established by interacting tissues has surfaced before in developmental biology. Child (1941) has outlined an extensive theory of morphogenesis in which an organizer induces a gradient of rates of differentiation in a developmental field (so-called physiological gradients). Coupled with the graded rates of differentiation is the assumption that there exists a hierarchy of preferred differentiations among possible cell types, which Child called dominance hierarchies. The most preferred type would commence to develop nearest the organizer but simultaneously secrete an inhibiting substance which eventually prevents the most preferred type from emerging some distance down the field, where differentiation is slower. In this region, cells would choose the next most preferable fate and in this manner, the pattern would be established. Rose (1957,1970) has given a related argument for cellular interaction during development. An alternative to the presence ofpreestablished, intrinsic physiological gradients is to postulate activating morphogens. In this development framework, the organizer could release notochord activator, stimulating differentiation of nearby tissue to notochord. Notochord could then release somite activator, and so on, resulting in a hierarchy of dorsal-to-ventral development. More recently, Meinhardt and Gierer (1980) have proposed a model for the generation of a pattern of several elements in which each succeeding type is induced by its predecessor. In this way, determination "cascades" down the developmental field. It is not clear how this particular scheme could explain the bipolar pattern in amphibian embryos, however. A bipolar pattern would presumably have to be a second locally stable steady-state which would be reached following a sufficiently large perturbation from the monotonic one. This seems to be mathematically possible but biologically unlikely. However, this type of model does allow for dynamical interaction between emerging

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

555

pattern parts and could potentially simulate the somite transplant experiments of Cooke (1983). One way to proceed in modeling development in amphibian embryos would be to combine the advantages of a regulated diffusion model (perfect size regulation and ability to generate a bipolar pattern) with the advantages of a sequential induction model (a mechanism for somite inhibition). We consider such a model in the next section.

2. A Cascading Development Model.

The mesodermal tissue in an amphibian embryo forms a roughly cylindrical jacket, one cell thick beneath the ectoderm at the time of gastrulation. During gastrulation six bands of tissue form along the dorso-ventral axis : notochord most dorsally, followed by bipolar, paired bands of somite and pronephros, and finally lateral plate tissue most ventrally. The embryo exhibits bilateral symmetry and thus we use one spatial dimension as an approximation in modeling the development. The developmental axis is an interval consisting of a semicircle extending from the dorsal to the ventral zones. Experimentally, removal of presumptive mesodermal tissue from the embryo results in a rescaling of the cell type pattern to fit the smaller developmental field. Transplantation of a donor blastoporal lip to the ventral side of a host causes the emergence of a bipolar pattern with similar total proportions of cell types as in undisturbed embryos. Transplantation of donor presumptive somite to the ventral side of a host yields a pattern relatively deficient of somite. A minimal model must account for all of these features of development. We envisage the following process during development. Early gastrula mesodermal cells monitor the concentrations of four primary morphogens which serve as activators for specific cell types. When the concentration of one of the morphogens exceeds a threshold value in some region of the developmental axis, differentiation to the corresponding cell type commences. U p o n differentiation, a given cell type releases the activating morphogen for the next cell type in the dorsal-to-ventral sequence. The activator diffuses through the system and in this manner differentiation cascades ventrally from the blastoporal lip. A distinction is made between the "interior" cell types, somite and pronephros, and the "boundary" cell types, notochord and lateral plate, located at the two ends of the developmental axis. The interior cell types are presumed to arrest their own development by secreting an inhibitor morphogen which reacts with the activator. The boundary cell types must have different developmentai courses. Since there is no precursor to start notochord, the activating morphogen for notochord is assumed to be independent of developmental events and to evolve in tandem with another independent species according to a Turing instability. Lateral plate, being the last cell type, does not need to arrest its own development because there is no cell type following it. Therefore it releases neither inhibitor of itself nor activator

556

K. YATES AND E. PATE

for another cell type. All morphogens have their diffusivities regulated by a control species chemical and no chemical leaks out of the system. The organizing tissue located at the dorsal end of the axis destroys the control species, while the ventral end is impermeable to all chemicals including the control species. With a constant production of the control species in all cells and constant diffusion of the control species throughout the organism, a monotonic gradient is set up which is proportional to the square of the axis length. By making all other diffusivities proportional to the concentration of the control species, the morphogen concentration profiles are guaranteed to be independent of axis length. The simulated development would then take the following steps. After a rapid equilibration of the control species, a monotonic gradient of notochord activator develops. Notochord tissue then arises on the dorsal end of the developmental axis, where notochord activator is above threshold. The notochord tissue subsequently releases somite activator which diffuses down the axis. Somite begins developing and releases both somite inhibitor and pronephros activator. As the inhibitor arrests somite differentiation, pronephros activator exceeds threshold and pronephros begins developing. Pronephros releases its own inhibitor and lateral plate activator. Finally, pronephros differentiation is halted by inhibition and lateral plate fills the remaining region of the developmental axis. Summarizing, as suggested by Cooke (1983), a developmental wave starts at the blastoporal lip and cascades ventrally through the mesoderm. Notochord appears first, with somite, pronephros, and lateral plate emerging sequentially. The model is expressed mathematically as a system of parabolic partial differential equations :

Wt = D w W ~ + g w U, = Dv(WUx):r

UV2-k2U

V~ = Dv(WVx)x+k~ UV 2+ k 2 U - k 3 V; (As)t = DAs[ W(As)x]x + gas N - h A s A s - i,~.IsAs

(Ap), = DAp[W(Ae)x]x +gApS-- hApAe- iAelpAp (At), = DAL[ W(Ac)x]x +gALP-- hA~AL ( Is) t = Dis[ W( [s) x]x -[-g ,s S - h zsls

(1~,), = Die[ W(Ie)~]x + g i p P - hzele, Nt = g N H ( V - Vc,

1)M

St = g s H ( A s - A s , c, 1)M

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

557

Pt = ffpH(AI,-- Ae,c, 1)M Lt = 9LH(AL--AL,c, 1)M Mt=-[N~+St+Pt+L,];

0
t>0.

(8)

The equations represent balance of mass relations for both chemical species and cell types. Note that W, U, V, As, Ap, AL, Is, Ie, N, S, P, L, and M are all functions of both x and t, and the x and t subscripts denote partial differentiation. W is the control species which regulates the diffusion of all of the other chemicals in the model. U and V are two interacting chemicals which are independent o f developmental events. V is used as the notochord activator in the simulations while U plays no direct role in tissue differentiation. As, Ae, and AL are activator chemicals for the somite, pronephros, and lateral plate cell types, respectively. Is and Ip are inhibitor chemicals for somite and pronephros, respectively. Cells are assumed to be one of five types and the variables used to describe their evolution represent fractions of all cells at a given (x, t) devoted to the respective type : N is the notochord fraction, S is the somite fraction, P is the pronephros fraction, L is the lateral plate fraction, and M is the undifferentiated fraction. Because the cell type variables are defined as fractions, the sum of N, S, P, L, and M for any (x, t) is normalized to 1. Dw, Du, Dv, D~s, DAe, DAr, D~s, and Dzp are diffusion coefficients, while gW, ki, fla x, gAp, flAir, 9I S, filp, h~s, hap, hAz, his,, hip, iAs, and iAF a r e all kinetic coefficients, fiN, fiX, ffP, and 9L are rates of differentiation and V,., As.c, Ap.c, and AL.,. are threshold values of the activating chemicals. The modeling of threshold activation is done via Heaviside step functions. The form H(c~, 1) represents an "on/off switch" function which takes the value of 0 for e < 0 and 1 for c~ > 0. Y is the length of the system, and x = 0 is correlated with the dorsal end of the developmental axis while x = 50 is correlated with the ventral end. The U, V, As, Ae, AL, Is. and Ie morphogens are all assumed to obey noflux boundary conditions at both ends, while W has a no-flux boundary condition at the ventral end but is assumed held zero at the dorsal end :

W, WUx, WVx, W(As)x, W(e)x, W(AL)~, W(Is)x, W(Ip)~ = 0, Wx, WUK, WV~, W(As)x, W(AF)x, W(ADx, W(Is)~, W(Ip)~ -- 0,

at x = 0, at x = 5 ~ (9)

No-flux b o u n d a r y conditions on an interval are the mathematical equivalent of reflecting boundary conditions at the endpoints of a two joined, symmetrical semicircles, and the latter is the literal geometry of the system. (No boundary conditions are needed for the cell types since, without diffusion terms, only initial conditions are needed to m a k e their equations welI-posed.)

558

K. YATES AND E. PATE

Under appropriate assumptions about the parameters (Othmer and Pate, 1980), W reaches steady-state well before the morphogens evolve significantly and is thus set at the steady-state in the actual simulations. U and V are set to (nonzero) constant steady-states. The rest of the m o r p h o g e n s - - A s , A,, AL, Is, and I p - - a r e assigned zero initial conditions. All cells are assumed undifferentiated at the outset of normal differentiation and thus M is set to 1 while N, S, P, and L are set to 0 :

W(x, O) = (gw/D w) (Lx-- x2/2) U(x, O) = kok2/(k~k 2 + k2k 2) V(x, O) = ko/k3 A s(x,O), A p(x, 0), A L(x, 0), Is(x, 0), Ip(x, O) = 0 N(x, 0), S(x, 0), P(x, 0), L(x, O) = 0

(lO)

M ( x , O) = 1.

The analysis of this system is facilitated by introducing dimensionless variables:

z = tk3,

z = x/~,

tc=k2/k3,

6 2 = kok,/k3, 2 3

Dw = Dw/(k3~-q~2),

[w, u, v, as, ap, a,, bs, bp] = [W, U, V, As, Ap, AL, Is, Ie]/x/k3/kl, [vc, as,c, ap,c, at.J = [V~, As.~, Ap.c, AL,~]/.~k3/k,,

[Du,Dr, Oa~,Oap, Oa,, Ob s, Obp] = [Du, Dr, DAs, DA., DA~, Dis, D j / ( ~ 2 kx/~k3), [gw, ga~,gap, ga,, gb.,, gbfl = [gw, gA~, gA,, gA~, g's' gly(k3~/k3/k,),

[r r

= [iAs, i j / x / k j k ~ ,

[ha. hap, ha,, he, hb~] = [hAs, hap, hAL, his, hip]/k3, [g., gs, gp, gt] = [gu, gs, gp, gL]/k3,

(1 l )

and, to standardize the use of lower case letters for the dependent variables in the dimensionless system,

[n,s,p,l,m] = [ N , S , P , L , M]. Note that 1/k3 is the time scale, ~ is the length scale, and x/k3/kl is the chemical concentration scale chosen to nondimensionalize these equations. The actual numerical choices used in the simulations are listed in the caption

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

559

to Fig. 1. The partial differential equations become : w~ = DwwzzWgw u~ = Du(wUz)z + 6 -- uv 2

-

-

t~lg

V~ = Dv(wvz)z + uv2 + tcu-- v (a,), = Das[W(a,)z]z + gasn -- h , , f l s - i,,flsbs (ap), = D,,,, [w(ap)j~ + 9,,pS- h,,ea p - ia~apbp (a~).~ = Da~[w(af)z]~ + ga,p--ha,at,

(bs)~ = Dbs[W(bs)z]~ +gb s - h~ b, (bp), = Db,,[w(b,.,)z]~ + gb,,P - hb,,bp n~ = gnH(v--vv, 1)m sT = g s H ( a ~ - a .... 1)m p~ = 9 p H ( a p - - a p . , 1)m l~ = 9lH(at--al, c, 1)m rn,, = - - [ n , + s~+p~+l~] ;

0
r>0.

(12)

The b o u n d a r y conditions convert to: w, WUz, wv~, W(as)z, W(ap)z, w(al)z, w(bs)~,w(bp)~ = 0,

atz = 0,

%,WUz, WV~,W(as)z,W(ap)z,W(at)~,w(b~)~,w(bp)z = 0,

atz = 1.

(13)

The initial conditions become: w(z, O) = (g~/Dw)(Z--Z2/2)

u(z, 0) = a/0c + a 2) v(z, O) = a a,(z, 0), ap(Z, 0), a,(z, 0), bs(z, 0), bp(z, O) = 0

n(z, o), s(z, O),p(z, o), t(z, o) = o m ( z , O ) = 1.

(14)

There are two fundamental hypotheses made regarding development. One is that undifferentiated cells (m) are monitoring the concentrations of the four activating chemicals (v, as, ap, and a3 and when any one of the latter increases above a threshold value, differentiation to the relevant cell type commences.

560

K. Y A T E S A N D E. P A T E

The other is that the differentiated cell types produce chemicals that dynamically affect the developmental course. Physically, equations (12)-(14) model the following process. The u and v chemicals form an independent chemical scheme utilized to initiate development of notochord. The actual kinetics used are not crucial to the conclusions as all that is needed is a monotonic profile for the notochord activator. A previously-characterized, positive-feedback scheme (Ashkenazi and (a)t.0

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LEGEND n V(NOT.ACTIVATOR)0 SOM.ACTIVATOR ~ 0 u ASOM INHIBITOR ~

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A CASCADING DEVELOPMENTMODEL FOR AMPHIBIAN EMBRYOS

561

O t h m e r , 1978) is e m p l o y e d here. T h e d e v e l o p m e n t a l sequence is p r e c i p i t a t e d b y p e r t u r b i n g slightly the spatially u n i f o r m b u t t e m p o r a l l y unstable steadystates used as initial c o n d i t i o n s for u a n d v. By p e r f o r m i n g a linear stability analysis o f the u-v e q u a t i o n s in (12)-(14), sufficient c o n d i t i o n s f o r the parameters c a n be f o u n d w h i c h m a k e these steady-state solutions : i) u n s t a b l e in time to small p e r t u r b a t i o n s , a n d ii) such t h a t o n l y the n = 1 m o d e (i.e. the first spatially n o n c o n s t a n t , m o n o t o n i c a l l y - s h a p e d t e r m in a F o u r i e r series e x p a n s i o n o f the d i s t u r b a n c e ) grows. T h e u a n d v c o n c e n t r a t i o n s thus evolve to o p p o s e d , m o n o t o n i c profiles as new, stable steady-states. A f t e r the n o t o c h o r d a c t i v a t o r (v) has r e a c h e d its m o n o t o n i c stable steady-state distribution, u n d i f f e r e n t i a t e d cells (m) begin differentiating to n o t o c h o r d (n) at rate ga,, w h e r e v e r v exceeds its t h r e s h o l d value, Vc. As n o t o c h o r d develops, it releases somite a c t i v a t o r (as) at a rate g~- This chemical diffuses d o w n the axis, with D~, being its diffusion coefficient, decays linearly with itself at a rate has, a n d is d e g r a d e d b y somite i n h i b i t o r (b~) at a rate i,. S o m i t e (s) develops at rate gs where as is a b o v e the t h r e s h o l d value o f a~,c. S o m i t e i n h i b i t o r (bs) is p r o d u c e d b y d e v e l o p i n g somite cells at rate gb~, decays linearly with itself at rate hbs, a n d diffuses with rate Dbs. S o m i t e tissue also stimulates the p r o d u c t i o n o f p r o n e p h r o s a c t i v a t o r (ap) a n d the latter develops in an a n a l o g o u s w a y to somite activator. P r o n e p h r o s (p) develops in r e s p o n s e to p r o n e p h r o s a c t i v a t o r (ap). As p r o n e p h r o s develops, it releases p r o n e p h r o s i n h i b i t o r (bp) which degrades p r o n e p h r o s activator. P r o n e p h r o s cells also p r o d u c e lateral plate a c t i v a t o r (a~) which induces lateral plate (l) d e v e l o p m e n t . (

Figure 1. (a-c) Development of notochord and relevant morphogens. Left end of horizontal axis corresponds to the dorsal zone and right to the ventral zone. Left vertical axes give cell type fractions, right vertical axes give concentrations of chemical species. All times are transformed to dimensioned time units. Symbols and shadings for chemical species and cell types are given at bottom of figure. (a) 0 hours. Initial conditions : notochord is set to 0 and the u and v morphogens are perturbed slightly from spatially homogeneous, unstable steady-states. (b) 8.75 hours. Notochord development in progress (c) 10 hours. Notochord development complete. (d-g) Simulation of somite development and relevant morphogens. Notochord, somite tissue, somite activator and somite inhibitor shown. (d) 10 hours. Note that panes (c) and (d) are at the same time with different morphogens shown. (e) 13.75 hours. (f) 17.5 hours. (g) 22.5 hours. Somite development complete. (h) 42.5 hours. Development of all cell types complete with final cell type distributions shown. No morphogens are shown. Parameters: Dw = 1.2, Oh. ~ 0.12, gw = 2.4, gb~ = 0.1, D~ = 0.83, hb~ = 0.04, 6 = 1.0, Db~ = 0.84, D~ = 0.05, gb, = 0.5, x = 0.01, hbp = 0.2, Da, = 0.06, y, = 5.0, g,, = 0.53, 9s = 1.5, h,s = 0.025, 9e = 0.5, ia, = 0.055, gt = 5.0, Do, = 0.036, vc = 2.65, gap = 0.072, as,,. = 0.4, h,, = 0.012, ap,c = 0.55, ias = 0.14, al,c= 3.5, Dot = 0.24, g,~ = 1.0, h, =0.005, k3~= 1200 s, Lie=0.1 cm, and kx/~kl3/kl= 1.0 (arbitrary units). Length scale implies horizontal axis may be interpreted as 1 mm in length. Time and length scales chosen so that dimensional diffusion coefficients correspond to maximum diffusion rates on the order of 7.0 x 10-6(cm2/sec), or less. =

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Finding exact solutions for the partial differential equation system in (12)(14) is not feasible: the system is nonlinear and contains discontinuous coefficient functions. Therefore approximate solutions were obtained numerically. A double-precision (16 decimal digits) F O R T R A N program employing the method of lines with centered, second-order differencing of the diffusion terms was utilized on the (dimensionless) partial differential equation system. The resulting ordinary differential equations were analyzed using a version of Gear's Method suitable for stiff systems of ordinary differential equations (Gear, 1971). A large number of spatial mesh points were used (typically 200230) because of the "moving discontinuity" built in to the growth terms in the governing equations for each cell type and for many of the morphogens. All computing was done on an IBM 3090 computer with the Conversational Monitor System (CMS) operating system. Summarizing the mathematical framework : initially w is set to steady-state, u and v are set to small perturbations of (nonzero) constant, unstable steadystates, and all other chemicals are set to zero. The cell types start out undifferentiated--m is set to one while all other cell types are set to zero. With appropriate choices of the parameters, u and v begin developing to opposing monotonic, steady-state profiles which in turn drives the evolution of the entire system.

3. Results. The implications of the model are illustrated by several simulations. First, it is necessary to show that the model is capable of establishing the correct mesodermal pattern observed in normally differentiating (hereafter called normal) amphibian embryos. Next, the size-regulation that occurs upon removal of ventral blastomeres at the early gastrula stage is confirmed. Then the experiment of transplanting an extra blastoporal lip on the ventral side of an early gastrula and obtaining a bipolar pattern is discussed. Finally, we simulate the experiment in which grafting of somite from a more developed gastrula to the ventral side of an early gastrula results in a pattern noticably deficient in somite. The structure of the model allows for the three experiments to be simulated by changing only appropriate boundary and/or initial conditions relative to normal development, while keeping all evolutionary parameters fixed. Using equations (12)-(14) the normal development simulation shown in Fig. 1 illustrates the interaction of emerging cell types and the sequential development hypothesized in the model. Shadings for the various cell types are given at the bottom of the figure. In Fig. la-c, we see the emergence of notochord along with the evolution of notochord activator v(U]) and the companion chemical u ( 9 Cell type fractions are given on the left vertical axis while morphogen concentrations are given on the right axis. Note that the horizontal axis has been scaled so that the units can be interpreted as

A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

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millimeters; the developmental field is simulated as being 1 mm long. The dorsal and ventral ends are correlated with z = 0 and z = 1, respectively. Though the dimensionless system of partial differential equations was used for numerical integration, all times given in the ensuing discussion will have been converted back to dimensioned time units, using k31 = 1200 seconds. Initially, the u and v concentrations are perturbed slightly away from their spatially homogeneous but unstable steady-states using a high-frequency, small amplitude spatial oscillation (Fig. la). After 8.75 hours, the v profile has risen above threshold (v~ = 2.65) at the extreme dorsal end of the axis and notochord differentiation has begun (Fig. lb). By 10 hours, u and v approximate new, monotonic, stable steady-state profiles and notochord development is complete (Fig. lc). Notochord occupies the most dorsal 4.5% of the axis. The stage is now set for somite differentiation. Notochord tissue, somite tissue, somite activator a~(O), and somite inhibitor bs(A), are depicted in Figs 1d-g. Differentiating notochord cells have been releasing somite activator (O) into the mesoderm (Fig. ld). Note that the time in Fig. ld is 10 hours, the same as in Fig. lc. For clarity of presentation only the morphogens relevant to somite differentiation are depicted. By 13.75 hours (Fig. le), the somite activator concentration rises above threshold (a,.~ = 0.4) and somite tissue has begun to differentiate. The somite tissue releases somite inhibitor (A) which is also visible at this time. After 17.5 hours (Fig. lf), a larger somite region has emerged but the growth of the somite activator profile has been slowed by the increase of somite inhibitor. By 22.5 hours (Fig. lg), a complete somite region has been established, filling 39.5% of the developmental axis. Further somite differentiation is not possible because the still-increasing somite inhibitor profile has arrested the somite activator profile. The developmental sequence continues in an analogous fashion. Somite tissue releases pronephros activator, sparking pronephros differentiation. As pronephros emerges, it releases its own inhibitor and lateral plate activator. Eventually, just as with somite tissue, pronephros formation is halted by inhibition and the remaining uncommitted tissue differentiates into lateral plate. For brevity, plots of pronephros and lateral plate simulated development have been omitted. Complete details can be found in Yates (1987). Figure lh illustrates the final dorsal-to-ventral sequence of notochord, somite, pronephros, and lateral plate cell types achieved. Note that the left vertical axis again demarcates cell type fractions and no morphogens are depicted. The simulated development is sequential. Notochord development is completed by 10 hours, somite by 22.5 hours, pronephros by 32.5 hours, and lateral plate by 42.5 hours. The proportions achieved in the final pattern are 4.5% notochord, 39.5% somite, 9.5% pronephros, and 46.5% lateral plate, consistent with experimental data for Xenopus laevis (Cooke, 1983).

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While the time scale for the model was set so that development finished in less than 48 hours and the normal developmental axis length is 1 mm, other length and time scales can be accommodated. The size reduction experiment can be simulated by simply changing the length scale, L~. Figure 2a shows the cell types achieved by setting the developmental length to 0.5 m m (time is 42.5 hours). The symbols for the cell types are as in Fig. l h. The proportions devoted to each cell type are exactly the same as in the normal development simulation. As discussed previously, size regulation of the model is guaranteed by the length-modulated diffusion rates. In carrying out the nondimensionalization of the governing equations (8)-(10) to obtain the dimensionless equations in (12)-(14), 5~ actually appears in two ways : explicitly in the diffusion rates and implicitly in the step function growth rates for the activators of somite (as), pronephros (ap), and lateral plate (a~), the inhibitors o f somite (b,) and pronephros (bp), and the cell types themselves (n, s, p, a n d / ) . But the key to making size-regulation depend solely on modified diffusion rates is that, since notochord activator (v) and its companion (u) are made independent of all cell types, their governing equations depend on 5f only

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through their diffusion terms. The presence of the control species (w) is therefore all that is required to size-regulate the u and v profiles. In essence, size-regulatory capability itself then "cascades" through the other governing equations. The complete developmental sequence and additional details are provided in Yates (1987). The practical implication of the hypothesized regulated diffusion is that diffusion is slowed for shorter developmental systems, thereby keeping the relative speed of diffusion the same and ensuring the same pattern for arbitrary developmental lengths. The transplanted blastoporal lip experiment is simulated by changing the boundary condition for the control species, w, on the ventral end. The organizer tissue has been assumed to rapidly degrade the control species, so the transplant corresponds mathematically to both boundaries having zero control species concentrations. As discussed in Pate (1984), a linear stability analysis of the u and v solutions reveals that the same parameters which yield monotonic profiles for u and v with the "one organizer" boundary conditions for w yield bipolar profiles for u and v with these new, "two organizer" boundary conditions for w. The developmental sequence proceeds from both ends toward the middle (Yates, 1987) and culminates in a mirror-symmetric pattern (Fig. 2 ; time is 42.5 hours). Notochord, somite, pronephros, and lateral plate form two half-sized miniatures with an axis of symmetry at the dorso-ventral midline. The total proportions of the developmental fietd devoted to each cell type are 4.5% notochord, 40% somite, 10% pronephros, and 45.5% lateral plate, again in good agreement with experimental results (Cooke, 1981). The final experiments analyzed were somite t,'ansplants. The transplants were simulated by adding 0.05, 0.1, and 0.15 mm to the ventral end of the developmental axis and setting the initial conditions of somite and undifferentiated tissue in this region to 1 and 0, respectively. Since somite tissue releases two morphogens, somite inhibitor and pronephros activator, a number of things could possibly happen when a zone of somite exists from the outset of development. A primary expectation would be for somite inhibitor to diffuse dorsally through the system, affecting the somite activator profile, and thus the amount of host tissue committed to somite. Another expectation would be that pronephros activator produced by the transplant could affect development. One possibility would be that pronephros activator diffuses down the system and enlarges the pronephros region (relative to normal development) induced by host somite tissue. Even more intriguing, if enough pronephros activator is released by the somite transplant, a zone of pronephros could be induced next to the transplanted somite. In fact, simulations suggest that all of these events occur in the somite transplant experiments and we dilineate the cases. We first compare somite development in the normal and transplant cases. Figures 3A-D, ~ d contrast how somite activator and somite inhibitor affect

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K. YATES A N D E. PATE

development in the normal and 0.15 mm somite transplant cases. In each figure, somite in the normal development and somite in the simulated transplant development are pictured but for the sake of clarity, notochord--the source of somite activator--is not. Figures 3 A - D depict somite activator (O) in the normal development simulation and somite activator ( 0 ) in the transplant simulation. Figures 3a-d depict somite inhibitor (/%) in the normal

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A CASCADING DEVELOPMENT MODEL FOR AMPHIBIAN EMBRYOS

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development simulation and somite inhibitor (A) in the transplant simulation. Note that the transplant and normal development simulations are overlayed on each pane. Each pair of figures (i.e. Figs 3A, 3a, Figs 3B, 3b, etc.) corresponds to the same time in simulated development. Simulations for the case of normal somite development are plotted on the spatial domain [0,1]. Simulations for the transplant experiment are plotted in the same pane on the spatial domain [0, 1.15] with the transplant occupying the right-hand 15%. This allows direct comparison of morphogen concentration profiles. Somite activator has not appeared after 2.5 hours (Fig. 3A) in either the normal or transplant simulation because its source, notochord, has not yet developed. However, we do see somite inhibitor emerging in the transplant simulation at this time (Fig. 3a) because it is being released at the right-hand end by the transplanted somite tissue. By 13.75 hours, somite development has commenced in both the normal and the transplant simulations (Fig. 3B). Notice that the somite activator profile in the transplant simulation is actually a little higher than in the normal simulation (Fig. 3B) at this point in time. The somite region in the transplant simulation is correspondingly a little larger than the normal somite region. This may seem counterintuitive until one considers the following fact: because the length of the developmental axis is increased from 1 to 1.15 mm in the transplant simulation, the notochord region in the transplant case will be slightly larger than in the normal case. This means that the source term for somite activator (notochord ; not pictured in figures) is slightly stronger in the development of the transplant simulation

Figure 3. Superimposed development of somite and s0mite activator (A-D) and somite inhibitor (a~l) in both normal and 0.15 mm somite transplant simulations. Left axes give cell fractions, right axes give chemical concentrations. Two simulations are plotted in each figure. Simulations for the case of normal somite development are plotted on the spatial domain [0, 1]. Simulations for the transplant experiment are plotted in the same pane on the spatial domain [0, 1.15] with the transplant occupying the fight hand 15%. This allows for direct comparison of morphogen concentration profiles. Symbols are defined at the bottom of the figure. Note that since two simulations are superimposed on individual panes, crosshatched regions denote overlap of normal and transplant somite. Panes (A,a) 2.5 hours. None of the dependent variables have moved from their initial conditions. No somite activator is visible in either simulation because notochord (not shown), the source of somite activator, has not yet evolved. Panes (B,b) 13.75 hours. Notochord (not shown) development is complete in both simulations and somite activator profiles are visible. Somite activator profile is larger in the transplant simulation than in the normal simulation and correspondingly more somite has developed in the transplant simulation than in the normal simulation. Panes (C,c) 17.5 hours. The normal somite region has overtaken in size the (host) somite region in the transplant simulation as described in text. Panes (D,d) 22.5 hours. Somite development is complete in both simulations. Parameters as in Fig. 1 with boundary conditions modified as described in text.

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than in normal development simulation. Figure 3b foreshadows why the early lead of the host somite region in the transplant simulation does not last. With somite inhibitor being produced both by the somite transplant and by the newly-formed host somite, a relatively large profile of somite inhibitor has emerged. In contrast, the only somite inhibitor in the normal development system is the small amount produced by the the newly-formed host somite. After 17.5 hours, the host somite region in the normal simulation is noticeably larger than the host somite region in the transplant simulation (Fig. 3C). The reason for this is that the much larger amount of somite inhibitor in the transplant system (Fig. 3c) has had time to significantly slow the growth of the somite activator profile (Fig. 3C). Somite development is complete in both simulations by 22.5 hours (Fig. 3D). In each case, sufficient somite inhibitor (Fig. 3d) is present at this time to stop the growth of the somite activator profiles (Fig. 3D) and thus arrest somite development. Let us now consider the entire developmental sequence in the 0.15 mm transplant simulation. Figures 4a-f illustrate the emerging cell types: notochord, somite, pronephros, and lateral plate. The initial state is shown in Fig. 4a; all cells are undifferentiated except the somite transplant occupying the ventral 0.15 mm of the system. By 8.75 hours, notochord activator (not shown) has evolved to its monotonic profile and notochord has emerged at the dorsal end of the axis (Fig. 4b). At 12.5 hours, something unexpected has occurred: pronephros development has begun next to the somite transplant (Fig. 4c). The ectopic pronephros develops because enough pronephros activator (not shown) is released by the somite transplant to exceed the threshold value. Also visible at this time is the beginning of the regular, host somite region. Development of both the ectopic pronephros region and the regular somite region is complete by 22.5 hours (Fig. 4d). Notice that, though no morphogen profiles are shown, what is happening in the undeveloped region of the system at this time is that both the transplant and the host regions of somite are releasing pronephros activator while the ectopic pronephros is releasing pronephros inhibitor. Figure 4e shows the result of these competing influences : a smaller zone of pronephros (as compared to normal development--see Fig. lh) has emerged. The pronephros region is smaller both because of the pronephros inhibitor produced by the ectopic pronephros and because the regular host somite region, the primary source of pronephros activator, is smaller than in the normal case. The remaining tissue is induced by the pronephros cells on both sides to become lateral plate (Fig. 4f; time is 32.5 hours). The actual proportions developed are: 5% notochord, 23.5% somite, 4.5% pronephros, 60% lateral plate, and 7% (ectopic) pronephros. (The percentages listed here and in the following are of the initially undeveloped, host tissue and exclude the amount of the somite transplant.) The final state of the cell types in the 0.1 mm and 0.05 mm transplant

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simulations are pictured in Figs. 4g and 4h, respectively. The cell types are depicted in the same manner as in Fig. 4f. The proportions achieved in the 0.1 m m transplant simulation (Fig. 4g) are 5% notochord, 29.5% somite, 7.5% pronephros, 53.5% lateral plate, and 4.5% ectopic pronephros. The proportions achieved in the 0.05 m m somite transplant simulation (Fig. 4h) are 5% notochord, 37% somite, 11.5% pronephros, and 46.5% lateral plate. To more easily compare the transplant simulations, the amount of host tissue devoted to each of the mesodermal cell types in each of the transplant cases and the normal development simulation is plotted in Fig. 5. Notice that the horizontal axis is the size of the somite transplant as a percentage of available host tissue, with the " 0 % " transplant being the normal development simulation. The vertical axis is the percentage of available host tissue developed to each cell type. Several interesting features deserve comment. The degree of somite inhibition is clearly correlated with the size of the transplant. The host somite developed occupies 37%, 29.5%, and 23.5% of available host tissue in the 0.05, 0.10, and 0.15 m m transplants, respectively. These results compare with the 39.5% somite region obtained in the normal development simulation and suggest that larger transplants cause smaller host somite regions to develop. The explanation for the phenomena would be that the larger somite transplants are greater sources of somite inhibitor and thus suppress somite formation in the host tissue to a greater degree than do smaller transplants. This is consistent with experimental data obtained by Cooke (1983). 0.6

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The effects of somite transplants on subsequent pronephros development are surprising and raise some interesting questions. The emergence of ectopic pronephros regions next to the two larger somite transplants (Figs. 4f and 4g) suggests that somite tissue, while lacking truly independent "organizer" properties, may well be capable of inducing pronephros tissue development. No ectopic pronephros developed in the smallest transplant simulation, the 4.5% of the host tissue nearest the 0.1 mm somite transplant became pronephros, and the 0.15 mm transplant induced an ectopic pronephros region filling 7% of available host tissue (Fig. 5). The results support the possibility that there is a critical mass of somite transplant (between 0.05 and 0.1 mm as measured in these simulations) above which ectopic pronephros tissue is induced. Indeed, some pronephros tissue was observed near the larger and better developed somite transplants performed by Cooke (1983). It is interesting to note the effects of somite transplants on pronephros development next to the somite regions in the host, in the zone normally fated to be pronephros at the dorso-ventral midline. Pronephros regions of 11.5%, 7.5%, and 4.5% of the available host tissue emerged in the 0.05, 0.10, and 0.15 mm somite transplant simulations, respectively (Fig. 5). These results compare with the 9.5% region obtained in the normal development simulation. In the two larger transplant cases, where ectopic pronephros regions emerged before regular host pronephros development began (Fig. 4d), smaller pronephros regions than in normal development arose. The reasons for this, as indicated earlier, are that pronephros activator contributed from the relatively distant somite transplants is more than offset by the smaller local pronephros activator sources (i.e. the smaller host somite regions) and by pronephros inhibitor produced by the ectopic pronephros inhibiting regular host pronephros formation. In the smallest transplant case, however, the regular pronephros zone is actually larger than in the normal development simulation (11.5% vs 9.5%). There are three factors combining which lead to this result. The primary source of pronephros activator, the host somite region, is not much smaller than it is in the normal case (37% vs 39.5%; Fig. 5). Also, additional pronephros activator is diffusing dorsally from the transplant. Finally, because no ectopic pronephros develops, there is no pronephros inhibitor in the system before regular pronephros development commences.

4. Discussion. A cascading model capable of simulating several diverse developmental phenomena in amphibian embryos has been presented. The capability for dynamical contributions by developing cell types to influence subsequent developmental events is explicitly built into the model. Only one set of initially specialized cells are hypothesized--the dorsal blastoporal lip. Unlike most previous models, each cell type is assumed to have its own activating morphogen. The assumption that the development of each cell type

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is initiated by the dorsally neighboring cell type suggests that a "wave" of development begins at the dorsal midline and cascades ventrally through the mesoderm (Cooke, 1983). By postulating a diffusion-regulating "control species" (Othmer and Pate, 1980) along with sequential pattern development (Meinhardt and Gierer, 1980), the ability to simulate size-regulated development and transplanted blastoporal lip experiments has been demonstrated. The inclusion of inhibiting morphogens enables the model to simulate the somite inhibition phenomenon. The further possibility that ventral somite transplants may induce ectopic pronephros tissue has been successfully simulated and these results suggest that a sufficiently large somite region is necessary for this induction. Other feedback signals may be operating in developing amphibian embryos and the structure of the model could easily embrace such signals. For example, it may be the case that pronephros tissue inhibits further pronephros differentiation. The model is capable of simulating such inhibition and this possibility is discussed further below. The results of the developmental simulations presented here suggest that further experimental work is needed to clarify the mechanisms involved in mesodermal tissue development. The mechanism of inhibition hypothesized in this model is a relatively simple o n e - - a cell type produces inhibitor which degrades the associated activator. Another possibility is that cell types themselves degrade their own activators. A model in which somite inhibits itself directly by breaking down somite activator was considered. Somite activator evolution would be governed by an equation of the form (compare with equation for As in (8)) :

(As)t = DAs[W(As)x]x +gAsN - h~sAs- iAsSAs. (15) As(x, t) is somite activator, OAs is a diffusion rate, W(x, t) is the control species, gas is a growth rate, h~s is a decay rate, iAs is an inhibition rate, N(x, t) is the notochord cell fraction, and S(x, t) is the somite cell fraction. The advantage of this type of inhibition is that no inhibiting chemical is assumed. However, an extensive analysis of this scheme suggested that such inhibition simply could not be made strong enough to obtain both an appropriately sized somite region in normal development simulations and significant inhibition of somite in transplant simulations. If such a mechanism is to work, inhibition must be weak enough during normal development that it does not stop somite activator too soon : approximately 40% of the developmental system should become somite. Yet when inhibition is made this weak, relative to realistic diffusion rates and the biologically observed time scale, the ventrally located somite transplant is too distant from the dorsal somite activator source (notochord ceils) to act as an effective sink for somite activator. The more complex inhibitory scheme used in the present model works because somite transplants do not inhibit passively by degrading somite activator from a distance, but actively by having somite inhibitor diffuse into the presumptive somite region

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in the host. Still another alternative for a mechanism of inhibition would be a two-switch development scheme. Differentiation to a given cell type would depend on an activator quickly evolving and exceeding a threshold while a slower-evolving, inhibitor morphogen is below a threshold. When the inhibitor rises above threshold, differentiation would be shut off. A theoretical question which should be addressed in consideration of a reaction-diffusion model for pattern formation is the sensitivity of the final pattern to small changes in parameters. In the course of investigating this model we observed, as have others, that the sensitivity of the size of a cell type region correlated strongly with the shallowness of the activator gradient near the threshold value. Due to the specific activator kinetics chosen, the profiles of all activator species have a decaying exponential form. In particular, the activator profiles are steepest near their sources (the dorsal cell type neighbor) and become increasingly shallow toward the ventral end. Thus the somite region is more sensitive to parameter changes than the much smaller notochord and pronephros regions. However, as can be seen from Fig. 1, the somite activator gradient at the ventral end of the somite zone remains sufficiently large to leave the model fairly insensitive, even at this location. The mesodermal development simulations presented here suggest several possible lines of experimental research. The zones of ectopic pronephros observed near somite transplants by Cooke (1983) and simulated by the model are particularly intriguing. A key question is whether the pronephros is of host origin, as suggested by the model, or of donor origin. If the former case is true, this would be strong evidence that somite tissue actively affects pronephros development. If the latter case is true, then ectopic pronephros in somite transplants must be donor tissue already programmed to become pronephros and thus its significance would be reduced to experimental artefact. More refined experiments may be possible with the advent of better cell labelling techniques. If somite transplants do in fact induce host tissue to become pronephros, it would be interesting to know if the ectopic pronephros differentiates before the normally-placed pronephros does, as the simulated results presented here suggest (Fig. 4d). Cooke (1983) mentions that the same type of inhibition established for somite may also occur for pronephros. A ventral transplant of pronephros could be easily simulated by the model. By changing pronephros initial conditions on the ventral end, one would predict that pronephros inhibitor would diffuse dorsally in the system and that less host pronephros would emerge. Indeed, in the simulations of somite transplants presented in Results, supernumerary pronephros induced by somite transplants did seem to inhibit later host pronephros development. Pronephros inhibiting itself may be an experimentally testable hypothesis. Size-regulating development is a fascinating phenomenon. For the many developmental systems known to possess size-regulating ability--e.g, hydra,

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slime mold, amphibian embryos--the mechanism for "size sensing" must be fundamental to any model of those systems. Evidence of size-regulation of mesodermal pattern parts in response to surgical removal of ventral blastomeres has already been noted (Cooke, 1981, 1982). But the fact that embryos of Xenopus laevis, the amphibian most commonly utilized by embryologists, vary by a size range of as much as three in normal development is also evidence for the existence of size-regulated development (Cooke and Webber, 1985b). Further evidence of a size-regulating mechanism in amphibian development is provided by experiments done on 2-cell embryos. First cleavage for normally developing Xenopus laevis embryos occurs along the head-to-tail plane of bisymmetry ; the embryo splits into right and left halves. When these two cells are separated experimentally, each develops independently to the tadpole stage. The relevant point is that each embryo has half of the normal mesoderm and the mesodermal cell types are normally proportioned (Kaguera and Yamana, 1983 ; Cooke and Webber, 1985b). The mechanism for size-regulation hypothesized in the model for amphibian embryo development may unfortunately be hard to confirm experimentally. Very few morphogens have actually been identified in developmental systems at the present time. This makes the further assumption of a chemical species regulating diffusion of these hypothesized morphogens a difficult one to test. Since an implication of regulated diffusion by a control species is that the rate of diffusion increases toward the ventral end of the system (Yates, 1987), experiments testing diffusivities at various sites along the dorsoventral axis might provide evidence for the existence of regulated diffusion. Possible ways in which a control species might influence diffusion include chemical regulation of gap junctions and activation of membrane proteins responsible for the transmission of morphogen-carried information within the cell. Recent experiments have suggested that diffusing chemicals are involved in amphibian embryo development. It has been well established that vegetal cells in the early embryo can induce mesodermal development in cell types not normally fated to do so (Grunz and Tacke, 1986). Vegetal pole cells and animal pole cells, neither of which normally contribute to mesodermal structures, were excised from an early blastula stage embryo and separated by a Nucleopore membrane. Later assay of the animal pole tissue revealed that mesodermal cells had developed, suggesting strongly that an inductive chemical signal was being sent by the vegetal cells to the animal cells. McCaig (1986) has shown that Xenopus presomitic myoblasts tend to elongate perpendicularly to the line of diffusion when placed in vitro near notochord tissue. This experiment suggests strongly that notochord may well be releasing a "somite activator" affecting in vivo development, as hypothesized in the model presented here. Another recent experiment involved injecting antibodies raised against the primary protein in rat liver gap junctions into one cell of an early amphibian

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embryo (Warner et al., 1984; Warner, 1985). These experimenters showed that, after injecting dye into the antibodies-treated cell, the spread of dye into neighboring cells was greatly reduced compared to the spread of dye injected into cells not treated with the antibodies. After further controls suggesting that neither the antibodies nor the dye had deleterious effects on cell metabolism, later assay of embryos with the injected antibodies showed defective development in the region populated by the progeny of the antibodies-treated cell. While certainly not conclusive, these experiments do provide evidence that morphogens are involved in the intercellular communication of developmental information. More experimental work elucidating how the biochemical environment of an organism affects development is definitely needed. A modified version of the cascading development model may be applicable to hydra. Hydra exhibit an extraordinary ability to size-regulate a five cell type pattern : pieces of tissue ten times smaller than normal hydra can develop into proportioned miniatures (Bode and Bode, 1984). This fact suggests that there is a specific mechanism which size-regulates development. Another very interesting feature exhibited by mature hydra is the ability to regenerate structures which have been excised. This "back pressure" development effectively means that for hydra, tissue differentiation is always labile. Tissue that has already differentiated to one type can be reprogrammed and converted to another type as the need arises. The cascading development model, tailored here for amphibian embryos, could be modified so that : (i) all cell types, not just undifferentiated cells, can respond to activating morphogens, (ii) a given cell type activates neighboring cell types in either direction, and (iii) a more complex hierarchy of activating and inhibiting morphogens enables the system to replace pattern parts missing from either or both ends of the system as well as any intermediate region. These assumptions would be necessary to allow already-differentiated tissue to "undifferentiate" and develop into a new cell type, and thus enable simulated fragments of hydra tissue to reproportion a new, miniature, complete hydra pattern. A survey of current research on amphibian development reveals evidence that the hypothesis of sequential development is plausible. It is likely that notochord differentiation occurs first in amphibian embryos (Forman and Slack, 1980; Cooke, 1981). Tissue fated to be notochord begins segregating from other mesoderm during gastrulation, and somite segregation follows shortly after this. Pronephros is not recognizable until much later, during the neural fold stage, and lateral plate is a primitively differentiated cell type (Nieuwkoop and Faber, 1967). While these developmental events are seen, it is not as clear when a given cell has irreversibly committed to a particular type. Undoubtedly the committment process is more gradual and continuous than the simple "on/off" switch mechanism postulated here. While some initial tendencies may be specified in mesoderm early in development (Kaguera

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and Yamaha, 1983, 1986; Cooke and Webber, 1985a, 1985b), such specification must be labile until at least the late blastula stage ( F o r m a n and Slack, 1980; Smith and Slack, 1983; Dale and Slack, 1987). A consensus of the current knowledge regarding amphibian embryo development would be that development proceeds as a loose sequence of inductions that enable the embryo to partially or completely recover from a wide variety of experimental interruptions. The proposed cascading development model, including chemical threshold signals for differentiation cues, m a y well be a reasonable approximation to the actual developmental process. Pattern development and underlying positional information questions continue to challenge developmental biologists. Undoubtedly, the true determination of developmental processes occurs at the genetic level. But the virtual certainty that within cells there is interaction between the intercellular chemical environment and the intracellular genetic environment makes chemical morphogens a plausible mechanism for providing developmental cues in developing systems. For the very few biochemical interactions that can be classified as well-understood, specificity of chemical compounds to particular tasks is typically found. This makes the suggestion of a multiple m o r p h o g e n model with cell-specific activating chemicals an appealing alternative to single morphogen models for patterns involving more than two cell types. The model presented simulates several diverse features of development in amphibian embryo mesoderm because it does not rely on strict positional information. By hypothesizing a dynamically interacting system of reaction-diffusion equations to regulate development, the model combines successful aspects o f positional information models with the added flexibility o f allowing emerging parts to provide feedback to the developing system.

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Received 18 O c t o b e r 1988 Revised 3 A p r i l 1989