J. theor. Biol. (2001) 208, 419}438 doi:10.1006/jtbi.2000.2229, available online at http://www.idealibrary.com on

A Model of Primitive Streak Initiation in the Chick Embryo

KAREN M. PAGE*-,PHILIP K. MAINI*?,NICHOLAS A. M. MONKA AND CLAUDIO D. STERN#

*Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles1, Oxford OX13¸B, ;.K., -Institute for Advanced Study, Olden ¸ane, Princeton, NJ 08540, ;.S.A., AMathematical Modelling and Genetic Epidemiology Group, Division of Genomic Medicine, ;niversity of She.eld, Royal Hallamshire Hospital, She.eld S10 2JF, ;.K. and #Department of Genetics and Development, Columbia ;niversity, 701 =est, 168th Street, New >ork, N> 10032, ;.S.A.

(Received on 29 June 2000, Accepted in revised form on 2 November 2000)

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction}di!usion models. In this paper, we propose a new activator}inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both di!usive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying para- meters of the model, two very di!erent types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation. ( 2001 Academic Press

1. Introduction 2000; Schnell et al., 2000). However, to our know- The primitive streak is the "rst axial structure to ledge, there are no mathematical models pro- appear during embryonic development in am- posed, as yet, to describe streak initiation. This is niotes. It forms the sca!old on which the main the aim of the present paper. We begin with body axis will be built. Its initiation is thus of a description of the relevant background biology. fundamental importance in amniote embryonic development. In the chick embryo, the process of 1.1. BACKGROUND BIOLOGY primitive streak initiation shows a remarkable capacity to regulate after experimental manipula- The embryo just prior to streak formation tion. This illustrates the fact that embryonic de- [stage XIII on the staging system of Eyal-Giladi velopment is not simply an automated process & Kochav (EG&K), 1976] is a bilaminar disc (the pre-determined by the genetic code, but is highly blastodisc), composed of a single-cell thick epi- adaptable. There are a number of models recently blast underlain by a looser layer, the hypoblast. proposed to describe the spatiotemporal dynam- The diameter of the disc is approximately 2 mm. ics of streak formation (see e.g. Painter et al., The epiblast consists of a central region, the area pellucida or central disc, surrounded by a peri- ? Author to whom correspondence should be addressed. pheral region underlain by large yolky cells,

0022}5193/01/040419#20 $35.00/0 ( 2001 Academic Press 420 K. M. PAGE E¹ A¸.

FIG. 1. (a) Bird's eye view of the blastodisc, showing the central disc (area pellucida), outer ring (area opaca) and marginal zone, which are present when the egg is laid at stage X. The position of Koller's sickle is also shown. (b) Cells move from posterior lateral parts of the area pellucida towards the central midline between stages X and XIV, and a thickening arises at the posterior edge of the epiblast at stage XIV-2 (the future primitive streak). called the area opaca. Between the two regions is performed to alter the position of this axis or a ring of cells called the marginal zone, see indeed to induce multiple axes. Fig. 1(a). Under normal circumstances the blastodisc de- At stage XIV (EG&K), cells migrate from lat- velops a single primitive streak, which marks the eral posterior parts of the area pellucida towards position of the future backbone of the embryo. the central midline [see Fig. 1(b)], accompanied When the blastodisc is bisected, however, each by a thickening of cells in the posterior of the half (provided it contains at least a portion of the upper layer. This thickening forms the beginnings marginal zone) develops its own streak (see of the primitive streak. The streak then elongates Spratt & Haas, 1960; Bachvarova et al., 1998). anteriorly (becoming thinner) towards the centre Thus, a simple cut is su$cient to trigger the of the blastodisc. It extends 60}75% of the way development of twin embryos. Up to four streaks across the area pellucida. A groove forms in can be produced when the blastoderm is cut into the middle of the streak and cells migrate from quarters, one in each segment. The streak nor- the epiblast at the sides of the streak, through the mally forms adjacent to the posterior marginal groove and into the cavity below. These cells zone and there is evidence (see Bachvarova et al., form the mesodermal and endodermal layers of 1998) that this structure causes streak induction. the embryo. A thickening, called Hensen1s node, All (or at least most) of the marginal zone has the forms at the anterior end of the streak and cells ability to initiate streak formation, but most of also migrate through this node to become head the circumference appears to do so only in the process and notochordal mesoderm. The node absence of a more potent inducer. (Indeed, if then starts to regress posteriorly, leaving a second posterior marginal zone is transplanted notochord structures in its wake. into the lateral margin of a host embryo, then often only one of the two posterior marginal zones present will initiate axis formation; Eyal- 1.2. THE REGULATION OF THE CHICK BLASTODISC Giladi, 1991.) Once the streak has started to form Although, under normal conditions, in the posterior, however, the rest of the marginal antero-posterior polarity is speci"ed fairly early zone seems to lose permanently its initiation po- on by the action of gravity (see Kochav & Eyal- tential and/or the host its capacity to respond, Giladi, 1971), the axis is not irreversibly deter- such that, even if the posterior marginal zone and mined (i.e. external cues may alter its orientation) streak are removed, no new streak will be in- until the appearance of the streak itself. In fact, itiated (Khaner & Eyal-Giladi, 1989; Eyal-Giladi many manipulation experiments have been & Khaner, 1989; Bachvarova et al., 1998). This PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 421 loss of potency/competence occurs almost simul- primitive streak, within approximately 3 hr, taneously with the emergence of the streak in the Skromne & Stern (1998). Given that the circum- posterior. This may be a coincidence and both ference of the marginal zone is about 6 mm and may be triggered by earlier events, but assuming that the di!usion coe$cients of likely morpho- that the loss of potency is caused by a signal gens are very small in vivo, it is unlikely that this emanating from the posterior at the time of could be explained by a di!usive or reac- streak emergence, the signal must spread very tion}di!usion mechanism (Page, 1999). rapidly throughout the marginal zone. In this paper, we present a novel model of The cues which lead to the initiation of the primitive streak initiation, which we believe can primitive streak and to the regulative behaviour explain naturally the key experimental observa- of the blastodisc are poorly understood. The tions and could have much wider applicability in identi"cation of candidate primitive streak in- embryonic development. The model is described ducers is an active area of research (Shah et al., in Section 2 and the results of a travelling wave 1997; Cooke et al., 1994; Ziv et al., 1990). analysis (detailed in Appendix A) are summarized in Section 3 showing that, for realistic parameter values, this model is capable of generating 1.3. MODELLING PRIMITIVE STREAK INITIATION propagating signals on the time-scale necessary The capacity for small portions of the chick to explain the available experimental results. In blastodisc to initiate the same number of streaks Section 4, we show that the model is consistent (one) as the whole blastodisc is di$cult to explain with the results of a number of experiments and using traditional models of embryonic pattern in Section 5 we use the model to make experi- formation, such as reaction di!usion and mentally testable predictions. We then discuss, in mechanochemical models. These models do not Section 6, the wider applicability of this type of naturally exhibit size regulation, the property of model to pattern formation. creating patterns which adapt to the size of the domain. In general, in these models, the number of pattern elements is proportional to the size of 2. The Model the domain. More complicated reaction}di!u- We propose a model mechanism in which mar- sion models can exhibit size regulation. For ginal zone cells produce a streak-inducing chem- example, Meinhardt (1993) considered a reac- ical at a rate which is graded*higher posteriorly tion}di!usion model with a gradient source term and lower anteriorly.* When the activator con- and showed that it exhibited size regulation, with centration reaches a threshold at a particular application to head formation in hydra. Othmer point in the domain, it causes the cells there to & Pate (1980) showed how a reaction}di!usion commit to di!erentiation, i.e. they are now des- model in which the di!usion coe$cients were tined to form part of the primitive streak. (It functions of the concentration of a control chem- should be noted that the cells of the posterior ical could exhibit in"nite scale invariance. How- marginal zone do not themselves become part of ever, none of these authors examined the the streak, but rather cells just inside the mar- time-scale on which patterns could plausibly ginal zone.) This commitment is irreversible. The form. Our studies, Page (1999), show that models committed cells produce an inhibitory factor at based on straightforward reaction di!usion are a constant rate. This inhibitor spreads as a wave likely to be too slow to be able to explain the throughout the domain preventing commitment regulation of the streak initiation process on of other cells in the epiblast. The inhibitor pre- a realistic time-scale. vents commitment either directly (when it ex- Another feature of streak formation which is ceeds a certain threshold, the cells do not commit di$cult to explain is the speed of the patterning process. When a blastodisc is cut into posterior and anterior halves, a localized patch of gene * The posterior end appears to contain a higher density of cells (Kochav & Eyal-Giladi, 1971), and hence we propose expression is initiated in the anterior portion of that production of the activator by these cells could lead to the marginal zone, marking the site of the future an appropriately graded production rate. 422 K. M. PAGE E¹ A¸. even in the presence of super-threshold activator) and position x, measured from the posterior mar- or by degrading the activator. ginal zone (i.e. x"0 at the posterior). Crucial to this model is a mechanism for the We consider "rst the model mechanism in rapid passage of the inhibitor throughout the which the inhibitor degrades the activator and domain. There is evidence that cells can com- propagates via di!usion. The model equations municate via the release of di!usible substances are as follows: (see Gurdon et al., 1994; Shimizu & Gurdon, "o !c !l # 1999; Slack, 1994) and that they can also signal uR (x) u uv DSuVV, (1) to their immediate neighbours by presenting proteins on their surfaces which can bind to " # !j # vR ka g(v) v DTvVV, receptors on the neighbours' surfaces (see e.g. Collier et al., 1996; Zimmerman et al., 1993). The q * 1ifmaxOR[u(x, )] u , production of spatial gradients and of travelling a(x, t)" * waves by the latter mechanism is investigated in G0 otherwise, Monk (1998). In the context of the cell-inhibitory model of this paper, we investigate travelling where o(x) is the rate of production of ac- waves occurring in both di!usive and juxtacrine tivator at the point x in the domain, c is the decay signalling systems and assess the ability of each rate of the activator, l is the reaction rate of the to explain the regulatory properties of the chick decay of the activator by the inhibitor, DS is the blastoderm. activator di!usion coe$cient, k is the rate at which the tissue made up of committed 2.1. MODEL ASSUMPTIONS cells produces an inhibitor, g(v) is a term describ- ing the self-activation of the inhibitor, j is In addition to the general description of the linear decay rate of the inhibitor, D is the the model outlined above, we assume the follow- T inhibitor di!usion coe$cient and u is the ac- ing: * tivator threshold required in order to trigger cell o E The inhibitor either (i) di!uses and activates its commitment. The production term (x) is graded own production locally, or (ii) activates its own and is largest in the posterior. The variable a(x, t) production in neighbouring cells by juxtacrine takes the value 1 if the cells at x have committed signalling. There may be a delay between sig- to di!erentiation at time t and otherwise takes nalling and production of the inhibitor. the value 0. This means that the "rst cells produce E The production rate of the inhibitor by the inhibitor only after they have experienced uncommitted cells is zero when the inhibitor a super-threshold level of activator. concentration is zero and increases with that If instead the inhibitor blocks cell commit- concentration, but always remains below a cer- ment directly then we drop the third term from l" tain maximal rate. the "rst equation ( 0) and the last equation E The activator and inhibitor both decay nat- becomes urally. q * E The inhibitor dynamics are fast relative to the 1ifmax3 [u(x, )] u , " O 2T(V R) * activator dynamics, such that the wave of inhi- a(x, t) (2) G0 otherwise, bition is rapidly triggered and spreads before much of the rest of the domain is activated. where ¹ (x, t)"+q(t, s.t. v (x, q)(v ,, is the T ** time period before t and before the point x has 2.2. MODEL EQUATIONS been inhibited from committing to di!erentiation We assume that the marginal zone is a thin and v is the threshold level of inhibitor at ** ring which can be modelled as a one-dimensional which cell commitment is blocked. domain with periodic boundary conditions. We If the inhibition propagates via juxtacrine sig- denote by u(x, t) and v(x, t) the activator and nalling and the inhibitor is con"ned to reside in inhibitor concentrations, respectively, at time t the marginal zone then the second equation PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 423 becomes We thus analyse the inhibitor equation in iso- lation and treat the term ka as a general localized " # N !p !j (vG)R kaG g(vG(t )) vG, (3) source term. (If the inhibitory wave is established and is su$ciently fast then a should remain zero where v (t) represents the concentration of inhibi- G except in the posterior.) tor at cell i at time t, aG(t) is one if cell i has committed to form streak at time t and zero otherwise, g is again a self-activation term, p is 3. Travelling Wave of Inhibition the delay between signalling and production of To study the travelling waves in isolation, we N more inhibitor and vG(t) represents the average consider the equations inhibitor concentration in the cells neighbouring cell i. In the one-dimensional case considered " # !j # N " # vR s(x) g(v) v DTvVV (4) here, vG (vG\ vG>). Note that the model is now spatially discrete. and In setting up these model equations, we have (v ) "s #g(vN (t!p))!jv (5) assumed that the activator and committed cells G R G G G reside in (or in the case of the cells, just in- for the di!usive and the juxtracrine cases, respec- side*experiments show that posterior marginal tively, where s(x) and s are the corresponding zone cells do not actually contribute to the G general localized source terms. streak, Bachvarova et al., 1998) the marginal zone The form of the travelling wave is governed of the chick blastodisc.- This is approximately by (i) the form of g, (ii) the nature of the coupling, a circular domain and so for the purposes of (iii) the domain geometry, and (iv) the parameter simulation we consider the "rst and third equa- values. We summarize our results below. Full tions of system (1) on a one-dimensional domain details are given in Appendix A. with periodic boundary conditions. Experimental evidence suggests (Spratt & Haas, 1960) that the (i) ¹he form of g: we have analysed the equa- inhibitor probably resides in the central disc as tion for two biologically plausible forms of well as the marginal zone. We initially analyse the g*Heaviside and Hill function form. Both sup- case when the second equation is on a one- port travelling waves. dimensional domain with periodic boundary (ii) ¹he form of the coupling: both types of conditions, but then consider the second equa- coupling can give rise to fast propagating waves. tion on a circular disc-shaped domain with zero (iii) Domain geometry: in the case of di!usive #ux boundary conditions (we suppose that the coupling the speed of propagation on a two- boundaries of this region are impermeable to the dimensional domain is slightly slower than that inhibitor). Initially, we have u(x, t)"v(x, t)" in one dimension. For the juxtracrine case the a(x, t)"0. speed can be in#uenced by the geometry of the We assume that o(x)'cu , ∀x. (This means * cell packing. that each point in the marginal zone has the (iv) Parameter values: all cases considered ex- potential to initiate a streak). It is thus clear that hibit su$ciently rapid travelling waves for biolo- the activator will reach threshold in the posterior gically realistic parameter values. and the inhibitor will then start to be produced. The pattern which results will depend crucially on whether a travelling wave of inhibition is 4. Numerical Simulations of the (full) Primitive established and, if so, on how fast the wave Streak Model travels in comparison with the rate at which the Once the inhibitory wave has been initiated in activator reaches threshold at points in the do- the posterior of the domain, the fate of the rest of main distant from the posterior. the domain depends on a &&race'' at each point between reaching a superthreshold level of ac- - There is strong evidence (Bachvarova et al., 1998; Spratt & Haas, 1960) that the streak-inducing signal comes from tivator and being inhibited by the approaching the marginal zone, see Section 1.2. wave. Consideration of the activator equation 424 K. M. PAGE E¹ A¸. indicates that, for realistic parameters, ac- implants (younger posterior marginal zones seem tivator di!usion can be neglected and hence to be more successful; Eyal-Giladi, 1991). that activator concentration ahead of the wave is approximately the same as if the inhibitor We assume that when these experiments were were everywhere zero. In the wake of the wave, performed on real blastodiscs, the manipulation even in the case of inhibitor degradation of occurred (stages X}XIII on the staging system of the activator, cells which have not become Eyal-Giladi and Kochav) before commitment committed by the time the wave reaches them had taken place in the posterior. have no possibility to do so. Thus, the commit- We assume, for ease of simulation, that the ment of a cell at a point x in the domain really inhibitor resides in the marginal zone rather than depends on whether the activator "rst reaches the central disc. This means that the waves of threshold there or whether the inhibitor wave inhibition take slightly longer to reach their tar- reaches it. Straightforward analysis shows that if gets and so slightly more of the cells will commit the inhibitory wave is established su$ciently to di!erentiation. On balance, experiments sug- quickly and moves fast enough, then the commit- gest that the inhibitor actually spreads through- ted cells will be con"ned to a small region in the out the central disc, but this makes only a small posterior. di!erence to the speed of propagation (as our We simulate the full model equations [system calculations in Appendix A show) and will only (1)] to see if realistic parameters give rise to an dramatically a!ect the result of the glass bead inhibitory wave which is su$ciently fast to con"ne implant experiment. commitment to a small region in the posterior, as We compare direct inhibition of cell commit- is suggested by experimental results. We also test ment with degradation of the activator and that the model correctly predicts the outcome of propagation of the inhibitory wave via a di!usive the following manipulation experiments: mechanism with propagation via juxtacrine signalling mechanisms. 1. Bisecting the blastodisc perpendicular to the First, we consider the system in which the anterior}posterior axis: experiments show that inhibitor di!uses and directly blocks cell commit- this normally results in a streak forming from the ment. The domain length (circumference of the posterior end of the posterior half (its normal blastodisc) is about 6 mm and we take the inhibi- position in the whole embryo), but also a streak tor di!usion coe$cient to be as large as is realis- forming from one of the lateral margins of the tic, approximately 1.0;10\ cm s\. If we non- anterior half. In some cases, the anterior half may dimensionalize such that the domain becomes form a streak from its anterior margin or two [0, 1] (the domain is periodic and the points streaks, one from each lateral margin (Spratt 0 and 1 correspond to the posterior of the mar-  & Haas, 1960). ginal zone, whilst the point  corresponds to the 2. Implanting glass beads to replace the poste- anterior) and with a time-scale of 1000 s, then the ; \ rior marginal zone: this generally results in streaks non-dimensional value of DT is about 3.0 10 . forming on either side of the beads, which are For purposes of illustration, we take the activator implanted simply to prevent healing of posterior di!usion coe$cient to be 10 times smaller (this is marginal zone (Eyal-Giladi, 1991). simply because we have taken the inhibitor di!u- 3. Implanting a second posterior marginal zone sion coe$cient to be as large as possible). We from a donor embryo into the lateral margin of take the threshold level of activator, u , and the * the host blastodisc: this often results in only one of threshold for self-activation of the inhibitor v * the posterior marginal zones initiating a streak. both to be 1.0 in our arbitrary concentration This competition between posterior marginal units. We take the threshold for inhibition of zones only occurs if the implant is su$ciently cell commitment, v , to be 0.5. In order for a ** close to the original marginal zone. Which of the travelling wave to be possible, we require posterior marginal zones is successful depends on k/j'v "1.0, and we also want the inhibitor * the size of the implant (the bigger it is the more kinetics to be much faster than the activator likely it is to succeed) and the relative ages of the kinetics. We choose k"500.0, j"10.0 and PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 425

FIG. 2. (a) Final steady-state committed cell pro"le, a(x), from a numerical simulation of the full model (1), with di!usive inhibitor which directly blocks cell commitment. Parameter values as given in the text. A discretization of 2001 points was used. (b) Inhibitor pro"le at times t"7500, 8000, 8500 and 9000 s in the simulation described in (a).

FIG. 3. Final committed cell pro"le in the anterior portion of the blastodisc in which the blastodisc was cut after a time t"5000 s. (a) The remaining portion is [0.01, 0.95], (b) the remaining portion is [0.35, 0.75], (c) the remaining portion is [0.2, 0.75]. Parameter values D"0.0001, k"10.0, j"1.0 and v "1.0. See text for details. * k"100.0. In dimensional terms, the inhibitor has inhibitor concentration pro"le at various times a half-life of 100 s. The activator dynamics should after determination occurs in the posterior. be much slower and in order for each point in the Figures 3(a)}(c) show the "nal steady-state domain to have the potential to initiate streak, pro"les of a when the blastodisc was cut at time we require o(x)'cu , ∀x. We choose c"0.1 t"5000 s in various di!erent places. Figure 3(a) * (corresponding to a half-life of around 3 hr) and corresponds to the glass beads experiment in which the posterior marginal zone is excised (and replaced with beads to prevent healing). The por- !x x( 0.2(1 ), , tions of the domain [0, 0.01] and [0.95, 1.0] were o(x)" G excised, so that the excised portion was not sym- 0.2xx*.  metrical about the anterior}posterior axis. Zero- #ux boundary conditions were imposed at the cut Figure 2(a) shows the value of a plotted against edges. Nevertheless, a streak formed on either x at time t"8000 s (x"0 and 1 correspond to side of the cut. Di!erentiation had occurred by the posterior). The form of a (corresponding to t"8000 s. Figures 3(b) and (c) show the "nal a small patch of determined cells in the posterior) pro"le of a in the anterior half of the blastodisc remains constant thereafter. Determination oc- when the posterior half is excised. In Fig. 3(b), the curs at t+7000 s+2 hr. Figure 2(b) shows the anterior half consists of the portion of the domain 426 K. M. PAGE E¹ A¸.

FIG. 4. (a) Activator pro"le and (b) inhibitor pro"le at times t"7000, 7500,2, 9000 s in a simulation of the system with adi!usive inhibitor which degrades the activator (l"1.0). Other parameters as in Fig. 3. See text for details.

[0.35, 0.75] and only one streak forms at x"0.75, to have the same values as in Fig. 2. We take k N ! which comes from the posterior-most part of the a self-activation term H(vG v ), where N " # * blastodisc. In Fig. 3(c), the anterior half is more vG (vG\ vG>). We divide the domain [0, symmetrical and larger, consisting of [0.2, 0.75], 6 mm] into 2000 cells. This assumes that the cells and two streaks form. The time taken before cells have diameter approximately 3 lm. Again, we committed to form a streak in the anterior half obtain cell commitment in exactly the same pla- was about 6000 or 7000 s after the cut was made. ces as in Figs 2 and 3. The committed regions are This is less than 2 hr. However, the time at which a few cells thick. commitment to form a streak in the anterior Thus, the model either with di!usive propaga- half takes place is later than it would do so in tion of the inhibitor or with juxtacrine signalling the posterior in an unmanipulated blastodisc. and either with inhibitor degradation of the ac- This agrees with the results of experimental tivator or with direct blocking of cell commit- manipulations. ment is capable of explaining normal initiation of We repeat the simulations of Figs 2 and 3, with one primitive streak in the whole blastoderm and the same parameters, but with inhibitor degrada- the regulation that occurs when the blastodisc is tion of the activator (l"1.0) rather than direct bisected or has its posterior marginal zone inhibition of cellular commitment. The sites of excised (for a similar example of a model for streak initiation are the same in all cases, al- generating size-regulating patterns, see Green though these sites are slightly narrower. How- & Cooke, 1991). It can also explain the relatively ever, we can alter the width of these sites by short time-scales in which these events occur. altering v or l. Small v and large l lead to The model also predicts that the anterior mar- ** ** narrow sites of committed cells. In Fig. 4, we ginal zone will permanently lose its ability to show the activator and inhibitor concentration induce a streak once the wave of inhibition from pro"les at di!erent times. This shows that the the posterior has entered the anterior half. This activator concentration ahead of the wave of clearly occurs after commitment in the posterior. inhibition is largely una!ected by the inhibitor Experiments suggest that the marginal zone loses (see Section 3) and that behind the wave it decays its potential to initiate a streak, even in the ab- rapidly. sence of the posterior marginal zone, once the We now look at the system with juxtacrine streak has started to form in the posterior, see signalling and no inhibitor di!usion. The ac- Khaner & Eyal-Giladi (1989), Eyal-Giladi & tivator di!usion seems irrelevant to the mecha- Khaner (1989) and Bachvarova et al. (1998). The nism of our model and so, for ease of simulation, initial appearance of the streak and the loss of " we assume DS 0. We take the other parameters potency of the marginal zone seem to be almost PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 427 exactly coincident to within experimentally de- that during this short-time window anterior seg- terminable limits. ments lose their streak-forming competence later If a second posterior marginal zone is im- than lateral segments. planted into the lateral margin of a host embryo, We propose an experiment in which a stage provided the rate at which cells in each posterior XIV anterior half, whose marginal zone has lost marginal zone produce activator is somewhat the capacity to initiate a streak, is grafted on to di!erent (for example, if activator production a stage X anterior half. According to the model, rate is age-dependent and the embryos are of the cells in the older anterior half should have slightly di!erent ages) and the separation of the been inhibited and should cause the production two posterior marginal zones is not too great of the inhibitor in the younger anterior half, thus then, according to the model, competition will preventing a streak from being initiated there. occur and only one streak will be induced. If the Finally, we propose an experiment which posterior marginal zones are too similar in their should be able to distinguish between a Turing production rate and level of activator concentra- mechanism of streak initiation (in which the ac- tion or if they are too far apart then two streaks tivator and inhibitor react and di!use and be- will form, as observed experimentally. come patterned via a di!usive instability) and the cell-inhibitory model with di!usive inhibitor, provided that di!usion in both model mecha- 5. Model Predictions nisms occurs through gap junctions. This is very The model presented here predicts that, in an plausible, since the epiblast of the blastodisc is embryo which is defective for the inhibitor, the a coupled epithelium (Stern & Mackenzie, 1983). whole of the marginal zone will induce streak (In the case of the Turing mechanism, it is ex- formation. The streaks in the mutant should form tremely likely that the activator would di!use in order, from posterior to anterior. through gap junctions, since it must somehow be The axin mutant mouse has a phenotype which con"ned to the marginal zone.) The experiment induces multiple streaks emanating from all re- involves the drug-induced blocking of gap junc- gions of the margin, see Zeng et al. (1997). (Axin tional communication, see Levin & Mercola blocks the Wnt signalling pathway, which has (1998, 1999). The crucial di!erence between the been proposed to have a role in streak induction models is that the activator di!usion is an impor- in the chick, see Cooke et al., 1994.) Since axin is tant component in the Turing mechanism, but an intracellular molecule, we propose that the not in the cell-inhibitory mechanism. In the Tur- inhibitor may be a secreted antagonist of Wnt or ing scenario, we predict that the activator and nodal signalling, such as frzb, Dkkl, cerberus or inhibitor would remain at their uniform crescent (Baranski et al., 2000; Zhu et al., 1999; steady-state values and not induce a primitive Pfe!er et al., 1999). streak, whereas in the cell-inhibitory model the The behaviour of our model is characterized activator would reach threshold at each point by processes operating on two distinct before any inhibitor was produced there. Hence, time-scales. Initially, there is a slow increase of streaks would form from all sites round the mar- activation across the whole region with a peak at ginal zone. the posterior, then, once the most posterior cells have committed to form primitive streak, there is initiation of a wave of inhibition that rapidly 6. Further Features of the Model sweeps across the whole domain. During the "rst We have shown that the model is capable of phase, there exists a posterior to anterior gradi- explaining many of the experimental results relat- ent of streak-initiating potential consistent with ing to streak initiation. We "nd also that it is experimental observations suggesting that lateral capable of exhibiting very di!erent behaviour segments are better inducers than anterior seg- and producing periodic patterns. We simulate the ments. During the second phase, there is a loss of model equations with a di!usive inhibitor and streak-forming competence which results from inhibitor degradation of the activator. We use the the wave of inhibition. Hence, the model predicts same parameter values as before, except that we 428 K. M. PAGE E¹ A¸.

FIG. 5. (a) Final committed cell pro"le in a numerical simulation of the full model with a di!usive inhibitor which degrades the activator. Parameters given in the text. A discretization of 2001 points (each point corresponds to a cell) was used. (b) A blown-up version of the pro"le on [0, 0.05]. set k"10.0 instead of 500.0. This means that committed and uncommitted cells. Nearer the a forward-propagating travelling wave of inhibi- anterior, committed cells become sparser until in tion cannot be set up. We use a discretization of the anterior there are no committed cells. Such 2001 points. Thus, each point corresponds patterns in which the frequency of committed roughly to a cell in the biological system. In cells decreases towards the anterior appear to be Fig. 5(a), we show the "nal steady-state pro"le of typical in relay systems. The patterns which form a. We consider each point in the domain to cor- are probably dependent on the juxtacrine coup- respond to a cell and so those points at which ling. Our system employs a very simple coupling. a"1 correspond to cells committed to di!erenti- For more sophisticated couplings, see Monk ation and those at which a"0 correspond to (1998), Owen & Sherratt (1998) and Owen et al. uncommitted cells. Figure 5(b) shows the pro"le (1999). Further investigation of the types of pat- on the section of the domain [0, 0.05]. This shows terns that could develop in similar systems in two that approximately one cell in every 20 becomes dimensions, with various forms of cellular coup- committed to di!erentiation. Although similar ling, would be interesting. Even with our simple patterns can be produced by a typical ac- coupling, however, the system produced "ne- tivator}inhibitor Turing system, the novelty here grained structures with period greater than two is that these patterns have been produced cells. This is a property that many juxtacrine by a size-regulating model on changing one models do not have (see Collier et al., 1996) and parameter. may have important biological applications. We repeat the simulation of Fig. 5 with direct blocking of cellular commitment (l"0). When v "0.5, approximately one cell in every four 7. Discussion ** commits to di!erentiation. With v "0.05, The initiation of the primitive streak in the ** a pattern similar to that in Fig. 5 is produced, chick embryo is a fundamental developmental with roughly every 20 cell committed to di!eren- event. The mechanisms involved in the initiation tiation. The spacing between committed cells in- are, as yet, poorly understood. Experimental ma- creases away from the posterior (region of nipulations of the chick blastodisc reveal that the greatest activator production). initiation process displays size regulation. In A simulation with a juxtacrine coupling, no most cases, an isolated portion of the blastodisc, inhibitor di!usion and inhibitor degradation of provided it contains part of the marginal zone, the activator (l"1.0) leads to a pattern in which will initiate exactly one streak. Although this there is a small patch of committed cells in property of size regulation can be exhibited by the posterior, followed by a patch of alternating modi"ed reaction}di!usion models (Meinhardt, PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 429

1993; Othmer & Pate, 1980) it is not clear if this we were able to calculate the wave speed analyti- can occur on a su$ciently fast time-scale for cally for the cases of both di!usive and juxtacrine primitive streak initiation. Models based on coupling. It is an open question as to how to straightforward di!usion or on reaction-di!usion determine wave speeds in juxtacrine signalling are likely to be too slow to be able to explain the systems with general self-activation. This carica- regulation of the streak initiation process on ture model serves to give insight into the import- a realistic time-scale (Page, 1999). In this paper, ance of di!erent parameters in determining the we sought an alternative model of primitive wave speed and to estimate the order of magni- streak initiation which could display its key fea- tude of possible wave speeds. We found good tures more naturally. We consider that this agreement between the analytical results and model could have wider applicability in embry- those of numerical simulations. onic development. Our simulations show that in both the di!usive The mechanism involves cells producing an and juxtacrine cases, more realistic forms of inducing agent which, on reaching a threshold, the self-activation term led to waves which causes cellular commitment to form part of the were slightly slower than, but with speeds of the primitive streak. Committed cells produce an in- same order of magnitude as those of the carica- hibitor which activates its own production in ture model. In two dimensions, the di!usive uncommitted cells and passes as a wave through- travelling wave was slower than its one-dimen- out the domain, preventing commitment at other sional counterpart very near to a localized sites. Such waves of inhibition are well source, but beyond a short distance from the documented in other areas of developmental source the di!erence became negligible. In the biology, such as calcium waves in fertilized juxtacrine system, the two-dimensional problem eggs*see, for example, Busa & Nuccitelli (1985); was more complex. The propagation of the wave Gilkey et al. (1978); Lane et al. (1987) and Ja!e was in#uenced by the positions of the cells and (1999). The model is intrinsically di!erent from the numbers of orientations of the nearest-neigh- a reaction}di!usion model in that cells have bour interactions. We found that with the simple a memory*their commitment is irreversible. An coupling used in our model, the wave speed was important feature of the model is that whilst considerably reduced compared to its one-di- straightforward di!usion of an inhibitor would mensional counterpart, because cells had to share be too slow to be biologically realistic (see Crick, their signalling between more neighbours and 1970), a di!usive (or indeed juxtacrine) travelling hence forwards signalling was reduced. We pos- wave can pass much more rapidly throughout tulate, however, that this might not be the case the domain. In a standard reaction}di!usion for other more complicated couplings. The model a travelling wave could not pattern the propagation of waves in a juxtacrine relay system blastodisc, since the "nal steady state left behind in two dimensions with various di!erent coup- the wave would be uniform and hence cells would lings warrants further investigation. ultimately reach the same state of di!erentiation Considering realistic parameters, we found everywhere. The cell memory property is thus that a di!usive inhibitory wave could probably a crucial element of the model proposed here. have a speed of up to about 10\ mm s\ and We began the analysis of the model by assess- hence pass from the posterior to the anterior of ing the speed of propagation of (inhibitory) the embryo in less than 5 min. In the case of the travelling waves in a system with self-activation juxtacrine relay model, a wave of inhibition could and either di!usive or juxtacrine coupling. We sweep across the domain in a similar time, pro- found that if a travelling wave arose then it had vided there exists a quick-release mechanism (see, a speed uniquely determined by the parameters e.g. Lyons et al., 1988) for the inhibitor. of the system. This suggests that mechanisms of Analysis of the activator equation and numer- this type could allow cells to communicate over ical simulations of the full model showed that if "xed distances on a predictable time-scale. For the speed of propagation of the inhibitory wave the case in which the self-activation of the inhibi- were fast enough, the cells would commit to tor takes a caricature Heaviside function form, di!erentiation only in a small region in the 430 K. M. PAGE E¹ A¸. posterior. Numerical simulations of the model of the system the same mechanism could have predict that it can explain the regulation that evolved to give rise to both size regulating and occurs in primitive streak initiation and that it "xed wavelength periodic patterns. It would be can explain initiation on a realistic time-scale. very interesting to investigate more thoroughly This is independent of whether the inhibitor the types of periodic patterns that can be produc- propagates via juxtacrine signalling or via di!u- ed by this system, especially in two dimensions, sion and of whether the inhibitor blocks cell looking also at the e!ects of di!erent couplings in commitment directly or does so by degrading the the juxtacrine case. In the one-dimensional sys- activator. tem, we found that the pattern spread out as The model gives rise to a number of testable a wave from the &&posterior'' (fastest activator- predictions, as detailed in the previous section. producing) end to the &&anterior'' end. This se- The streak-inducing agent itself has not yet quential formation of pattern occurs in many been positively identi"ed, although a number of biological patterning processes. Examples in- candidates have been found*perhaps the most clude the formation of stripes on the back of the promising being the Vg1 protein, acting together Alligator mississippiensis, somites in amniotes with Wnt. and chick feather primordia. The model has the ability to give rise to a single In general, we have found that the novel model structure in domains of various sizes on proposed in this paper can give rise to two highly time-scales which are appropriate for biological di!erent types of pattern on biologically realistic development. We propose that this feature makes time-scales and could be applicable to a very this model relevant to many other biological sys- wide range of developmental patterning events. tems. Many other instances occur in development KMP would like to thank the Wellcome Trust for in which the emergence of a structure inhibits funding under the Prize Studentship Scheme in Math- formation of further (identical) structures else- ematical Biology, Grant Number 047910. She would where in the "eld of interest. For example, the also like to acknowledge support from the Leon Levy freshwater polyp Hydra normally has only and Shelby White Initiatives Fund, the Florence one head and foot, but if bisected regenerates Gould Foundation, the J. Seward Johnson, Sr. Chari- table Trusts, the Ambrose Monell Foundation and the the appropriate body part at the cut end of each Alfred P. Sloan Foundation. half (see Gierer & Meinhardt, 1972; Meinhardt, NAMM gratefully acknowledges the support of the 1993; Sherratt et al., 1995). Another fascinating Royal Society and BBS CDS's, research on this topic example is the initiation of limbs in the tailbud was funded by the National Institute of Health stage salamander Ambystoma maculatum. In this (GM 56656) case, each limb disc normally produces one limb, but if the limb disc is split into vertical segments REFERENCES by placing thin barriers which prevent reunion, BACHVAROVA, R. F., SKROMNE,I.&STERN, C. D. (1998). then each segment produces a distinct limb. Induction of primitive streak and Hensen's node by the A similar phenomenon was observed in a pond in posterior marginal zone in the early chick embryo. Devel- opment 125, 3521}3534. Santa Cruz, CA, in which many-legged frogs and BARANSKI, M., BERDOUGO, E., SANDLER, J. S., DARNELL, salamanders were found. It was discovered that D. K. & BURRUS, L. W. (2000). The dynamic expression this was due to an infestation of parasitic pattern of frzb-1 suggests multiple roles in chick develop- ment. Dev. 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GUMBINER,B.M.&COSTANTINI, F. (1997). The mouse solutions of fused locus encodes axin, an inhibitor of the Wnt signalling pathway that regulates embryonic axis formation. Cell 90, 181}192. g(v)"jv. (A.2) ZHU, L., MARVIN, M. J., GARDINER, A., LASSAR, A. B., MERCOLA, M., STERN,C.D.&LEVIN, M. (1999). Cerberus There are no biological data to guide us to the regulates left}right asymmetry of the embryonic head and heart. Curr. Biol. 9, 931}938. form that g(v), the inhibitor production term, ZIMMERMAN, G. A., LORANT, D. E., MCINTYRE,T.M. should take. Therefore, we consider the three &PRESCOTT, S. M. (1993). Juxtacrine intercellular signall- typical functional forms that are commonly used ing*another way to do it. Am. J. Resp. Cell Mol. Biol. 9, 573}577. as production terms in biology. ZIV, T., SHIMONI,Y.&MITRANI, E. (1990). Activin can In the case represented by Fig. A1(a), 0 is the generate ectopic axial structures in the blastoderm ex- only homogeneous steady state and is stable. In plants. Development 115, 689}694. this case, the inhibitor will just decay away from the source. For the form of g(v) illustrated in Fig. A1(b), the zero steady state is unstable and APPENDIX A any slight perturbation will cause convergence to A1. Speed of the Wave in a Di4usive System the non-zero steady state. Travelling waves do occur in systems with this type of production We consider the equation term, but they will only do so if the level of inhibitor away from the source is exactly zero. If v "s(x)#g(v)!jv#D v , (A.1) there is a noisy background of inhibitor concen- R T VV tration then this will grow to the non-zero homo- geneous steady-state level. Hence, this case does where s(x) is a general localized source term, on not exhibit the necessary robustness required for the domain [0, ¸] with zero-#ux boundary con- the problem being studied here and we will ditions. To study travelling waves, we really need ignore it in the subsequent analysis. to consider an in"nite domain. However, if we For the form of g represented by Fig. A1(c), we consider a su$ciently large domain with look for travelling wave solutions of the form zero-#ux boundary conditions, such that the v"v(z), where z"x!ct, c is the speed of the P PR P boundaries do not a!ect the dynamics of the wave and v(z) 0asz , v(z) v as P!R travelling wave, then we can simulate travelling z . v is the stable non-zero homogeneous wave pro"les numerically. We assume that the steady-state value. These boundary conditions e!ect of the source term is simply to trigger imply that far ahead of the wave, the inhibitor a travelling wave of inhibition and that the e!ects concentration is zero, while far behind the front, of the domain boundaries are small except in the inhibitor concentration is at the steady state their immediate neighbourhood. We hence look given by the largest intersection of the line for travelling wave solutions of eqn (A.1), without y"g(v) with the line y"jv. In travelling wave the source term, on an in"nite domain. coordinates, the equation is then Equation (A.1), without the source term, has # # !j " homogeneous steady states which are the positive DTvXX cvX g(v) v 0. (A.3)

FIG. A1. Possible forms of g(v), the self-activation for the inhibitor. v is the concentration of inhibitor; j is its decay rate. PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 433

j For given DT, and g of the form in Fig. A1(c), we wave solution is given by can show, using standard phase-plane analysis, that there will be a solution for a unique travel- v eK(X\X), z'z , *  ling wave speed c. v" k k (A.7) For a given self-activation function, g, there is # ! K(X\X) ( G Av Be , z z, a maximal value of j given by j * j where z is determined by the initial conditions and T  m m  (g(v)!jv)dv"0, (A.4)  is the negative root and  the positive root of  Dm#cm!j"0. (A.8) for which the inhibitory wave will propagate forward. Numerical simulations of the corresponding To obtain an estimate for the speed of propa- partial di!erential equation model for v con"rm gation, we approximate g(v) by a Heaviside this analytical prediction for the wave speed function. This is equivalent to assuming that the (Page, 1999). uncommitted cells produce inhibitor at a "xed rate only when the background concentration A1.1. HILL-FUNCTION SELF-ACTIVATION OF exceeds a threshold. We discuss more realistic THE INHIBITOR forms of g later, although in some cases a sharp If the inhibitor enhances its own production by threshold response to cell signalling may be binding to a receptor on the cell surface, then the realistic, but this usually involves protein syn- rate of self-activated production of the inhibitor thesis (Ferrell & Machleder, 1998; Papin should re#ect the level of receptor occupancy. We & Smith, 2000). assume for now that the inhibitor production With Heaviside function self-activation, rate is simply proportional to the level of receptor eqn (A.3) becomes binding. As the kinetics of receptor binding is # #k ! !j " likely to be rapid then the level of receptor bind- DTvXX cvX H(v v ) v 0, (A.5) L # L * ing will be approximated by kv /(k v ), where n, the Hill coe$cient, is the number of molecules where of inhibitor that bind simultaneously to one re- ceptor and k and k are constants that depend 1, v'v ,   * H(v!v )" on the rate constants of the inhibitor}receptor * G0, v(v . binding (see Hill, 1910; Segel, 1991; Murray, * 1989). Thus, we consider a self-activation term of this Hill function form. For n large, the Hill func- We assume that v (the non-zero homogeneous steady-state value of v),k/j'v . This means tion is similar to a Heaviside function. We now * that there is a stable non-zero homogeneous compare the ability to generate travelling waves " and their speeds and forms in systems having steady state v v. We assume that, in the travel- P PR P a self-activation term which takes the form of ling wave solution, v 0asz and v v as zP!R. a Hill function of order 2 or 3 with the corre- We "nd that the wave speed is given by sponding properties of the system with Heaviside function dependence. If the self-activation terms takes the form of a Hill function of order 1 then v 1 c"(D j !1! . (A.6) the zero steady state of the inhibitor concentra- T v A * v B tion is unstable, see Fig. A1(b), and any noise will !1 v trigger convergence to the non-zero steady state, * so we assume this form is not relevant. A very In particular, we can see that the wave will large n is unlikely from a physical point of view move forward i! v '2v , which is equivalent [although there have actually been found cases in  * to T[k (v!v )!jv]dv'0. The travelling which the apparent Hill coe$cient, calculated  H * 434 K. M. PAGE E¹ A¸. from the dose}response curve, is as high as The agreement is better for n"3. The numerical n"35, see Ferrell & Machleder (1998), but this simulations suggest a travelling wave speed of high apparent coe$cient seems to rely on protein c"0.019 for n"2 and c"0.023 for n"3 com- synthesis-dependent positive feedback, which pared to an analytical prediction (which agrees would probably be slow] and anyway we expect with the numerically calculated value) of extremely good agreement with the Heaviside c"0.027 for Heaviside function self-activation. solution, since the form of the Hill function tends Hence, the Hill function self-activation leads to towards a Heaviside function form as nPR. a travelling wave of inhibition which is slower In order to compare like parameters with like than that initiated with Heaviside function in the Heaviside and Hill function equations we self-activation. consider self-activation terms with the same When we simulate the Hill function equation inhibitor production rate at very high inhibitor with n"2 or 3 and parameter values v "2.0, * concentration and the same inhibitor concentra- k"5.0, j"1.0 and D"0.001 the inhibitor con- tion at the point of maximum slope of g. The Hill centration collapses to zero. In both these cases, function corresponding to kH(v!v ) is hence there is no non-zero homogeneous steady state. * k(vL/(KvL #vL)), where K"(n#1)/(n!1). This For a non-zero steady state to exist for n"2, we * choice of matching is somewhat arbitrary, but need v ((1/2(3)(k/j); however, this still does * gives functional forms which are fairly similar. not ensure that the travelling wave solution We simulate the Hill function equations, with propagates forwards. This will happen if n"2 and 3, for the parameter values v "1.0, v (0.265k/j [using condition (A.4)], this being * * k"10.0, j"1.0 and D"0.0001, chosen such only slightly less than the value of v required for * that, according to our analysis for Heaviside a non-zero homogeneous steady state. For n"3, function self-activation, a forward-propagating these values are more di$cult to compute, but, as travelling wave solution exists and such that the an approximation, we require v (0.37k/j for * di!usion coe$cient is large enough to allow reas- a forward propagating wave. These values com- onably fast, accurate computation. We also pare with the condition v (k/j for the Heavi- *  match the analytical solution of the Heaviside side function equation. Thus, the more receptor function equation (dotted line) at t"16, to com- binding sites involved in the inhibitor}receptor pare the forms of the solutions (see Fig. A2). For binding, i.e. the higher the value of n and the n"2, the solutions are similar except behind the more like a Heaviside function the self-activation, wave, where the di!erence is largely due to di!er- the more robustly the system initiates a forward ent non-zero homogeneous steady-state values. propagating wave of inhibition.

FIG. A2. Numerical solutions of the inhibitor equation with Hill function self-activation (==) and analytical form of the solution with Heaviside self-activation ( ))))*see text for details. Parameters as in Fig. 3*exact form of Hill function given in text. Output times shown in the "gure. (a) Hill function of order 2. (b) Hill function of order 3. In both cases, a spatial discretization of 2001 points was used. PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 435

A1.2. REALISTIC PARAMETER VALUES In our primitive streak initiation model, we want only one streak to be initiated in any con- nected piece of blastodisc. This will happen if the travelling wave of inhibition is su$ciently fast. From both our simulations and our analytical form of the wave speed in the Heaviside function inhibitor equation, we see that the inhibitory wave speed increases with the ratio v/v , where "k j * v / . We thus consider how large this ratio could plausibly be. As the ratio gets large, a non-zero homogeneous steady state will always exist and will take a value approximately equal to v . Thus, v /v corresponds approximately to the   * ratio of the saturation concentration of inhibitor to the threshold value at which the non-di!eren- FIG. A3. Numerical solutions of the inhibitor equation tiated cells begin to produce inhibitor. It has been with Hill function self-activation (==) and Heaviside shown in certain morphogenetic systems that self-activation (**) at times t"40, 80, 120, 160, 200 and cells respond to threshold levels of chemicals 240 s. Realistic parameter values of D"1.0;10\ cm s\, k"10.0 concentration units s\, j"0.1 s\ and v "1.0 which are very far from the saturation levels [see * concentration unit are taken on a periodic domain of length Dyson & Gurdon (1998), although it should be 4 mm. A spatial discretization of 8001 points was used. noted that this paper refers to low receptor occu- pancy and not to a saturation level given in terms of kinetic parameters]. Here we shall take a gen- length 4 mm, with the parameters taking their erous estimate of the maximum likely value of bounding values (see Fig. A3). We see that in the this ratio as being 100; less generous estimates Heaviside function case, the wave takes less than would predict slower waves of inhibition. We 200 s to reach the anterior of the domain (we assume that the half-life of the inhibitor in vivo consider the middle of our simulation to corres- will be at least 10 s, leading to a maximum value pond to the posterior and the edges to corres- of j of 0.1 s\. Since we are not interested in the pond to the anterior) and in the Hill function case absolute values of the inhibitor concentration, we less than 240 s. This shows very good agreement take v to be 1 in some arbitrary units. This with the analytical estimate. * means that k)10.0 concentration units per sec- ond. We assume that the inhibitor is a protein of average mass and that its di!usion coe$cient will A1.3. INHIBITORY TRAVELLING WAVES IN TWO be )1.0;10\ cm s\. DIMENSIONS If we allow all the parameters to take their Experiments involving manipulation of the bounding values then we obtain an analytical blastodisc suggest that the wave of inhibition wavespeed for the system with Heaviside should pass through the central disc rather than self-activation of approximately 10\ mm s\. be con"ned to the marginal zone. We are inter- Thus, the inhibitory wave would pass from the ested therefore in whether the propagation of posterior of the blastoderm to the anterior in less a wave in two dimensions di!ers signi"cantly in than about 5 min. speed from that in one dimension. The eikonal Since the ratio v /v is large we expect the equation (Grindrod, 1996) shows that in the case  * behaviour of the system with Hill function of radially symmetric propagation, the wave self-activation to be similar to that with Heavi- propagates at rate which is an amount D/r slower side self-activation, although the wave will be than the one-dimensional speed, where D is the a little slower. We simulate the system numer- di!usion coe$cient of the inhibitor and r is the ically with Heaviside function and Hill function radial coordinate. Biologically, realistic di!usion of order 2 self-activation on a periodic domain of coe$cients are very small and so at a very short 436 K. M. PAGE E¹ A¸. distance from the source of the wave the di!er- Substituting this into eqn (A.9) gives ence between the one- and two-dimensional speeds should become negligible. Thus, if we as- "!j #k  #q!p (vG)R vG H[vG(t ) sume that the source is not perfectly localized < (and so outside the source r 1/D), then the time #v (t!q!p)!v ]. (A.10) taken for the inhibitory wave to propagate in one  G * dimension should give us reasonable estimates We assume that initially v "0 for i'0 and so v for the two-dimensional case. Numerical simula- G G will always be zero in front of the wave (i.e. before tions on a two-dimensional domain con"rm this v reaches threshold), for i'0. Thus, in this result (Page, 1999). G\ simple scenario, only the signal from the We conclude, therefore, that a cell-inhibitory left-hand side matters. model in which the inhibitor di!uses in the classic Now when ()v (t#q!p)'v ,(v ) "! jv Fickian sense can exhibit wave speeds su$ciently  G * G R G #k,so fast to account for the inhibition of streak initia- tion observed experimentally. k " ! \HR vG j Be . (A.11) A2. Wave Speed in a Juxtacrine Relay System

A2.1. ANALYSIS OF THE CASE WITH HEAVISIDE This must match with the zero solution when v (t#q!p)"v . Let the time when the match FUNCTION SELF-ACTIVATION  G * occurs be denoted by t (i), then We assume that the level of signalling depends * k on the average of the inhibitor concentration in ! \H(R\R (G)) ' " j(1 e * ), t t (i), the cell's immediate neighbours and that the in- vG(t) * hibitor decays linearly. The model equations are G 0, t(t*(i), as follows: v (t (i)#q!p)"v . (A.12) "!j  G * * (vG)R vG The "rst and third of these give us #k [v (t!p)#v (t!p)!v ], H  G\  G> * 1 1 q"p# ln , (A.13) k j A1!(2v /v )B " P!R *  vG(t) j as i for any "xed t, "k j q where v / . The speed is then s/ . This is in P PR good agreement with numerical simulations of vG(t) 0asi for any "xed t, (A.9) eqn (A.9). (Note, for j;k/v , the speed is almost * independent of j, with q+p#2v /k.) where is the Heaviside function, v is the con- * H G As with the di!usive travelling wave, more centration of inhibitor in cell i, v is the threshold * realistic Hill function self-activation of the inhibi- concentration which causes a cell to start produ- tor leads in general to a slower wave. For cing inhibitor and p is the delay between signall- some parameter values, a travelling wave may be ing to a cell and the upregulation of the signal established in the case of Heaviside function presented by that cell. The "rst equation has " self-activation, but not for Hill function self- uniform steady-state solutions, vG(t) 0 and "k j ∀ activation (Page, 1999). vG(t) / , i and t. For a travelling wave in a cellular system, we want the concentration p in cell i at time t to be the same as that in A2.2. THE EFFECT OF cell i!1 at a certain "xed delay before, i.e. Analysis of the juxtacrine relay inhibitor equa- " !q ∀ vG(t) vG\(t ), i and t. This gives a tion with Heaviside self-activation indicates that wavespeed of s/q, where s is the cell separation. for "xed j, k and v there is a maximal speed * PRIMITIVE STREAK INITIATION IN CHICK EMBRYO 437 which occurs for p"0. The time taken, q, for the For ease of simulation, we assume that p"0. wave to pass from one cell to the next is given Our previous results suggest that a non-zero by p simply adds a delay to the time taken for the wave to move from one cell to the next. Figure A4(a) shows the position of the wave- q(p)"q(0)#p. (A.14) front at t"5, t"10,2, t"45 (by t"50 it has covered the entire domain) in a numerical simula- Thus, a delay between inhibitor signalling to tion of the relay inhibitor equation on a square the cells and the production of more inhibitor grid of 101;101 cells with Heaviside self-activa- translates simply into a delay in the wave of tion and parameter values k"10.0, j"1.0 and inhibition passing from one cell to the next. Nu- v "1.0. The cells are indexed by pairs (i, j) * merical simulations indicate that the same rela- denoting their positions in the two directions tion between q and p holds (approximately) when parallel to the edges of the square. The inhibitor the self-activation takes the Hill function form concentration initially has value k/j for cells with (data not shown). j)10 and 46)i)56. The geometry of the lat- tice means that the wavefront approximately

A2.3. INHIBITORY TRAVELLING WAVES IN TWO preserves its initial square form. The speed of the DIMENSIONS travelling wave is less than half of that in the one-dimensional case. This reduction in speed is The dynamics of juxtacrine signalling systems largely due to the fact that the cells now share in two dimensions are dependent on the positions their signalling between four neighbours rather of the cells and the number and orientation of than two and this reduces the amount of signall- nearest-neighbour interactions. In the following ing received by a cell at the front from its neigh- simulations, we assume that cells are arranged in bour behind the front. a square lattice and that they signal only to those Figure A4(b) shows the position of the wave- neighbours with which they share a side. We also front at t"5, t"10,2, t"50 in a simulation of assume that a cell presents one-quarter of its the relay inhibitor equation with Hill function quota of inhibitor on each side. Thus, the level of order 2 self-activation. The other details of the inhibitor presented to the cell at position (i, j)is simulation are as for Fig. A4(a). In this case, the given by wavefront becomes much more rounded (this seems to be due to the fact that it is not sharp).  # # ! # # # ! The travelling wave is again much slower than in (v(i 1, j) v(i 1, j) v(i, j 1) v(i, j 1)). (A.15) one dimension.

FIG. A4. Position of wavefronts at 5 time unit intervals in numerical simulation of the relay inhibitor equation on a domain of 101;101 cells. (a) Heaviside function self-activation, k"10.0, j"1.0 and v "1.0 as in Fig. 3. For details, see text. (b) As * (a) but with Hill function order 2 self-activation. 438 K. M. PAGE E¹ A¸.

we assume that p"0 and allow for an additional time of approximately 100p, where p is the actual delay, for the wave to propagate. The position of the wavefront at 10 s intervals is shown in Fig. A5. Initially, it is assumed that only cells adjacent to the posterior marginal zone are ac- tivated and so we set v(i, j)"k/j for j"1 and 45)i)55 and v(i, j)"0 elsewhere. The wave takes less than 60 s to propagate throughout the domain. We can assume, since the wave propa- gates at approximately constant speed, that it will take less than 5 min to propagate across a domain of 500;500 cells. FIG. A5. Position of wavefronts at 10 s intervals in a p numerical simulation of the relay inhibitor equation with The magnitude of the delay depends on the Hill function order 2 self-activation on a domain of mechanism by which cells upregulate their sur- 101;101 cells. &&Realistic'' parameter values !j"0.1 s\, face inhibitor concentration in response to recep- k"10.0v s\. For details see text. * tor signalling (see Monk, 1998). A mechanism involving gene transcription could cause a delay of as much as several minutes (too slow for the purposes of this model). Release of inhibitor from A2.4. REALISTIC PARAMETERS intracellular vesicles is likely to take the order of As in the case of the di!usive wave, we try to a minute (or a little less). This again would prob- assess how fast a wave of inhibition might propa- ably be too slow and certainly these delays would gate in the juxtacrine signalling system. As dominate the time taken for the wave to propa- before, we take parameter values j"0.1 s\, gate. A faster mechanism would be the conver- v "1 in some concentration units and sion of inhibitor from an inactive to an active * k"10.0v s\. We simulate the system with Hill form via protein cleavage (see Lyons et al., 1988). * function order 2 self-activation on a square do- This could take less than a second and hence the main. The blastodisc has a diameter of about inhibitory wave could propagate across the do- 2 mm which corresponds to roughly 500 cell main in around 10 min. lengths. We simulate on a smaller domain of These calculations suggest that a juxtacrine 101;101 cells, in order to limit the time taken to signalling mechanism is also able to propagate perform the simulations. For ease of simulation fast inhibitory waves.