A pattern formation mechanism to control spatial organization in the embryo of Drosophila melanogaster

J. theor. Biol. (1988) 132, 277-306

A Pattern Formation Mechanism to Control Spatial Organization in the Embryo of Drosophila melanogaster B. N. NAGORCKA

CSIRO, Division of Entomology, GPO Box 1700, Canberra ACT 2601, Australia (Received 21 December 1987) It is known that cells are already committed to a particular segment at the cellular blastoderm stage during embryogenesis of Drosophila melanogaster. Recently, several segmentation genes have been observed to be expressed in a sequence of banded spatial patterns in the syncytial blastoderm, prior to the formation of the cellular blastoderm. It is demonstrated in this paper that a two component reaction-diffusion (RD) system with net production function~ which are antisymmetric with respect to the uniform steady-state values, is capable of producing a sequence of seven spatial patterns in the syncytial blastoderm. The sequence of patterns obtained exhibit a strong preference for banded or striped patterns. The first pattern is a simple anteroposterior gradient while the second is a gradient in the dorsoventral direction. The next five patterns are a sequence of banded patterns which exhibit frequency doubling, i.e. the number of bands in each pattern tend to be double the number in the previous pattern. The predicted pattern sequence is comparable to that observed in the expression of some segmentation genes. It is suggested that a pattern formation mechanism based on such an RD system may exist in the embryo where it produces a sequence of prepatterns to regulate the expression of various segmentation genes leading ultimately to a segmented embryo. There is sufficient spatial information in the sequence of banded prepatterns for the segments to be unique. 1. Introduction

Morphological aspects o f the embryonic development of Drosophila melanogaster have now been described in considerable detail ( C a m p o s - O r t e g a & Hartenstein, 1985). The events following fertilization which give rise to the syncytial blastoderm (Foe & Alberts, 1983) are o f particular interest here. Each of the first seven nuclear division cycles following fertilization occurs synchronously resulting in a group of 128 nuclei with their associated cytoplasm located centrally in the egg. Each nucleus is surrounded by a yolk-free region of cytoplasm (the p r o t o p l a s m ) and the combination is called an energid. The periplasm at the periphery of the egg is also visibly different from the yolk (Sonnenblick, 1950) and may well contain at least some proteins not found in the yolk (Foe & Alberts, 1983, p. 64). The groups o f energids which build up within the egg during the first seven division cycles, form a syncytium often depicted as a prolate spheroid reflecting the shape of the egg itself (Fig. 1). Following the seventh division, most of the energids begin to migrate out towards the periphery of the egg leaving behind - 2 6 "yolk-nuclei". The migrating energids form a connected domain, assumed here to be the surface o f a prolate spheroid, and referred to as the presumptive syncytial blastoderm. The energids forming the 277

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C FtG. 1. A schematic diagram (based on Fig. 1 in Foe & Alberts (1983)) indicating the geometrical arrangement of the energid~, consisting of nucleii (black discs) and their associated cytoplasm (clear region around black discs), during nuclear division cycles 7-14 in the Drosophila embryo. The energids divide with a high degree of synchrony. Cycle numbers are given beside each embryo. The energids cluster together to form a connected region known as the syncytium. The stippled region is the yolk and the clear region at the periphery is a thin layer of yolk-free cytoplasm known as the periplasm. Some energids become cellularized at the posterior pole during cycle 10 to form pole cells, which continue to divide although out of synchrony with energids forming the syncytial blastoderm. The nucleii which do not migrate to the periphery but remain behind in the yolk are called yolk-nucleii.

presumptive syncytial blastoderm continue to divide synchronously during the eighth and ninth nuclear division cycles. Following the division phase o f each o f these two cycles, the number o f energids doubles and the size o f the presumptive syncytial blastoderm also suddenly increases (Foe & Alberts, 1983). As a consequence o f the size increase of the blastoderm the density of energids forming the presumptive syncytial blastoderm tends to remain unchanged. At the end of the ninth cycle, the energids have reached the vitelline membrane of the egg and no further increase in size o f the syncytial blastoderm is possible. Therefore, following each of the nuclear division cycles 10, 11, 12 and 13, which also occur with a high degree of synchrony, the density of the energids near the surface of the egg doubles. At the end of a relatively long 14th cycle, the energids cellularize as they are surrounded by the infolding plasma membrane, thus forming a cellular blastoderm. Although segments do not appear as clear morphological units until sometime after the cellular blastoderm has formed, clonal analysis (reviewed by Lawrence,

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1981) and transplantation studies (Illmensee, 1978) indicate that cells are already committed to a specific segment at the cellular blastoderm stage. Furthermore, it has been shown that ligation of Drosophila embryos (Vogel, 1977), as well as embryos of other insect species (reviewed by Sander, 1981), causes a section of the embryo to be lost. The size of the section lost in Drosophila decreases as the time at which the embryo is ligated is increased. On this basis, it has been suggested that differentiation to a spatial region occurs in a sequence of steps, prior to the cellular blastoderm stage, which progressively divide the embryo into finer and finer subdivisions. However, the actual location and pattern of the committed somatic cells have been most clearly defined through fate mapping studies (Hartenstein et al., 1985). A fate map of the cellular blastoderm (Fig. 2) indicates that the cells which eventually form the segments map to a series of bands around the blastoderm. The aim of the work described in this paper is to investigate the possibility that a reaction-diffusion

Segments MdlMxILblT1[T21T31A1[A21A3iA41ASIA61A71A8 A9 Compartments Parasegments I 1 I 2 I 3 I 4151617

]-head region

I S I S I101111121131141

Tail region

FIG. 2. The spatial location and relationship of segments, parasegments and anterior (A) and posterior (P) compartments are shown with reference to (a) an embryo which has almost completed germ band shortening, and (b) an embryo at the cellular blastoderm stage (Campos-Ortega & Hartenstein, 1985). The mandibular (Md), maxillarly (Mx), labial (Lb), thoracic (T1-T3), and abdominal (AI-A10) segments are indicated. Hatched areas in (b) are regions which invaginate at gastrulation. The l-head region in (b) is the region of the head or anterior blastoderm which invaginates in a complicated series of movements primarily during germ band shortening and dorsal closure. Other abbreviations are:- am: anterior midgut; as: amnioserosa; ch clypeolabrum; es: esophagus; ms: mesoderm; ph procephalon; ph: pharynx; pro: posterior midgut; pr: protodeum.

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system may be a pattern formation mechanism which regulates gene expression prior to the formation of the cellular blastoderm, providing spatial information required for segmentation. Turing (1952) first suggested that a class of biochemical reactions, now known simply as reaction-diffusion systems (denoted in this paper as RD systems), may have the potential to act as mechanisms supplying spatial information during embryonic development. The simplest RD system is composed of just two substances (or morphogens) which react with each other and diffuse through the tissue containing them. As will be shown in this paper, the spatial information produced by an RD system is frequently in the form of wave-like spatial patterns in the distribution of the morphogens rather than a unique set of positional values (Wolpert, 1971) for the concentration. Kauffman et al. (1978), on the basis of a linear approximation to the RD system equations, suggested that the RD sys:em would give rise to a sequence of spatial patterns in the early stages of development of the Drosophila embryo. The sequence of patterns they derived using a linear approximation to the RD system, can in fact provide a basis for dividing the blastoderm into a number of sections roughly corresponding to segments (Kauffman, 1981). However, Bunow et al. (1980) solved the RD system equations numerically in order to test this suggestion and were unable to produce the sequence of patterns predicted by Kauffman et al. (1978). Furthermore, their results seem to indicate that a spatial pattern consisting of a series of waves along the anteroposterior (AP) axis of a prolate spheroid (i.e. a banded pattern) would be an extremely unlikely stable solution of the RD system. If the RD system is to provide the spatial information necessary for segmentation, it is essential that it is able to produce a sequence of banded patterns. The need for this is clear from recent observations of the spatial expression of genes such as fushi-taratzu ( f t z ) (Hafen et al., 1984) and hairy (h) (Ingham et al., 1985), which are known to influence segmentation. Both these genes are, for example, expressed in spatial pattern of seven "stripes" along the AP axis, as shown in Fig. 3, and belong to the pair-rule class of segmentation genes (N~isslein-Volhard et al., 1982). Meinhardt (1986), on the other hand, has proposed quite a different explanation for the segmentation of the Drosophila embryo which still makes use of the spatial pattern-forming properties of the RD system. He uses one RD system to set up a simple gradient along the AP axis. On the basis of this gradient and a set of four thresholds he divides the syncytial blastoderm into five "cardinal regions"--along the AP axis. The division of the embryo into dorsal and ventral regions is considered to occur via a separate mechanism. Division of the embryo along the AP axis into finer units, namely, segments, parasegments, and compartments is achieved using a further two 2-component RD systems, which must be linked in some way, or alternatively, a single 4-component RD system. The mechanism proposed by Meinhardt (1986) appears, in total, significantly more complex than the mechanism envisaged by Kauffman et al., since it involves the equivalent of four separate pattern formation mechanisms (allowing one for the dorsoventral division). It is also difficult to envisage how the Meinhardt proposal would explain the sequence of spatial patterns observed in the expression of several segmentation genes (discussed in

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h

A P axis

FIG. 3. The 7-band patterns of expression of the pair-rule genes hairy (h) and fushi-tarazu (ftz) are shown in a lateral view of a Drosophila embryo at the cellular blastoderm stage. The anteroposterior axis (AP) is indicated; anterior is to the left.

section 5) which exhibit "frequency doubling", i.e., the number of bands doubles in each successive pattern. A third approach suggested by Russell (1985) requires two independent RD systems and a mechanism (which was not defined) to interpret the phase angle between the independent wave-like patterns produced by the two RD systems. Russell assumed that the spatial patterns produced by the two RD systems may be approximated by trigonometric functions of the distance along the AP axis. The wavelength of the variation along the AP axis is set equal to one segment in the case of the wild type. Variation of the phase angle due to changes in the uniform steady state values or wavelengths of the RD systems provide some interesting comparisons with two classes of segmentation genes. Of particular interest is the capacity of the proposed mechanism to correctly predict whether the epithelium will regenerate or duplicate sections of tissue during intercalation following surgical manipulation. However the mechanism is not able to account for the development of a series of unique segments or to account for many of the phenotypes associated with the gap and co-ordinate classes of segmentation genes (Niisslein-Volhard et al., 1982). In this paper, we reconsider the capacity of a single RD system to produce a sequence of patterns during the early stages of development in Drosophila which are able to act as prepatterns controlling the differentiation of energids in the blastoderm. The sequence of patterns must be able to account for the spatial expression of the full range of segmentation genes which are known to be required for the normal segmentation of Drosophila. It will be shown in section 4 that at least one class of RD systems does exist, which avoids the difficulties encountered by Bunow et al. (1980) and produces a sequence of banded patterns. The essential features of a mechanism for pattern formation based on such an RD system are

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described in section 2. A linear analysis of the RD system is also presented (section 3). The desired sequence of patterns is obtained after certain restrictions are placed on the non-linearities of the RD system (section 4), causing the results to stand in complete contrast to the findings of Bunow et al. (1980). The pattern sequence obtained also exhibits a "frequency doubling" effect, and it is suggested in section 5 that such a pattern sequence may indeed account for the spatial expression of many segmentation genes, and act as a basis for segmentation of the Drosophila embryo into a set of unique segments.

2. A Prepattern Mechanism for Segmentation It is proposed that the prepattern mechanism for segmentation consists of just one RD system, which for simplicity is assumed to be composed of two substances, denoted here as U and V. It is convenient, although not necessary, to assume that U and V are confined to react and diffuse within the syncytium made up of spatially connected energids. The more general case where U and V may also react and diffuse outside of the syncytium is discussed briefly later. On the basis of the description of embryonic development of Drosophila m. given in the introduction, it appears that the syncytium during the nuclear division cycle 7 (and perhaps also cycles 5 and 6) may be approximated by a solid prolate spheroid, denoted here as I'l. The possible behaviour of the RD system in l'l is discussed briefly in the Appendix. The more important stage for the prepattern mechanism proposed here begins at the end of cycle 7 when most energids begin a synchronous stepwise migration outwards, forming at the beginning of the process the presumptive syncytial blastoderm (Fig. 1). The presumptive syncytial blastoderm will be represented as the surface of a prolate spheroid and will be denoted by 8~. The nucleii which do not migrate outwards are the yolk-nucleii; they remain quite separate from 8[1 and may be ignored. Although the size of the presumptive syncytial blastoderm increases during the stepwise migration, i.e. during cycles 8 and 9, for convenience, the shape is assumed to remain unchanged. At the end of cycle 9 the energids have reached the outer membrane so that the outward migration ceases and the syncytial blastoderm is formed. Therefore during the remaining cycles, namely cycles 10-14, neither the shape nor the size of 8fl, now the syncytial blastoderm, may undergo any further change. Hence, the density of nucleii forming the syncytial blastoderm must double at the completion of each of the nuclear division cycles 10-14.

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During cycles 8-14, let U and V diffuse at the rates du and dr, respectively, within 8D. U and V also react with each other so that at any point (x, y) on 81) the net production rates of U and V are given by f ( U(x, y), V(x, y)) and g( U(x, y), V(x, y)), respectively, where U(x, y) and V(x, y) are the concentrations of U and V at the point (x, y). The RD system equations (i.e. the mass balance equations)

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for U and V at (x, y) On 8tq, therefore, are given by

OU(x, y) _ duV2 U(x, y ) + f ( U(x, y), V(x, y)), Ot

(1) o V(x, y) _ dvV2 V(x, y) + g(U(x, y), V(x, y)). ot These equations have a steady state solution U(x, y) = Uo and V(x, y) = Vo, where Uo where Vo are constants such that f ( Uo, Vo) - g( Uo, Vo) -= 0. Provided do, dv and the parameters defining the reaction rates Kuu = Of/ OU, Kuv = Of/ OV, Kvu = Og/ OU, and Kvv = Og/O V at (/do, I/o) obey certain broad constraints, the state (Uo, Vo) is unstable to small perturbations in U and V. The subspace of the parameter space defined by these broad constraints is called the Turing space (Murray, 1982). Given that the parameters defining the reaction and diffusion rates lie within the Turing space, spatial patterns (i.e. inhomogeneities) in the distribution of U(x,y) and V(x, y) on 8 ~ may then arise spontaneously depending on the size and shape of 8fL T H E E F F E C T OF THE D E N S I T Y O F N U C L E A R M A T E R I A L

Unlike previous applications of RD systems to the problem of segmentation (Kauffman et al., 1978; Russell, 1985; Meinhardt, 1986), the RD system equations (1) will be modified here by requiring the expressions for the net production of U and V to be a function of the density, n, of nuclear material (and/or its precursors) over 8fL It is known that DNA is replicated during the initial period (interphase) of each of the first fourteen nuclear division cycles (Edgar & Schubiger, 1986). Presumably other components of the cytoplasm of the newly formed energids are also replicated during this period. In practice, therefore, it may be possible to regard n simply as the density of energids. However, since energid density itself does not appear to affect pattern formation, at least in situations where nucleii either fail to divide or undergo precocious division (Sullivan, 1987), we will continue to define n as the density of nuclear material. New production functions, F and G, are defined as follows, F(U, V, n ) = K ( n ) f ( U , V) G( U, V, n) = K(n)g( U, V). The diffusion rates are also considered to depend on n; the dependence being given by the function D(n). Therefore new diffusion rates, Du and Dr, are defined according to the expressions Du(n) = D(n)du

Dv(n)=D(n)dv. Assuming that n is uniform over 8fl, the RD system equations now become 0 U(x, y) _ Du (n)V 2U(x, y) + F( U(x, y), V(x, y), n) Ot

a V(x, y) Ot

Dv(n)V 2V(x, y)+ G( U(x, y), V(x, y), n).

(2)

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It follows that the reaction rates estimated at the uniform steady-state (Uo, Vo) are also dependent on n and are equal to the elements of the matrix K ( n ) ~ , where

y{= ( Kuu \Kvu

Kuv'~ Kvv/"

The particular example to be considered in detail in this paper is one where

K(n)=n and

D(n)=n-' In other words, it is proposed that rates of production a n d / o r degradation of U and V per unit area o f 6 f / a r e simply scaled up or down according to the density of nuclear material (and other energid components) in the (presumptive) syncytial blastoderm. It is also proposed that Du and Dv will decrease as those components of energids which impede diffusion, such as microfilaments associated with the nuclear material and the nuclear material itself, increase in density. In fact it will become clear from the results and discussion presented in sections 4 and 5, that what is really required is that the ratio K(n)/D(n) be proportional to n 2. In this case, the characteristic spatial wavelength o f the RD system, AM, (i.e. the expected mean distance between adjacent maxima or minima in U(x, y) and V(x, y ) - - d e f i n e d in the Appendix) will be proportional to n -~. Hence, if n doubles at the end of a nuclear division cycle, Am will be halved. This is the basis o f the phenomena referred to here as "frequency doubling", which is essential to the RD system, defined above, if it is to explain the formation of a set of unique segments in insects.

DIFFUSION

AND

REACTION

OUTSIDE

THE SYNCYTIAL

BLASTODERM

As mentioned above, it is not necessary to consider U and V to be confined to 6f/. U and V may be transported from ~51~into the yolk, or from the yolk into 6f/, provided the concentrations of U and V in the yolk tend to be maintained at uniform values characteristic of the yolk, Uy and Vv for example. In particular, the reaction and diffusion rates of U and V in the yolk must not fall within the Turing space so that there is no tendency for inhomogeneous distributions of the U and V to arise spontaneously within the yolk. The net transport of U and V into 6f/ can then be explicitly incorporated into the net production functions F and G in terms of ( U(x, y) - Uv) and ( V(x, y) - ~v ). Provided the parameters defining the reaction rates, which now explicitly include the net transport of U and V into 6f~, and diffusion rates of U and V within 611, still fall within the Turing space, spatial patterns in the distribution of U(x, y) and V(x, y) may be expected. Such spatial patterns may also extend into the yolk region where U(x, y) and V(x, y) must tend towards the values Uy and Vy with distance away from 6f/.

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3. The Prepattern Sequence Predicted on the Basis of a Linear Analysis

It is frequently useful to carry out a linear analysis of the non-linear RD system equations (2) in order to gain some appreciation of the number and type of spatial patterns which might be expected to appear at a particular stage of development. The steps involved are outlined in the Appendix. The linear analysis suggests that spatial patterns which develop spontaneously in the distribution of U and V on 8f~ may have the form of a linear combination of wavefunctions, ~ , . . , having wavenumber k where

~,.. = S * , ( c , r1) cos (m4,).

(3)

n and ~b are the spheroidal co-ordinates and c =fk. 2 f is the distance between the focii of 8£/ (Fig. 4). S * , ( c , rl) are prolate angular wavefunctions (Appendix). The subscript m specifies the number of full-waves of variation as a function of ~b, where 0 -< ~b<-2zr, while the quantity n - m, where n is also a subscript, may be regarded as the number of half-waves of variation as a function of distance along the AP axis. Thus, ~o, is a simple AP gradient, while ~,~ is a simple gradient in the dorsoventral (DV) direction. For a (presumptive) syncytial blastoderm of a particular size and density of nuclear material per unit surface area, the spatial pattern in U and V is expected to be determined by a linear combination of growth modes. The growth modes are the subset of ~ , , , whose wavelengths (wavenumbers) lie within a narrow range of

Y z

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-1.0 Anterior I 100%

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I

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I

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FIG. 4. T h e surface of a prolate s p h e r o i d is defined by the e q u a t i o n s , x = a~'rl, y = ax/i~ :2 - 1)(1 - 0 2) sin ~b and z = ax/(¢ 2 - 1 ) ( 1 - 7/2) cos ~b, where ~:2 = r 2 / ( r 2 _ 1) and r = a/b. The focii are located at ( + f , 0, 0), where f = a~"-t. T h e relationship between egg length (EL) a n d "0 is also s h o w n . Strictly s p e a k i n g , egg length, as defined here in terms of r/, s h o u l d be referred to as s y n c y t i u m length, since the R D s y s t e m e q u a t i o n s are solved over the surface of a prolate s p h e r o i d representing the syncytial b l a s t o d e r m . However, the term u s e d by experimentalists is always egg length. O n l y d u r i n g nuclear division cycles 8 a n d 9 are the two significantly different (see Fig. 1).

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AM(kM) defined by (Appendix) 27r

(4a)

AM = ~ M

where

K n /Kw÷, v

k ~ : 2--~n ) \--~-v

--~-v /

oc n 2.

(4b) (4c)

The extent of the narrow range about AM is given by the quantity Ak~ defined in the Appendix. THE PREPATTERN D U R I N G NUCLEAR D I V I S I O N CYCLE 8

Many expressions could be chosen for f and g satisfying the broad constraints, given in the Appendix, required to ensure that spatial patterns will arise spontaneously. Initially, a very simple form for f and g will be chosen here, namely

f(U, V ) = U2V - V + A g(U, V ) = - U 2 V + B .

(5)

These reaction terms were first proposed by Gierer & Meinhardt (1972), and in a more general form by Schnakenberg (1979). The parameter values A = 0.25, B = 0.75, du = 0.227 and dv = 5-0 lie within the Turing space and were chosen so that Uo = 1, k ~ = 1, and Ak~ =0.3. The shape of 81~ is given by the ratio ~'= a/b, which was set to 2.7 to reflect the shape of the egg itself (Sonnenblick, 1950). The size of the presumptive syncytial blastoderm during cycle 8 was set in arbitrary units such that (fkM)2 = c 2 = 3.7. With this choice of c 2 and other parameter values, only one growth mode is possible during cycle 8, namely qSot (see Appendix), i.e. a simple gradient along the AP axis as shown in Fig. 5. The linear analysis therefore suggests the prepattern in U and V during cycle 8 will have the same form as ~o~. (The situation prior to cycle 8 is discussed in the Appendix. It will be assumed here that no prepatterns arise prior to cycle 8.) THE PREPATTERN D U R I N G N U C L E A R DIVISION CYCLE 9

Following nuclear division at the end of cycle 8, the presumptive syncytiai blastoderm suddenly increases in size and so appears much larger during cycle 9. The size change can be estimated using the observations o f Foe & Alberts (1983) assuming the shape, i.e. z, remains unchanged. Their observations suggest that the surface area of 8f~ approximately doubles following the 8th nuclear division. Hence, the density of nuclear material in ~Sfl remains unchanged. However, it can be shown (Appendix) that the large change in size is sut~cient to cause the AP pattern, present during cycle 8, to cease being a growth mode. Therefore it is expected to become unstable and a new pattern to appear. Only one growth mode is possible following

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FIG. 5. T h e wavefunctions expected to determine the spatial pattern in U during cycles 8 and 9 are ~ot =S*~(c, ~1) a n d qttl = S*~(c, 7)cos ~b, respectively. The wavefunctions required to dominate the spatial pattern in U over a restricted 77 range during cycles 10 and 11 are ~ o 2 = S ' 2 ( c , 71) and q'o4 = S*(c, rl), respectively. The prolate angular wavefunctions, S*,,,,(c, rt), in each case are plotted here as a function of ft.

the size increase, namely the wavefunction ~ o , which is a simple gradient in the DV direction. Once again, the same maternal (pre-existing) gradient which determines the polarity of the AP prepattern will also determine the polarity of the DV prepattern since it is assumed to contain a small component in the DV direction. T H E P R E P A T T E R N D U R I N G N U C L E A R DIVISION C Y C L E 10

Following nuclear division at the end of cycle 9 the size of 8F~ increases once again. However, this time the energids migrate to the periphery of the egg so that the outwards migration is halted. As a consequence, the surface area of 8~ cannot double to accommodate the increased number of energids, and hence the density of energids and nuclear material forming the syncytial blastoderm must increase. In order to obtain patterns consistent with the observed banded expression of some segmentation genes (e.g. Fig. 3), and to be consistent with the fate map in Fig. 2, it is proposed that the RD system is switched off in much of the I-head and tail regions (shown in Fig. 2) when the energids reach the periphery of the egg. The 1-head region is the anterior portion of the blastoderm which eventually involutes to form a number of the internal organs and tissues of the larval head (CamposOrtega & Hartenstein, 1985). That part of the I-head region where the RD system is switched off, is defined to lie between 100% and 80% of EL; and that part of the tail region where the RD system is switched off will be defined, for the moment, to lie between 0% and 5% of EL. The simplest approach, adopted here to begin

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with, is to assume that U and V are fixed at certain values in the head and tail regions and to solve the linearized equations only in the region from 5% to 80% of EL (details are given in section 4). The size and density changes which occur at the end of cycle 9, are such that the DV prepattern, present during cycle 9, ceases to be a growth mode. Hence the DV prepattern is expected to become unstable and to change to a new pattern. It can be shown that more than one growth mode is now possible. One of the growth modes, ~o2, is plotted as a function of 77 in Fig. 5. In order to explain the spatial expression of some segmentation genes it would be necessary for ~o2 to dominate the form of the prepattern which arises during cycle 10. THE

PREPA'I'TERN

DURING

NUCLEAR

DIVISION

CYCLE

11

Following nuclear division at the end o f cycle 10, the energid density doubles since the size of 8f~ remains unchanged. Therefore, both n and the wavenumber kM also double during the early part of cycle 11 so that the cycle 10 prepattern is expected to become unstable. It can be shown that multiple growth modes are now possible (Table A1 in the Appendix). One of these, ~o4, is shown in Fig. 5. q%4 is the growth mode required to dominate the prepattern at cycle 11 in order to explain the spatial expression of some segmentation genes (see section 5). The dominance of ~o2 during cycle 10 and ~o4 during cycle 11 cannot be demonstrated on the basis of the linear analysis alone due to the presence of multiple growth modes. Numerical solution of the non-linear equations (2) is necessary to determine what form the prepattern will take. In fact, the same situation occurs at the beginning of each of the cycles 12-14 when both the n and kM double. In each case, the existing prepattern is expected to become unstable and a new set of growth modes are expected to determine the new prepattern. The common feature of cycles 11-14, and to a very large extent that of cycle 10 as well, is the frequency doubling effect of n in eqn (4c).

4. Numerical Solution of RD System Equations To confirm the patterns in Fig. 5 predicted for cycles 8 and 9 and, in particular, to determine the patterns expected to arise during cycles 10-14 when multiple growth modes are possible, the non-linear RD system equations (2) have been solved numerically. The numerical method chosen is described fully by Mooney (1986). Initially, the non-linear net production terms defined in section 3 by eqn (5), will be used. The initial state (i.e. the initial distribution of U and V) in all calculations, except for cycle 8, is taken to be the pattern obtained in the previous cycle. In the case of cycle 8 the initial state is a small maternal gradient, i.e. and AP gradient with a small DV component added to introduce some asymmetry. The maximum amplitude of the maternal gradient is chosen to be 1% of Uo and V0. For cycles 8 and 9 the RD system are solved over the full surface of 8f~ and the solutions in the form o f spatial distributions of in U('O, ~b) are shown in Fig. 6(a) and (b). It can be seen that for both cycles 8 and 9, the solutions, i.e. the spatial

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FIG. 6. The distribution of U over the surface of a prolate spheroid, 8D, is displayed in the form of contour plots over the "0~bplane (Fig. 4). The distributions shown in Fig. 6 (a)-(d) were obtained by solving the RD equations using the net production functions defined in eqn (2). The distributions shown in Fig. 6 (e)-(h) were obtained using the antisymmetric net production functions (eqn (6)). In each case, the RD system equations were solved on the surface of 8D choosing the size (f) and density of nuclear material (n) of 8~ appropriate to nuclear division cycle 8 (Fig. 6(a) and (e)), cycle 9 (b and f), cycle 10 (c and g), and cycle 11 (d and h). The r/ range was restricted in cycles 10 and 11 since the RD system is assumed to be switched off in the I-head and tail regions (Fig. 2, section 3) once the energids reach the periphery of the egg. The initial distributions of U (and V) for the calculations during cycles 9, 10 and 11 were taken to be the distribution of U (and V) obtained in the previous cycle. In the case of cycle 8, the initial distribution was a small maternal gradient (also added to the initial distribution in the case of cycle 9, Fig. 6(f)). It is clear that both RD systems produce an AP gradient during cycle 8 (Fig. 6(a), (e)) and a DV gradient during cycle 9 (Fig. 6(b), (f)). However, only the antisymmetric net production functions show a strong tendency to produce bands around the embryo during cycles 10 and 11 (Fig. 6(g), (h)). p a t t e r n s o b t a i n e d , a r e c o n s i s t e n t with t h o s e e x p e c t e d o n t h e b a s i s o f t h e l i n e a r a n a l y s i s (Fig. 5). This is n o t s u r p r i s i n g s i n c e d u r i n g e a c h o f t h e s e t w o cycles t h e r e is o n l y o n e p o s s i b l e g r o w t h m o d e . As s t a t e d in the d i s c u s s i o n c o n c e r n i n g n u c l e a r d i v i s i o n cycle 10 in s e c t i o n 3, it is p r o p o s e d t h a t o n c e t h e e n e r g i d s r e a c h t h e p e r i p l a s m a n d the o u t e r m e m b r a n e o f the egg at t h e e n d o f c y c l e 9, the R D s y s t e m is s w i t c h e d off in t h e I - h e a d a n d tail regions. Strictly s p e a k i n g , the R D system s h o u l d still be s o l v e d o v e r the full s u r f a c e o f 8 D u s i n g different sets o f r e a c t i o n a n d d i f f u s i o n p a r a m e t e r s in t h e I - h e a d a n d tail regions. H o w e v e r , to s i m p l i f y the p r o b l e m , t h e R D s y s t e m is s o l v e d in t h e m i d d l e r e g i o n o n l y w h e r e t h e R D s y s t e m is u n c h a n g e d a n d w h e r e , u n l i k e t h e I - h e a d a n d tail r e g i o n s , s p o n t a n e o u s d e v e l o p m e n t o f p a t t e r n s is still p o s s i b l e . T h e r e f o r e it is a s s u m e d t h a t in the I - h e a d a n d tail r e g i o n s t h e r e is a s t r o n g t e n d e n c y f o r U a n d V to be h e l d at fixed v a l u e s , c h o s e n h e r e to b e significantly b e l o w t h e i r u n i f o r m s t e a d y - s t a t e values. I n i t i a l l y , the fixed v a l u e s o f U a n d V at the i n t e r f a c e with t h e

290

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I-head and tail regions will be set close to the anticipated minima in U and V (i.e. 0.7 Uo and 0-7 Vo in Fig. 6). Under these conditions, the patterns obtained during cycles 10 and 11 are those shown in Fig. 6(c) and (d). They are clearly a mixture of two or more linear modes (Appendix). The result in the case of cycle 10 is a little surprising since the solution contains a strong component of a linear mode which is not one of the expected growth modes (Table A1, Appendix). The solution at cycle 11, on the other hand, is primarily a mixture of two of the possible growth modes (Table 1). Even when the initial state in cycles 10 and 11 is taken to be o f the form o f the growth modes ~o2 and ~o4, respectively (i.e. the modes required to dominate the solutions at cycles 10 and 11), the stable patterns shown in Fig. 6(c) and (d) still develop. Thus, the solutions at cycles 10 and 11 clearly demonstrate a preference for a pattern which is a combination o f the possible linear modes, rather than one in which a single growth mode will dominate. In this respect, the results obtained here are similar to those of Bunow et al. (1980), who used a different form for the non-linear net production terms. Indeed, we obtained similar results to those in Fig. 6(a)-(d) using yet another form for the net production terms, namely, the expressions proposed by Kauitman et al. (1978). Unless expressions for the net production functions can be found which exhibit the necessary strong preference for striped or banded patterns then the conclusion must be the same as that of Bunow et al. (1980), i.e. that the pattern-forming capacity of the RD system is very unlikely to form a basis for segmentation. THE

FORMATION

OF STRIPED

PATTERNS

We have explored numerically the pattern-forming properties o f net production functions which are antisymmetric with respect to a sign change in both u = U - U0 and o = V - V0. Such antisymmetric functions were, in fact, found to exhibit a very strong preference for striped patterns. The mathematical properties of this class of functions is still being investigated and will be reported elsewhere. It is sufficient for the purposes of this paper to consider just one example of the antisymmetric class of functions. The example to be used has deliberately been chosen to be similar to the expressions of Kauffman et al. (1978), which, as stated above, do not show a preference for striped patterns. The non-linear terms in the Kauffman et al. equations are a form of the Hill equation, well known in biochemistry for describing a sigmoidal curve o f enzyme activity as a function of substrate concentration. We have simply replaced the Hill expression by a function involving tanh to ensure the antisymmetry condition is satisfied, so that the net production functions become f(U, V)=-AU+B[l÷tanh

(s(V-

V0))]

g( U, V ) = - C U + D [ I + a tanh (s( V - Vo))].

(6)

Basically, the tanh function is a sum of polynomials, consisting o f odd powers o f V - V0, acting to limit the amplitude o f the linear growth modes and to control the interaction between them. Other forms for the net production function could have been chosen but eqn (6) is sufficient for our purposes and allows a comparison with other results derived using the equations o f Kautiman et ai. Once again the parameter

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values A = B = 2 3 . 1 , C = D = 1.3, a = 0 . 8 , s =3-0, du = 2 0 and dv = 1 were chosen so that Uo = V0 = 1, k 2 = 1 and Ak~ = 0.3. The sequence of solutions obtained for cycles 8-11 using the antisymmetric net production functions in eqn (6) can be seen in Fig. 6(e)-(h). They show clearly the very strong preference to form striped or banded patterns in complete contrast to the results in Fig. 6(a)-(d). In fact, so strong is this preference, that it is necessary to assume that the maternal gradient, which contains a small DV component, is still present at cycle 9 in order to obtain the DV pattern in Fig. 6(f); otherwise the DV pattern will not form, given that the initial state is the purely AP pattern obtained for cycle 8. It should be noted that the cycle 10 and cycle 11 patterns in U remained basically the same when the value-fixed boundary conditions were set as 0-5 of the uniform steady state values, rather than 0.7 as in Fig. 6. Using the net production functions in eqn (6), the calculations were extended to cycles 12, 13 and 14. In each new cycle frequency doubling occurs due entirely to the dependence of kM on n, causing an apparent split in the stripes o f the previous pattern. In each case, the pattern may be considered to be "perfectly" striped in the sense that the wave-like variation in U (and in V) is restricted to the one co-ordinate, 77 (i.e. along the AP axis). Hence, the results are not shown as a two-dimensional contour plot over the r/4~ plane as in Fig. 6, but rather U is plotted as a function of egg length (or 71) in Fig. 7. The r/ range is restricted to lie between 80% and 15% for these calculations since some of the highly banded expressions o f segmentation genes do not extend into the posterior 15% of EL (discussed in the next section). The results for cycles 8-11 are also shown in Fig. 7. With the exception o f the cycle 14 pattern in U, the solutions obtained did not alter significantly when small changes were made to the fixed boundary conditions. However, the cycle 14 pattern was found to be sensitive to small changes in the boundary conditions so that a pattern with one half-wave of variation more or one half-wave less may also develop instead of the cycle 14 pattern shown in Fig. 7. The linear analysis suggests that other striped patterns are also possible at cycles 12 and 13. However, the initial state, i.e. the pattern of the previous cycle, largely removes these possibilities in the case o f cycles 12 and 13. It may also be possible to remove the degeneracy o f cycle 14 by choosing different reaction and diffusion rates to define the RD system in the I-head and tail regions, and extending the solutions for U and V into these regions. Calculations are currently being carried out to determine the solutions over the whole blastoderm now that preference for striped patterns has been established.

5. Spatial Patterns in the Expression of Segmentation Genes FREQUENCY DOUBLING AND "'PAIR-RULE'" GENES Transcription o f zygotic genes, including the pair-rule genes, is known to begin during the l l t h nuclear division cycle (Edgar & Schubiger, 1986). Hence, if the spatial distribution of U during cycle 10, for example, is to regulate the expression of various pair-rule genes, then it will be necessary for the concentration o f U to

292

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FIG. 7. The RD system equations (2) with antisymmetric net production functions (6) were solved for each o f the nuclear division cycles 8-14. With the exception of the DV pattern obtained in cycle 9, the patterns in U appeared to be perfectly banded or striped hence only the variation in 7/ is displayed here for each cycle. The position of the segments a n d parasegments was assigned on the basis o f the cycle 12 and cycle 13 patterns in U and the expression of the pair-rule genes h, opa (runt), ftz and eve (as described in section 5, see also Fig. 9). The dotted curves for cycles 10 and 11 are the solutions obtained when the boundary conditions at the interface with the i-head region were changed so that U = 0 . 5 Uo.i.e. the same as for cycles 12-14. The dashed section of the curves in cycles 10-14 are schematic only a n d emphasize the fact that the calculations have yet to be extended into the 1-head and tail regions.

influence their transcription indirectly. One possibility would be for U to control the affinity of certain DNA-binding proteins, of maternal origin, to bind with each other forming complexes such as dimers. The dimers may subsequently bind to D N A and regulate transcription. If the half-life of the dimers is ~ 1 0 minutes, or one cycle, then the spatial patterns in U at each cycle will cause a similar pattern to arise in the distribution of the dimers. However, the localized concentrations of the dimers may arise too late in the nuclear division cycle to bind to D N A and influence the transcription rate of particular genes during that cycle. The transcription rate may only be influenced in the subsequent cycle. The transcripts themselves may

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accumulate over a period of time, assumed here to be - 1 5 minutes or one to two cycles, determined by the degradation rate of the transcripts, as well as the half-life of dimers existing as DNA-bound proteins. A combination of these delays would make it possible for the cycle 10 pattern in U (or perhaps earlier patterns in U, depending on the length of the delays) to regulate the expression of zygotic genes. In fact some delay between the appearance of a spatial pattern in U and the appearance of a similar pattern in the distribution of particular transcripts may be expected for each cycle. If we consider a situation where the DNA-binding protein tends to form dimers in regions of the syncytium where the concentration of U is below the value Uo, then a sequence of spatial patterns in the distribution of the dimers reflecting those of U shown in Fig. 7 will arise. If the dimers also bind to the regulatory region of a gene such as the ftz gene, for example, and inhibit its expression, then the spatial distribution of the ftz transcripts in the syncytial blastoderm will also appear as a sequence of spatial patterns, predicted using Fig. 7 to be the pattern sequence shown in Fig. 8. The predicted pattern sequence begins with the cycle 10 pattern in U. Each pattern consists of a set of bands around the syncytial blastoderm corresponding to regions where U exceeds Uo and so will be referred to as the Um,~ patterns. The observed sequence of patterns in the distribution o f f t z transcripts is also plotted in Fig. 8(a) immediately below the Umaxpatterns. There appears to be good qualitative agreement between the two pattern sequences after allowing for a delay of about 20 minutes from the time at which the pattern in U arises. Both the observed and the predicted pattern sequences exhibit frequency doubling, i.e. each band of one pattern tends to be replaced by two bands in the subsequent pattern. The major differences between the observed and predicted pattern sequences are (1) that ftz transcripts are not observed anterior of the line at 65% EL, which corresponds approximately to the middle of the maxillary segment, and (2) the final splitting of the bands in the 7-band pattern is not observed; instead the 7-band pattern is maintained (Weir & Kornberg, 1985). The banded spatial patterns of expression of two other pair-rule genes, namely, even-skipped eve and paired prd (Niisslein-Volhard et al., 1982) are also observed to undergo frequency doubling (Fig. 8(b) and (c)) (Macdonald et al., 1986; Harding et al., 1986; Kilchherr et al., 1986). Both these genes are expressed in a very clear 7-band pattern during the 14th cycle. In both cases, the 7-band pattern changes to a 14-band pattern at a slightly later time. However, in the case o f p r d this is achieved by a splitting of six of the six bands with the most anterior band remaining unchanged. A new larger band is also added posteriorly. In the case of eve, it is achieved by the appearance of seven new bands in between the existing bands, which narrow, and by adding one band posteriorly. Both changes can be accounted for on the basis of the Umaxsequence of patterns by associating eve expression with the patterns of Umi, (i.e., the bands or regions where U < U0) and associating the prd patterns with those of Um~. This is consistent with the observations made at the 7-band stage, which show that the spatial expression of prd is highly correlated with ftz expression (Kilchherr et al., 1986) while the spatial expression of eve is highly anti-correlated with ftz expression (Macdonald et al., 1986). There also seems to

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FIG. 8. The location and width of the bands in which U > U 0 is shown (dotted areas) for the patterns in the distribution of U given in Fig. 7, obtained during cycles 10-14. This sequence of Ureax patterns is plotted on a segment m a p in all four parts of the diagram. The segment m a p was constructed on the basis that the presumptive segments on the blastoderm are o f equal width and lie between 77% EL and 15% EL. The sequence of U,,a~ patterns is compared with the sequence of spatial patterns observed (black areas) during various nuclear division cycles (indicated in the left-hand column; 14e, 14m and 141 indicate the early, middle and late part of cycle 14, respectively) in the expression of the pair-rule genes (a) ftz, (b) eve, (c) prd, and (d) h. The hatched areas in the cycle 14(I) patterns o f eve and prd indicate b a n d s which are weakly expressed, at least temporarily, relative to the alternating bands. Other hatched areas represent uncertainties in the extent o f s o m e observed bands. Clearly there is a delay between the time (division cycle) at which a particular pattern in U is expected a n d the appearance of the corresponding pattern in the expression of the various pair-rule genes. Allowing for this delay, Urn, ~ patterns appear to be correlated with those observed in f t z and prd but anticorrelated with the patterns in eve expression. It should be noted that the declining height o f s o m e b a n d s s h o w n in (b) (cycle 13) indicates that a significant AP gradient has been observed in the expi'ession o f eve. An AP gradient also appears to be present in the expression of the prd b a n d s although this has not been s h o w n in (c). The expression of eve during cycle 14e is not uniform, but no clear b a n d e d pattern has been reported, hence no pattern is s h o w n in (b) for cycle 14e.

be significant anti-correlation in some earlier patterns in the distribution of eve transcripts (Fig. 8(b)). Unfortunately there is no clear banded pattern following the 2-band eve pattern at cycle 13 even though the expression of eve is not uniform (Macdonald et al., 1986); hence this has been omitted in Fig. 8(b). A strong AP gradient in the expression of eve, indicated in Fig. 8(b), tends to make the observations difficult to interpret. An AP gradient is also seen in the expression of prd. The bandg of the 14-band pattern in both eve and prd are observed to alternate in intensity. The section of cuticle deleted in some of the phenotypes o f eve (Fig. 8(b)) and prd (Fig. 8(c)) tends to correspond to the stronger bands o f the 14-band pattern and so occurs in every second segment (Niisslein-Volhard et al., 1982). However, in the stronger eve alleles sections of the cuticle from every segment are deleted (Niisslein-Volhard et al., 1985). In the case o f f t z , the deletion pattern corresponds to a large extent with the 7-band patterns (Fig. 8(a); Niisslein-Volhard et al., 1982).

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Therefore, it seems that the 7-band pattern o f f t z and eve expression and the 14-band patterns of eve and prd expression, are necessary for normal development. On the other hand, no role has, as yet, been established for the earlier patterns in the various sequences. It is suggested here that the earlier patterns may simply reflect the nature of the pattern formation mechanism, e.g. the RD system as proposed in section 2. SEGM ENTATION

The pair-rule gene hairy (h) is also observed to be expressed in a' 7-band pattern (Ingham et al., 1985; Ish-Horowicz et al., 1985). However, in contrast with the other pair-rule genes discussed, h shows no sign of frequency doubling. In fact the 7-band pattern seems to be the only banded pattern of expression exhibited by h (Fig. 8(d)). Furthermore, the seven bands partially overlap the seven ftz bands (Ish-Horowicz et al., 1985) and may be expected to partially overlap the seven eve bands as well. The situation is depicted in Fig. 9. It is clear, therefore, that the seven h bands do not coincide with the maxima or the minima of U during any of the nuclear division cycles. The seven h bands do, however, lie precisely between the maxima and the minima of U at cycle 12. Another pair-rule gene called odd-paired (opa) has a deletion pattern complementary to that of h (Niisslein-Volhard et al., 1982) and so may prove to be expressed in a 7-band pattern which is also complementary to that of h (Fig. 9). On the other hand, the pair-rule gene runt may be the gene which will prove to be expressed in a complementary pattern to that of h. To simplify the discussion we will assume here that opa is the ~omplementary gene. If this is correct then opa expression will be aligned with the maxima and minima of U in the distribution of U expected at cycle 12. On this basis, it is suggested that the maxima and/or minima of U actually promotes the transcription of opa, which may begin during cycle 11 but only becomes significant during cycle 12. If, in addition, there is an interaction between the h and opa genes via a diffusible product of both such that opa promotes its own transcription as well as the transcription of h, while h inhibits its own transcription and the transcription of opa, then spontaneous development of complementary banded patterns in h and opa is possible. These conditions for h and opa mean that together they form an RD system, as already suggested by Meinhardt (1986). The reaction and diffusion parameters would need to lie within the Turing space and to define a suitable value for AM tO ensure that a 7-band pattern in both h and opa arises spontaneously. The difference in the proposal here, compared to that of Meinhardt, is that the opa bands will be aligned with the maxima and minima of U in its distribution at cycle 12. The alignment also establishes the phase of the h and opa patterns with respect to the 7-band patterns of ftz and eve, which are determined by the cycle 13 pattern in the distribution of U. The predicted relationship between the 7-band patterns of h, opa, ftz and eve is, therefore, the same as that shown in Fig. 9. The 7-band spatial patterns in Fig. 9 are, in principle, sufficient to determine a series of segments, parasegments and compartments. This subdivision of the blastoderm could be achieved by the following interactions:

296

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FIG. 9. (a) The observed spatial relationship between the 7-band patterns of expressior~ofthe pair-rule genes h, eve and .ftz along with the anticipated for opa (or perhaps runt) is illustrated. It is suggested that the seven h bands determine seven "'even" or '" +'" segments, while a complementary set of seven opa bands determines seven "odd" or " - ' " segments. The seven ftz and eve bands then split each segment into an anterior (A) and a posterior (P) compartment. Therefore two types of A and P compartments are created, namely A+, A- and P+, and P-. The fourteen bands of en expression, which are associated with the posterior compartments, initially alternate in intensity suggesting at least the temporary existence of two types of posterior compartment. (b) The effect offtz (and eve) in determining the A and P compartments is opposite in even and odd segments. It can be imagined that the complementary pattern of h and opa expression acts as a switch which, in effect, inverts the slope of the cycle 13 pattern in U (controlling the 7-band ftz/eve expression) to produce in every second segment (dashed curves) a saw-tooth pattern. The saw-tooth pattern provides a basis for dividing segments into A and P compartments and establishing segment borders where there is a large discontinuity. (1) the e x p r e s s i o n o f f t z a n d opa t o g e t h e r c o m m i t cells to form a p o s t e r i o r c o m p a r t m e n t o f the " + " type, d e n o t e d P + ; (2) the e x p r e s s i o n o f eve a n d opa t o g e t h e r c o m m i t cells to form a n a n t e r i o r c o m p a r t m e n t also of the " + " type, d e n o t e d , A+; (3) the e x p r e s s i o n o f eve a n d h t o g e t h e r c o m m i t cells to form a p o s t e r i o r c o m p a r t m e n t o f the " - " type, d e n o t e d P - ; (4) the e x p r e s s i o n o f f t z a n d h together c o m m i t cells to form a n a n t e r i o r c o m p a r t m e n t also o f the " - " type, d e n o t e d A - . These i n t e r a c t i o n s establish two types of A a n d P cells. E v i d e n c e for the existence of two types o f p o s t e r i o r c o m p a r t m e n t cells comes from the e x p r e s s i o n o f the pair-rule gene e n g r a i l e d ( e n ) . en is k n o w n to be expressed in the p o s t e r i o r c o m p a r t m e n t s of all s e g m e n t s b u t the e x p r e s s i o n is initially m o r e i n t e n s e in the even t h a n the o d d p o s t e r i o r c o m p a r t m e n t s (Weir & K o r n b e r g , 1985) as s h o w n in Fig. 9(a). Thus, o n c e the correct spatial e x p r e s s i o n o f the p a i r - r u l e genes f t z , eve, h a n d opa is a c h i e v e d t h r o u g h the cycle 12 a n d 13 p a t t e r n s in U, a n i n t e r a c t i o n b e t w e e n these genes is still r e q u i r e d to regulate the e x p r e s s i o n o f other zygotic genes, such as en,

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so that t h e ' P and A compartments in all segments are correctly determined. The interactions defined above assume that each o f the seven bands of opa and h expression extend across a full segment and thus establish two different types of segment, denoted here as + and - (or even and odd). The cycle t3 pattern in U then divides each segment into a posterior and an anterior compartment depending on whether U is above Uo (when f t z is expressed) or U is below Uo (when eve is expressed). In an even segment, f t z causes cells to develop as posterior (P) cells while eve causes cells to develop as anterior (A) cells. In odd segments, the effect o f f t z and eve is opposite to their effect in even segments. (It is on this basis that segments and parasegments have been drawn in relation to the cycle 12 and cycle 13 patterns of U in Fig. 7.) Thus, the complementary spatial expression of h and opa can be thought of as the basis for a switch designed to invert the slope of the cycle 13 distribution of U in every second segment, yielding the saw-tooth pattern in Fig. 9(b). It is suggested that the distribution of the products of one or more of the "segment polarity" genes (Nfisslein-Volhard et al., 1982; Niisslein-Volhard & Wieschaus, 1980), or genes which they regulate, may reflect the saw-tooth pattern and establish the polarity of the segment. Segment boundaries may then form between A and P cells where there is also a large difference in the concentration of the gene product(s) which record the segment polarity gradient (Fig. 9(b)). The advantage of such a scheme is that it is consistent with the conditions under which segment boundaries are observed to regenerate in grafting and wounding experiments (Wright & Lawrence, 1981). An alternative scheme for the formation of segment boundaries would be to introduce a three-compartment segment such as that defined by Meinhardt (1986). Although the above scheme is sufficient, in principle, to determine segments, parasegments and compartments, an accurate description of the segmentation process will undoubtedly require the 14-band patterns of eve, prd and several other pair-rule genes. Further experimental results concerning the function and interaction of the pair-rule genes are required before the role played by each of the genes will be understood in detail. UNIQUE SEGMENTS

While the 7-band patterns o f expression shown in Fig. 9 contain sufficient spatial information to establish a series of segments or parasegments, which, in principle at least, may alternate in character, they are unable to provide a basis for a group of two or more segments (or parasegments) to develop along a unique path relative to a neighbouring group of two segments (or parasegments). A divergence in the developmental path of neighbouring groups of two parasegments can, however, be achieved through the pattern in the distribution of U at cycle 12. This is possible because, at cycle 12, the regions where U is above Uo (or where U is below U0) encompass two parasegments. Likewise, the pattern in U at cycle 11 is able to direct a group of four segments along a different developmental path to a neighbouring group of four segments. Clearly the phenomena of frequency doubling is potentially a very efficient mechanism for progressively subdividing the insect blastoderm. It is because of frequency doubling that the sequence of patterns in U, when considered

298

B.N. N A G O R C K A

in total, are c a p a b l e o f assigning a u n i q u e character to each segment, p a r a s e g m e n t and c o m p a r t m e n t . While in principle s o m e o f the patterns in U are able to direct g r o u p s o f segments or parasegments along different d e v e l o p m e n t a l paths, in practice, g r o u p s o f segmentation genes called gap genes and co-ordinate genes are k n o w n to play a m a j o r role in directing the d e v e l o p m e n t o f groups o f energids in the s y n c y t i u m at a coarser spatial level than one segment. Some o f the co-ordinate genes seem to influence pattern f o r m a t i o n either anterior to the first a b d o m i n a l segment (Fr/Shnhofer & Niisslein-Volhard, 1986) or posterior to the m e t a t h o r a x ( M a c d o n a l d & Struhl, 1986). O t h e r co-ordinate genes clearly control the dorsoventral pattern ( A n d e r s o n & Niisslein-Volhard, 1986). It is possible that the patterns in U at cycles 8 and 9 m a y be required to regulate the expression o f s o m e o f the co-ordinate genes. If this is true then it w o u l d be necessary for at least s o m e o f the c o - o r d i n a t e genes to be transcribed maternally, since transcription o f zygotic genes only begins during cycle 11. In fact, almost all co-ordinate genes are f o u n d to be maternal genes. A description o f the possible role o f the R D system in regulating the expression o f the co-ordinate genes will be presented elsewhere. The spatial expression a n d p h e n o t y p e s o f the gap genes are quite complex. The observed spatial expression o f h u n c h - b a c k (hb) a n d Kriippel (Kr) (J~ickle et aL, 1986) are s h o w n in Fig. 10 along with the segments deleted, or partially deleted, by

cad (pair-rule deletion)

1.2 1-0

0.8

bcd (deletion)

Cycle 8

_ _

hb

1.2 1.0 [U] 0.8

Cycle 10

hb /

0-6 Cycle

11

1-2 1.0

0.8 0.6 IO0

I

1

I

75

50

25

%

- 11.0

I

-0-5

egg

0

length I

t

o

0.5

I

1.0

FIG. 10. The cycle 8 pattern in U (from Fig. 7) is compared here with the group of segments partially or totally deleted as a result of mutations in the co-ordinate genes caudal (cad) and bicoid (bcd). The transcripts and products of cad are known to become distributed according to an anteroposterior gradient at about the time of cycle 8. The regions deleted as a result of mutations in the gap genes knirps (kn) and giant (gt) are compared to the cycle 10 and I I patterns in U. The banded regions of expression of two other gap genes, hunchback (hb) and Kri.ippel (Kr) are also shown with respect to the cycle 10 pattern in U.

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ORGANIZATION

IN

D. M E L A N O G A S T E R

299

the gap genes knirps (kn) (Niisslein-Volhard & Weischhaus, 1980) and giant (gt) (Petschek et al., 1987). Despite the complexity, there appears to be some agreement between the /-/max patterns and the Umin patterns during cycles 8, 10 and 11, and the regions influenced by the gap genes (Fig. 10). The interaction between the gap genes themselves as well as their interaction with other segmentation genes such as the co-ordinate group of genes, needs to be considered in some detail before the function(s) of each gene and the role played by an RD system in regulating these genes can be understood. It should be noted that, in general, the various patterns in the distribution of U may only be required to set up an initial spatial pattern in the expression of a number of the segmentation genes. Interaction between these segmentation genes would still be required in order to establish sharp boundaries in their spatial domains of expression, and also to ensure that certain segmentation genes will reflect a particular pattern in U rather than the full sequence of patterns. A more detailed description of the possible role of the RD system in regulating the expression of both the gap and the co-ordinate genes will be presented elsewhere.

6. Conclusion and Discussion

Previous studies to determine the spatial patterns produced by an RD system were unable to demonstrate that anon-linear RD system is capable of producing the kind of banded spatial patterns required to regulate the expression of segmentation genes and so to act as a basis for segmentation of insects. The numerical results presented in section 4 now demonstrate that it is indeed possible for an RD system to produce a sequence of banded spatial patterns in the syncytial blastoderm of the insect embryo. The major features of the RD system employed here, which were not present in any of the earlier attempts to demonstrate its capacity to produce the desired patterns, are: (1) that the reaction and diffusion~rates depend on the density of nuclear material, n, in such a way that the characteristic spatial wavelength of the RD system, AM, is inversely proportional to n; and (2) that the net production functions of the RD system are antisymmetric with respect to the uniform steady-state as defined in section 3. The patterns produced on the syncytial blastoderm by an RD system having these features consist of a sequence of seven patterns. The reaction and diffusion parameters were chosen so that the first pattern is a simple gradient in the AP direction, which arises during cycle 8. The second pattern is then a simple gradient in the DV direction appearing during cycle 9. Each of the next five patterns appears as a set of bands spaced along the AP axis. The number of bands in each pattern tends to be double that of the preceding pattern. This "frequency doubling" effect is caused by the doubling of n at the beginning of each nuclear division cycle and the influence this has on the reaction and diffusion rates (see (1) above and section 2). It is the frequency-doubling effect on the pattern sequence which provides the basis for: (a) an explanation of the observed sequence of spatial patterns in the genes ftz, eve and prd ;

300

B.N. NAGORCKA

(b) the spatial relationship between the 7-band patterns of h, opa, ftz and eve, and hence the segmentation of the embryo into a series of odd and even segments each composed of an anterior and posterior compartment; and (c) for the assignment of a unique developmental path to each segment and compartment. In addition, it has been observed that zygotic expression does not begin until cycle 11 (Edgar & Schubiger, 1986). Therefore, it is reasonable to expect that maternal gene products will be required to interpret the patterns in U expected during cycles 8, 9 and 10. In fact almost all of the co-ordinate class of segmentation genes are maternal genes, and some may be required to interpret simple AP and DV gradients, i.e. the patterns in U expected during cycles 8 and 9. However, as stated in section 5, the cycle 10 pattern in U is seen in the expression of some zygotic genes, e.g. ftz. This has prompted the suggestion that there exist maternal genes which are expressed uniformly through the blastoderm producing DNAbinding proteins which are required to interpret the spatial pattern in U during cycle 10 (and perhaps earlier cycles). The DNA-binding proteins form complexes according to the spatial pattern in U and are available subsequently to bind to and regulate the expression of zygotic genes such as ftz and eve. Since no maternal mutants with a complete pair-rule phenotype have been observed, thus far, it is possible that the same maternal genes may be required to regulate the expression of two or more complementary segmentation genes such as ftz and eve or perhaps h and opa. Under these circumstances, a mutation of such a maternal gene would probably produce little or no larval cuticle at all. It should be possible to screen such mutants to determine if they are required for the correct expression of zygotic genes which constitute spatial complements. A major area not considered at all in this paper is the sensitivity of pattern formation to changes in the size and shape of the syncytial blastoderm caused, for example, by ligation. A number of studies have been made of the effect of ligation on embryo development (Vogel, 1977; Sander, 1981). It appears that in Drosophila (Vogel, 1977) a complete larva is still formed in an egg iigated at the anterior or the posterior end at a very early stage. The egg is effectively reduced in length by 15%. The mechanism proposed here would predict the development of an almost normal larva in this situation provided the effect of the small maternal gradient established during oogenesis is not completely lost. As a very simple example to demonstrate this, consider the egg to be in the form of a cylinder (Fig. 11). Suppose that the size of the cylinder is altered by removing a piece of the cylinder from one end at or before cycle 8 or 9 when the energids are still migrating to the periphery of the egg, so that no energids are removed from the major egg fragment and no energids enter the minor egg fragment (see the early ligation path in Fig. 11). In this situation, the area of the cylinder representing the egg will decrease with cylinder length, causing the density of nuclear material to increase relative to the normal embryo once the energids reach the periphery of the egg. Since n is inversely proportional to the cylinder length, A~ will decrease in proportion to the cylinder length, as illustrated in Fig. 11. It follows that each of the banded patterns expected

SPATIAL

ORGANIZATION

cycle 8 or 9 or earlier~

cycle

301

I N D. M E L A N O G A S T E R

e

ear

"

~°

"i

"

d

10

i"

cycle 12

£2 + P.a ~ P-0

~~'~! ) ) ',,L. n increased

1oo

o

EL(%)

n unchanged

D

oo

o

EL(%)

1oo

0

EL(%)

FIG. 11. Illustrated are the effects of an early ligation ipre cycle I0) and a late ligation (at or post cycle 10) of the Drosophila egg n, on the density of nuclear material (dots represent nucleii) as well as the size and shape of the syncytium. Viewing the egg as a cylinder, it can be seen that early ligation causes n to increase, since n varies inversely with cylinder length. It follows that the banded patterns produced by the R D system will compress in proportion to the cylinder length as s h o w n by the cycle 12 distribution in U. Late ligation, on the other hand, divides the syncytium and the patterns. In this case new b o u n d a r y conditions (not shown) must be applied at the point or plane of ligation.

in the larger fragment during cycles 10-14, will tend to be a compressed version of that predicted in the unligated egg, so that a normal larva will tend to develop. An accurate prediction, however, will require a more realistic blastoderm shape to be considered along with the establishment of the I-head and tail regions. On the other hand, if the egg is iigated at - 5 0 % EL at, or soon after, the time when the presumptive syncytial blastoderm forms (i.e. the late iigation path in Fig. 11), then part of the blastoderm will be in each of the two egg fragments. It may be that each of the fragments initially contains a syncytium in the form of a hemi-spheroid, for example. Under these circumstances, the proposed mechanism would predict that only a part of the normal larva would develop in each egg fragment (in complete contrast to the criticisms of Meinhardt (1982) concerning the proposal of Kauffman et al. (1978)). Furthermore, new boundary conditions (not indicated in Fig. 11) must be applied at the point of ligation, and may cause a localized change in the pattern giving rise to a gap in the embryo. The gap would be expected to vary in size depending on the value of AM, i.e., depending on the stage of the cycle at which the egg is ligated. There are many observations of the altered embryonic development following ligation and, in order to test the proposed

302

a . N . NAGORCKA

mechanism, it will be necessary to carry out further calculations designed to simulate, as far as possible, the various ligation experiments. Part of this work was carried out in the Centre for Mathematical Biology, University of Oxford, with support from the Science and Engineering Research Council of Great Britain (Grant Gr/D/13573). Helpful discussions with Rob Saint and Rick Tearle, from the Division of Entomology, CSIRO, are also gratefully acknowledged. REFERENCES ABRAMOWITZ, H. & STEGUN, I. A. (1972). Handbook of Mathematical Functions. New York: Dover Pub. Inc. ANDERSON, K. V. & NOSSLEIN-VOLHARD, C. (1986). In: Gametogenesis and the earl), embryo (Gall, J. G., ed.). P. 177. New York: Alan R. Liss Inc. BUNOW, B., KERNEVEZ, J., JOLY, G. & THOMAS, D. (1980). J. theor. Biol. 84, 629. CAMPOS-ORTEGA, J. A. & HARTENSTEIN, V., (1985). The embryonic development of Drosophila melangaster. Berlin Heidelberg: Springer-Verlag. EDGAR, B. A. & SCHUBtGER G. (1986). Cell 44, 871. FOE, V. E. & AIberts, B. M. (1983). J. Cell. Sci. 61, 31. FROHNHOFER, H. G. & NOSSLEIN-VOLI-IARD, C. (1986). Nature 324, 120. GIERER, A. & MEINHARDT, H. (1972). Krvbernetik 12, 30. HAFEN, E., KUROIWA, A. & GEHRING, W. J. (1984). Cell 37, 833. HARDING, K., RUSHLOW, C., DOYLE, H. J., HOEY, T. & LEVINE, M. (1986). Science 233, 953. HARTENSTEIN, V., TECHNAU, G. M., & CAMPOS-ORTEGA, J. A. (1985). Roux's Arch. Dev. Biol. 194, 213. It.LMENSEE, K. (1978). In: Genetic Mosaics and Cell Differentiation Gehring, W. J., ed. p. 51. Berlin, Heidelberg, New York : Springer-Verlag. INGHAM, P. W., HOWARD, K. R., & IsH-HoRowlcz, D. (1985). Nature 318, 439. ISH-HOROWICZ, D., HOWARD, K. R., PINCHIN, S. M. & INGHAM, P. W. (1985). ColdSpring Harbour Syrup. Quant. Biol. 50, 135. J,~CKLE, H., TAUTZ, D., SCHUH, R., SEIFERT, E. & LEHMANN, R. (1986). Nature 324, 668. KAUFFMAN, S. A. (1981). Phil' Trans. R. Soc. Lond. B295, 567. KAUFFMAN, S. A., SHYMKO, R. M. & TRABERT, K. (1978). Science 199, 259. KILCHHERR, F., BAUMGARTNER, S., BOPP, D., FREI, E. & NOLL, M. (1986). Nature 321, 493. LAWRENCE, P. A. (1981). Cell 26, 3. MACDONALD, P. M., INGHAM, P. & STRUHL, G. (1986). Cell 47, 721. MACDONALD, P. M. & STRUHL, G. (1986). Nature 324, 537. M EINHARDT, H. (1982). Models of Biological Pattern Formation. London:Academic Press. MEINHARDT, H. (1986). J. Cell Sci. Suppl. 4, 357. MOONEY, J. R. (1986). Math. Comp. Sire. 28, 209. MURRAY, J. D., (1982). J. theor. Biol. 98, 143. NAGORCKA, B. N. & MOONEY, J. R. (1982). J. theor. Biol. 98, 575. NUSSLEIN-VOLHARD, C., KLUOING, H. & Jt3RGENS, G. (1985). Cold Spring Harbour Syrup. Quant. Biol. 50, 209. N0SSLEIN-VOLHARD, C. & WIESCHAUS, E. (1980). Nature 287, 795. NUSSLEIN-VOLHARD, C., WIESCHAUS, E. & JIdRGENS, G. (1982). Verh. Dtsch. Zool. Ges., p. 91. PETSCHEK, J. P., PERRIMON, N. & MAHOWALD, A. (1987). Dev. Biol. 119, 175. RUSSELL, M. A. (1985). Dev. Biol. 108, 269. SANDER, K. (1981). In Progress in Developmental Biology (Sauer, H. W., ed.). Fortschritte Zool. 26, 101. Stuttgart : Gustav Fischer Verlag. SCHNAKENBERG, J. (1979). J. theor. Biol. 81, 389. SONNENBLICK, B. P., (1950). In: Biology of Drosophila (Demerec, M., ed.) p. 62. New York and London : Hafner Pub. Co. SULLIVAN, W. (1987). Nature 327, 164. TURiNG, A. M. (1952). Phil, Trans. R. Soc. B237, 37. VOGEL, O. (1977). Roux's Arch. Dev. Biol. 182, 9. WEIR, M. P. & KORNBERG, T. (1985). Nature 318, 433.

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WOLPERT, L. (1971). Curt. Top. Dev. Biol. 6, 183. WRIGHT, D. A. (~¢ LAWRENCE, P. A. (1981). Dev. Biol. 85, 317.

APPENDIX

Linear Analysis of the RD System An indication of the spatial patterns which may be produced by the RD system defined by eqn (2) can be obtained by solving the linearized form of eqn (2) with respect to the uniform steady-state (Uo, Vo), and considering the stability of the state (Uo, Vo). Linearization of eqn (2) with respect to u and v, where u = U - Uo and v = V - Vo, yields --= at

where • =

D(n)dV2dp+

(.)

o)

d=

v '

(A1)

K(n)?7~

dv

and ~t? is defined in section 2. Equation (A1) is to be solved on the prolate spheroid 8f~ with major axis a and minor axis b and the distance between focii equal to 2 f (Fig. 4). The solutions to eqn (A1) may be written as a linear combination of the wavefunctions ~mn (Abramowitz & Stegun, 1972) where ~It,.. = ~ e'~'qt,..

(A2)

cos

= ol e ~ ' S * . ( c , rl)si. (m~b)

where rI and ~b are the spheroidal co-ordinates (Fig. 4.), and c =fk. a is a constant two vector specifying the relative magnitudes o f the u and v components. ~ , . . are wavefunctions of the Laplacian having wavenumber k, i.e. (V2+ k : ) ~ m . = 0 on 81L It follows that the wavefunctions S * . ( c , rl) must satisfy the equation

d--~ (1-rl2)"~ S*'(c''rt)

+

2

m2

S~,,(c, -r/) = 0

(A3)

where Am. = c2~:2- m 2 / ( ~ : 2 - 1) and ~r2 = a 2 / ( a 2 - b2). During nuclear division cycles 8 and 9, eqn (A3) is solved over - 1 <- rI <- + 1. Hence the S*,,(c, rl) are equivalent to the angular prolate spheroidal functions, Sin., defined by Abramowitz & Stegun (1972). However, during cycles 10-14, the RD system is assumed to be switched off in the I-head and tail regions of ~f~ (section 3) and, as a first approximation, eqn (A3) is solved only over a restricted r/ range during these cycles. Initially it will be assumed that U and V decline to values close to their anticipated minima at the interface with the l-head and tail regions. In fact, U and V are fixed to values at, or significantly below, their minimum values when solving the coupled non-linear equations (2). To solve the linearized equation (A1), on the other hand, zero flux boundary conditions are applied in order to force the ~ , . . to attain their minimum values at the limits of the 77 range.

304

B.N.

On substituting a i r

NAGORCKA

into equation (A1) we have that

( - D(n)dk2+ K(n)Y{- r r l ) a

=0

(A4)

from which it follows that tr(k 2) will be positive for a positive finite range of k 2 provided the elements of D(n)d and K(n)Y[ satisfy certain broad constraints, described in detail by Murray (1982) (see also Nagorcka & Mooney, 1982). These constraints define the Turing space. It can be shown that the positive finite range of k 2 in which tr(k-') > 0 is specified by the limit k0a and k~ defined by koJ = k ~

ak~,,

(A5)

where

K(n) ~Kuu+Kvv~, k ~ = 2 D ( n ) \--d-~-u --~-v]

Ak~ and

K(n) ~/ Q D(n) dudv

Kvu)2+-vvKvv

The set o f wavefunctions ~lIfmn having wavenumbers which lie within the range ko ~ k~ will be referred to as growth modes since ty > 0 for each of these wavefunctions. It may be concluded from the linear analysis that the uniform steady-state is unstable to perturbations having the form of a linear combination of the growth modes which are expected to increase in amplitude. Previous studies of RD systems suggest that, to a first approximation, the non-linear terms in the net production functions often act simply to limit the amplitude o f the growth modes and to determine the proportion of each growth mode in the final stable solution. It follows that in situations where only one growth mode is possible, the stable spatial pattern which forms will be given to a very large extent by that growth mode. However, where more than one growth mode is involved, it will normally be necessary to solve the non-linear equations (2) in order to determine the weighting of each growth mode present in the stable spatial pattern which develops. In either case, the stable solution of the non-linear RD system is expected to be a wave-like pattern having a wavelength close to the value AM =2rr/kM. AM is therefore referred to as the characteristic wavelength of the RD system. It is clear from the above discussion and eqn (A2) that the spatial patterns in the distribution of U and V are expected to be complementary. Hence, only the spatial pattern in the distribution of U will be referred to in the following. Furthermore, the assumption that a maternal gradient containing a small component in the dorsoventral direction exists in the egg, independent of the RD system but interacting with it, is a basis for removing the degeneracy with respect to the ~b co-ordinate. Hence, only the cos (mq5) dependence in eqn (A2) needs to be considered here.

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I N D. M E L A N O G A S T E R

The ~ , . . which define the spatial dependence of a growth mode ~ m . during cycle p, will be denoted as g ~ . . The X.,,~ P and their associated wavenumbers k~., expressed as the square of the dimensionless quantity fkP..,., have been estimated for nuclear division cycles p = 8-10 (Table A1). The values offkM, which vary for each nuclear cycle due to changes in the size, f, of 81) and/or changes in the density, n, of nuclear material per unit area of 8f~, are also given in Table AI. A second set of growth modes with spatial dependence given by the wavefunctions ,~P..., which satisfy the boundary conditions , ~ , , ( = ~ , . . ) = 0 at 77=0.6 and 7 / = - 0 . 9 , are also listed in Table A1, since the ) ~ . may form a significant component of any solution of the non-linear equations (2) which satisfy fixed boundary conditions at the limits of the 7/ range. The tilde over ~P... (and ~ . , . in Table AI) indicates that these wavefunctions satisfy a different set of boundary conditions to those of the X~n (and ~m.). TABLE A1

The table lists the observed distance, Ab, of nucleii from the periphery of the egg (Foe & Alberts, 1983), the estimated minor axis, 2b (Fig. 4), of the (presumptive) syncytial blastoderm (2b = 150- 2Ab) and the estimated density, n, of nuclear material forming the blastoderm (set to unity at cycle 8 ) f o r each of the nuclear division cycles p. The wavenumbers km and k~,, scaled by the quantity f (the focus of the prolate spheroidal blastoderm ), and the growth modes of the linearized R D system, xPn and ~ , (defined in the text), were also calculated for each cycle, p, given that the ratio of the major axis to the minor axis of the blastoderm is 2.7. p

Ab (o.m)

2b (~.m)

n

(fkM) 2

(fk~,,,)2

X~,,,

~,,,

8 9 10 11

35 18 2.5 2.5

80 114 145 145

1 1 1.24 ( = N ) t 2N

3.7 7.4 18.3 73.1

3.4 8.4 16.2 60.0

~0t ~tl ~02 ~o4

-2~t2 ~ 1 , , . ~24, xF25, ~33, xt/34

12 13 14

2.5 2.5 2.5

145 145 145

4N 8N 16N

292 1170 4680

t. N is the density o f nuclear material at cycle 10.

The single growth modes listed for cycles 8 and 9 are the only possible growth modes for these cycles. Equation (A3) is solved only over a restricted rl range during cycles 10-14, since the RD system is switched of[ in the presumptive head and tail regions of the syncytial blastoderm (section 3). To determine the potential growth modes during cycles 10 and 11, zero flux boundary conditions were applied at "0 = 0-9 and r/= -0.6, thus ensuring that U and V decline to their minimum values at the interface between the I-head and tail regions. Additional growth modes with spatial wavefunctions of the form Xmn "p are also possible for cycles 10 and 11, and these are indicated in Table A1. Multiple growth modes are also expected for cycles 12-14. It is essential for reasons given in the text that the RD system be chosen so

306

a . N . NAGORCKA

that only one growth mode of the form Won, dominates the spatial pattern in U during each of the cycles 10-14.

Nuclear Division Cycles 1-7 The number o f energids is sufficient at least by cycle 7, and perhaps earlier during cycles 5 and 6, to group together as a syncytium in the form o f a solid prolate spheroid ~. In this case, unlike cycles 8-14, the RD system equations (2) must be solved in three dimensions. In addition, any terms describing the transport of U and V between ~ and the yolk cannot be incorporated into f and g but must be included explicitly as a boundary condition defining the flux of U and V on the surface of fl. Therefore, if transport terms are involved, the RD system equations will be quite different at cycle 7 and earlier than those applicable during cycles 8-14. With such a difference, it is certainly possible to choose parameter values so that no spatial patterns arise prior to cycle 8. Alternatively, if U and V are confined to the syncytium (i.e. there are no transport terms) then it can be shown that no spatial pattern is expected to arise prior to the cycle 8 pattern due to the smaller size of ~ during cycle 7 relative to the size of 8 ~ at cycle 8. Under the circumstances, the situation during cycles 1-7 will not be explored further in this paper.