= ’ka ai j fi j + kD bi j ’ d f fi j + P f (bi j ), (7.3)

dt

dbi j

= ka ai j fi j ’ kD bi j ’ kI bi j .

dt

For the values of the rate constants in the above equations, Sherratt

and coworkers used the experimental data for epidermal growth fac-

tor and its receptor (Waters et al., 1990). For the speci¬c forms of the

functions P a (bi j ) and P f (bi j ) see Owen et al. (2000). The symbol in

Eqs. 7.3 denotes an average over nearest neighbor cells and is the

re¬‚ection of juxtacrine signaling. For example,

ai j ≡ 1 (ai, j’1 + ai, j+1 + ai’1, j + ai+1, j ). (7.4)

4

The terms on the right-hand side represent the total number of ligand

molecules on the surfaces of cells (in two dimensions) adjacent to the

cell at the position (i, j) in Fig. 7.5. It is assumed that the ligands are

uniformly distributed along the four lateral faces of a cell attached

to its neighbors.

3. Once a homogeneous (i.e., spatially uniform) steady-state solu-

tion, denoted by ah , fh , bh , of the system given by Eqs. 7.3 is deter-

mined (it is easy to show that at least one such nontrivial solution

exists), linear stability analysis around this solution, along the lines

described in the Appendix to Chapter 5, can be performed (for de-

tails see Wearing et al., 2000). The authors introduced the expres-

sions ai j = ah + ai j , fi j = fh + f i j , bi j = bh + bi j , where the quantities

˜ ˜

˜

with wavy lines are assumed to be small perturbations and in general

inhomogeneous (i.e., varying over space). Inserting these forms into

Eqs. 7.3, using the fact that ah , fh , bh satisfy the original equations

with the left-hand sides set to zero, and retaining only terms linear

in the small quantities (i.e., linearizing the equations around the ho-

mogeneous solutions), a set of equations is generated for the spatially

7 PATTERN FORMATION: SEGMENTATION, AXES, AND ASYMMETRY 171

varying quantities ai j , f i j , bi j , whose solutions are sought in the form

˜˜˜

ai j = a e»t cos kr or ai j = a e»t sin kr (and similarly for f i j and bi j ; r =

˜ ˜

˜ ˜ ˜ ˜

(i, j) is a compact notation for the cell position). Here a is a constant,

˜

» is the temporal growth rate and k is a wavenumber. (Actually the

solutions can be written in the complex-number form ai j = a e»t+ikr ,

˜ ˜

√

where i = ’1; in view of the relationship e = cos kr + i sin kr , this

ikr

is just a convenient way to incorporate simultaneously the forms con-

taining cos kr and sin kr .)

For spatial pattern formation, it is required that the homogeneous

solution ah , fh , bh be (i) stable under homogeneous perturbations and

(ii) unstable under inhomogeneous perturbations. The ¬rst case cor-

responds to k = 0, » < 0, whereas for the second case k = 0 and

» > 0. For a complex system of this sort, linear stability analysis typi-

cally identi¬es solutions for several possible wavenumbers, which are

called ˜˜modes.” Furthermore, for each mode there is a functional re-

lationship between k and the corresponding ». The fastest-growing

mode will be the one that eventually prevails, so it is this mode that

is identi¬ed with the developing pattern. Which of the modes is the

fastest growing depends on the parameters in Eqs. 7.3.

In principle, Eqs. 7.3 determine all possible particular patterns of

the dynamical system, without any stability analysis. However, ¬nd-

ing the inhomogeneous solutions of such a complex system presents

a formidable mathematical challenge. Linear stability analysis is a

systematic way to explore the wide range of possibilities. Once a

particular inhomogeneous solution describing a pattern is identi¬ed

by linear stability analysis, one may attempt to locate the region in

parameter space where this solution is stable using numerical meth-

ods, as was done by Sherratt and coworkers (Owen et al., 2000). (Note

that linear stability analysis can only establish the possibility of a par-

ticular spatial pattern, because the amplitude (e»t ) of the mode that

characterizes the pattern grows in time and thus the linear stability

analysis itself breaks down eventually.) The system de¬ned by Eqs. 7.3

contains ¬ve parameters in addition to the two functions P a (bi j ) and

P f (bi j ), and thus such an analysis is highly nontrivial.

In Fig. 7.6 we reproduce a pattern generated by the model of Sher-

ratt et al. that corresponds to speci¬c forms of the functions P a (bi j )

and P f (bi j ).

Mesoderm induction by diffusion gradients

The nonuniform distribution of any chemical substance, whatever

the mechanism of its formation, can clearly provide spatial informa-

tion to cells. Let us assume, for example, that a population of cells

is capable of assuming more than one differentiated fate -- ectoderm

vs. mesoderm, for example -- or differentially adhesive versions of the

same cell type confronting each other across a compartmental bound-

ary. We have seen above how cell-autonomous or juxtacrine mechan-

isms are capable of generating such spatially patterned con¬gura-

tions. Exposing an array of cells to varying amounts of an external

172 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

signaling molecule is another, even simpler, way to make them behave

differently from one another. And molecular diffusion is probably the

most physically straightforward and reliable way of producing a range

of levels of a substance across a spatial domain.

For at least a century, embryologists have considered models

for pattern formation and its regulation that employ diffusion gra-

dients. Only in the last decade, however, has convincing evidence

been produced that this physical mechanism is utilized in early de-

velopment. The prime evidence comes from studies of mesoderm

induction, a key event preceding gastrulation (see Chapter 5) in

the frog, Xenopus. As noted above, important early steps in meso-

derm induction in sea urchins occur by a Notch--Delta-based jux-

tacrine mechanism. In contrast, analogous events in Xenopus occur

through paracrine inductive signaling by secreted factors. ˜˜Animal

caps” (prospective ectoderm) of Xenopus blastulas, when explanted

alone, become epidermis and express all the molecular and mor-

phologic features of skin. The prospective fates of the vegetal pole

explants are the endoderm of the gut, as well as extra-embryonic tis-

sue. Nieuwkoop juxtaposed these two types of explants and demon-

strated that the vegetal poles induced the animal caps to become

muscle instead of skin (Nieuwkoop, 1969).

The following kinds of study strongly suggested that a paracrine

mechanism was at work (see the review by Green, 2002). Various

soluble factors of the TGF-β superfamily (e.g., activin, BMPs, Nodal)

and the FGF family could substitute for the vegetal pole cells (i.e.,

the ˜˜vegetal pole organizer”) in the experiments of Nieuwkoop. In-

terference with receptors for these factors in animal cap cells, or the

administration of protein inhibitors of their activities, such as Noggin

or Chordin, attenuated induction. Different doses of activins and FGFs

elicited different cell-type-characteristic patterns of gene expression

in animal cap cells. Finally, both TGF-β (McDowell et al., 2001) and FGFs

(Christen and Slack, 1999) can diffuse over several cell diameters.

None of this proves beyond question that the simple diffusion of

morphogens between and among cells, rather than some other, cell-

dependent, mechanism, actually establishes the gradients in question.

Kerszberg and Wolpert (1998) for example, asserted that the capture of

morphogens by receptors impedes diffusion to the extent that stable

gradients can never arise by this mechanism. They proposed that mor-

phogens are instead transported across tissues by a ˜˜bucket brigade”

mechanism in which a receptor-bound morphogen on one cell moves

by being ˜˜handed off” to receptors on an adjacent cell. This could

be accomplished, for example, by repeated cycles of exocytosis and

endocytosis, referred to as ˜˜planar transcytosis” (Freeman, 2002).

While arguments can be made in either direction (i.e., for dif-

fusion or for the bucket-brigade mechanism) on the basis of simple

models and thought experiments, sometimes the best way of answer-

ing a question of this sort is to set up a mathematical model with

a realistic degree of complexity (one that would defy intuitive exam-

ination or straightforward analytical solution) and solve it numeri-

cally. This is what Lander and coworkers (2002) did, to address the

7 PATTERN FORMATION: SEGMENTATION, AXES, AND ASYMMETRY 173

Time = 0 hours Time = 40 hours Time = 160 hours

3000 6000

Fig. 7.6 An extended quasi-periodic spatial pattern that results from lateral induction

in the model of Sherratt and coworkers (Owen et al., 2000), de¬ned by Eqs. 7.3. Pattern

development is induced by small random perturbations about the homogeneous

equilibrium applied throughout a 30 — 30 grid of cells. Only the density of

bound-ligand“receptor complexes (molecules per cell, see the color scale) is shown. The

¬nal pattern has a characteristic wavelength (with some irregularities), which depends

crucially on the strength of feedback, as quanti¬ed by the functions P a (bi j ) and P f (bi j ):

the stronger the feedback the longer the characteristic wavelength. The simulation

results correspond to the following parameter set in Eqs. 7.3: ka = 3 — 10’4

molecules’1 min’1 , kD = 0.12 min’1 , kI = 0.019 min’1 , da = 6 — 10’3 min’1 ,

d f = 0.03 min’1 . The feedback functions are given for all cell positions (i , j ) by

P a (b) = C 1 b/ (C 2 + b) and P f (b) = C 3 + C 4 b 3 / C 5 + b 3 , with C 1 = 110,

3 3

C 2 = 2500, C 3 = 90, C 4 = 7.4, and C 5 = 5450. The periodicity of the pattern in the

¬gure is in qualitative agreement with the results of linear stability analysis. (Reprinted

from Owen et al., 2000, with permission from Elsevier.)

question whether the simple extracellular diffusion of morphogens

combined with receptor binding can plausibly set up developmental

gradients. The biological system that they modeled was the Drosophila

wing imaginal disc (see Chapter 5), where quantitative estimates of

the spreading of morphogens are available (Entchev et al., 2000; Tele-

man and Cohen, 2000). However, their conclusions may also apply

to other systems, such as mesoderm induction in Xenopus (see above)

and vertebrate limb development (see Chapter 8). Since the model of

Lander et al. (2002), as well as others (for axis formation and left--right

asymmetry), discussed later in this chapter, is based on generalized

reaction--diffusion systems, ¬rst we brie¬‚y outline the standard math-

ematical analysis used to study such systems.

Reaction“diffusion systems

The rate of change in the concentrations c i , i = 1, 2, . . . , N , of N

interacting molecular species is determined by their reaction kinetics

and expressed in terms of ordinary differential equations

dc i

= F i (c 1 , c 2 , . . . , c N ). (7.5)

dt

174 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

The explicit forms of the functions F i in Eq. 7.5 depend on the details

of the reactions. We have seen examples of such equations in Chap-

ter 3 (cf. Eqs. 3.1), Chapter 4 (cf. Eqs. 4.3) and Chapter 5 (cf. Eqs. 5.1).

Spatial inhomogeneities also cause time variations in the concentra-

tions even in the absence of chemical reactions. If these inhomo-

geneities are governed by diffusion then, in one spatial dimension,

‚c i ‚ 2c i

= Di 2 . (7.6)

‚t ‚x

Here D i is the diffusion coef¬cient of the ith species. (Because the

concentrations now depend continuously on both x and t we need

to use symbols for partial differentiation, as explained in Box 1.1.)

In general, both diffusion and reactions contribute to the change

in concentration at any point in the reaction domain, and the time

dependence of the c i is governed by reaction--diffusion equations:

‚c i ‚ 2c i

= D i 2 + F i (c 1 , c 2 , . . . , c N ). (7.7)

‚t ‚x

Generation of morphogen gradients: the analysis of Lander

and coworkers

Lander and coworkers (Lander et al., 2002) applied the above formalism

in their attempt to resolve the controversy concerning whether mor-

phogen gradients are set up by diffusion or by more elaborate mech-

anisms (e.g., the bucket-brigade mechanism; Kerszberg and Wolpert,

1998). They considered the generation of a gradient of the protein

Decapentaplegic (Dpp), a morphogen expressed in the Drosophila wing

imaginal discs. The authors chose this system because Dpp had previ-

ously been visualized in living embryos by introducing a genetic con-

struct, in which Dpp was fused to green ¬‚uorescent protein (Entchev

et al., 2000; Teleman and Cohen, 2000). They could thus compare their

model™s predictions with experimental results directly. Below we sum-

marize the assumptions of the model and the main conclusions of

its analysis.

1. The wing disc represents an essentially two-dimensional system

in which the morphogen sources consist of a linear array of cells in

the center of the disc. Since Lander et al. were primarily interested

in the formation of gradients perpendicular to this array (along the

anteroposterior axis in the case of Dpp), they formulated their model

in one dimension (Fig. 7.7). They assumed that morphogens are intro-

duced at a constant rate v at one end of the system and absorbed at a

distance 100 µm away (xmax ), representing a distance of about 40 cell

diameters.

2. As morphogen molecules diffuse in the intercellular space, they

interact with receptors on the cells™ surfaces and establish ligand--

receptor complexes, which form and decay with respective rates kon

and koff . The model also allowed for the internalization and subse-

quent degradation of the morphogen--receptor complexes at a rate

kdeg .

7 PATTERN FORMATION: SEGMENTATION, AXES, AND ASYMMETRY 175

Morphogen

production

x = x max

x=0 x=0

Morphogen transport

Fig. 7.7 Representation of morphogen ¬elds in the model of Lander et al. (2002). In

the left-hand panel a stripe of cells produces a morphogen (orange) that spreads