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lowing way. For a given autoregulatory gene, let m(t) be the number
of mRNA molecules in a cell at time t and let p(t) be the number of
the corresponding protein molecules. The rates of change of m and p
are then assumed to obey the following equations:

d p(t)
= am(t ’ T p ) ’ bp(t), (7.1)
dt

dm(t)
= f ( p(t ’ T m )) ’ cm(t). (7.2)
dt
Here the constants b and c are the decay rates of the protein and
its mRNA, respectively, a is the rate of production of new protein
molecules, and f ( p) is the rate of production of new mRNA molecules.
A

gene
her1

Tm1
delay
mRNA


T p1
delay
protein




B with protein synthesis at normal rate

120 1200
mRNA




protein
80 800
40 400

200 400 600 800 time (min)

C
with protein synthesis attenuated x 1/10




200 400 600 800 time (min)



with protein synthesis at normal rate, allowing for noise
D




200 400 600 800 time (min)



with protein synthesis attenuated x 1/10, allowing for noise
E




200 400 600 800 time (min)
Fig. 7.4 Cell-autonomous gene-expression oscillator for zebra¬sh somitogenesis. (A)
Molecular control circuitry for a single gene, her1, whose protein product acts as a
homodimer to inhibit her1 expression. In the case of a pair of genes (i.e., her1 and her7)
the analogous circuit would contain an additional branch with coupling between the two
branches. (B) Computed behavior for the system in A (de¬ned by Eqs. 7.1 and 7.2), in
terms of the numbers of mRNA molecules per cell in red and protein molecules per cell
in blue. Parameter values were chosen appropriate for the her1 homodimer oscillator
(see the form of the function f ( p) in the text) on the basis of experimental results:
a = 4.5 protein molecules per mRNA molecule per minute; b = c = 0.23 molecules
(Cont.)
7 PATTERN FORMATION: SEGMENTATION, AXES, AND ASYMMETRY 167


The function f ( p) is assumed to be a decreasing function of the
amount of protein. (The form used by Lewis and Monk is
k
f ( p) = ,
1 + p2 / p0
2


with constants k and p0 , to represent the action of an inhibitory
protein, assumed to be a dimer. The results of simulations turned
out to be quite insensitive to the speci¬c form of f ( p).)
The above time-delay differential equations were numerically
solved for Her1 and He7 (for which Lewis, drawing on experimental
results, was able to estimate the values of all the model parameters in
Eq. 7.1 and 7.2). The solutions indeed exhibit sustained oscillations in
the concentration of Her1 and Her7, with predicted periods close to
the observed ones (Fig. 7.4). The important conclusions from this anal-
ysis are that no oscillations are possible if delay is not incorporated
(i.e., if T m = T p = 0) and that the oscillators are quite insensitive to the
rate of protein synthesis (i.e., to the value of a, Eq. 7.1). Furthermore,
Lewis showed that incorporation into the model of the inherently
noisy nature of gene expression (by adding stochastic effects to the
deterministic equations 7.1 and 7.2) reinforces continued oscillations
(Fig. 7.4). (Without noise, oscillations are eventually damped, which
would upset normal somite formation beyond the ¬rst few.)
The sequence of blocks of tissue generated by the clock and wave-
front mechanism become somites only after a physical boundary
forms at their interfaces. In the quail, the expression of the gene
Lunatic fringe, a modulator of Notch signaling, at the position ’1 (i.e.,
one somite length caudal to the most recently formed intersomitic
morphological boundary) initiates ¬ssure formation at the boundary
between the PSM and what will become the posterior half of the
next-forming somite (Sato et al., 2002).
As will be seen below, the generation of left--right asymmetry in
the vertebrate embryo utilizes many of the signaling molecules em-
ployed in somitogenesis, e.g., FGFs, the Wnt and Notch pathways.
Dynamical pattern-forming systems using shared diffusible compo-
nents would interfere with each other in the absence of special ways

per minute, corresponding to protein and mRNA half-lives of 3 minutes; k = 33 mRNA
per diploid cell per minute, corresponding to 1000 transcripts per hour per gene copy in
the absence of inhibition; p0 = 40 molecules, corresponding to a critical concentration
of around 10’9 M (moles per litre) in a 5 µm diameter cell nucleus; Tm ≈ 20.8 min;
Tp ≈ 2.8 min. (C) Decreasing the rate of protein synthesis (to a = 0.45) causes little or
no effect on the period of oscillation. All the other parameters are the same as in B. (D)
The computed behavior for the system in A when the noisy nature of gene expression is
taken into account. To model stochastic effects an extra independent parameter is
introduced, the rate constant koff for the dissociation of the repressor protein (i.e.,
Her1) from its binding site on the regulatory DNA of its own gene (i.e., her1). Results
are shown for koff = 1 min’1 , corresponding to a mean lifetime of 1 min of the
repressor bound state. (E) The same as in D except for the rate of protein synthesis,
which is as in C. Parameter values not mentioned explicitly in C“E are the same as in B.
(Reprinted from Lewis, 2003, with permission from Elsevier.)
168 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


of protecting against this. It is signi¬cant, therefore, that mammals,
birds, and ¬sh all employ the diffusible lipid molecule retinoic acid,
in a mechanism that resists the tendency of somites on opposite sides
of the body to go out of register as they are forming and left--right
asymmetry is simultaneously being established (Vermot et al., 2005;
Vermot and Pourqui´, 2005; Kowakami et al., 2005).
e


Epithelial patterning by juxtacrine signaling
As discussed above, juxtacrine signaling is one of the inductive mech-
anisms that can lead to pattern formation by the production of
new cell types. In juxtacrine communication, signaling molecules
anchored in the cell membrane bind to and activate receptors on
the surface of immediately neighboring cells, thus typically leading
to patterning on the scale of a few cells.
In the discussion of the segmentation model of Lewis (2003), Notch
signaling was employed to establish synchrony among neighboring
cells of a single type. More typically, the Notch--Delta system is used to
specify different fates (i.e., the commitment to speci¬c types) in adja-
cent cells of a developmentally equivalent population. An example of
this is the differentiation of gonadal cells in the nematode Caenorhab-
ditis elegans. The predifferentiated cells can differentiate into either
an anchor cell (AC) or a ventral uterine precursor cell (VU). The choice
between AC or VU, however, depends on the interaction between the
Notch-type receptor, LIN-12, of one cell with its Delta-like ligand, LAG-
2, on the adjacent cell. Activation of LIN-12 in one cell forces that cell
to adopt the VU fate, whereas cells in which the action of LIN-12 is
suppressed adopt the AC fate (Greenwald, 1998).
In other cases, nonequivalent precursor cells communicate
through Notch--Delta signaling to progress to the next stage of dif-
ferentiation. When this occurs, intrinsic or extrinsic factors confer a
bias to one of two neighbors, and this is then sharpened or consoli-
dated by Notch--Delta interactions. This was the hypothesized role of
Notch and Delta in the Kerszberg--Changeux model for neurulation
discussed in Chapter 5. It is also the role played by this juxtacrine
pair in the induction of mesoderm at the beginning of sea urchin
gastrulation. Speci¬cally, between the eighth and ninth cleavage in
the sea urchin Strongylocentrotus purpuratus the micromeres (see Chap-
ter 5; Fig. 5.7) begin to express Delta, which triggers the speci¬cation
of the adjoining inner ring of cells (bearing a Notch receptor and re-
ferred to as Veg2 cells; see Fig. 5.7), to a non-skeletogenic mesodermal
fate (Sherwood and McClay, 2001). In amphibians (Green, 2002; see be-
low) and mammals (Beddington and Robertson, 1999), the equivalent
gastrulation-related inductive steps appear to occur via paracrine,
rather than juxtacrine, mechanisms.
Initial evidence suggested that Notch™s function was mainly to re-
press the acquisition of a new state of differentiation by cells that
carry the Notch protein (Artavanis-Tsakonas et al., 1999): the Notch
7 PATTERN FORMATION: SEGMENTATION, AXES, AND ASYMMETRY 169


receptor-bearing cell is inhibited from assuming a differentiated state,
while the Delta ligand-bearing cell is free to do so. Consequently, the
Notch--Delta pathway was thought to act only via lateral inhibition.
More recently, however, this simple model has come into con¬‚ict with
evidence for longer-range effects mediated by Notch-dependent posi-
tive differentiation signals (Rook and Xu, 1998; Pourqui©, 2000).
A model constructed by Collier et al. (1996) for Notch--Delta signal-
ing, which made use only of lateral inhibition, gave rise to pattern
over a range of one or two cells. Here we summarize the model of
Sherratt and coworkers (Owen et al., 2000; Wearing et al., 2000; Savill
and Sherratt, 2003), which predicts more realistic patterns via jux-
tacrine communication on a longer scale on the basis of a positive
feedback mechanism (Owen et al., 1999; see also Webb and Owen,
2004). The model, like some of those presented in Chapters 3 and
5, employs ˜˜linear stability analysis” to predict transitions between
alternative dynamical states. (See the Appendix to Chapter 5 for an
example of how this works).

Juxtacrine signaling: the model of Sherratt and coworkers
Juxtacrine signaling can lead to pattern formation via either lateral
inhibition or lateral induction (i.e., activation). (Inhibition and acti-
vation are the main outcomes of what we referred to as inductive
signaling earlier in this chapter. Often, however, the term ˜˜induc-
tion” is used synonymously with activation, and we will occasionally
follow this usage in what follows.) Experimental evidence suggests
that the Notch--Delta ligand--receptor interaction (see the model for
neurulation in Chapter 5) can produce both inhibition (Haddon et al.,
1998) and induction (Huppert et al., 1997). The model of Sherratt et al.
(Owen and Sherratt, 1998; Owen et al., 2000; Wearing et al., 2000) pro-
vides a general mechanism for lateral induction through juxtacrine
communication in an epithelial sheet, which the authors propose as a
prototype for the Notch--Delta system. Embryonic epithelial patterns
that may arise in this fashion include the veins in the Drosophila
wing, the feather germs (primordia) in avian skin, and stem cell clus-
ters in the epidermis (Owen et al., 2000; Savill and Sherratt, 2003).
Below we summarize the general features of the model.
1. Consider a two-dimensional cellular sheet, a portion of which
is depicted in Fig. 7.5. Let ai j (t), fi j (t), and bi j (t) denote respectively
the number of ligand molecules, free receptors, and bound receptors
on the surface of the cell at the intersection of row i and column j
(Owen et al., 2000).
2. Sherratt and coworkers used a simple kinetic scheme to model
the time variation of ai j (t), fi j (t), and bi j (t). Ligands and free receptors
can reversibly associate to produce bound ligand--receptor complexes
(at a rate ka ) or decay (with respective decay constants da and d f ).
Bound ligand--receptor complexes can dissociate (at a rate kD and thus
produce new ligands and receptors at the same rate) or be internalized
(at a rate kI ). The distinguishing feature of the model is a positive feed-
back mechanism: apart from the dissociation of the bound complexes,
170 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



j ’2 j ’1 j j+ 1 j+ 2
Fig. 7.5 Part of the cellular grid
used in the simulation of Sherratt
and coworkers (Owen et al.,
i ’1
2000). Each square represents a
single cell in an epithelial sheet.
Neighboring cells are connected
via Notch“Delta ligand“receptor i
pairs. The rows and columns of
cells are labeled by the indices
i and j , respectively. The position i+1
of a cell is uniquely determined by
the pair i j .


there is an additional source of ligands and receptors with a rate of
production that increases with the level of occupied receptors (gov-
erned by the functions P a (bi j ) and P f (bi j ), which are both increasing
functions of bi j ). In mathematical form,
dai j
= ’ka ai j fi j + kD bi j ’ da ai j + Pa (bi j ),
dt

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. 33
( 66 .)



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