= (1 ’ A 2 ’ B 2 )A ’ B ,

dt (3.2)

dB

= A.

dt

Here there is an unstable ¬xed point at (0, 0). Interestingly, the circle

of unit radius, A 2 + B 2 = 1, surrounding the unstable solution is a

stable closed orbit. Note that the time derivatives do not vanish on

the circle.

Whereas any small displacement of the system state away from

an unstable ¬xed-point solution will take it still farther away, if the

system ¬nds itself on a stable closed orbit, a circle in the case of

the above equations, any small displacement will bring it back to the

same circle (but not necessarily to the same point). Indeed, if the

system begins in any initial state inside or outside the circle it will

eventually wind its way onto it and then remain on it forever (Fig. 3.1).

Such orbit is called a ˜˜limit cycle”: along it the system periodically

revisits each point. To prove that the unit circle is a stable limit cycle

of the dynamical system de¬ned by Eqs. 3.2 requires the application

of linear stability analysis. However, it is easy to see that it is indeed

a solution of Eqs. 3.2, in the sense that the equations map the circle

onto itself. The trigonometric identity 1 = cos2 t + sin2 t implies that

if A and B satisfy the equation for the unit circle, their time depen-

dence, for the particular case in which A(0) = 1 and B (0) = 0, can

be written as A(t) = cos t and B (t) = sin t. Inserting these expressions

into Eqs. 3.2 (noting that the derivative of sin t is cos t, whereas the

derivative of cos t is ’sin t), we arrive at the desired result.

56 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

Trajectory Stable fixed point Flow

Saddle

Saddle

point

point

Unstable

fixed point Limit

cycle

Unstable

fixed point

Fig. 3.1 Possible trajectories (¬‚ow lines) of the point representing the state of a

dynamical system (the “system state”). The different scenarios are shown in a vector

¬eld superimposed on a two-dimensional phase plane. In addition to the overall

geometry of the vector ¬eld (gray ¬‚ow lines, open arrowheads), the ¬gure shows

representative trajectories (red, green, and blue) leading to special points: a stable ¬xed

point or attractor, to which all local trajectories converge; an unstable ¬xed point or

repeller, from which all local trajectories diverge; a saddle point, attracting some local

trajectories and repelling others; and a stable limit cycle (a closed orbit surrounding an

unstable point) upon which all local trajectories converge. (In an unstable limit cycle, not

pictured, the closed orbit, from which all local trajectories diverge, surrounds a stable

¬xed point.) All these behaviors have analogues in dynamical state spaces of higher

dimensionality. Regions of state space in which the local dynamical behaviors differ are

known as “basins of attraction” (see the upper panel) and are divided from each other

by separatrices, lines in the phase plane or, in an n-dimensional state space, surfaces of

fewer than n dimensions.

The examples in Eqs. 3.1 and 3.2, which were chosen for their

heuristic value rather than for their plausibility as models for bio-

chemical interactions, show three possible dynamical behaviors for a

two-component system (i.e., a single stable or unstable ¬xed point and

a limit cycle). More generally, a system can have more than one stable

node or orbit. Systems of more than two variables can even wander

around in a not strictly recurrent fashion in a limited region of the

state space (a ˜˜strange attractor”). In cases in which there is more than

3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 57

one stable behavior, each will have its own ˜˜basin of attraction,” that

set of system states for which the trajectories passing through them

will wind up at that particular node or orbit (Fig. 3.1). A boundary

curve between adjacent basins of attraction is called a separatrix. A

consequence of this situation is that there will be some distinct, but

close, system states that will be in the basins of attraction of different

nodes, causing the dynamical system to behave like a ˜˜switch.” A sim-

ilar fundamental property of a nonlinear system, known as chaotic

behavior, means that even when the system starts its evolution from

nearly identical states it may terminate at unpredictable endpoints.

We have considered only systems with two interacting compo-

nents. Living cells produce tens of thousands of different proteins

and contain a myriad of sugars, amino acids, nucleotides, and other

small molecules. The state space of a real cell has thousands of di-

mensions, and the vector ¬eld describing the system™s behavior is

spectacularly more complicated than the two-dimensional case. For-

tunately, in many cases only a small number of interacting compo-

nents are needed to determine the cell™s fate at critical stages of de-

velopment -- other components are either uncoupled from the ˜˜main

event” or change on such different time scales that they can be treated

as constant, or considered only with regard to their steady-state val-

ues. In what follows we will describe cases in which the oscillatory

or switching behaviors of cellular dynamics can drive developmental

decisions.

Oscillatory processes in early development

In our description of the mechanics of cleavage in Chapter 2 we as-

sumed that at a particular point in the lifetime of the zygote, cy-

tokinesis is initiated by a signal that arises from the mitotic appara-

tus (the astral signal). We did not discuss what the initiating signal

might be, however. It is clear that this signal must be periodic in na-

ture, since once the initial cleavage division is complete each of the

daughter cells undergoes the same process, and so on for the result-

ing daughters, until the egg cytoplasm is subdivided into the hun-

dreds or thousands of cells of the blastula, depending on the species.

Experiments with extracts of fertilized eggs have indicated that the

basis of this cyclical control process is a biochemical oscillator, since

it is capable of occurring spontaneously in a soluble mixture of egg

cytoplasmic components, with no organized cytoskeleton or nuclei

present (Murray and Hunt, 1993).

Before we can consider the mechanism by which the cell cycle is

controlled during blastula formation and later development, we must

deal with one peculiarity of cell division prior to fertilization. Re-

call from the previous chapter that each cell in a diploid organism

contains two versions (maternal and paternal) of each chromosome.

Moreover, once DNA synthesis has occurred in preparation for cell

division the cell contains two copies of each of these two versions.

58 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

Because fertilization must lead to a zygote that is itself a diploid cell,

the generation of the egg and sperm (˜˜oogenesis” and ˜˜spermatoge-

nesis”) must involve a cell division step that reduces the number of

chromosomes to one version of each in each daughter cell. This is

called the ¬rst meiotic division, or meiosis I (see Fig. 9.1). After this,

one daughter cell (the other, called the ˜˜polar body,” receives very lit-

tle of the cytoplasm and has no further developmental role) contains

two physically attached copies (sister chromatids) of a single version

of each chromosome. This cell then undergoes a division in which the

sister chromatids of the replicated chromosomes (Fig. 2.1) are parti-

tioned into another polar body and a daughter cell which therefore

now contains only one copy of one version of each chromosome. This

division step, called meiosis II, is mechanically identical to mitosis, in

which sister chromatids generated during the S phase are similarly

separated. Fertilization, which brings together the chromosomes of

the egg and the sperm, yields a cell, the zygote, which thus contains

two versions of each chromosome.

The frog, Xenopus laevis, has provided an experimentally useful

system for studying the control of the cell cycle in early development.

During oogenesis (Chapter 9) in the frog, once the immature egg,

or oocyte, reaches about 1 mm in diameter, the DNA undergoes one

round of replication and then arrests before the ¬rst meiotic division.

After that the oocytes are stimulated by the hormone progesterone,

produced in the mother™s body, to undergo the two meiotic divisions,

arresting once again, this time at the metaphase of meiosis II (meiosis

in mammals, such as the human, is under similar hormonal control).

After fertilization, the egg nucleus completes meiosis II and fuses with

the sperm nucleus. The zygote then undergoes 12 rapid, synchronous,

mitotic cycles to form a hollow blastula of 4096 cells. At this stage,

called the midblastula transition, cell division in the embryo slows

down and gastrulation movements begin.

The 14 (two meiotic and 12 mitotic) cell divisions that produce the

frog blastula are triggered by an M-phase-promoting factor (MPF), a

protein kinase (an enzyme that phosphorylates other proteins) con-

sisting of two subunits, Cdc2 (the catalytic subunit) and cyclin B (the

regulatory subunit). MPF phosphorylates an array of proteins involved

in nuclear-envelope breakdown, chromosome condensation, spindle

formation, and other events of meiosis and mitosis. Whereas Cdc2 is

present at a constant level throughout the cell cycle, the concentra-

tion of cyclin B and thus the MPF activity varies in a periodic fashion,

rising to a peak value just before M-phase (see Chapter 2) and drop-

ping to a basal value as a cell exits M-phase (Fig. 3.2).

Although MPF controls nuclear properties, the oscillation in its

levels can occur in the absence of nuclei. Indeed, in frog embryos

no transcription of any genes (including those specifying the compo-

nents of MPF) in zygotic or blastomere nuclei occurs until the mid-

blastula transition. In nucleus-free cytoplasmic extracts of immature

frog eggs there are spontaneous oscillations of MPF with a period of

about 60 min. If sperm nuclei are added to the extract these nuclei

3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 59

Fertilization

Progesterone

MPF activity

Interphase Metaphase

arrest arrest

Oocyte Meiosis I Meiosis II Fertilized First mitosis Second mitosis

egg

Fig. 3.2 Variation in MPF activity in the meiotic and mitotic cycles of frog eggs.

Between the oocyte stage and meiosis I, a reduction division leaves the egg in a haploid

state with one member of each homologous pair of chromosomes and most of the

cytoplasm (one homologous pair is shown). The other members of each pair are

allocated to the ¬rst polar body (pictured as a small empty circle). During meiosis II

(which is completed after fertilization) the egg divides once more, this time separating

identical replicated copies of each remaining chromosome into the zygote and the

second polar body. Fertilization restores the diploid state, thus now two versions (i.e., a

homologous pair) of each chromosome are present (see also Fig. 9.1). For illustrative

purposes, the chromosomes are represented in a condensed state even though the cells

are mainly in interphase during these events. Once the egg is fertilized and starts cleaving

(the ¬gure shows the ¬rst two cleavages) periodic MPF ¬‚uctuations continue through the

12 mitotic divisions that produce the frog blastula. (After Borisuk and Tyson, 1998.) See

the main text for a discussion of the relationship of progesterone and fertilization to

interphase and metaphase arrest.

undergo periodic mitoses whenever the MPF concentration is high.

Biochemical analysis (Murray and Kirschner, 1989) showed that the

cyclin protein is periodically degraded in the extract and resynthe-

sized in a manner that depends upon the presence of its mRNA.

Mitotic control in frog eggs: the Tyson“Novak model

It is reasonably straightforward to devise a simple set of coupled

nonlinear differential equations along the lines of Eqs. 3.1 and 3.2

for the breakdown and synthesis of cyclin in a cell-free extract and

to propose the resulting limit-cycle oscillation as a model for cell-

cycle control in the cleaving frog egg (Norel and Agur, 1991; Tyson,

1991; Goldbeter, 1996). But the cell-free system differs in several im-

portant features from the intact egg, and it is the properties of

the latter that any realistic model must explain. In the ¬rst place, the

free-running MPF cycle in the extract is about twice as long as the

MPF-cell-cycle period in intact eggs (Murray and Kirschner, 1989), sug-

gesting that additional important processes are at work in the living

system.

60 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

One important control variable that acts differently in the cell-free

extract and in the intact egg is the phosphorylation and dephosphory-

lation of Cdc2. In the extract, Cdc2 goes through cycles of phosphory-

lation and dephosphorylation on one of its tyrosine residues, whereas

after the ¬rst cleavage in the intact egg Cdc2 is no longer found in a

tyrosine-phosphorylated form (Ferrell et al., 1991). It is likely that the

dynamics of mitotic control in the early embryo, which includes mul-

tiple steady states in addition to sustained oscillations, is dependent

on the feedback regulation of Cdc2 phosphorylation.

Taking into account what is known experimentally from both the

cell-free and intact systems, Tyson, Novak, and their coworkers have

presented a more complex model than a simple limit cycle for the gen-

eration of active MPF in the early frog embryo. This model comprises

nine coupled nonlinear differential equations (for the concentration

of nine regulatory proteins), governing, among other factors, the regu-

lation of cyclin B levels and the state of Cdc2 phosphorylation (No-

vak and Tyson, 1993; Borisuk and Tyson, 1998) (see Fig. 3.3). Using

computational methods they found solutions to these equations cor-

responding to several physiological or experimental states.

(i) A steady state with low MPF activity (as in the immature

oocyte prior to progesterone activation).

(ii) A steady state with high MPF activity (as in the mature egg