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dA
= (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

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