<<

. 13
( 66 .)



>>

prior to fertilization).
(iii) A limit-cycle oscillation of MPF concentration accompanied
by cyclic changes in the phosphorylation state of Cdc2 hav-
ing maxima between peaks of MPF. (This feature of the cell-
free extract must characterize the system dynamics under
some parameter choices. However, prediction by the model of
realistic Cdc2 dynamics in the egg requires parameter choices
that suppress this behavior once cleavage begins).
(iv) A limit-cycle oscillation of MPF concentration with little ty-
rosine phosphorylation of Cdc2 (as in the cleaving egg).
The numerous concentrations, rate constants, and other parameters
that enter into such a complex system are often poorly characterized
and continually undergoing experimental evaluation and reevalua-
tion. Therefore it is important to know whether behaviors seen with
different parameter values are tied to those particular values or are
˜˜robust” (that is, capable of maintaining their integrity) in the face of
changes. By exploring the structure of the system™s vector ¬eld using
numerical methods, it is possible to test features of the model despite
empirical uncertainties.
Borisuk and Tyson (1998) performed a ˜˜bifurcation analysis” of
the Tyson--Novak model using the AUTO computer program, which
was speci¬cally designed to handle such problems (Doedel and Wang,
1995). Bifurcation refers to an abrupt change in the type or number
of attractors due to variation in one or more parameters, which leads
to qualitative alterations in the system™s behavior. Figure 3.4 gives an
example of one of the bifurcation diagrams obtained. In this case,
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 61




Active MPF
Wee 1



Cdc25



PP CAK PP CAK

Wee 1

Protein
synthesis
Cdc25 Cdc2




Fig. 3.3 Relationships among regulatory proteins in the cell-cycle model of Borisuk
and Tyson (1998). Active MPF consists of a molecule of Cdc2 (blue rectangle) that is
phosphorylated on threonine-161 (T), but not on tyrosine-15 (Y), and dimerized, i.e.,
associated, with a molecule of cyclin (green oval). The yellow circles represent phosphate
groups that are attached by the kinases Wee 1 (acting on tyrosine) and CAK (acting on
threonine) and are removed by protein phosphatases operating on tyrosine (Cdc25) and
threonine (PP, most likely the protein phosphatase 2C; Cheng et al., 1999; De Smedt
et al., 2002). Active MPF promotes the phosphorylation of Wee 1 and Cdc25, which
respectively inactivates (’) and activates (+) these enzymes and indirectly promotes the
degradation of cyclin (broken-line arrow). (After Borisuk and Tyson, 1998.)




MPF activity is plotted against the rate constant k1 for cyclin synthe-
sis, which is one of the ˜˜adjustable” parameters of the system. There
is a range of values for this rate constant that dictate distinct sta-
ble steady states (depending on k1 ) for MPF activity and a range of
values that dictate limit-cycle behavior. These limit cycles also vary
in their properties, both period and amplitude (maximum MPF activ-
ity minus minimum MPF activity) being functions of k1 . Thus while
details such as the absolute value of MPF concentration cannot be pre-
dicted with certainty, the qualitative behavior of the system is robust
to fairly wide changes in the parameters. Borisuk and Tyson (1998)
also carried out bifurcation analysis with two simultaneously varying
parameters and came to similar conclusions with respect to robust-
ness.
The system of equations in the Tyson--Novak model also exhibits
unstable steady states and limit cycles: at particular points in the bi-
furcation diagram there is a qualitative change in the character of the
solution with, for example, a small change in k1 causing the system
to jump from a stable steady state to a limit cycle (Fig. 3.4). This is
known as a Hopf bifurcation (Strogatz, 1994). Because changes in a pa-
rameter can be triggered from outside (e.g., by exogenous chemicals),
such bifurcations provide a model for alterations in the behavior of
62 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



A
0.35 38




MPF ACTIVITY
0.25




0.15




0.05

10’2 10’1
10’3 1
Rate constant for cyclin synthesis (k1)

B
0.35
MPF ACTIVITY




0.15




0.05



0.0025 0.0035 0.0045 0.0055
Rate constant for cyclin synthesis (k1)

Fig. 3.4 One-parameter bifurcation diagram for the Tyson“Novak model. The
character of the biological state changes as a function of the rate constant k1 for cyclin
synthesis. Solid red line, stable steady state; broken red line, unstable steady state; mauve
dots, stable limit cycle; blue dots, unstable limit cycle; black square, Hopf bifurcation.
At the Hopf bifurcations the system abruptly moves from a stable steady state to an
unstable steady state or limit cycle. Consider the value of k1 at the black vertical line.
For this particular value of k1 , the MPF concentration varies periodically between the
two points at the ends of the line, the upper and lower being its maximum and minimum
values, respectively. The encircled number indicates the period of oscillation. All the
pairs of dots above and below the red line should be interpreted analogously. Panel B is a
horizontal enlargement of part of panel A. (After Borisuk and Tyson, 1998.)


the egg when the immature oocyte is activated by progesterone or
the mature egg is fertilized.
Does a physical model of the frog-egg mitotic oscillator need to
be so complicated that it consists of nine differential equations and
26 adjustable parameters? Borisuk and Tyson pointed out that while
simpler models can exhibit robust oscillations, the full range of dy-
namical behaviors seen in the frog egg, which also includes multiple
steady states, seem to depend on many of the positive and negative
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 63


feedback loops known to exist in the MPF regulatory system. And the
story becomes even more complex when we consider the cell cycle
in cells of the later embryo (e.g., the post-midblastula-transition frog
embryo) or of more mature tissues.
The cell divisions of the cleaving egg occur with essentially no
time gaps between mitosis and the round of DNA synthesis that fol-
lows it, or between the ensuing DNA synthesis and mitosis. In con-
trast, the division of mature cells occurs with several checkpoints
at which the cell cycle is halted. These checkpoints are overcome by
the irreversible signals ˜˜Start,” in which cyclin synthesis is induced
and its degradation inhibited, and ˜˜Finish,” in which the anaphase-
promoting complex (APC) is activated (Novak et al., 1999). APC de-
grades the proteins that tether the sister chromatids together and
targets cyclins for degradation (Zachariae and Nasmyth, 1999). Other
complexities involve the existence of more than one family of cyclins
and numerous enzymes of the cyclin-dependent kinase (Cdk) class,
of which Cdc2 is only one (Kohn, 1999). As shown by Tyson, Novak,
and their coworkers (Sveiczer et al., 2000; Tyson and Novak, 2001),
the overall effect of the interactions among these components is to
change the cell cycle from a free-running process, based on a chem-
ical oscillation such as that seen in the egg, to a recurrent process
based on a sequence of stationary states in which the cell is conveyed
from one point to another by a series of stage-speci¬c triggers. These
two ways of generating a cell cycle, each of which occurs at appropri-
ate stages of development, have been referred to as the ˜˜clock” and
the ˜˜domino” mechanisms (Murray and Kirschner, 1989).


Multistability in cell-type diversi¬cation
The hypothesis that the cell cycle is controlled by the sort of dynam-
ical system studied by physicists implies that when a cell divides it
inherits not just a set of genes and a particular mixture of molecular
components but also a dynamical system in a particular dynamical
state. A limit-cycle oscillator like the one thought to underlie the cell
cycle has a continuum of dynamical states that de¬ne a stable orbit
surrounding an unstable node. A particular dynamical state on such
an orbit corresponds to a particular phase of the limit cycle, for in-
stance the phase at which MPF activity is at its minimum. If a given
dynamical state of the cell-cycle regulatory network can be inherited,
this should also be possible in other biochemical networks. Elowitz
and Leibler (2000) used genetic engineering techniques to provide
the bacterium Escherichia coli with a set of feedback circuits involving
transcriptional repressor proteins in such a way that a biochemical
oscillator not previously found in this organism was produced. In-
dividual cells displayed a chemical oscillation with a period longer
than the cell cycle. This implied that the dynamical state of the ar-
ti¬cial oscillator was inherited across cell generations (if it had not,
no periodicity would have been observed). Because (unlike in the MPF
64 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


example described above) the biochemical oscillation was not tied
to the cell cycle, newly divided cells in successive generations found
themselves at different phases of the oscillation.
The ability of cells to pass on dynamical states (and not just ˜˜infor-
mational” macromolecules such as DNA) to their progeny has impor-
tant implications for developmental regulation, since the continuity
and stability of a cell™s biochemical identity is a key factor in the per-
formance of its role in a fully developed organism. Inheritance that
does not depend directly on genes is called ˜˜epigenetic inheritance”
and the biological or biochemical states inherited in this fashion are
called ˜˜epigenetic states.” The ability of cells to undergo transitions
among a limited number of discrete, stable, epigenetic states and to
propagate such ˜˜decisions” from one cell generation to the next is
essential to the capacity of the embryo to generate diverse cell types.
Let us consider a dynamical system that, instead of exhibiting
limit-cycle behavior, displays alternative stable steady states. Like the
examples above, a dividing cell might transmit its particular system
state to each of its daughter cells, but it is also possible that some
internal or external event accompanying cell division could push one
or both daughter cells out of the basin of attraction in which the
precursor cell resided and into an alternative state. Thus, just as a
limit cycle triggers particular biological activities (mitosis, entering
interphase) by its distinct dynamical states (i.e., the phases of the oscil-
lation), multistable dynamical systems can similarly provide the phys-
ical basis for developmental changes in cell state -- what were referred
to as ˜˜determination” and ˜˜differentiation” earlier in this chapter.
Not every molecular species needs to be considered simulta-
neously in modeling a cell™s transitions between alternative biochem-
ical states. Changes in the concentrations of the small molecules
involved in the cell™s ˜˜housekeeping” functions, such as energy
metabolism and amino acid, nucleotide, and lipid synthesis, occur
much more rapidly than changes in the pools of macromolecules
such as RNAs and proteins. The latter represent the cell™s ˜˜gene ex-
pression” pro¬le, the changes in which can therefore be considered
against an average ˜˜metabolic background.” Moreover, a cell™s active
genes can be partitioned into different categories, and this leads to
a simpli¬cation of the gene regulatory dynamics. Many of the cell™s
housekeeping genes (see above) are kept in the ˜˜on” state during the
cell™s lifetime. The pools of these ˜˜constitutively active” (i.e., always
˜˜on”) gene products can be considered constant, so that their con-
centrations enter into the dynamic description of the developing em-
bryo as ¬xed parameters rather than variables. In a similar fashion,
those housekeeping genes whose expression varies in a cyclical fash-
ion keyed to the cell cycle are typically not dynamically connected
to the genetic circuitry that determines cell type. It is primarily the
developmentally regulated genes that are important to consider in
analyzing determination and differentiation. And of these genes, the
most important are those whose products control the activity of other
genes.
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 65




Epigenetic multistability: the Keller autoregulatory
transcription-factor-network model
Transcription factors are proteins responsible for turning genes on
and off. They do this by binding to speci¬c sequences of DNA called
˜˜enhancers.” Enhancers are usually (but not always) located next to
the ˜˜promoter,” itself a regulatory DNA sequence directly adjacent
to a gene™s transcription start site. Frequently the term ˜˜promoter”
is used as short-hand for the true promoter plus the enhancer se-
quences. We will follow this practice in the present text. Important

<<

. 13
( 66 .)



>>