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PLoS Comput Biol. 2009 March; 5(3): e1000303. Published online 2009 March 13. doi: 10.1371/journal.pcbi.1000303. | PMCID: PMC2646127 |
Copyright Manu et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited. Canalization of Gene Expression and Domain Shifts in the
Drosophila Blastoderm by Dynamical Attractors Manu,#1 Svetlana Surkova,#2 Alexander V. Spirov,1 Vitaly V. Gursky,3 Hilde Janssens,4 Ah-Ram Kim,1 Ovidiu Radulescu,5 Carlos E. Vanario-Alonso,1 David H. Sharp,6 Maria Samsonova,2 and John Reinitz1* 1Department of Applied Mathematics and Statistics, and Center for
Developmental Genetics, Stony Brook University, Stony Brook, New York, United
States of America 2Department of Computational Biology, Center for Advanced Studies, St.
Petersburg State Polytechnical University, St. Petersburg, Russia 3Theoretical Department, The Ioffe Physico-Technical Institute of the
Russian Academy of Sciences, St. Petersburg, Russia 4EMBL/CRG Research Unit in Systems Biology, CRG – Centre de
Regulació Genòmica, Barcelona, Spain 5Institute of Mathematical Research of Rennes, University of Rennes,
Rennes, France 6Theoretical Division, Los Alamos National Laboratory, Los Alamos, New
Mexico, United States of America Stanislav Shvartsman, Editor Princeton University, United States of America Received October 30, 2008; Accepted January 23, 2009. The variation in the expression patterns of the gap genes in the blastoderm of
the fruit fly Drosophila melanogaster reduces over time as a
result of cross regulation between these genes, a fact that we have demonstrated
in an accompanying article in PLoS Biology (see Manu et al.,
doi:10.1371/journal.pbio.1000049). This biologically essential process is an
example of the phenomenon known as canalization. It has been suggested that the
developmental trajectory of a wild-type organism is inherently stable, and that
canalization is a manifestation of this property. Although the role of gap genes
in the canalization process was established by correctly predicting the response
of the system to particular perturbations, the stability of the developmental
trajectory remains to be investigated. For many years, it has been speculated
that stability against perturbations during development can be described by
dynamical systems having attracting sets that drive reductions of volume in
phase space. In this paper, we show that both the reduction in variability of
gap gene expression as well as shifts in the position of posterior gap gene
domains are the result of the actions of attractors in the gap gene dynamical
system. Two biologically distinct dynamical regions exist in the early embryo,
separated by a bifurcation at 53% egg length. In the anterior region,
reduction in variation occurs because of stability induced by point attractors,
while in the posterior, the stability of the developmental trajectory arises
from a one-dimensional attracting manifold. This manifold also controls a
previously characterized anterior shift of posterior region gap domains. Our
analysis shows that the complex phenomena of canalization and pattern formation
in the Drosophila blastoderm can be understood in terms of the
qualitative features of the dynamical system. The result confirms the idea that
attractors are important for developmental stability and shows a richer variety
of dynamical attractors in developmental systems than has been previously
recognized. C. H. Waddington predicted in 1942 that networks of chemical reactions in embryos
can counteract the effects of variable developmental conditions to produce
reliable outcomes. The experimental signature of this process, called
“canalization,” is the reduction of the variation of the
concentrations of molecular determinants between individuals over time.
Recently, Waddington's prediction was confirmed in embryos of the fruit
fly Drosophila by observing the expression of a network of
genes involved in generating the basic segmented body plan of this animal.
Nevertheless, the details of how interactions within this genetic network
reduced variation were still not understood. We use an accurate mathematical
model of a part of this genetic network to demonstrate how canalization comes
about. Our results show that coupled chemical reactions having multiple steady
states, or attractors, can account for the reduction of variation in
development. The variation reduction process can be driven not only by chemical
steady states, but also by special pathways of motion through chemical
concentration space to which neighboring pathways converge. These results
constitute a precise mathematical characterization of a healing process in the
fruit fly embryo. Canalization refers to the constancy of the wild type phenotype under varying
developmental conditions [1]– [4]. In order to explain
canalization, C. H. Waddington hypothesized that there must only be a finite number
of distinct developmental trajectories possible, since cells make discrete fate
decisions, and that each such trajectory, called a chreod, must be
stable against small perturbations [5]. One aspect of
canalization, the buffering of phenotypic variability against genotypic variability
in wild type, has received considerable experimental [2], [6]– [10] and theoretical [11]– [13] attention. The
phenomenon of canalization of genotypic and environmental variation was seen by
Waddington as a consequence of the underlying stability of developmental
trajectories, an idea supported by theoretical analysis [13]. But this central idea
of Waddington's has heretofore received little attention in real
developmental systems because of a lack of relevant quantitative molecular data. The
further investigation of Waddington's hypothesis is of great importance
because it provides a scientific connection between the reliability and invariance
of the formation of cell types and tissues in the face of underlying molecular
variability, as we now explain. Quantitative molecular data permitting the study of developmental canalization are
now available for the segment determination process in Drosophila
[14]. The
segmented body plan of the fruit fly Drosophila melanogaster is
determined when the embryo is a blastoderm [15] by the segmentation
genes [16]. Quantitative spatiotemporal gene expression data
show that the maternal protein gradients and the early expression patterns of the
zygotic gap and pair-rule genes vary a great deal from embryo to embryo [14], [17]. The variation of the expression patterns of the gap
and pair-rule genes decreases over time so that it is significantly lowered by the
onset of gastrulation at the end of cellularization ( [14], ). The observed reduction of variability
over time in the segmentation gene system suggests that the developmental trajectory
of the Drosophila embryo is stable against perturbation. The
characterization of the stability properties of the developmental trajectory is
central to our understanding of the mechanisms that underlie canalization [3]. In the case of the gap genes, we have shown elsewhere [18] that variation reduction
relative to the maternal gradient Bicoid (Bcd) occurs because of gap gene cross
regulation. Using a gene circuit model of the gap gene network [18]– [22] we
identified specific regulatory interactions responsible for variation reduction
in silico and verified their role in canalization
experimentally. Importantly, the model reproduces the observed low variation of gap
gene expression patterns [18], which provides an opportunity to analyze the
properties of the system that give rise to developmental stability. These results raise two generic problems that occur in the analysis of complex
numerical models. First, even if the model describes a natural phenomenon
faithfully, understanding the natural phenomenon is only achieved when the
model's behavior can be understood as well. The complexity of the model,
unsurprising in terms of the underlying complexity of the biological system itself,
poses a significant challenge to understanding model function. Second, any model is
an approximation to the actual mechanisms operating in an organism. The
model's behavior must be robust to perturbation, since organisms develop
and function reliably even though the underlying mechanisms are subject to a wide
variety of perturbations and stresses. There is extensive molecular variability
among cells and embryos ( [14], [17], [23]– [26]; reviewed in [27]) and yet
there is functional identity between equivalent cell types or conspecific
individuals. René Thom tried to resolve this apparent contradiction between the
constancy of biological function and the variability in biological substructure by
proposing a qualitative topological view of the trajectories of dynamical models
[28].
The term “topology” is used here to refer to properties of
developmental trajectories that are invariant under continuous deformation. The
preservation of these properties ensures the robustness of model behavior, while a
qualitative view often leads to an intuitive understanding of complex mechanisms.
One such robust property is an attractor state, or a stable steady state of a
dynamical system, that attracts all trajectories in some neighborhood of itself.
Attractor states are locally stable under small perturbations of the dynamical model
[29],
and for this reason it has been proposed that cell fates are attractors [11], [13], [30]– [33]. The presence of an attractor state in the phase space of a system implies that there
exists a region of phase space, called the basin of attraction, in which all
trajectories approach the attractor asymptotically [34], [35]. This suggests that an
attractor is the kind of qualitative robust property that could explain the
stability of trajectories, and hence canalization. There are, however, three
important considerations to keep in mind when using attractors to describe the
Drosophila blastoderm. First, the reduction of variation due to
attractors is only guaranteed at late times, but the reduction in the variation of
the gap gene expression patterns takes place over about 100 minutes prior to
gastrulation. The reduction of variation before gastrulation is biologically
essential as the expression patterns of engrailed and
wingless, which form the segmentation prepattern, have a
resolution of one nucleus and are created by the precise overlap of pair-rule and
gap domains [14], [36]. Furthermore, at about the time of gastrulation
the embryo undergoes the midblastula transition [37], [38] at which time a
qualitative change occurs in the genetic control of the embryo. Second, in general
there can be more than one attractor in the phase space [39]– [43]. Thus,
the basins of attraction need to correspond to biological initial conditions and be
large enough to ensure robustness. Finally, the set of attractors found must succeed
not only in explaining canalization but also the morphogenetic properties of the
system. One such property is the anterior shift of gap gene domains located in the
posterior region [14], [21], [44]. These shifts are biologically significant and
are difficult to reconcile with stable point attractors. In this paper we show that the variation reduction of gap gene expression patterns is
a consequence of the action of robust attracting states. We further show that the
complex patterning of the gap gene system reduces to the three qualitative dynamical
mechanisms of (1) movement of attractors, (2) selection of attractors, and (3)
selection of states on a one dimensional manifold. The last of the three mechanisms
also causes the domain shifts of the gap genes, providing a simple geometric
explanation of a transient phenomenon. In the Gap Gene Circuits section we briefly describe the gene circuit model; see
[18] for
a full description. For each nucleus in the modeled anteroposterior (A–P)
region, we identified the attractors in the gap gene phase space, calculated the
trajectories, the basins of attraction and other invariant sets such as one
dimensional attracting manifolds (Stability Analysis of the Trajectories of the Gap
Gene System section). The stability of the trajectories was tested by varying the
initial conditions within a biological range, based on gene expression data, that
represents the variability of early gap gene expression. We plotted the attractors
and several trajectories corresponding to different initial conditions to make phase
portraits that show the global qualitative behavior of the system. Finally, we
studied how the phase portraits changed as A–P position was varied to
infer qualitative pattern formation mechanisms. The biological conclusions about
canalization and pattern formation arising from the dynamical characterization are
presented in the Mechanisms of Canalization and Pattern Formation section. Gap Gene Circuits The gene circuit used in this study models the spatiotemporal dynamics of the
protein expression of the gap genes hunchback
( hb), Krüppel ( Kr),
giant ( gt), and knirps
( kni) during the last two cleavage cycles (13 and 14A)
before gastrulation [37] in the Drosophila
blastoderm. The protein products of these genes localize to nuclei [45]– [48] so that the state
variables are the concentrations of the proteins in a one dimensional row of
nuclei along the A–P axis of the blastoderm. The concentration of the  protein in the  nucleus at time  is denoted by  . In the model we considered a region, from 35% to
92% egg lenth (EL) along the A–P axis, which corresponds
approximately to the region of the blastoderm fated to form the segmented part
of the adult body [49], [50]. The gap genes are expressed in broad domains (; [14]) under the control
of maternal cues. The anterior maternal system acts primarily through the
protein gradient Bcd [51]– [53] which is
essentially stationary and has an exponential profile (; [14], [51], [54]) during the modeled
time period. The posterior maternal system is represented by the maternal Hb
gradient (; [55]– [57]). The terminal
system regulates gap gene expression by activating tailless
( tll) and huckebein ( hkb)
[58]– [61]. The terminal system
is represented in the model by the Tll gradient, which is expressed posterior to
80% EL in the modeled region during cycles 13 and 14 ( [14]
and Fig.
S1B). tll is considered upstream of the gap genes since
its expression pattern is unchanged in gap gene mutants [62]. The concentration
of Bcd in nucleus  is denoted by  and was determined using Bcd data from a representative cycle
13 embryo by an exponential fit, so that  (see [18] for details). The concentrations of Tll and
another upstream regulator, Caudal (Cad) [63], [64],
were determined by interpolating average data in time [18]. The concentrations
of Tll and Cad are denoted by  respectively, with an explicit dependence on time, since these
gradients are not stationary ( Fig. S1). The dynamical equations governing  are given by where  in a gene circuit with  genes and  nuclei. The first term on the right hand side of Eq. (1)
represents protein synthesis, the second one represents protein transport
through Fickian diffusion and the last term represents first-order protein
degradation. The diffusion constant,  varies inversely with squared internuclear distance, and  is the degradation rate. The synthesis term is set to zero
during the mitosis preceding the thirteenth nuclear division as synthesis shuts
down [65]. Following this mitosis, the nuclei are divided
and daughter nuclei are given the same state as the mother nucleus.  is the maximum synthesis rate, and  is a sigmoidal regulation-expression function. The first term
in the argument of  represents the transcriptional cross regulation between the
gap genes and the genetic interconnectivity is specified by the matrix  . Positive elements of  imply activation while negative ones imply repression. The
regulation of the gap genes by Bcd is represented in the second term and  is the regulatory strength. The regulation of the gap genes by
upstream time-varying inputs is represented in the third term and  is the number of such inputs. There are two such inputs in
this model, Cad and Tll, and the elements of the matrix  have the same meaning as those of  . The last term,  , represents the effect of ubiquitous transcription factors and
sets the threshold of activation. The initial conditions for Hb are specified using cleavage cycle 12 data. Cycle
12 data are a good approximation to the maternal Hb gradient since the zygotic
expression of hb appears to begin in cleavage cycle 13 [17]. The initial conditions for Kr, Gt, and Kni are
taken to be zero, since their protein expression is first detected in cycle 13
[14], [61], [66]– [68]. The gene circuit's parameters were determined by performing a
least-squares fit to a time series of averaged gap gene data [14]
using the Parallel Lam Simulated Annealing algorithm (see Methods). This time
series has nine points (time classes; see Table S1),
one in cycle 13 and the rest in cycle 14A. The output of the gene circuit () fits the data
() well and its
network topology ()
is consistent with previous results (see [18] for discussion and
parameters). Stability Analysis of the Trajectories of the Gap Gene System In order to characterize the stability of the trajectories of the gap gene system
in terms of qualitatively robust features like attractors, we apply the tools of
dynamical systems theory [34], [69]. Since the gene
circuit has  variables (Gap Gene Circuits section) its state is represented
as a point in an  -dimensional concentration space, or phase space. In general
the concentrations of gap proteins change with time, and hence, a solution of
the gene circuit is a curve in this phase space. The gene circuit can also have
solutions which do not change with time. Such a solution, called an equilibrium
or steady state solution, is represented as a single point in phase space. The
positions of the equilibrium solutions in phase space and their stability
properties determine the stability of a general time varying solution of the
gene circuit. The reader not familiar with linear analysis near an equilibrium
point should see Protocol S2 for a pedagogical description of equilibria and their
stability in two dimensions. Diffusionless approximation In the gap gene circuit used in this study (Gap Gene Circuits section), there
are 58 nuclei and 4 gap genes, giving rise to a dynamical system having 232
dimensions. Such a large number of dimensions pose a significant challenge
to visualizing the results of the phase space analysis. In order to make the
analysis tractable, we made the approximation that there is no diffusion,  in Eq. (1), while keeping all other parameters in Eq. (1)
at their original values [18]. This uncouples the nuclei and the system
of  coupled ODEs reduces to a set of  independent systems of  ODEs. Eq. (1) thus becomes where the dependence of the concentrations of gap gene
proteins, Bcd and time varying external inputs on the location of nucleus  along the A–P axis is denoted by  . In the absence of diffusion, the model still gives the correct sequence of
gap gene domains (); and the border positions are close to their values in the
presence of diffusion ( Table S2). In contrast to the circuit
with diffusion however, the domains have sharp boundaries and have little or
no overlap ().
The general agreement between the circuits with and without diffusion allows
the analysis of the reduced system and supports the result from earlier work
[21] that diffusion is not required for making gap
gene expression patterns. In Eq. (2), the terms  represent the anterior and terminal maternal systems. They
parametrize the set of solutions possible in each nucleus as a function of
A–P position. From the set of solutions specified by the
concentrations of Bcd, Cad, and Tll, a particular solution is chosen by the
initial conditions specified by the concentration of maternal Hb (Gap Gene
Circuits section; ). Therefore the posterior maternal system specifies position in a
manner distinct from the anterior and terminal systems. We further
simplified the analysis by neglecting the effects of Tll in Eq. (2), giving The gap gene expression patterns () produced by Eq. (3) agree with
a model that includes tll (Eq. 2) until 71% EL,
therefore the simplified analysis is valid in the region from 35%
EL to 71% EL. Equilibria, stability, and bifurcations We considered each nucleus in cleavage cycle 13 and its posterior daughter in
cycle 14A since the anterior daughter differs only slightly from the
posterior one in the concentrations of Bcd and Cad. For simplicity we refer
to mother-daughter pairs by the daughter nuclei, which are located at odd
numbered positions from 35% to 71% EL. For each
nucleus we determined the equilibria and, based on maternal Hb data (), the family of
solutions corresponding to the observed range of initial conditions. The
concentration of Cad (  ) varies with time ( Fig. S1A), implying that the equilibria
in the phase space of a nucleus also change with time. We are interested in
studying the properties of the solutions during late cycle 14A, since the
variation in gap gene expression levels is least in that time period (). Also, the
stationary approximation for Bcd is valid until time class T6 (~16
mins before gastrulation; Table S1) as Bcd concentration decreases
two-fold afterward [14]. For these reasons, we calculated
equilibria at time class T6 ( Protocol S1: Eq. S3). The trajectories
were calculated using time varying Cad data until time class T6 and with T6
Cad data thereafter ( Protocol S1: Eq. S1) so that their late
time behavior corresponded to the calculated equilibria. The equilibria were calculated using the Newton-Raphson method [70], [71] and classified
according to their stability ( Protocol S3). We calculated several
trajectories with different initial conditions to test variation reduction
and stability, and continued the integration of Eq. (S1) ( Protocol
S1) to very late times in order to visualize asymptotic behavior.
Since the segments are determined by the time of gastrulation, significant
variation reduction must occur beforehand for the precise specification of
position. Hence, we distinguish the biological behavior from asymptotic
behavior graphically. We found that each nucleus has multiple attractors and
that its trajectory can potentially approach any one depending on the
initial condition. It was necessary, therefore, to characterize the basins
of attraction [34], [69] of the
attractors. We exploited the fact that only Hb has non-zero initial
conditions (Gap Gene Circuits section) to characterize the basins as
intervals on the Hb axis (see Protocol S3 for details of calculation).
We also calculated one dimensional unstable manifolds [72] of saddle
points to better understand the transient behavior of solutions ( Protocol
S3). The analysis of the phase space for different nuclei revealed several
A–P positions at which the set of equilibria changes. This
suggests that the dynamical system is undergoing bifurcations resulting from
the changing values of its parameters, the concentrations of Bcd and Cad, as
A–P position is varied. Since we used Cad data in Eq. (3), it is
also possible that these bifurcations are artifacts caused by fluctuations
in the data. We verified that the changes observed in the analysis at
discrete odd numbered positions were actual bifurcations by varying the
concentrations of Bcd and Cad, parametrized by the A–P position,
continuously ( Protocol S4). This was done by
interpolating the Cad data with a cubic polynomial ( Fig.
S3). Note that the Bcd concentration profile has already been
parametrized as a function of position (  ; Gap Gene Circuits section), and can be varied
continuously. The continuous analysis validated all the bifurcations at
discrete locations ( and Fig. S4; circles) save for two. These two
bifurcations (;
boxes) could not be observed in the analysis at discrete locations as they
occur consecutively between two nuclei and leave the set of equilibria
unchanged. Although the bifurcations occur at particular values of the
concentrations of Bcd and Cad, subsequently we will refer to them by
A–P position for simplicity. We further characterized the type of bifurcations ( Protocol
S4). Though many types of bifurcations are possible only one type,
the saddle-node (see Fig. S2 for illustration in two
dimensions), occurs in the gap gene system. In the four-dimensional gap gene
system any creation or annihilation of a pair of equilibria that differ in
the sign of one eigenvalue is a saddle-node bifurcation ( and Fig.
S4). Mechanisms of Canalization and Pattern Formation Based on the analysis in the previous section the region of interest, from
35% EL to 71% EL, can be divided into an anterior and a
posterior region ()
having distinct modes of canalization and pattern formation. The two regions are
separated by a saddle-node bifurcation that occurs at 53% EL (), that is, at the peak
of the central Kr domain. We next demonstrate that in the anterior region (), which extends from the peak of the
third anterior gt domain to the peak of the central
Kr domain, the state of a nucleus at gastrulation is close to a
point attractor. The trajectories are stable by virtue of being in the basin of
attraction of the nucleus's attractor state and hence canalize. Pattern
formation occurs by the selection of one state from many in a multistable phase
space. The concentrations of the Bcd and Cad gradients control pattern formation
in the anterior by determining the sizes of the basins and the positions of the
attractors, while maternal Hb concentration selects a particular attractor by
setting the starting point in its basin. Previous experimental [73] and theoretical [19] work suggested
that Bcd and maternal Hb patterned the anterior of the embryo synergistically;
our results identify specific roles for Bcd and Hb in anterior patterning. The posterior region extends from the peak of the central Kr
domain to the peak of the posterior gt domain () and its nuclei have
phase spaces with very different properties. In this region, the state of the
nucleus is far from any attractor state at gastrulation. Instead the state of a
nucleus is close to a one-dimensional manifold and canalization is achieved due
to attraction by this manifold. Even though the phase space is multistable, the
biological range of maternal Hb concentrations in the posterior region place all
nuclear trajectories in one basin of attraction. As a consequence, the modes of
pattern formation operative in the anterior cannot function in the posterior.
Maternal Hb patterns the posterior by determining the position on the attracting
manifold which a particular trajectory reaches by the time of gastrulation.
These results reveal the mechanism by which maternal Hb acts as a morphogen in
the posterior [73]– [75] and also explain
the dynamical shifts of gap gene domains [14], [21], a
significant biological property of the posterior region. We begin the presentation of detailed results by describing the phase spaces of
typical nuclei in the two regions, highlighting mechanisms for canalization and
pattern formation. An equilibrium is labeled by either  (point attractor) or  (saddle equilibrium), denoted by a superscript, with
subscripts denoting the number of eigenvalues having positive or negative real
parts. For example  denotes the second saddle equilibrium in the modeled region
which has one eigenvalue with positive real part and three with negative real
parts. Equilibria are also given descriptive names based on which proteins are
at high levels ( on) ignoring the proteins that are at low
levels. For example, if a point attractor is at hb-on,
Kr-off, gt-on, and
kni-off, it is referred to as the
“ hb,gt-on” attractor. The anterior region shows the phase
portrait of the nucleus at 37% EL from the anterior region. Since
the phase space is four-dimensional, a three-dimensional projection onto the
Hb-Kr-Gt axes is shown (see Fig. S6 for other projections). There are
three attractors,  ( hb,gt-on),  ( hb,Kr-on), and  ( Kr-on) shown in blue. Two saddle
equilibria,  , are shown in red. Saddle equilibria that are unimportant
for the dynamics are not shown (see Protocol S5 and Table
S4). In order to show the family of trajectories allowed by the
values of Bcd and Cad concentrations in this nucleus, trajectories are
plotted with starting points distributed uniformly on the Hb axis between
0–100, corresponding to the observed range of maternal Hb in the
modeled region [14]. Time is represented as a color gradient
along the trajectories, with the start of cycle 13 green, and gastrulation
red. Trajectories are blue after gastrulation to indicate their asymptotic
behavior. The red-to-blue transition on the trajectories occurs very close to the
attractors ()
implying that the nucleus is close to equilibrium at gastrulation. It is
apparent by visual inspection that the initial variation of the trajectories
is reduced dramatically by the onset of gastrulation, that is, trajectories
are stable due to the point attractor. To verify this property numerically,
we constructed a four-dimensional box such that its dimensions corresponded
to the range of gap protein concentrations observed in cycle 13 and its
volume represented initial variation. We then calculated the time evolution
of the volume of this box using the model (; see Protocol
S8 for details of calculation). Variation in the concentration of
each protein shrinks by a factor of ~20 on average by gastrulation,
confirming the stability of the developmental trajectory. The concentration of maternal Hb, depending on which basin of attraction it
lies in, selects a particular attractor state. The boundaries between the
basins of attraction of the three attractors are indicated by arrows in
(see Protocol
S5 for details). The concentration of maternal Hb in the nucleus at
37% EL puts it in the basin of the hb,gt-on (  ) attractor ( Table 1). This analysis correctly accounts for the observed gap
gene expression in the nucleus, which lies at the peak of the third anterior
gt domain and in the anterior hb
domain ().
Posterior to this nucleus and up to 45% EL, all nuclei remain in
the basin of the  attractor ( Table 1).  changes its position in the phase space from
hb,gt-on at 37% EL to hb-on at
43% EL ( Table
1 and ).
The posterior border of the third anterior gt domain
therefore forms by the movement of  in the phase space. Since the Bcd and Cad concentrations
are the only parameters being varied, the movement of an attractor is one
mechanism by which Bcd patterns the anterior. | Table 1Basins of point attractors in the anterior region and their
selection by the value of maternal Hb concentration. |
The nucleus at 47% EL is in the basin of the
Kr-on (  ) attractor, unlike the nuclei to its anterior, which are
in the basin of hb-on ( Table 1). The switch from
hb-on to Kr-on causes the formation of the
posterior border of the anterior hb domain and the anterior
border of the central Kr domain between 45% EL
and 47% EL (). This happens due to 1) an increase in the size of the
basin of the Kr-on attractor and 2) a decrease in the
concentration of maternal Hb (second and fifth columns of Table 1). As is the case
with the movement of the  attractor, the first effect is due to the changing value
of Bcd concentration. Hence, Bcd and maternal Hb pattern the anterior region
in tandem [19], [73]; Bcd has
the role of setting attractor positions and the extent of basins and
maternal Hb selects a particular basin. A bifurcation separating the anterior from the posterior The nuclei remain in the basin of the Kr-on attractor up to
53% EL, where a saddle-node bifurcation annihilates
Kr-on (). Although there are other bifurcations in the A–P
region considered ( Table S3), only this bifurcation affects
the dynamics of the gap gene system in a significant way. Posterior to this
bifurcation the remaining two attractors hb-on (  ) and hb,Kr-on (  ) persist to the end of the region being analysed. No new
attractors that might correspond to gap gene expression domains of the
posterior appear, suggesting that the mechanisms of canalization and pattern
formation are different from the anterior. The posterior region shows the
Hb-Kr-Kni projection (see Fig. S8 for others) of the phase portrait
of the nucleus at 57% EL in the posterior region. There are two
attractors  ( hb-on) and  ( hb,Kr-on). Two saddle equilibria,  , are shown, see Protocol S6, Fig.
S7, and Table S4 for others. The basin of
attraction of  is very small ((43.99, 44.04)) and is shown with an arrow.
The basin of attraction of  is divided into two intervals by the basin of  . Trajectories right of the arrow (“direct
interval”) reach close to  by gastrulation. Trajectories from the “indirect
interval” to the left of the arrow are not close to the attractor
at gastrulation as can be seen from the red-to-blue transitions. The direct
trajectories are nonbiological since the concentration of maternal Hb () in the posterior
region nuclei places all biological trajectories in the indirect interval. Although the biological indirect trajectories are not in the vicinity of an
attractor at gastrulation, they appear to be converging to a single
trajectory in the phase space. This trajectory can be visualized by
following the blue (post-gastrulation) segments of the trajectories in . Therefore the
trajectories are demonstrating stability, which we verified numerically by
calculating the time evolution of a volume representing initial variation
(). The
average variation in the concentration of each protein reduces by a factor
of ~10 by gastrulation. In order to understand this puzzling
stability further, we calculated the one-dimensional unstable manifolds of
the saddle equilibria  . The unstable manifold of  , is shown as a translucent magenta tube in . This manifold is
precisely the trajectory to which all indirect trajectories are converging.
Furthermore, the trajectories are close to this manifold by gastrulation
(red-to-blue transitions). Hence this attracting manifold plays the same
role in the posterior as attractors do in the anterior, and is responsible
for the stability of trajectories and canalization (see also Protocol
S7). The attracting manifold  is also important for pattern formation in the posterior
region. The positions and stability of equilibria in the phase space are
qualitatively the same in all nuclei of the posterior region. This is a
reflection of the relative constancy of Bcd, which is shallow due to
exponential decay with position, and Cad, which has almost uniform
expression in the posterior region (; [14]). The
invariance of the qualitative properties of the phase space implies that Bcd
does not provide positional information in the posterior, a result arrived
at by different means in other work [18], [21]. In order to understand how pattern formation occurs in the posterior, it is
important to note two properties of the phase space (). First, the attracting manifold  traverses all the posterior region states
( Kr-on, kni-on, gt-on,
and intermediate values; see ), before reaching  . Second, the state achieved by an indirect trajectory at
gastrulation has a continuous dependence on initial Hb value. High values of
maternal Hb (20–40) in a nucleus lead to a Kr-on
state, intermediate levels (12–20) lead to a
kni-on state and lower values (4–8) lead to a
gt-on state. Since maternal Hb decreases monotonically
with A–P position (), these two properties lead to the formation of gap gene
expression patterns in the posterior region. In fact, changing only the
maternal Hb concentration in a single posterior region nucleus (Bcd
concentration is fixed), produces correct gap gene expression patterns in
the posterior region (). This explains the experimental result that maternal Hb alone can
pattern the embryo, if the anterior and terminal systems are absent,
producing the posterior region expression patterns over the length of the
whole embryo [75]. There is a further decrease in
variability within the subintervals of maternal Hb concentration that lead
to a particular gap gene state at gastrulation (see Protocol
S7 and Fig. S9 for details). These results
establish that maternal Hb is the morphogen in the posterior and patterns
the region by selecting states on the attracting manifold  . Finally, the attraction of trajectories by the manifold  provides a mechanism for dynamical shifts of gap gene
domains [21]. We illustrate this mechanism with the
nucleus through which the Kr posterior border and the Kni anterior border
pass as they shift anteriorly (). shows the trajectory of the nucleus at 59% EL. The
trajectory starts on the Hb axis, and is attracted to  , reaching close to the kni-on state at
gastrulation. However in approaching  , it goes through intermediate states. First Kr increases,
and then Kni increases with a concomitant reduction in Kr (see ). We note that the  trajectory is a result of asymmetric repression of
anterior domain genes by posterior domain proteins [21]. A nucleus in
the Kr-on state goes to the Kni-on state because Kni's repression
of Kr is stronger than that of kni by Kr.
A similar mechanism applies to nuclei at the Kni posterior and Gt anterior
boundaries (data not shown). A discrete [6], [7] and buffered response to perturbations is the
hallmark of a canalized developmental system. Without recourse to molecular data,
Waddington sought to explain these two properties of the response by postulating
certain favored stable developmental trajectories which he called chreods. Our
results (see for summary)
show that dynamical systems with multiple attracting states possess both of these
properties. Small perturbations are damped because of phase space volume contraction
driven by attractors. A discrete response to larger perturbations is a consequence
of the discontinuous boundaries between the basins of attraction of a multistable
system or of bifurcations. Using a model based on gene expression data, we can
conclude that the trajectory of the gap gene system is a chreod. The initial high variation of gap gene expression may arise from early events
governed by stochastic laws. Previous observations indicate that the first nuclei in
which gap gene transcription is activated are selected probabilistically [68], [76].
Moreover, the gradients of Hb and Cad proteins are formed by translational
repression from the Nanos and Bcd gradients respectively [57], [77], [78] under conditions of
relatively low molecular number [23], which is likely to lead to intrinsic
fluctuations [79]. Our results show that a deterministic
description of gap gene dynamics is sufficient to account for the reduction of
initial variation regardless of its source. It is evident however that there are at
least two other types of variation that the system might be subject to. First, a
natural population will have genotypic variation which, in the framework of the
model, would be reflected in the variation of its parameters. Second, gap gene
expression itself is likely to be a stochastic process rather than a deterministic
one. Notwithstanding this fact, there is no evidence in Drosophila
for the coupling of molecular fluctuations to phenotypic fluctuations as seen in
prokaryotes [80], suggesting that molecular fluctuations are buffered
in some sense. We emphasize that an attractor is stable against small perturbations
of the model itself [29], and hence is a model property that is preserved
to an extent if there is genotypic variation in a population or if errors are
introduced by stochastic gene expression. However, further study of both of these
aspects of canalization is required in order to more fully understand their role. With regard to pattern formation in the blastoderm, the prevailing theory is that the
border positions of downstream genes are determined at fixed values or thresholds of
the Bcd gradient [23], [52]. This idea cannot, however, account for either
the low variability of downstream gene border positions [14], [17], [18], or the
dynamical shifts of domains in the posterior [14], [21]. Fixed threshold
specification also cannot explain precise placement of the borders in the posterior
since the low molecular number of Bcd in the nuclei implies a high level of
molecular noise [23], [81]. In the dynamical
picture (), contrary to
the threshold view, Bcd ceases to have a role in positional specification posterior
to the peak of the Kr domain since, posterior to this position, the
geometry of the phase space does not change qualitatively with A–P
position. Instead, maternal Hb acts as a morphogen, obviating the problems arising
from a low molecular number of Bcd. Maternal Hb has long been recognized as a
morphogen [74], [75] for the posterior region but the mechanism with
which it specifies the posterior region pattern was not clear. As is the case with
Bcd, a threshold-based theory for positional specification by Hb [82] is
incomplete and requires the postulation of thresholds that can be modified by their
targets. The qualitative dynamics provides a viable mechanism for posterior
patterning. The attracting manifold  is the geometric manifestation of asymmetric repression between
the gap genes in reverse order of gap gene domains,  . The initial Hb concentration determines which neighborhood of the
manifold the trajectory traverses as it is reaches the manifold:
Kr-on, kni-on, or gt-on. In other
words, posterior patterning works by triggering particular feedback loops in the gap
gene network based on maternal Hb concentration. This mechanism also accounts for
domain shifts, a property particular to the posterior region, since the trajectories
mimic the geometry of the manifold as they approach it. The dynamical analysis of the gap gene system provides a simple and integrative view
of pattern formation in the blastoderm (). The existence of distinct anterior and posterior patterning
systems was inferred from the effect of maternal mutations on larval cuticle
phenotype and was subsequently characterized in terms of the effects of the Bcd
[52], [73], [83], [84] and
maternal Hb gradients [56], [57], [77]. But where and how is the control of patterning
transferred from Bcd to maternal Hb? Our analysis shows that the hand-off occurs at
the A–P position where the Kr-on attractor is annihilated
through a saddle-node bifurcation, implying a sharp rather than gradual transfer.
With knowledge of the two dynamical regimes, the complex spatiotemporal dynamics of
the gap gene system can be understood in the simple terms of three mechanisms:
movement of attractors through phase space, selection of attractors by initial
conditions, and the selection of states on an attracting manifold (). Finally, we mention the advantage of having the unexpected mechanism of a one
dimensional manifold for canalization and patterning. The Bcd concentration is a
bifurcation parameter of the dynamical equations. If there were specific attractors
corresponding to each gap gene state, with bifurcations creating and annihilating
them successively as the Bcd concentration is varied, the molecular noise in Bcd
[23]
would give rise to “jitter” or rapid switching between
attractors. The manifold with its smooth dependence on maternal Hb is qualitatively
robust to such fluctuations. In a connectionist model of cognition [85],
one dimensional unstable manifolds connecting a sequence of saddle points have been
proposed as a means of representing transient brain dynamics. The gap gene phase
space is a low dimensional projection of the high dimensional phase space of all the
molecular determinants in the blastoderm. It may well be that the attractors found
in our analysis are actually saddle points in the high dimensional phase space and
are way points, with manifolds connecting them, rather than final end points. The methods used to obtain and characterize the quantitative data are as described in
earlier work [14]. All gene expression levels are on a scale of
0–255 chosen to maximize dynamic range without saturation. The numerical
implementation of the gene circuit equations is as described [18], [21]. The gap gene circuit
was fit to integrated gap gene data [14] using Parallel Lam
Simulated Annealing (PLSA) [86], [87]. PLSA minimizes the root mean squared (RMS)
difference between model output and data. For each nucleus, data were available at
nine time points ( Table S1). Search spaces, penalty function, and other annealing
parameters were as described [22], [88]. The circuit analyzed in detail had an RMS score
of 10.76, corresponding to a proportional error in expression residuals of about
4–5%. Equilibria were determined by the Newton-Raphson method as described in Protocol S3.
One-dimensional unstable manifolds of hyperbolic equilibria were calculated by
solving the ODEs using the Bulirsch-Stoer [71] method with starting
points in the unstable eigenspace of the equilibria [72]. The basin
boundaries on the Hb axis were calculated by finding starting points for
trajectories that reach saddle points with one positive eigenvalue ( Protocol S3).
The time evolution of volume phase space was calculated as described ( Protocol S8).
The methods used to calculate the equilibria branches and to determine the type of
bifurcations are described in Protocol S4. Protocol S1 Hybrid nonautonomous and autonomous equations. (0.03 MB PDF) Protocol S2 Equilibria and bifurcations in two dimensions. (0.02 MB PDF) Protocol S3 Equilibria, stability, one-dimensional manifolds, and basins of attraction. (0.04 MB PDF) Protocol S4 Continuous analysis and bifurcations. (0.03 MB PDF) Protocol S5 Saddle equilibria, bifurcations, and basins in the anterior region. (0.02 MB PDF) Protocol S6 Saddle equilibria, bifurcations, and basins in the posterior region. (0.02 MB PDF) Protocol S7 Reduction of variation in maternal Hb. (0.02 MB PDF) Protocol S8 The calculation of volume changes over time. (0.03 MB PDF) Figure S1 Integrated data for time-varying inputs. The data are from cleavage cycles 12
(C12), 13 (C13), and 14 (T1–T8). (A) Cad. (B) Tll; T4 and T5
curves are underneath the T6 curve at the peak of the posterior domain.
Shaded area shows modeled region. (0.21 MB TIF) Figure S2 Equilibria and bifurcations in two dimensions. (0.06 MB TIF) Figure S3 Interpolation of time class T6 Cad profile for continuation analysis. The
interpolant (black curve) is the cubic polynomial
−0.0075x3+0.2264x2+2.3611x+9.8004. (0.15 MB TIF) Figure S4 Other equilibria branches determined by the continuous analysis. Saddle
equilibria having one or two eigenvalues with positive real part are red or
brown respectively. The y-axis is the projection of
equilibria positions on the Kr axis. The x-axis is the
bifurcation parameter, the A–P position x. (A)
The equilibria S31,3 and
S42,2, showing their
bifurcation at 36.96% EL. (B)
S71,3 and
S82,2 are created at
53.32% EL, and there are no further
bifurcations at more posterior positions. (0.14 MB TIF) Figure S5 Bifurcations in the anterior region. Hb-Kr-Gt projection of equilibria
diagrams at (A) 35% EL, (B) 37% EL, (C) 43%
EL, and (D) 45% EL. The axes originate from
(−10,−10,−10), and have
length 250 in relative concentration units. The
xy-plane is shown in gray. To aid perception of depth,
shadows from a light source directly above the xy-plane are
rendered as dark gray traces on the xy-plane. Equilibria
are represented by spheres of radius 10. Point attractors
are blue and saddle equilibria having one or two eigenvalues with positive
real part are red or brown respectively. Red arrows in panel A point to
saddles, S31,3 and
S42,2, that disappear through
a saddle-node bifurcation between 35% EL and 37% EL.
In panels B and C, the A10,4
attractor goes from hb,gt-on state to
hb-on state. Red arrows in panel D point to two saddles,
S51,3 and
S62,2 created by a saddle
node bifurcation between 43% EL and 45% EL.
S51,3 and
A30,4 disappear through a
saddle node bifurcation at 53% EL that separates the anterior and
posterior regimes. (1.80 MB TIF) Figure S6 All four three-dimensional projections of the phase portrait at
37% EL. (A) Hb-Kr-Gt projection; red arrows are basin boundaries.
(B) Hb-Kr-Kni projection. (C) Hb-Gt-Kni projection. (D) Kr-Gt-Kni
projection. The axes, xy-plane, and equilibria are as in
Fig.
S5. 10 trajectories are shown with starting points
equally distributed on the Hb axis between
0– 100. Time is represented
as a color gradient along the trajectories, with start of cycle 13 as green,
and gastrulation as red; trajectories are blue after gastrulation. The
temporary reversals in trajectories are mitoses, during which the
trajectories move toward the origin. (2.22 MB TIF) Figure S7 Bifurcations in the posterior region. Hb-Kr-Kni projection of phase portraits
at (A) 53% EL and (B) 55% EL. The axes,
xy-plane, and equilibria are as in Fig.
S5. See Table S3 for bifurcation parameter values
determined by the continuous analysis. Black arrows point to saddles,
S71,3 and
S82,2, that are created via a
saddle-node bifurcation between 53% EL and 55% EL. (0.93 MB TIF) Figure S8 All four three-dimensional projections of the phase portrait at
57% EL. Axes, xy-plane, and equilibria are as in
Fig.
S5. All saddle equilibria are not shown (see Fig.
S7). The unstable manifold of saddle
S21,3,
U2 is shown as a translucent magenta tube of
radius 5. 10 trajectories are shown in panels A, C, and D,
while 25 are shown in panel B. (A) Hb-Kr-Gt projection. Red arrow shows the
separation of the indirect route trajectories from direct route ones. (B)
Hb-Kr-Kni projection. Red arrow shows the separation of the indirect route
trajectories from direct route ones. (C) Hb-Gt-Kni projection. (D) Kr-Gt-Kni
projection. U2+ traverses
the anteroposterior progression of gap gene states in the posterior
region— Kr-on to kni-on to
gt-on. (2.69 MB TIF) Figure S9 Tolerance to variation in maternal Hb. The range of initial conditions (B,
error bars) for which modeled gap gene expression patterns have the same
expression level variation as gap gene data in T8 (A). Error bars are ranges
of concentrations, and percentage variation is the ratio of range to mean.
(A) The variation in expression levels at the Kr peak is 30%
(yellow bar), at the Kni peak is 35% (red bar), and at the Gt
peak is 50% (black bar). (B) The tolerance range for maternal Hb
is shown at three A–P positions (Kr,
kni, and gt peaks). Maternal Hb profile is
shown in red. The tolerance to initial variation is
150% at Kr peak,
85% at kni peak and
100% at gt peak. (0.46 MB TIF) Table S1 Time classes (0.02 MB PDF) Table S2 Position of gap gene boundaries in the circuits with and without diffusion. (0.01 MB PDF) Table S3 Comparison of bifurcation parameter values determined in the discrete
analysis with the values determined in the continuous analysis. (0.02 MB PDF) Table S4 Summary of all equilibria, the A–P region they exist in, and their
function. (0.02 MB PDF) We thank R. K. Lewis, J. P. Gergen, J. Jaeger, K. Krishan, C. Martinez, and A. M.
Samsonov for valuable discussions and several anonymous reviewers for helpful
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