Institute for Adaptive and Neural Computation, School of Informatics,
University of Edinburgh, Edinburgh EH1 2QL, UK
* Present address: Department of Anatomy and Neurobiology, Washington University
School of Medicine, 660 S. Euclid, St Louis, MO 63110, USA
Author for correspondence (e-mail:
eglen{at}pcg.wustl.edu)
Accepted 29 August 2002
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SUMMARY |
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Key words: Retinal mosaics, Cell fate, Cell death
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INTRODUCTION |
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To date, the best known example of cell fate mechanisms controlling the
formation of regular cell spacing in the retina is the lateral inhibition of
cell fates in Drosophila eye
(Frankfort and Mardon, 2002).
In this system, several molecular signalling pathways, including Delta-Notch
signalling, determine one cell from a pool of precursor cells to become an R8
photoreceptor. After each R8 cell has been determined, the surrounding cells
take on different fates to make a full complement of photoreceptors within an
ommatidium. It has recently been suggested
(Frankfort and Mardon, 2002
)
that vertebrate ganglion cell development follows a similar path, given that
they are the first cells to be produced, and that they inhibit surrounding
cells from also becoming retinal ganglion cells (RGCs), perhaps mediated via
Delta-Notch signalling (Austin et al.,
1995
; Waid and McLoon,
1998
). In chick retina, as soon as RGCs are identifiable, they are
arranged regularly, although it is not known how many types of RGCs will be
formed from this initial population
(McCabe et al., 1999
). The
effects of these lateral inhibition mechanisms upon spatial regularity have
not yet been investigated experimentally.
A related issue in retinal development is how complementary types of cells
might form. In particular, and ß RGCs form two complementary
types, on- and off-centre, depending on whether the cell responds to onsets or
offsets of light. This physiological classification is mirrored by an
anatomical distinction in mature RGCs: the dendrites of on-and off-centre RGCs
stratify in different layers of the inner plexiform layer (IPL)
(Famiglietti and Kolb, 1976
).
However, at early stages of development, the dendrites of an RGC initially
bistratify into both sublayers of the IPL, before eventually losing dendrites
in one sublayer of the IPL (Bodnarenko and
Chalupa, 1993
). It is still not known whether the on- and
off-centre
(or ß) RGCs are generated independently, or whether a
cell first becomes an RGC, then maybe an
RGC, and finally becomes
either an on- or off-centre
RGC. For example, Jeyarasasingam et al.
(Jeyarasasingam et al., 1998
)
have suggested that on- and off-centre cells may be generated and spatially
positioned independently of each other in a random manner, but then postnatal
cell death helps improve the regularity of each mosaic by removing
inappropriately placed cells. Alternatively, once an initial population of
bistratified RGCs has been produced, environmental interactions, such as
competition for inputs from bipolar cells, may decide the fate of a ganglion
cell to become either on- or off-centre
(Kirby and Steineke, 1996
).
These competing hypotheses for formation of on- and off-centre RGC mosaics
have not yet been evaluated in detail.
One approach to studying these various hypotheses for mosaic formation is
to use theoretical models to explore the implications of different mechanisms
upon development. Several theoretical models of mosaic formation have been
produced. Exclusion zone models (where cells are randomly positioned, subject
only to the constraint that they do not come within some minimal distance of
other cells) have successfully replicated the distribution of retinal mosaics
(Galli-Resta et al., 1997;
Galli-Resta et al., 1999
).
ß RGC mosaics could also be replicated by assuming each cell is displaced
randomly from a regular hexagonal mosaic
(Zhan and Troy, 2000
). These
models capture the statistical properties of the mosaics, but do not explain
any particular biological mechanisms. By contrast, models employing more
biological mechanisms have recently been produced. The possibility that fish
photoreceptors self organise by swapping location with neighbouring cells of
different types has been studied theoretically
(Mochizuki, 2002
). In
addition, we have previously investigated the role of dendritic interactions
in guiding lateral cell movement underlying the formation of mosaics
(Eglen et al., 2000
). For a
fuller review of modelling in this field, see Eglen et al.
(Eglen et al., 2002
).
Only Tohya et al. (Tohya et al.,
1999) have studied the effects of cell fate upon retinal mosaic
formation. In this study, they showed that the spatial arrangement of
different types of fish photoreceptor can be generated by allowing the fate of
each cell to change according to the fate of neighbouring cells. However, this
model assumed that initially cells are initially positioned in a perfect
square lattice. This regular geometry may be appropriate for modelling fish
photoreceptors, but most retinal mosaics are far less regular than seen for
the fish photoreceptors. Even in other models where cell fate has been
studied, cells are positioned in regular hexagonal arrays
(Goodyear et al., 1995
;
Collier et al., 1996
;
von Dassow et al., 2000
). To
our knowledge, the effect of cell fate upon the spacing of different types of
cells in irregular arrays has not been studied, although the effect upon the
relative numbers of cells has been investigated
(Honda et al., 1990
;
Tanemura et al., 1991
). Our
current work therefore extends these previous approaches by investigating the
effects of cell fate mechanisms in irregular arrays. In the first part of this
paper, we explore the effect of cell fate processes upon mosaic regularity of
both the undifferentiated and differentiated populations. In the second part
we compare two mechanisms, cell fate and cell death, to see whether they alone
or together are suitable to generate the observed distributions of on- and
off-centre RGCs. Some of this work has been previously presented in abstract
form (Eglen and Willshaw,
2000
).
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MATERIALS AND METHODS |
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![]() | (1) |
Cell fate mechanisms
Two different styles of simulating cell fate were investigated here, namely
a one-step method (Honda et al.,
1990) and an iterative method
(Collier et al., 1996
). Both
are lateral inhibitory methods, in which some local process decides whether an
initially undifferentiated cell should eventually acquire one of two fates. If
the cell fate process is competitive, then typically a cell that wins the
competition acquires primary fate, and a cell that loses the competition
adopts secondary fate. In the iterative method, each cell continuously updates
its decision to acquire primary or secondary fate; by contrast, in the
one-step methods, the fate decision is made just once.
In both the one-step and iterative methods, we need to decide which cells
compete with each other during the cell-fate process. Here, we assume each
cell competes with its neighbours to acquire primary fate. As cells are not
positioned in a regular grid across the retinal surface, we have used the
Voronoi tessellation (Fortune,
1987)
(http://cm.bell-labs.com/netlib/voronoi/index.html)
to decide whether two cells are neighbours. The Voronoi tessellation divides
the retinal surface into non-overlapping polygons, one per cell. The Voronoi
polygon of a cell represents the region of space that is closest to that cell
(see Fig. 1A, for an example).
Two cells are then neighbours of each other if they share an edge of a Voronoi
polygon. Rather than suggesting that the retina computes the Voronoi
tessellation to discover its neighbours, we simply use it here as a good first
approximation as to which cells might communicate with each other during
development.
|
The one-step methods for lateral inhibition of cell fate
(Honda et al., 1990;
Tanemura et al., 1991
) are
simpler to implement than the iterative methods. Honda et al.
(Honda et al., 1990
) devised
several related one-step methods, differing in only the ranking technique
used. Once an initial undifferentiated population of cells has been created
with the exclusion zone model, the Voronoi tessellation
(Fortune, 1987
) is calculated,
and the neighbours of each cell found. The cells are then ranked according to
some criterion. Each cell is sequentially selected in order of its rank; if it
is undifferentiated, it acquires primary fate and all its neighbours adopt
secondary fate. Otherwise, if the cell already has primary or secondary fate,
the fate of the cell is unchanged. Tanemura et al.
(Tanemura et al., 1991
)
investigated four different ranking techniques:
To model lateral inhibition in a more realistic way than the one-step
methods, we used the iterative method presented by Collier et al.
(Collier et al., 1996) to
simulate Delta-Notch signalling. In this formulation, the primary and
secondary cell fates are represented by high levels of Delta and Notch,
respectively. High levels of Delta in one cell induce higher Notch expression
in its neighbours, which in turn decreases the expression of Delta in these
cells. Consequently, expression of these two factors tends to be driven to
opposing extremes. This signalling mechanism is formalised by assuming each
cell i has two variables, ni and
di, that represent the level of Notch and Delta,
respectively. (In this form, each variable is dimensionless and varies between
[0,1].) Initially, ni and di are
chosen from a uniform random distribution in the range [0.9,1]. They are then
updated by the following coupled differential equations:
![]() | (2) |
![]() | (3) |
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Fig. 1 illustrates the difference in the two types of neighbourhoods. The differential equations [(2) and (3)] were solved numerically until the activity of all cells had reached stable values (see Computational details). At this point, each cell i acquired primary fate if di>0.9, otherwise it acquired secondary fate.
Cell death
The mechanisms that underlie cell death in the retina are not clearly
understood. Even in other systems, there are few theoretical models
investigating cell death (for a review, see
Clarke, 2002). Jeyarasasingam
et al. (Jeyarasasingam et al.,
1998
) suggest that cell death may help mosaic formation by
removing those cells that are inappropriately placed. However, this case has
not yet been formalised. To allow us to investigate the impact of cell death
upon mosaic formation, we investigated three possible mechanisms of cell
death. These methods do not correspond to any known biological processes of
cell death, but allow us to begin to investigate the importance of cell death
in mosaic formation. When choosing which cell should die, one of three methods
was used.
If more than one cell shares the same minimum distance or Voronoi polygon area, one of those cells is randomly selected to die. Also, for methods 1 and 2, once a cell has been deleted, the nearest-neighbour distances and Voronoi polygon areas are recalculated for the remaining cells.
Cells are deleted one by one until a fixed percentage of cells have been
deleted. Up to 20% of ganglion cells may die postnatally
(Jeyarasasingam et al., 1998
),
although here we have tested the effects of up to 40% cell death. When
investigating the role of cell death upon the formation of on- and off-centre
RGC mosaics, cells were deleted independently from each mosaic.
Evaluating mosaic regularity
Mosaic regularity was evaluated with two complementary measures. The
regularity index is the ratio of the mean to the standard deviation of the
distances from each cell to its nearest-neighbour of the same type
(Wässle and Riemann,
1978). The higher the regularity index, the more regular the
mosaic; values greater than 2 typically indicate a non-random distribution
(Cook, 1996
). The second
measure is the packing factor (Rodieck,
1991
), which quantifies how well a set of disks of a given radius
are packed. The packing factor ranges from 0 for a random distribution to 1
for a perfectly hexagonal arrangement. The formulae required for computing the
packing factor are more detailed than the regularity index; full details are
given by Rodieck (Rodieck,
1991
). Packing factors were calculated using 20 bins, each 5 µm
wide. Experimentally observed retinal mosaics typically have regularity
indexes of 4-9, with packing factors around 0.1-0.4. Both the regularity index
and packing factor are dimensionless. For the cell fate mechanisms, we also
measured the cell count ratio: the number of secondary fate cells divided by
the number of primary fate cells.
Computational details
When computing the Voronoi tessellation
(Fortune, 1987), toroidal
wraparound conditions were imposed so that cells at the borders were
considered neighbours of each other. This was done so that each cell had
roughly an equal number of neighbours. When computing the regularity index, we
measured the nearest-neighbour distance of cells within the central region
(all cells that were not within 100 µm of an edge). Any artifacts
introduced by the toroidal wraparound were reduced using this technique. For
the packing factor measurement, we ignored all cells within 100 µm of an
edge, and instead used the correction factor suggested by Rodieck
(Rodieck, 1991
) to handle edge
effects.
The system of 2N differential equations given by (2) and (3) were solved
numerically using the ordinary differential equation solver LSODE from Octave
(www.octave.org).
This system has an unstable homogeneous steady state (when all Notch levels
are equal, and all Delta levels are equal), as seen during times 5-20 in
Fig. 1C,D. Any small
perturbation of this steady state then self amplifies into one of two stable
inhomogeneous steady states (Collier et
al., 1996) when a cell is driven to have high Delta activity and
low Notch activity (or vice versa; seen in
Fig. 1C,D for times beyond 30).
Hence, the simulations were run until the network reached the stable
inhomogeneous steady state.
Computer simulations and analysis were performed in Octave and R
(Ihaka and Gentleman, 1996).
To test the robustness of the models, each simulation was run repeatedly from
different initial conditions by generating different initial cell positions in
the exclusion zone model. The simulations are summarised here by plotting
error bars denoting the mean±1 s.d. of each measure. Some error bars
are smaller than the symbol depicting the mean, and hence are not visible.
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RESULTS |
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Cell count ratio
Given that each cell typically has five to seven Voronoi neighbours, and
that each secondary fate cell typically, but not always, has two primary fate
neighbours, the estimated cell count ratio is expected to be 2.5-3.5:1. The
cell count ratio was measured for the different cell fate mechanisms
(Fig. 3). For the one-step
mechanisms, our results agreed with previous work
(Honda et al., 1990;
Tanemura et al., 1991
): for a
given packing intensity, the cell count ratio was highest for the largest
method, and lower for the smallest method. This is expected, as the area and
number of edges of a Voronoi polygon are positively correlated [see
Fig. 6 by Tanemura et al.
(Tanemura et al., 1991
)].
Hence, if cells with larger polygon areas are picked first, they will
influence a high number of neighbouring cells to acquire secondary fate. The
random method produced cell count ratios around 3:1, in between the largest
and smallest method. The left-right method and all-neighbours inhibition
produced similar cell count ratios, typically around 2.6:1. By contrast,
nearest-neighbour inhibition produced cell count ratios around 1:1. This is as
expected as the cells usually, but not always, compete with just one other
cell, and so for every primary fate cell, there will typically be one
secondary fate cell.
|
|
To test whether the packing intensity, , of the initial
undifferentiated population influenced the cell count ratio, we compared the
cell count ratio when
=0.0 with
=0.5 for each cell fate method using
the two-sample Wilcoxon test. For the largest, smallest and random one-step
methods, the cell count ratio was significantly higher when
=0.5 than
when
=0.0 (P<10-6 in each case). This trend was
weaker, but still significant, for the left-right method (P=0.023)
and for all-neighbours inhibition (P=0.01). Finally, there was no
increase in cell count ratio with packing intensity for the nearest-neighbour
inhibition method (P=0.90). Hence, the cell count ratio is less
sensitive to variations in
for the iterative than the one-step
methods.
Regularity index and packing factor
The effects of the different cell fate mechanisms upon the regularity index
of the mosaics are summarised in Fig.
4. All four one-step mechanisms and the all-neighbours inhibition
produced similar effects upon the regularity index. When the packing intensity
of the undifferentiated mosaic was small, the regularity of the primary fate
mosaic was higher than the regularity of both the undifferentiated and
secondary fate mosaics. Hence, cell fate mechanisms can produce a regular
primary fate, but not secondary fate, mosaic from an irregular array of
undifferentiated cells. However, once the packing intensity of the
undifferentiated mosaic exceeded a critical value (=0.3-0.4), the
regularity of both primary and secondary fate mosaics was always less than the
regularity of the undifferentiated mosaic. With one exception (
=0.5 in
the largest method), the primary fate mosaic was always more regular than the
secondary fate mosaic. Therefore, if the initial population of
undifferentiated cells is already regular, cell fate produces less regular
primary and secondary fate mosaics.
|
The behaviour of nearest-neighbour inhibition was slightly different
(Fig. 4B). First, regardless of
packing intensity, the regularity of primary and secondary fate mosaics was
always similar. For low packing intensities (=0.0-0.1), both primary and
secondary fate mosaics were more regular than the undifferentiated mosaic.
Second, the critical value of packing intensity at which undifferentiated
mosaics were more regular than both primary and secondary fate mosaics was
lower (typically when
=0.2) than the threshold for the other cell fate
mechanisms.
The effect of packing intensity upon the packing factor
(Rodieck, 1991) of the
undifferentiated, primary fate and secondary fate mosaics was similar to the
effect upon the regularity index (Fig.
5). At low packing intensities, the packing factor of the primary
fate mosaic was higher than both the undifferentiated and secondary fate
mosaic. Once the packing intensity exceeded some critical value, the
undifferentiated mosaic had a higher packing factor than either the primary or
secondary fate mosaic. Again, this critical value was lower for
nearest-neighbour inhibition (
=0.2-0.3) than the other methods
(
=0.4-0.5).
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On- and off-centre mosaics
Some populations of retinal cells can naturally be divided into two
complementary types of roughly the same number of cells. The best known
examples are probably the classification of and ß RGCs into on-
and off-centre types (Wässle et al.,
1981a
; Wässle et al.,
1981b
). Could such mosaics be generated from an initial
bistratified population simply by cell fate? Out of the cell fate mechanisms
tested here, only nearest-neighbour inhibition produced cell count ratios
around 1:1, matching those observed experimentally
(Table 1). However, by
comparing the regularity index of the experimental mosaics
(Table 1) with the results from
nearest-neighbour inhibition, we have shown it to be unlikely that lateral
inhibition alone is sufficient to generate the spatial distributions observed
in adult. By contrast, the regularity indexes measured from the cholinergic
amacrine cells (Diggle, 1986
)
are closer to those produced by nearest-neighbour inhibition with a packing
intensity of 0.1. It is therefore unlikely that cell fate mechanisms alone are
responsible for generating the adult distributions of on- and off-centre RGCs
from a random undifferentiated population.
|
Cell death and mosaic formation
The results from the previous section indicate that lateral inhibition of
cell fate alone cannot generate the pattern of on- and off-centre RGC mosaics
from an irregular population of bistratified RGCs. Recent experimental
findings suggest that the pattern of on- and off-centre -RGCs in the
cat could be sculpted by postnatal cell death, and that the pattern, but not
the magnitude, of this cell death is activity dependent
(Jeyarasasingam et al., 1998
).
From these results, the authors suggested that 20% cell death is sufficient to
transform irregular mosaics into the regular adult pattern. To support this
claim, cells were deleted from computer-generated random mosaics and shown to
match the adult pattern. However, the deleted cells were chosen by eye, rather
than by following any rules (Leo Chalupa, personal communication). Hence, to
test this hypothesis, we needed to test simple rules for choosing which cells
to die from a random population.
Fig. 6 shows the results of using three methods of cell death upon mosaic regularity. Unsurprisingly, the distance method produces the mosaics with the highest regularity at each step we are simply removing the smallest value from the distribution of nearest-neighbour distances. Deleting cells according to their Voronoi area also improves the regularity index, but to a lesser degree than deleting by distance (P=0.007 after 20% cells deleted; P<0.001 after 40% cells deleted; both tests two-sample Wilcoxon test). However, when regularity is assessed using the packing factor, deleting cells by the area method produces more regular mosaics than the distance method (P=0.035 after 20% cells deleted; P<0.001 after 40% cells deleted; both tests two-sample Wilcoxon test). By contrast, deleting cells simply at random does not improve mosaic regularity, as assessed by both the regularity index and packing factor.
Fig. 6 also demonstrates
that the regularity index and the packing factor measure different aspects of
mosaic regularity, as the two measures are not perfectly correlated. If the
two measures were highly correlated, we would expect that the cell death
method that maximises the packing factor also maximises the regularity index.
However, Fig. 6 shows that the
regularity index is maximised by deleting cells according to nearest-neighbour
distance, whereas the packing factor is maximised by deleting cells according
to their Voronoi area. This result is unsurprising, as the regularity index
measures the distance of each cell to only its nearest-neighbour, whereas the
packing factor measures the distance to many neighbouring cells, as it is
based on the autocorrelation of cell positions. However, the data in
Fig. 6 are based upon deletions
from simulated mosaics. The degree of correlation between regularity index and
packing factor may be better tested upon real mosaics, similar to the way the
correlation between regularity index and dispersion index was investigated
[see Fig. 6B by Cook
(Cook, 1996)].
To test whether cell death is sufficient to create the spatial distribution
of on- and off-centre ganglion cells, three related hypotheses were examined
concerning how the on- and off- centre mosaics might arise, before any cells
then die. The first hypothesis (independence) was suggested by Jeyarasasingam
et al. (Jeyarasasingam et al.,
1998): on- and off-centre cells are generated independently in a
random distribution. The second hypothesis (random division), suggested by
Cook and Chalupa (Cook and Chalupa,
2000
), is that there is an initial population of bistratified RGCs
which presumably form an irregular mosaic. Each RGC then randomly decides to
become either on- or off-centre. The third hypothesis (inhibition), also
suggested by Cook and Chalupa (Cook and
Chalupa, 2000
), is like the random division hypothesis, except
that neighbouring RGCs tend to become opposite types, rather than being
assigned on-or off-centre at random. To implement this hypothesis, we used
nearest-neighbour inhibition, and arbitrarily assume that primary fate cells
become on-centre, and secondary fate cells become off-centre. Hence, the
independence hypothesis suggests that on- and off-centre RGCs are generated
independently of each other, whereas the random division and inhibition
hypotheses state that the on- and off-centre cells are produced by subdivision
from some initial population of bistratified RGCs. For each hypothesis, once
the on- and off-centre populations were created, up to 40% of cells from each
population were then killed using the distance cell death method.
The consequences of these hypotheses are shown in Fig. 7. Under the independence hypothesis, 20% cell death linearly increases the regularity of both on- and off-centre RGC mosaics from around 2.5 to 3.6 (Fig. 7A). To produce regularity indexes matching those found experimentally (Table 1), typically up to 40% cell death is required. The final regularity index of each mosaic is of course dependent on the regularity of the initial populations of on- and off-centre cells. If the initial populations of on- and off-centre cells is mildly regular (Fig. 7B), less cell death, typically 30%, is required to match the experimental regularity indexes. However, this somewhat defeats the purpose: the aim was to transform an irregular distribution to a regular distribution, rather than to start with a regular distribution. The random division hypothesis produced similar results (Fig. 7C). Given an initial undifferentiated population of low regularity, randomly assigning each cell as on- or off-centre produces two irregular mosaics. Cell death does improve the regularity of each mosaic, but in a similar pattern to that seen in Fig. 7A.
|
By contrast, under the inhibition hypothesis (Fig. 7D), even if the initial population of bistratified RGCs is irregular, lateral inhibition produces mildly regular mosaics, similar in regularity to those created in Fig. 7B (compare data points in B,D before any cell death). Furthermore, the rate at which regularity increases per cells deleted is highest under the inhibition hypothesis, such that 20-30% cell death produces mosaics with regularity indices matching those found experimentally (Table 1).
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DISCUSSION |
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The results may be somewhat sensitive to the exact nature of the cell fate
mechanism. However, we achieved similar results for several one-step
mechanisms (Honda et al.,
1990) and an iterative mechanism
(Collier et al., 1996
),
indicating the results are somewhat robust. The main difference between the
cell fate methods tested here is that the one-step methods tend to have a
higher ratio of secondary to primary fate cells. One-step methods may also be
more prone to `fractures', whereby two neighbouring cells acquire primary fate
if more than one initiation site is used
(Goodyear et al., 1995
).
In principle then, cell fate mechanisms, perhaps mediated by Delta-Notch
signalling, could produce a regular array of early differentiating cells from
an undifferentiated population (Frankfort
and Mardon, 2002). However, without knowledge of the regularity of
the initial population of undifferentiated cells, detailed comparisons with
the early regularity observed in chick RGCs
(McCabe et al., 1999
) are not
possible. However, given the high regularity index of these early
differentiating RGCs (around 5.0), our results would suggest that either the
initial population must be mildly regular (around 3.0 to produce a primary
fate population of 5.0, see Fig.
4A when
=0.1) or that the cell fate mechanism involves
interactions among cells that are not all immediate neighbours. Furthermore,
when comparing the regularity of mosaics produced by nearest-neighbour
inhibition with on- and off-centre RGCs, we conclude that cell fate alone is
unable to account for their regular arrangement. Cell death alone
(Jeyarasasingam et al., 1998
)
also cannot replicate the on- and off-centre RGC patterns, unless cell death
is much higher than the 20% reported in postnatal cat. However, the cell fate
and cell death mechanisms can be used together to transform one irregular
mosaic into two regular mosaics, with regularity indexes similar to those
observed experimentally.
The development of the on- and off-centre ganglion cells remains a mystery.
Our work here suggests that competition between neighbouring RGCs
(Kirby and Steineke, 1996) is
insufficient by itself to replicate the mature pattern of RGC mosaics.
However, we have shown here that another mechanism, cell death, could follow
after lateral inhibition to produce the adult regularity. This suggests that
environmental factors are sufficient to determine the fate (on- or off-centre
in this case) of RGCs. However, the existence of early recognition between
same-centre RGCs raises the possibility that these cells are molecularly
distinct from an early age, and so may never be part of the same
undifferentiated population (Lohmann and
Wong, 2001
). In addition, in most species [except rat (see
Peichl, 1991
)], there are
slightly more off-centre than on-centre
-cells
(Table 1). Such a bias towards
more off-centre cells is not explainable in this model, as cell count ratios
were typically around 1:1 for nearest-neighbour lateral inhibition.
The models presented in this paper allow us to assess the relative
influence of cell fate and cell death upon mosaic formation. However, these
mechanisms are somewhat simplistic still, especially concerning cell death.
Theoretical models of developmental cell death are still relatively simple
(Clarke, 2002) and are an area
for future research. We have used simple mechanisms that probably overestimate
the influence that cell death can have upon mosaic formation. However,
particularly if they are overestimates, the results still show that 20% cell
death alone (Fig. 7B,C) is
insufficient to generate regular mosaics. Future cell death mechanisms could
be based upon competition for neurotrophic resources from presynaptic cells,
such that cells receiving insufficient resources are more likely to die. In
addition, cell death is unlikely to be a universal method for improving
regularity. In postnatal rat, cell death among cholinergic amacrine cells is
not thought to affect mosaic regularity
(Galli-Resta and Novelli,
2000
). Note also that most retinal cell death occurs earlier in
development, before the period of mosaic formation, and is suggested instead
to relate to the formation of connections in the target area of the RGC axons
(O'Leary et al., 1986
).
We have limited ourselves to communication between nearest-neighbouring
cells. Lateral inhibition could also involve interactions between
non-neighbouring cells, perhaps mediated by the filopodia or cytonemes
observed in some developing systems
(Bryant, 1999). In this paper,
we have concentrated on modelling formulations proposed for Delta-Notch
signalling, which seems appropriate given experimental evidence
(Austin et al., 1995
). Future
work could incorporate other types of cell communication to see their effect
on mosaic formation.
Another limitation is that when comparing the simulated mosaics with
experimental data, we have used only the simplest of measures, the regularity
index. More sensitive measures, such as those based on distributions of
Voronoi polygon area, have recently been used when comparing model and
simulated mosaics (Galli-Resta et al.,
1999; Zhan and Troy,
2000
). However, in those cases the models were phenomenological,
rather than mechanistic, as the models were not testing specific biological
mechanisms. It is unlikely that the models presented here exactly replicate
the spatial distributions observed experimentally (such as those referenced in
Table 1) for two reasons.
First, we have looked at the role of just two mechanisms, cell fate and cell
death, and ignored others, such as lateral migration and non-uniform retinal
growth, which may also be involved to produce the overall distribution.
Second, the ideal experimental comparison would be against tissue early in
development, rather than adult tissue. Unfortunately, the lack of distinct
molecular markers for on- and off-centre ganglion cells makes it difficult to
collect such data, either early in development or at maturity.
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ACKNOWLEDGMENTS |
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