1 Department of Neuroscience, Georgetown University Medical Center, 3900
Reservoir Road NW, Washington, DC 20007, USA
2 Department of Physics, Georgetown University, 37th and O Streets NW,
Washington, DC 20057, USA
3 Queensland Brain Institute, Department of Mathematics and Institute for
Molecular Bioscience, University of Queensland, St Lucia, QLD 4072,
Australia
* Author for correspondence (e-mail: g.goodhill{at}uq.edu.au)
Accepted 5 August 2005
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SUMMARY |
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Key words: Axon guidance, Chemotaxis, Computational model, Nerve growth factor, Dorsal root ganglion
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Introduction |
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A key aspect of the chemotropic response of many types of cells is
adaptation, which has been defined as the continual adjustment of the baseline
against which further increases in concentration are compared
(Bray, 2001). This has been
particularly well studied in the context of bacteria (e.g.
Macnab and Koshland, 1972
;
Koshland et al., 1982
;
Barkai and Leibler, 1997
). It
is thus a reasonable hypothesis that growth cones might also implement
adaptation to gradients, which would mean that they display a similar
sensitivity to small concentration differences of an external ligand over a
wide range of background concentrations of the ligand. This would allow growth
cones to be guided over a greater distance by a single gradient
(Goodhill, 1998
;
Goodhill and Urbach, 1999
)
than growth cones that do not adapt. Perhaps the most explicit proposal of
this hypothesis has been made by Ming et al.
(Ming et al., 2002
). Using
gradients established by a pipette, they tested the change in responsiveness
of Xenopus spinal growth cones to a steep gradient when a sudden
change in background concentration was introduced. In each case, the
concentration of ligand at the growth cone was about 0.1 nM, the change in
concentration across 10 microns was about 10%
(Zheng et al., 1994
), and the
sudden changes in background concentration were about 0.1 nM (for ease of
comparison we have converted the ng/ml units originally quoted to molar
units). There was thus at most an approximate twofold increase in the absolute
concentration of ligand at the growth cone. Under these conditions a rapid
initial desensitization was observed whereby the growth cone could no longer
respond to the 10% gradient, which was followed by a more prolonged period of
resensitization (Ming et al.,
2002
). The timecourse of desensitization and resensitization has
recently been investigated in more detail by Piper et al.
(Piper et al., 2005
).
However, the experiments of Ming et al.
(Ming et al., 2002) examined
only the relatively short-term temporal dynamics of the response of growth
cones to small step changes in concentration. They did not explicitly test the
hypothesis that growth cones display a similar sensitivity to small
concentration differences across a wide range of background concentrations. A
recent experiment has addressed this more directly by examining the long-term
response of axons in a three-dimensional collagen gel environment to gradients
of precisely controlled steepness and shape
(Rosoff et al., 2004
).
Explants of rat dorsal root ganglia (DRGs) were grown for 2 days in gradients
of nerve growth factor (NGF) of a steepness of 0.2% per 10 microns at
background concentrations varying from 0.0001 nM to 100 nM, and the degree of
asymmetry in the outgrowth was quantified. A simple interpretation of the
adaptation hypothesis would predict that the guidance response should be
roughly constant over this range, as the stimulus (fractional change in
concentration across the growth cone) is the same in each case. However, this
is not what was observed: instead the guidance response peaked in the range
1-10 nM, and declined at both higher and lower concentrations, falling to zero
at the two ends of the concentration scale.
We decided to rigorously test whether this result is indeed consistent with
a non-adapting gradient detection mechanism by constructing a computational
model of growth-cone sensing and movement. Using the data of Rosoff et al.
(Rosoff et al., 2004) to
constrain the parameters of the model, we show that it is possible to produce
a close quantitative match between model and data without invoking explicit
adaptational mechanisms. In addition, we find that both spatial and temporal
averaging of the stochastic receptor-binding signal are required to produce
the exquisite level of sensitivity observed experimentally. Spatial and
temporal averaging can be interpreted as corresponding, respectively, to the
spatial spread of signaling effects downstream from receptor binding, and to
the finite time over which these signaling effects decay. For spatial
averaging the model predicts than an effective range of roughly one-third of
the extent of the growth cone is optimal, whereas for temporal averaging a
timescale of about 3 minutes is required for the model to reproduce the
experimentally observed sensitivity.
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Materials and methods |
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Model description
At each time step in the model, the probability pi for
receptor i to be bound is given by
pi=Ci/(Ci+KD),
where Ci is the external ligand concentration at the
position of receptor i and KD is the dissociation
constant for the receptor-ligand complex. This gives a highly noisy
measurement of differences in concentration around the growth cone that arise
from the presence of an external gradient. The experimental data of Rosoff et
al. (Rosoff et al., 2004)
suggests that growth cones must be averaging concentration measurements to
achieve their exquisite sensitivity to gradients, as the noise present in an
instantaneous measurement of the local concentration is much larger than the
gradient signal itself. In the model, we include both spatial and temporal
averaging before each movement event is initiated (see below). In temporal
averaging, the recent history of the binding state of the receptor is pooled
according to a decaying function of time, which we assume to be a
half-gaussian of variance
t. For small
t, only the immediate binding state of each
receptor is considered; for large
t, a long history
of binding is averaged.
t trades off noise in
concentration measurements against temporal locality: a large
t increases the accuracy of the concentration
measurement but decreases its temporal resolution so that it cannot respond to
rapidly changing signals, whether due to changes in the concentration field
itself or to movement of the growth cone within a fixed concentration field.
In spatial averaging, the binding of a spatially distributed set of receptors
is pooled to determine a binding density at each point. This is achieved in
the model by convolving the receptor-binding density (from either a single or
temporally averaged measurement) with a gaussian function of variance
s. For small
s, just a
few neighboring receptors are averaged; for large
s, a significant proportion of all of the receptors
on the growth cone are included in the average.
s
trades off noise in concentration measurements against spatial locality: a
large
s increases the accuracy of the concentration
measurement, but decreases the spatial precision in a concentration
measurement, which is key to measuring a gradient. For very large
s, all spatial locality is lost, and the binding at
each point equals the average concentration across the entire growth cone.
Once a receptor-binding density as a function of angle around the growth
cone has been calculated, the growth cone picks the direction in which this is
maximum. We do not model how this occurs, but there are several possibilities
for performing this type of amplification (e.g.
Parent and Devreotes, 1999;
Meinhardt, 1999
;
Iglesias and Levchenko, 2002
)
(see also Discussion). Because axons tend to grow in straight lines
(Bray, 1979
;
Katz, 1985
), we allowed the
axon to change direction only slightly in response to this gradient signal:
the new angle of growth is (1-
) times the old direction plus
times the new direction, where
is close to zero. The growth cone then
takes a small, constant, step forward. We refer to
as `momentum'. Note
that there is no explicit adaptation to external ligand levels in this model,
as in for instance Barkai and Leibler
(Barkai and Liebler, 1997
).
Equations of the model
The concentration of ligand ) at
position
in the gradient, where î and
are unit vectors
perpendicular and parallel, respectively, to the direction of the gradient, is
given by:
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Comparison with experimental data
Examples of the explant data generated by Rosoff et al.
(Rosoff et al., 2004) are
shown in Fig. 1A,B. Rosoff et
al. (Rosoff et al., 2004
)
quantified the response of axons to gradients by the `guidance ratio': the
number of pixels representing neurites on the up-gradient side of the explant
U was compared with that on the down-gradient side D, with
the guidance ratio defined as (U-D)/(U+D).
This was calculated for digital images of the explants at 640 by 480
resolution.
In order to match our computational simulations to these experimental
results, we generated digital images of simulated explants at the same
resolution. Simulation parameters were chosen to match the mean experimental
values for average neurite number and length, explant radius (all simulated
explants were circles), and growth cone diameter. The starting point for each
axonal trajectory was a random position within the explant, and the initial
direction was random. To represent the likelihood that not all neurites in the
experimental condition are actually responsive to NGF, i.e. that some neurites
do not express TrkA (Gallo et al.,
1997), we assumed that only a proportion of simulated neurites per
explant were actually responsive to the gradient. Non-responsive growth cones
in the simulations grew simply according to random fluctuations in receptor
binding with no gradient. Simulated explants were grown in identical gradients
to those used experimentally, i.e. exponential with a fractional change 0.1,
0.2 or 0.4% over 10 microns, at an absolute concentration ranging from 0.0001
nM to 100 nM. Images of neurite trajectories were generated using the Matlab
`plot' command with the `linewidth' parameter chosen to match the width of
neurites observed experimentally. The guidance ratio was then calculated as
for the experimental explants. We also investigated a more controlled method
of generating pixels to represent the trajectories, based on an optics-based
model for how images were generated experimentally by Rosoff et al.
(Rosoff et al., 2004
).
However, this produced guidance ratios that were not significantly different
from those produced by the `plot' method.
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An appropriate value for the momentum is constrained by the maximum
rate at which growth cones can turn. In the model, this is set by the timestep
t, as well as by
. A strong gradient
(sufficient to dominate binding noise) perpendicular to the direction of
travel of the axon will produce a signal at
/2. In a single timestep, the
change in the angle of the axon will be
/2. The growth cone will
turn 45 degrees towards the direction of the gradient after approximately
(
/4)/
/2=1/(2
) steps. Thus, the fastest observable
turning time, Tmin, is equal to
t/(2
) seconds. Changes in the timestep must
therefore be accompanied by proportional changes in
to keep a constant
response time. Experimentally Tmin is in the order of 10
minutes, so
=30/1200
0.03 for
t=30
seconds.
The momentum is also constrained by the degree to which axonal trajectories
meander. In the absence of a gradient, the growth cone executes a random walk
in angle space. Each timestep, the orientation of the growth cone changes by
some amount uniformly distributed in
[-
/2,
/2]. Consider first the case with no temporal
averaging. By the central limit theorem, after N statistically
independent steps a collection of axons will display a distribution of changes
in orientation with a spread given by
.
Experimentally, even after 36 hours (approximately 2000 minutes), most of the
axons have not changed direction by more than
/2. Thus,
,
or
.
To keep the trajectory meandering constant,
must change proportional
to the square root of the timestep. When temporal averaging is used, the
chosen orientation in successive timesteps is not statistically independent.
The longer the temporal averaging time, the more likely it is that the
position of maximum binding stays in approximately the same position. Roughly,
the number of statistically independent steps is reduced from N to
N/
t, and the average step size is
increased from
/4 to
t
/4.
From the arguments above,
.
Thus, the meandering increases with the square root of the temporal averaging
time.
For spatial averaging, we found there was a specific width
s that maximized the sensitivity of the growth cone
(see Results), and it was this optimal value that we used in subsequent
simulations. The duration of temporal averaging,
t,
was determined as described in the Results. Simulations were coded in Matlab,
and each explant took approximately 10 hours to run on a 3GHz Pentium 4 PC
running Linux. The simulations shown here thus represent a total of
approximately 2 years of CPU time.
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Results |
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Before matching the model to biological data it is important to understand
the individual effect of some of the key parameters in the model.
Fig. 2 shows how growth cone
sensitivity, as measured by the guidance ratio (see Materials and methods),
varies with the proportion of non-responsive neurites
(Fig. 2A), the width of spatial
averaging (Fig. 2B), and the
duration of temporal averaging (Fig.
2C). In Fig. 2B,
the width of spatial averaging, s, is expressed as
a percentage of the circumference of the semicircular growth cone. As
s increases the average is taken over more
receptors and thus the noise in the local concentration measurements is
reduced. Conversely, increasing
s implies less
specificity in exactly where the concentration is measured, making the
detection of a gradient more difficult. It can be seen that these two effects
trade off to produce a peak in sensitivity at about
s=5%. Because the total effective width of
averaging is roughly 6
s
(3
s either side of the central point), this
corresponds to a total width of about one-third of the distance around the
growth cone. As was expected, in Fig.
2C, the longer the temporal averaging (the higher
t) the greater the sensitivity. We found that a
value of
t=2 was required to match the sensitivity
of real axons (see below). This corresponds to looking back in time a total of
about 3
t=6 timesteps or 3 minutes. Beyond this the
axon trajectories become more convoluted than those seen experimentally (data
not shown), because a chance bias in binding on one side of the growth cone
persists for many minutes.
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Secondly, the model can shed light on the relative importance of trophic
versus tropic effects on explant asymmetry in NGF gradients. Could it be that
the higher concentration of NGF on the up-gradient side of the explant simply
promotes more growth on that side, causing a positive guidance ratio without
actual guidance? Although this was discussed in Rosoff et al.
(Rosoff et al., 2004), the
model allows this question to be addressed more quantitatively. Explants were
simulated with neurite growth cones that responded to the average
concentration, as determined from the receptor binding, but were insensitive
to variations in binding across their spatial extent. Neurite growth rate was
now assumed to vary linearly with absolute concentration. We used the largest
rate of change of outgrowth with concentration measured experimentally by
Rosoff et al. (Rosoff et al.,
2004
), which occurred in the range 1 nM-3 nM. To maximize the
possibility of seeing an effect, we simulated the steepest gradients
investigated experimentally (0.4% per 10 µm), allowed 100% of simulated
neurites to be responsive to the gradient, and took the absolute concentration
to be KD for maximum sensitivity. We found a guidance
ratio of -0.00097±0.0133, showing that no significant guidance effect
was induced by differential growth. The general point is that total outgrowth
does not vary with concentration fast enough in the data of Rosoff et al.
(Rosoff et al., 2004
) to
produce the observed asymmetry in the explants, given that the total change in
concentration across an explant in these experiments is small.
In general, the response depends on both gradient steepness and absolute
concentration. The experimental data of Rosoff et al.
(Rosoff et al., 2004) presents
only two one-dimensional cuts through a two-dimensional surface of growth cone
response, one along the steepness axis and one along the concentration axis.
Having tuned the parameters of our model to match the data along these cuts as
above, we then simulated the shape of the entire two-dimensional surface
(Fig. 4). The most notable
prediction is that the peak of response becomes broader as the gradient
steepness is increased. When quantitative measurements of the actual
concentration profiles of putative guidance molecules in vivo are available,
the response surface represented in Fig.
4 can act as a guide to the feasibility of these gradients for
actually guiding axons during development. This surface can also be used to
predict the gradients required to promote the guided regrowth of axons in
regeneration experiments.
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Discussion |
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It is clear from results using the pipette assay (e.g.
Ming et al., 2002) that
desensitization and resensitization of growth cones does occur. Although the
results we have presented here and in Rosoff et al.
(Rosoff et al., 2004
) do not
directly address these desensitization/resensitization processes, they provide
evidence that such processes may not be implementing adaptation in the sense
of a continual adjustment of the baseline against which further increases in
concentration are compared (Bray,
2001
). Desensitization/resensitization probably occur on a more
rapid timescale than that over which growth cones experience significant
concentration changes due to gradients in vivo
(Piper et al., 2005
), and
growth cones probably do not often experience sudden changes in concentration
over their entire surface in vivo. Another study sometimes cited in support of
growth cone adaptation is Rosentreter et al.
(Rosentreter et al., 1998
).
This examined the response of retinal ganglion cells to gradients of tectal
cell membrane density, and the authors argue that temporal retinal axons grew
up a fixed increment of concentration, independent of both gradient steepness
and starting concentration. The latter result suggests that retinal growth
cones may `adapt' in the sense of terminating their growth once they encounter
a certain increase in ligand concentration relative to the concentration at
which they first encountered the ligand. However, this is adaptation in a
different sense to that discussed in this paper. Rosentreter et al.
(Rosentreter et al., 1998
) did
not address changes in the ability of growth cones to detect gradients as a
function of concentration, as we have done here.
We find that response to a gradient is maximized when receptor binding
statistics are pooled spatially over a distance of about one-third of the
extent of the growth cone, and that temporal averaging of binding statistics
on a timescale of the order of 3 minutes is required to match the sensitivity
observed experimentally. The notion of there being an optimal
s is one that should apply in any system performing
chemotaxis by comparing concentrations across its spatial extent. However, as
far as we are aware this is the first time attention has been brought to this
phenomenon. A rough calculation gives insight into the optimal value for
s as follows:
Consider a one-dimensional sensing device that is a line split up into
n equal compartments, each containing about the same number of
receptors. Imagine this device is attempting to sense a gradient with
steepness m per compartment, i.e. a total concentration change across
the device of nm. Assume receptor binding is pooled over k
compartments in from each end of the sensing device. The total concentration
change between the midpoints of these two pools is now only
(n-k)m. Conversely, the error in a concentration measurement
within each of these pools has now been reduced by a factor of
1/
(Berg and Purcell, 1977
). What
is the optimal number of compartments over which to average to minimize the
steepness of the gradient that can be sensed? Following Berg and Purcell
(Berg and Purcell, 1977
),
threshold detection of the gradient occurs when:
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The intuitive idea is that pooling binding statistics over a spatial region increases the signal-to-noise ratio for the concentration measurement, but decreases the spatial specificity of the measurement. These two competing effects trade off to give the optimal pooling range of one-third of the extent of the sensing device. Although this argument is rough, it illustrates analytically the plausibility of an optimal spatial scale for the pooling of receptor statistics.
A number of theoretical models have attempted to address the principles
involved in growth cone movement (reviewed by
Goodhill and Urbach, 2003).
For the case of no external ligand gradient, careful quantitative analyses
have been performed of axonal trajectories (e.g.
Katz et al., 1984
;
Katz, 1985
) and cytoskeletal
dynamics including filopodia (Buettner,
1995
; Odde and Buettner,
1998
), and models have been proposed to capture these general
properties (Katz and Lasek,
1985
; Li et al.,
1994
; Hely and Willshaw,
1998
). Although models of the behavior of axons in the presence of
external gradients exist (Gierer,
1987
; Robert and Sweeney,
1997
; Hentschel and Van Ooyen,
1999
; Goodhill and Urbach,
1999
; Goodhill et al.,
2004
), it has been hard, until now, to compare these
quantitatively with experimental data, because suitably controlled
measurements have not been available. In particular, the two principal methods
used until recently for establishing diffusible gradients in vitro (reviewed
by Guan and Rao, 2003
) are
limited in this regard. In the pipette assay (e.g.
Zheng et al., 1994
;
Nishiyama et al., 2003
) a
defined gradient exists, but the response measured is only whether the growth
cone turns over a short period of time, rather than a complete trajectory. In
the 3D collagen-gel assay (e.g. Lumsden
and Davies, 1983
; Charron et
al., 2003
), the gradient present is not known, and is probably not
stable over time (Goodhill,
1997
; Goodhill,
1998
). The assay introduced by Rosoff et al.
(Rosoff et al., 2004
) has
alleviated this problem, and allows the precise quantitative comparisons with
theoretical results of the present paper. The model presented here is related
to the model of axonal gradient sensing proposed by Goodhill et al.
(Goodhill et al., 2004
). In
that model, growth cone filopodia were explicitly represented, and guidance
was achieved by the preferential generation of filopodia on the up-gradient
side (determined as here by noisy receptor-binding measurements) of the growth
cone. The growth cone then turned towards the average direction of the
filopodia. A comparison of the predictions of that model with the data of
Rosoff et al. (Rosoff et al.,
2004
) reveals an insufficient sensitivity to small gradients (data
not shown). In addition to a lack of spatial and temporal averaging, the
unavoidably probabilistic nature of filopodia generation in the model
contributes a source of noise that acts as a fundamental limitation on
sensitivity. The present model is somewhat more abstract, as it does not
represent filopodia explicitly.
How might spatial and temporal averaging be implemented biologically? A
simple interpretation of spatial pooling of receptor binding statistics is
that each receptor-binding event initiates a small release of a downstream
signaling molecule, which then spreads out by diffusion so that its effects
are combined with release from nearby receptor binding. However, in our model
this would require the signaling molecules to diffuse a distance of
3s
0.15x
x5 µm
2 µm in
about 3
t
3 minutes, which gives a diffusion
constant of the order 10-10 cm2/second. This seems too
small for a freely diffusing molecule in the cytosol, but is plausible for
diffusion of molecules in the membrane
(Goodhill, 1998
;
Goodhill and Urbach, 1999
). An
alternative explanation is therefore that the spatial spread of the effects of
receptor binding are mediated through a chain of intermediate signaling
components, some of which may be bound to the membrane or cytoskeletal
components so that they move and react with each other relatively slowly. This
fits with the picture emerging from experimental data on signal transduction
inside growth cones (reviewed by Song and
Poo, 2001
; Guan and Rao,
2003
). A simple interpretation of the duration of temporal
averaging is that it corresponds with the decay time of one component of the
transduction pathway. However, as for spatial averaging, a more likely
possibility is that it is the net effect of several different components that
provide inertia to the system.
The present computational results show that adaptation mechanisms are not
required to reproduce the most quantitative data currently available on
long-term axon guidance by gradients. If growth cones do not adapt, why might
they be different from, for instance, bacteria in this regard? One possibility
is that they simply do not require adaptation. Current data suggest that
single gradients only guide axons for a maximum of about 1 mm in the case of
diffusible factors, or about 1 cm in case of substrate bound factors (such as
ephrins in the tectum). These distances are consistent with a response over
only about 2 to 3 orders of magnitude of absolute concentration
(Goodhill and Baier, 1998;
Goodhill, 1998
;
Goodhill and Urbach, 1999
),
which as we have shown here can be achieved without adaptation. Axonal
trajectories tend to be broken up into numerous short segments involving
intermediate targets (Tessier-Lavigne and
Goodman, 1996
). In the highly complex environment of the
developing nervous system this may be a more robust strategy than relying on
single gradients extended over long distances, and might also provide more
combinatorial possibilities for sorting subpopulations of axons
(Goodhill, 2003
). Although
adaptive mechanisms are common in biology, they may have only evolved in cells
undergoing chemotaxis when there was a pressing need for guidance over long
distances by single gradients.
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ACKNOWLEDGMENTS |
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