A mathematical model of axon guidance by 
diffusible factors 
Geoffrey J. Goodhill 
Georgetown Institute for Cognitive and Computational Sciences 
Georgetown University Medical Center 
3970 Reservoir Road 
Washington DC 20007 
eof fOiccs. eoretown. edu 
Abstract 
In the developing nervous system, gradients of target-derived dif- 
fusible factors play an important role in guiding axons to appro- 
priate targets. In this paper, the shape that such a gradient might 
have is calculated as a function of distance from the target and the 
time since the start of factor production. Using estimates of the 
relevant parameter values from the experimental literature, the 
spatiotemporal domain in which a growth cone could detect such 
a gradient is derived. For large times, a value for the maximum 
guidance range of about 1 mm is obtained. This value fits well 
with experimental data. For smaller times, the analysis predicts 
that guidance over longer ranges may be possible. This prediction 
remains to be tested. 
1 Introduction 
In the developing nervous system, growing axons are guided to targets that may be 
some distance away. Several mechanisms contribute to this (reviewed in Tessier- 
Lavigne & Goodman (1996)). One such mechanism is the diffusion of a factor 
from the target through the extracellular space, creating a gradient of increasing 
concentration that axons can sense and follow. In the central nervous system, such 
a process seems to occur in at least three cases: the guidance of axons from the 
trigeminal ganglion to the maxillary process in the mouse (Lumsden & Davies, 
1983, 1986), of commissural axons in the spinal cord to the floor plate (Tessier- 
Lavigne et al., 1988), and of axons and axonal branches from the corticospinal tract 
to the basilar pons (Heffner et al., 1990). The evidence for this comes from both in 
vivo and in vitro experiments. For the latter, a piece of target tissue is embedded in 
a three dimensional collagen gel near to a piece of tissue containing the appropriate 
160 G. J. Goodhill 
population of neurons. Axon growth is then observed directed towards the target, 
implicating a target-derived diffusible signal. In vivo, for the systems described, 
the target is always less than 500/m from the population of axons. In vitro, where 
the distance between axons and target can readily be varied, guidance is generally 
not seen for distances greater than 500 - 1000/m. Can such a limit be explained 
in terms of the mathematics of diffusion? 
There are two related constraints that the distribution of a diffusible factor must 
satisfy to provide an effective guidance cue at a point. Firstly, the absolute concen- 
tration of factor must not be too small or too large. Secondly, the fractional change 
in concentration of factor across the width of the gradient-sensing apparatus, gen- 
erally assumed to be the growth cone, must be sufficiently large. These constraints 
are related because in both cases the problem is to overcome statistical noise. At 
very low concentrations, noise exists due to thermal fluctuations in the number of 
molecules of factor in the vicinity of the growth cone (analyzed in Berg & Purcell 
(1977)). At higher concentrations, the limiting source of noise is stochastic varia- 
tion in the amount of binding of the factor to receptors distributed over the growth 
cone. At very high concentrations, all receptors will be saturated and no gradient 
will be apparent. The closer the concentration is to the upper or lower limits, the 
higher the gradient that is needed to ensure detection (Devreotes & Zigmond, 1988; 
Tessier-Lavigne & Placzek, 1991). The limitations these constraints impose on the 
guidance range of a diffusible factor are now investigated. For further discussion 
see Goodhill (1997; 1998). 
2 Mathematical model 
Consider a source releasing factor with diffusion constant D cm2/sec, at rate q 
moles/sec, into an infinite, spatially uniform three-dimensional volume. Initially, 
zero decay of the factor is assumed. For radially symmetric Fickian diffusion in 
three dimensions, the concentration C(r, t) at distance r from the source at time t 
is given by 
C(r,t) = 4 rerfc (1) 
(see e.g. Crank (1975)), where erfc is the complementary error function. The per- 
centage change in concentration p across a small distance Ar (the width of the 
growth cone) is given by 
Ar [1 + r e -r/4ot 
P = ---  erfc(r/) (2) 
This function has two perhaps surprising characteristics. Firsfly, for fixed r, 
decreases with t. That is, the largest gradient at any distance occurs immediately 
after the source starts releasing factor. For large t, Ipl asymptotes at Ar/r. Secondly, 
for fixed t < oc, numerical results show that p is nonmonotonic with r. In particular 
it decreases with distance, reaches a minimum, then increases again. The position 
of this minimum moves to larger distances as t increases. 
The general characteristics of the above constraints can be summarized as follows. 
(1) At small times after the start of production the factor is very unevenly dis- 
tributed. The concentration C falls quickly to almost zero moving away from the 
source, the gradient is steep, and the percentage change across the growth cone 
p is everywhere large. (2) As time proceeds the factor becomes more evenly dis- 
tributed. C everywhere increases, but p everywhere decreases. (3) For large times, 
C tends to an inverse variation with the distance from the source r, while Ipl tends 
A Mathematical Model of Axon Guidance by Diffusible Factors 161 
to Ar/r independent of all other parameters. This means that, for large times, 
the maximum distance over which guidance by diffusible factors is possible scales 
linearly with growth cone diameter Ar. 
3 Parameter values 
Diffusion constant, D. Crick (1970) estimated the diffusion constant in cytoplasm 
for a molecule of mass 0.3 - 0.5 kDa to be about 10 -6 cm 2/sec. Subsequently, a 
direct determination of the diffusion constant for a molecule of mass 0.17 kDa in 
the aqueous cytoplasm of mammalian cells yielded a value of about 3.3 x 10 -6 
cm 2/sec (Mastro et al., 1984). By fitting a particular solution of the diffusion equa- 
tion to their data on limb bud determination by gradients of a morphogenetically 
active retinoid, Eichele & Thaller (1987) calculated a value of 10 -7 cm 2/sec for this 
molecule (mass 348.5 kDa) in embryonic limb tissue. One chemically identified 
diffusible factor known to be involved in axon guidance is the protein netrin-1, 
which has a molecular mass of about 75 kDa (Kennedy et al., 1994). D should 
scale roughly inversely with the radius of a molecule, i.e. with the cube root of its 
mass. Taking the value of 3.3 x 10 -6 cm 2/sec and scaling it by (170/75,000) /a 
yields 4.0 x 10 -7 cm 2/sec. This paper therefore considers D = 10 -6 cm 2/sec and 
D = 10 -7 cm2/sec. 
Rate of production of factor q. This is hard to estimate in vivo: some insight can 
be gained by considering in vitro experiments. Gundersen & Barrett (1979) found 
a turning response in chick spinal sensory axons towards a nearby pipette filled 
with a solution of NGE They estimated the rate of outflow from their pipette to 
be 1 /d/hour, and found an effect when the concentration in the pipette was as 
low as 0.1 nM NGF (Tessier-Lavigne & Placzek, 1991). This corresponds to a q of 
3 x 10 -ll riM/sec. Lohof et al. (1992) studied growth cone tuming induced by 
a gradient of cell-membrane permeant cAMP from a pipette containing a 20 mM 
solution and a release rate of the order of 0.5 pl/sec: q = 10 -5 nM/sec. Below a 
further calculation for q is performed, which suggests an appropriate value may 
be q = 10 -7 nM/sec. 
Growth cone diameter, At. For the three systems mentioned above, the diameter 
of the main body of the growth cone is less than 10 /m. However, this ignores 
filopodia, which can increase the effective width for gradient sensing purposes. 
The values of 10/m and 20/m are considered below. 
Minimum concentration for gradient detection. Studies of leukocyte chemotaxis 
suggest that when gradient detection is limited by the dynamics of receptor bind- 
ing rather than physical limits due to a lack of molecules of factor, optimal detec- 
tion occurs when the concentration at the growth cone is equal to the dissociation 
constant for the receptor (Zigrnond, 1981; Devreotes & Zigmond, 1988). Such stud- 
ies also suggest that the low concentration limit is about 1% of the dissociation con- 
stant (Zigrnond, 1981). The transmembrane protein Deleted in Colorectal Cancer 
(DCC) has recently been shown to possess netrin-1 binding activity, with an order 
of magnitude estimate for the dissociation constant of 10 nM (Keino-Masu et al, 
1996). For comparison, the dissociation constant of the low-affinity NGF receptor 
P75 is about 1 nM (Meakin & Shooter, 1992). Therefore, low concentration limits of 
both 10 - nM and 10 -2 nM will be considered. 
Maximum concentration for gradient detection. Theoretical considerations sug- 
gest that, for leukocyte chemotaxis, sensitivity to a fixed gradient should fall off 
symmetrically in a plot against the log of background concentration, with the peak 
at the dissociation constant for the receptor (Zigrnond, 1981). Raising the con- 
162 G. J. Goodhill 
centration to several hundred times the dissociation constant appears to prevent 
axon guidance (discussed in Tessier-Lavigne & Placzek (1991)). At concentrations 
very much greater than the dissociation constant, the number of receptors may be 
downregulated, reducing sensitivity (Zigmond, 1981). Given the dissociation con- 
stants above, 100 nM thus constitutes a reasonable upper bound on concentration. 
Minimum percentage change detectable by a growth cone, p. By establishing 
gradients of a repellent, membrane-bound factor directly on a substrate and mea- 
suring the response of chick retinal axons, Baier & Bonhoeffer (1992) estimated p to 
be about 1%. Studies of cell chemotaxis in various systems have suggested optimal 
values of 2%: for concentrations far from the dissociation constant for the receptor, 
p is expected to be larger (Devreotes & Zigmond, 1988). Both p = 1% and p = 2% 
are considered below. 
4 Results 
In order to estimate bounds for the rate of production of factor q for biological 
tissue, the empirical observation is used that, for collagen gel assays lasting of the 
order of one day, guidance is generally seen over distances of at most 500 /m 
(Ltrmsden & Davies, 1983, 1986; Tessier-Lavigne et al., 1988). Assume first that this 
is constrained by the low concentration limit. Substituting the above parameters 
(with D = 10 -7 cm2/sec) into equation 1 and specifying that U(500tm, 1 day) = 
0.01 nM gives q  10 -9nM/sec. On the other hand, assuming constraint by the 
high concentration limit, i.e. C(500/m, 1 day) = 100 nM, gives q  10 -5 nM/sec. 
Thus it is reasonable to assume that, roughly, 10 -9nM/sec < q < 10 -5 nM/sec. 
The results discussed below use a value in between, namely q = 10 -7 ruM/sec. 
The constraints arising from equations 1 and 2 are plotted in figure 1. The cases of 
6 2 7 2 
D - 10- cm /sec and D = 10- cm /sec are shown in (A,C) and (B,D) respec- 
tively. In all four pictures the constraints C = 0.01 rum and C = 0.1 rum are plotted. 
In (A,B) the gradient constraint p = 1% is shown, whereas in (C,D) p = 2% is 
shown. These are for a growth cone diameter of 10/m. The graph for a 2% change 
and a growth cone diameter of 20/m is identical to that for a 1% change and a 
diameter of 10 /m. Each constraint is satisfied for regions to the left of the rele- 
vant line. The line C = 100 nM is approximately coincident with the vertical axis 
in all cases. For these parameters, the high concentration limit does not therefore 
prevent gradient detection until the axons are within a few microns of the source, 
and it is thus assumed that it is not an important constraint. 
As expected, for large t the gradient constraint asymptotes at Ar/r = p, i.e. 
r = 1000/m/orp = 1% and r = 500/m forp = 2% and a 10/m growth cone. That 
is, the gradient constraint is satisfied at all times when the distance from the source 
is less than 500 /m for p = 2% and Ar = 10 /m. The gradient constraint lines 
end to the right because at earlier times p exceeds the critical value over all dis- 
tances (since the formula for p is non-monotonic with r, there is sometimes another 
branch of each p curve (not shown) off the graph to the right). As t increases from 
zero, guidance is initially limited only by the concentration constraint. The maxi- 
mum distance over which guidance can occur increases smoothly with t, reaching 
for instance 1500 /m (assuming a concentration limit of 0.01 ruM) after about 2 
hours for D = 10 -6 crn2/sec and about 6 hours for D = 10 -7 cm2/sec. However 
at a particular time, the gradient constraint starts to take effect and rapidly reduces 
the maximum range of guidance towards the asymptotic value as t increases. This 
time (for p = 2%) is about 2 hours for D = 10 -6 cm 2/sec, and about one day for 
D = 10 -7 cm 2/sec. It is clear from these pictures that although the exact size of 
A Mathematical Model of Axon Guidance by Diffusible Factors 163 
A B 
C 
3.0 
2.5 
2.0 l_ 
1.5- 
1.0- 
0.5- 
0.0 
0 
4.0 
3.5 
3.0 
2.5 
2.0 
1.5 
1.0 
0.5 
0.0 
0 
,1' 
500 
' 4.0 
--- C=O.01nM E 3.5 
 .. C = 0.1nM i: 
 p = 0.01 3.0 
L-.''I I' ..... F'' I 
1000 1500 2000 2500 
Distance (microns) 
--- C=0.01nM 
 -. C: 0.1nM 
-- p = 0.02 
500 1000 1500 2000 2500 
Distance (microns) 
D 
2.5 
2.0 r- 
1.5 
J 
1.0- 
0.5- 
0.0 
0 
4.0 
3.5 
3.0 
2,5 
2.0 
1.5 
1.0 
0.5 
0.0 
0 
--- C=0.01nM 
 .. C=0.1nM 
-- p = 0.01 
- L - .;, ' -' "1 ..... ' I I 
500 1000 1500 2000 2500 
Distance (microns) 
--- C = 0.01nM ' 
,.. C=0.1nM 
-- p = 0.02 
500 1000 1500 2000 2500 
Distance (mcrons) 
Figure 1: Graphs showing how the gradient constraint (solid line) interacts with 
the minimum concentration constraint (dashed/dotted lines) to limit guidance 
range, and how these constraints evolve over time. The top row (A,B) is forp = 1%, 
the bottom row (C,D) for p = 2%. The left column (A,C) is for D = 10 -s cm2/sec, 
the right column (B,D) for D = 10 -7 cm 2/sec. Each constraint is satisfied to the 
left of the appropriate curve. It can be seen that for D = 10-Scm 2/sec the gradient 
limit quickly becomes the dominant constraint on maximum guidance range. In 
contrast for D = 10 -7 cm 2/sec, the concentration limit is the dominant constraint 
at times up to several days. However after this the gradient constraint starts to 
take effect and rapidly reduces the maximum guidance range. 
164 G. J. Goodhill 
the diffusion constant does not affect the position of the asymptote for the gradient 
constraint, it does play an important role in the interplay of constraints while the 
gradient is evolving. The effect is however subtle: reducing D from 10 -6 cm 2/sec 
to 10 -7 cm 2/sec increases the time for the C' = 0.01 rum limit to reach 2000/m, but 
decreases the time for the C' = 0.1 nM limit to reach 2000 m. 
5 Discussion 
Taking the gradient constraint to be a fractional change of at least 2% across a 
growth cone of width of 10 /m or 20 /m yields asymptotic values for the max- 
imum distance over which guidance can occur once the gradient has stabilized 
of 500 /m and 1000 /m respectively. This fits well with both in vitro data, and 
the fact that for the systems mentioned in the introduction, the growing axons are 
always less than 500 /m from the target in vivo. The concentration limits seem 
to provide a weaker constraint than the gradient limit on the maximum distances 
possible. However, this is very dependent on the value of q, which has only been 
very roughly estimated: if q is significantly less than 10 -7 nM/sec, the low concen- 
tration limits will provide more restrictive constraints (q may well have different 
values in different target tissues). The gradient constraint curves are independent 
of q. The gradient constraint therefore provides the most robust explanation for 
the observed guidance limit. 
The model makes the prediction that guidance over longer distances than have 
hitherto been observed may be possible before the gradient has stabilized. In the 
early stages following the start of factor production the concentration falls off more 
steeply, providing more effective guidance. The time at which guidance range is a 
maximum depends on the diffusion constant D. For a rapidly diffusing molecule 
(D ,, 10-6cm2/sec) this occurs after only a few hours. For a more slowly diffusing 
molecule however (D  10-7cm2/sec) this occurs after a few days, which would 
be easier to investigate in vitro. In vivo, molecules such as netrin-1 may thus be 
large because, during times immediately following the start of production by the 
source, there could be a definite benefit (i.e. steep gradient) to a slowly-diffusing 
molecule. Also, it is conceivable that Nature has optimized the start of produc- 
tion of factor relative to the time that guidance is required in order to exploit an 
evolving gradient for extended range. This could be especially important in larger 
animals, where axons may need to be guided over longer distances in the devel- 
oping embryo. 
Bibliography 
Baier, H. & Bonhoeffer, E (1.992). Axon guidance by gradients of a target-derived 
component. Science, 255, 472-475. 
Berg, H.C. and Purcell, E.M. (1977). Physics of chemoreception. Biophysical Journal, 
20, 193-219. 
Crick, E (1970). Diffusion in embryogenesis. Nature, 255, 420-422. 
Crank, J. (1975). The mathematics of diffusion, Second edition. Oxford, Clarendon. 
Devreotes, EN. & Zigmond, S.H. (1988). Chemotaxis in eukaryotic cells: a focus on 
leukocytes and Dictyostelium. Ann. Rev. Cell. Biol., 4, 649-686. 
Eichele, G. & Thaller, C. (1987). Characterization of concentration gradients of a 
morphogenetically active retinoid in the chick limb bud. J. Ceil. Biol., 105,1917- 
1923. 
A Mathematical Model of Axon Guidance by Diffusible Factors 165 
Goodhill, G.J. (1997). Diffusion in axon guidance. Eur. J. Neurosci., 9, 1414-1421. 
Goodhill, G.J. (1998). Mathematical guidance for axons. Trends. Neurosci., in press. 
Gundersen, R.W. & Barrett, J.N. (1979). Neuronal chemotaxis: chick dorsal-root 
axons tum toward high concentrations of nerve growth factor. Science, 206, 
1079-1080. 
Heffner, C.D., Lumsden, A.G.S. & O'Leary, D.D.M. (1990). Target control of collat- 
eral extension and directional growth in the mammalian brain. Science, 247, 
217-220. 
Keino-Masu, K., Masu, M., Hinck, L., Leonardo, E.D., Chan, S.S.-Y., Culotti, J.G. & 
Tessier-Lavigne, M. (1996). Deleted in Colorectal Cancer (DCC) encodes a nettin 
receptor. Cell, 87, 175-185. 
Kennedy, T.E., Serafini, T., de al Torre, J.R. & Tessier-Lavigne, M. (1994). Netrins are 
diffusible chemotropic factors for commissural axons in the embryonic spinal 
cord. Cell, 78, 425-435. 
Lohof, A.M., Quillan, M., Dan, Y, & Poo, M-m. (1992). Asymmetric modulation of 
cytosolic cAMP activity induces growth cone tuming. J. Neurosci., 12, 1253- 
1261. 
Lumsden, A.G.S. & Davies, A.M. (1983). Earliest sensory nerve fibres are guided to 
peripheral targets by attractants other than nerve growth factor. Nature, 306, 
786-788. 
Lumsden, A.G.S. & Davies, A.M. (1986). Chemotropic effect of specific target ep- 
ithelium in the developing mammalian nervous system. Nature, 323, 538-539. 
Mastro, A.M., Babich, M.A., Taylor, W.D. & Keith, A.D. (1984). Diffusion of a small 
molecule in the c}ftoplasm of mammalian cells. Proc. Nat. Acad. Sci. USA, 81, 
3414-3418. 
Meakin, S.O. & Shooter, E.M. (1992). The nerve growth family of receptors. Trends. 
Neurosci., 15, 323-331. 
Tessier-Lavigne, M. & Placzek, M. (1991). Target attraction: are developing axons 
guided by chemotropism? Trends Neurosci., 14, 303-310. 
Tessier-Lavigne, M. & Goodman, C.S. (1996). The molecular biology of axon guid- 
ance. Science, 274, 1123-1133. 
Tessier-Lavigne, M., Placzek, M., Lumsden, A.G.S., Dodd, J. & Jessell, T.M. (1988). 
Chmotropic guidance of developing axons in the mammalian central ner- 
vous system. Nature, 336, 775-778. 
Tranquillo, R.T. & Lauffenburger, D.A. (1987). Stochastic model of leukocyte 
chemosensory movement. J. Math. Biol., 25, 229-262. 
Zigmond, S.H. (1981). Consequences of chemotactic peptide receptor modulation 
for leukocyte orientation. J. Cell. BioI., 88, 644-647. 
