{"title": "A Mathematical Model of Axon Guidance by Diffusible Factors", "book": "Advances in Neural Information Processing Systems", "page_first": 159, "page_last": 165, "abstract": "", "full_text": "A mathematical model of axon guidance by \n\ndiffusible factors \n\nGeoffrey J. Goodhill \n\nGeorgetown Institute for Cognitive and Computational Sciences \n\nGeorgetown University Medical Center \n\n3970 Reservoir Road \nWashington DC 20007 \n\ngeoff@giccs.georgetown.edu \n\nAbstract \n\nIn the developing nervous system, gradients of target-derived dif(cid:173)\nfusible factors play an important role in guiding axons to appro(cid:173)\npriate targets. In this paper, the shape that such a gradient might \nhave is calculated as a function of distance from the target and the \ntime since the start of factor production. Using estimates of the \nrelevant parameter values from the experimental literature, the \nspatiotemporal domain in which a growth cone could detect such \na gradient is derived. For large times, a value for the maximum \nguidance range of about 1 mm is obtained. This value fits well \nwith experimental data. For smaller times, the analysis predicts \nthat guidance over longer ranges may be possible. This prediction \nremains to be tested. \n\n1 Introduction \n\nIn the developing nervous system, growing axons are guided to targets that may be \nsome distance away. Several mechanisms contribute to this (reviewed in Tessier(cid:173)\nLavigne & Goodman (1996\u00bb. One such mechanism is the diffusion of a factor \nfrom the target through the extracellular space, creating a gradient of increasing \nconcentration that axons can sense and follow. In the central nervous system, such \na process seems to occur in at least three cases: the guidance ofaxons from the \ntrigeminal ganglion to the maxillary process in the mouse (Lumsden & Davies, \n1983, 1986), of commissural axons in the spinal cord to the floor plate (Tessier(cid:173)\nLavigne et al., 1988), and ofaxons and axonal branches from the corticospinal tract \nto the basilar pons (Heffner et al., 1990). The evidence for this comes from both in \nvivo and in vitro experiments. For the latter, a piece of target tissue is embedded in \na three dimensional collagen gel near to a piece of tissue containing the appropriate \n\n\f160 \n\nG. J Goodhill \n\npopulation of neurons. Axon growth is then observed directed towards the target, \nimplicating a target-derived diffusible signal. In vivo, for the systems described, \nthe target is always less than 500 J..lm from the population ofaxons. In vitro, where \nthe distance between axons and target can readily be varied, guidance is generally \nnot seen for distances greater than 500 - 1000 J..lm. Can such a limit be explained \nin terms of the mathematics of diffusion? \nThere are two related constraints that the distribution of a diffusible factor must \nsatisfy to provide an effective guidance cue at a point. Firstly, the absolute concen(cid:173)\ntration of factor must not be too small or too large. Secondly, the fractional change \nin concentration of factor across the width of the gradient-sensing apparatus, gen(cid:173)\nerally assumed to be the growth cone, must be sufficiently large. These constraints \nare related because in both cases the problem is to overcome statistical noise. At \nvery low concentrations, noise exists due to thermal fluctuations in the number of \nmolecules of factor in the vicinity of the growth cone (analyzed in Berg & Purcell \n(1977\u00bb. At higher concentrations, the limiting source of noise is stochastic varia(cid:173)\ntion in the amount of binding of the factor to receptors distributed over the growth \ncone. At very high concentrations, all receptors will be saturated and no gradient \nwill be apparent. The closer the concentration is to the upper or lower limits, the \nhigher the gradient that is needed to ensure detection (Devreotes & Zigmond, 1988; \nTessier-Lavigne & Placzek, 1991). The limitations these constraints impose on the \nguidance range of a diffusible factor are now investigated. For further discussion \nsee Goodhill (1997; 1998). \n\n2 Mathematical model \n\nConsider a source releasing factor with diffusion constant D cm2/sec, at rate q \nmoles/sec, into an infinite, spatially uniform three-dimensional volume. Initially, \nzero decay of the factor is assumed. For radially symmetric Fickian diffusion in \nthree dimensions, the concentration C(r, t) at distance r from the source at time t \nis given by \n\nC(r, t) = -4 D erfc r;-r:;-; \nv4Dt \n\nq \n7r r \n\nr \n\n(1) \n\n(2) \n\n(see e.g. Crank (1975\u00bb, where erfc is the complementary error function. The per(cid:173)\ncentage change in concentration p across a small distance D.r (the width of the \ngrowth cone) is given by \n\nD.r [ \n\nr \n\np = --;:- 1 + J7rDt erfc(r/J4Dt) \n\ne-r2/4Dt 1 \n\nThis function has two perhaps surprising characteristics. Firstly, for fixed r, Ipi \ndecreases with t. That is, the largest gradient at any distance occurs immediately \nafter the source starts releaSing factor. For large t, Ipi asymptotes at D.r Jr. Secondly, \nfor fixed t < 00, numerical results show that p is nonmonotonic with r. In particular \nit decreases with distance, reaches a minimum, then increases again. The position \nof this minimum moves to larger distances as t increases. \nThe general characteristics of the above constraints can be summarized as follows. \n(1) At small times after the start of production the factor is very unevenly dis(cid:173)\ntributed. The concentration C falls quickly to almost zero moving away from the \nsource, the gradient is steep, and the percentage change across the growth cone \np is everywhere large. (2) As time proceeds the factor becomes more evenly dis(cid:173)\ntributed. C everywhere increases, but p everywhere decreases. (3) For large times, \nC tends to an inverse variation with the distance from the source r, while Ipi tends \n\n\fA Mathematical Model of Axon Guidance by Diffusible Factors \n\n161 \n\nto 6.r/r independent of all other parameters. This means that, for large times, \nthe maximum distance over which guidance by diffusible factors is possible scales \nlinearly with growth cone diameter 6.r. \n\n3 Parameter values \n\nDiffusion constant, D. Crick (1970) estimated the diffusion constant in cytoplasm \nfor a molecule of mass 0.3 - 0.5 kDa to be about 10- 6 cm2 /sec. Subsequently, a \ndirect determination of the diffusion constant for a molecule of mass 0.17 kDa in \nthe aqueous cytoplasm of mammalian cells yielded a value of about 3.3 x 10- 6 \ncm2 / sec (Mastro et aL, 1984). By fitting a particular solution of the diffusion equa(cid:173)\ntion to their data on limb bud determination by gradients of a morphogenetically \nactive retinoid, Eichele & Thaller (1987) calculated a value of 10-7 cm2 /sec for this \nmolecule (mass 348.5 kDa) in embryonic limb tissue. One chemically identified \ndiffusible factor known to be involved in axon guidance is the protein netrin-1, \nwhich has a molecular mass of about 75 kDa (Kennedy et al., 1994). D should \nscale roughly inversely with the radius of a molecule, Le. with the cube root of its \nmass. Taking the value of 3.3 x 10-6 cm2 /sec and scaling it by (170/75,000)1/3 \nyields 4.0 x 10- 7 cm2 /sec. This paper therefore considers D = 10-6 cm2 /sec and \nD = 10-7 cm2 /sec. \nRate of production of factor q. This is hard to estimate in vivo: some insight can \nbe gained by considering in vitro experiments. Gundersen & Barrett (1979) found \na turning response in chick spinal sensory axons towards a nearby pipette filled \nwith a solution of NGF. They estimated the rate of outflow from their pipette to \nbe 1 /LI/hour, and found an effect when the concentration in the pipette was as \nlow as 0.1 nM NGF (Tessier-Lavigne & Placzek, 1991). This corresponds to a q of \n3 x 1O- 11 nM/sec. Lohof et al. (1992) studied growth cone turning induced by \na gradient of cell-membrane permeant cAMP from a pipette containing a 20 mM \nsolution and a release rate of the order of 0.5 pI/sec: q = 10-5 nM/sec. Below a \nfurther calculation for q is performed, which suggests an appropriate value may \nbe q = 10- 7 nM/sec. \nGrowth cone diameter, 6.r. For the three systems mentioned above, the diameter \nof the main body of the growth cone is less than 10 /Lm. However, this ignores \nfilopodia, which can increase the effective width for gradient sensing purposes. \nThe values of 10 /Lm and 20 /Lm are considered below. \nMinimum concentration for gradient detection. Studies of leukocyte chemotaxis \nsuggest that when gradient detection is limited by the dynamics of receptor bind(cid:173)\ning rather than physical limits due to a lack of molecules of factor, optimal detec(cid:173)\ntion occurs when the concentration at the growth cone is equal to the dissociation \nconstant for the receptor (Zigmond, 1981; Devreotes & Zigmond, 1988). Such stud(cid:173)\nies also suggest that the low concentration limit is about 1 % of the dissociation con(cid:173)\nstant (Zigmond, 1981). The transmembrane protein Deleted in Colorectal Cancer \n(DeC) has recently been shown to possess netrin-1 binding activity, with an order \nof magnitude estimate for the dissociation constant of 10 nM (Keino-Masu et aI, \n1996). For comparison, the dissociation constant of the low-affinity NGF receptor \nP75 is about 1 nM (Meakin & Shooter, 1992). Therefore, low concentration limits of \nboth 10-1 nM and 10-2 nM will be considered. \nMaximum concentration for gradient detection. Theoretical considerations sug(cid:173)\ngest that, for leukocyte chemotaxis, sensitivity to a fixed gradient should fall off \nsymmetrically in a plot against the log of background concentration, with the peak \nat the dissociation constant for the receptor (Zigmond, 1981). Raising the con-\n\n\f162 \n\nG. 1. Goodhill \n\ncentration to several hundred times the dissociation constant appears to prevent \naxon guidance (discussed in Tessier-Lavigne & Placzek (1991\u00bb. At concentrations \nvery much greater than the dissociation constant, the number of receptors may be \ndownregulated, reducing sensitivity (Zigm0nd, 1981). Given the dissociation con(cid:173)\nstants above, 100 nM thus constitutes a reasonable upper bound on concentration. \n\nMinimum percentage change detectable by a growth cone, p. By establishing \ngradients of a repellent, membrane-bound factor directly on a substrate and mea(cid:173)\nsuring the response of chick retinal axons, Baier & Bonhoeffer (1992) estimated p to \nbe about 1 %. Studies of cell chemotaxis in various systems have suggested optimal \nvalues of 2%: for concentrations far from the dissociation constant for the receptor, \np is expected to be larger (Devreotes & Zigmond, 1988). Both p = 1% and p = 2% \nare considered below. \n\n4 Results \n\nIn order to estimate bounds for the rate of production of factor q for biological \ntissue, the empirical observation is used that, for collagen gel assays lasting of the \norder of one day, guidance is generally seen over distances of at most 500 I'm \n(Lumsden & Davies, 1983,1986; Tessier-Lavigne et al., 1988). Assume first that this \nis constrained by the low concentration limit. Substituting the above parameters \n(with D = 10-7 cm2/sec) into equation 1 and specifying that C(500pm, 1 day) = \n0.01 nM gives q :::::: 10-9 nM/ sec. On the other hand, assuming constraint by the \nhigh concentration limit, i.e. C(500pm, 1 day) = 100 nM, gives q :::::: 10-5 nM/sec. \nThus it is reasonable to assume that, roughly, 10-9 nM/ sec < q < 10-5 nM/ sec. \nThe results discussed below use a value in between, namely q = 10-7 nM/sec. \nThe constraints arising from equations 1 and 2 are plotted in figure 1. The cases of \nD = 10-6 cm2/sec and D = 10- 7 cm2/sec are shown in (A,C) and (B,D) respec(cid:173)\ntively. In all four pictures the constraints C = 0.01 nM and C = 0.1 nM are plotted. \nIn (A,B) the gradient constraint p = 1 % is shown, whereas in (C,D) p = 2% is \nshown. These are for a growth cone diameter of 10 I'm. The graph for a 2% change \nand a growth cone diameter of 20 I'm is identical to that for a 1% change and a \ndiameter of 10 I'm. Each constraint is satisfied for regions to the left of the rele(cid:173)\nvant line. The line C = 100 nM is approximately coincident with the vertical axis \nin all cases. For these parameters, the high concentration limit does not therefore \nprevent gradient detection until the axons are within a few microns of the source, \nand it is thus assumed that it is not an important constraint. \nAs expected, for large t the gradient constraint asymptotes at D.r Ir = p, i.e. \nr = 1000 I'm for p = 1% and r = 500 I'm for p = 2% and a 10 I'm growth cone. That \nis, the gradient constraint is satisfied at all times when the distance from the source \nis less than 500 I'm for p = 2% and D.r = 10 I'm. The gradient constraint lines \nend to the right because at earlier times p exceeds the critical value over all dis(cid:173)\ntances (since the formula for p is non-monotonic with r, there is sometimes another \nbranch of each p curve (not shown) off the graph to the right). As t increases from \nzero, guidance is initially limited only by the concentration constraint. The maxi(cid:173)\nmum distance over which guidance can occur increases smoothly with t, reaching \nfor instance 1500 I'm (assuming a concentration limit of 0.01 nM) after about 2 \nhours for D = 10-6 cm2/sec and about 6 hours for D = 10- 7 cm2/sec. However \nat a particular time, the gradient constraint starts to take effect and rapidly reduces \nthe maximum range of guidance towards the asymptotic value as t increases. This \ntime (for p = 2%) is about 2 hours for D = 10-6 cm2/sec, and about one day for \nD = 10- 7 cm2 / sec. It is clear from these pictures that although the exact size of \n\n\fA Mathematical Model of Axon Guidance by Diffusible Factors \n\n163 \n\n500 \n\n1000 \n\nA \n\nC \n\nCi) \n>- 4.0 \n'\" \n~ \nCD 3.5 \nE \ni= \n\n3.0 \n\n2.5 \n\n2.0 \n\n1.5 \n\n1.0 \n\n0.5 \n\n0.0 \n\n0 \n\nCi) \n>- 4.0 \n'\" \n~ \nCD 3.5 \nE \ni= \n\n. 3.0 \n\n2.5 \n\n2.0 \n\n1.5 \n\n1.0 \n\n0.5 \n\n0.0 \n\n0 \n\n... \n\" . C=O.lnM \n\nC=O.OlnM \n\n-\n\np=O.OI \n\n1500 \n\n2000 \n\n2500 \nDistance (microns) \n\n... C = O.OlnM \n., . C = O.lnM \np=0.02 \n-\n\n. , ' \n\n.. ' \n\n1500 \n\n2000 \n\n2500 \nDistance (microns) \n\nCi) \n>- 4.0 \n'\" \n~ \nCD 3.5 \nE \ni= \n\n3.0 \n\n2.5 \n\n2.0 \n\n1.5 \n\n1.0 \n\n0.5 \n\n0.0 \n\n0 \n\nB \n\nD \n\n4.0 \n\n~ \n'\" \n~ \nCD 3.5 \nE \ni= \n\n3.0 \n\n2.5 \n\n2.0 \n\n1.5 \n\n1.0 \n\n0.5 \n\n0.0 \n\n0 \n\n\"'-J \n\n... C=O.OlnM \n... C = O.lnM \n-\n\np= 0.01 \n\n\" , \n\n, .. \" \n\n....... \" . \n\n\" \n\n500 \n\n1000 \n\n1500 \n\n2000 \n\n2500 \nDistance (microns) \n\n... C = O.OlnM \n., . C=O.lnM \n-\n\nP =0.02 \n\n.. ' .\" \n\n!~ ......... \n\n.. \n\n.' \n\" \n\n\" \n\n\" \n\n500 \n\n1000 \n\n1500 \n\n2000 \n\n2500 \nDistance (microns) \n\nFigure 1: Graphs showing how the gradient constraint (solid line) interacts with \nthe minimum concentration constraint (dashed/dotted lines) to limit guidance \nrange, and how these constraints evolve over time. The top row (A,B) is for p = 1%, \nthe bottom row (C,D) for p = 2%. The left column (A,C) is for D = 10-6 cm2 / sec, \nthe right column (B,D) for D = 10-7 cm2 /sec. Each constraint is satisfied to the \nleft of the appropriate curve. It can be seen that for D = 1O-6cm2 / s~c the gradient \nlimit quickly becomes the dominant constraint on maximum guidance range. In \ncontrast for D = 10-7 cm2 / sec, the concentration limit is the dominant constraint \nat times up to several days. However after this the gradient constraint starts to \ntake effect and rapidly reduces the maximum guidance range. \n\n\f164 \n\nG. 1. Goodhill \n\nthe diffusion constant does not affect the position of the asymptote for the gradient \nconstraint, it does play an important role in the interplay of constraints while the \ngradient is evolving. The effect is however subtle: reducing D from 10-6 cm2 /sec \nto 10- 7 cm2 /sec increases the time for the C = 0.01 nM limit to reach 2000 J..tm, but \ndecreases the time for the C = 0.1 nM limit to reach 2000 f..lm. \n\n5 Discussion \n\nTaking the gradient constraint to be a fractional change of at least 2% across a \ngrowth cone of width of 10 f..lm or 20 J..tm yields asymptotic values for the max(cid:173)\nimum distance over which guidance can occur once the gradient has stabilized \nof 500 f..lm and 1000 f..lm respectively. This fits well with both in vitro data, and \nthe fact that for the systems mentioned in the introduction, the growing axons are \nalways less than 500 f..lm from the target in vivo. The concentration limits seem \nto provide a weaker constraint than the gradient limit on the maximum distances \npossible. However, this is very dependent on the value of q, which has only been \nvery roughly estimated: if q is significantly less than 10-7 nM/sec, the low concen(cid:173)\ntration limits will provide more restrictive constraints (q may well have different \nvalues in different target tissues). The gradient constraint curves are independent \nof q. The gradient constraint therefore provides the most robust explanation for \nthe observed guidance limit. \nThe model makes the prediction that guidance over longer distances than have \nhitherto been observed may be possible before the gradient has stabilized. In the \nearly stages following the start of factor production the concentration falls off more \nsteeply, providing more effective guidance. The time at which guidance range is a \nmaximum defends on the diffusion constant D. For a rapidly diffusing molecule \n(D >::::: 1O-6cm jsec) this occurs after only a few hours. For a more slowly diffusing \nmolecule however (D >::::: 10-7 cm2 jsec) this occurs after a few days, which would \nbe easier to investigate in vitro. In vivo, molecules such as netrin-1 may thus be \nlarge because, during times immediately following the start of production by the \nsource, there could be a definite benefit (i.e. steep gradient) to a slowly-diffusing \nmolecule. Also, it is conceivable that Nature has optimized the start of produc(cid:173)\ntion of factor relative to the time that guidance is required in order to exploit an \nevolving gradient for extended range. 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