420 Diffusion in Embryogenesis by FRANCIS CRICK Medical Research Council Laboratory of Molecular Biology, Hills Road, Cambridge It has been a great surprise and of considerable import- ance to find that most onic fields seem to involve distances of less than 100 cells, and often less than 60. Professor Lewis Wolpert Wuen I read this sentence I was delighted because it ‘seemed to confirm some conclusions I had come to on purely theoretical grounds’. It is an old idea that “gradients” are involved in embryological development “—in fact, C. M. Child* in 1941 wrote a whole book on the subject. Many of the gradients to which Child referred ‘seem more likely, in retrospect, to be the resulte of -development rather than its cause. An outsider to -embryology has the impression that in recent years igradionta have become a dirty word. This is partly .because of the failure to isolate unambiguously the ‘molecules involved, whose concentration is presumed to constitute the gradient, and partly because a feeling has ‘grown up that diffusion is not a fast enough mechanism for establishing gradients. In this article I aim to show that this fear is unfounded, and, on the contrary, that the known facts, sparse as they are, fit rather well to a mechanism based on diffusion. The problem can be stated in this way: what is the maximum distance over which a steady concentration gradient could plausibly be set up in the times available during the development of the embryo ? The obvious model for setting up a simple gradient is illustrated in Fig. 1. At one end of a line of cells one postulates a source—a cell which produces the chemical (which I shall call a morphogen) and maintains it at s constant level. At the other end the extreme cell acts as @ sink: that is, it destroys the molecule, holding the concentration at that point to a fixed low level. The morphogen can diffuse from one cell to another along the line of cells. After a time the system approaches a dynamic equilibrium, and it is easy to show that if the effective diffusion constant is everywhere the same, the .concentration gradient will be linear. Of course, real embryological structures will have three dimensions, but if for convenience we restrict ourselves to sheets of cells such as, for example, the insect epi- dermis‘ or the developing amphibian retina’ the problem becomes two-dimensional. The source can be considered to be a line of cella (the line being perpendicular to the paper in Fig. 1) and similarly for the sink, thus reducing the problem to one dimension. It is not difficult to caloulate how long it would take to ‘set up such a system, supposing that both the source and the sink are turned on at time zero. Diffusion is a random walk process, and the dimensions of the diffusion constant, | D, are L*T-' (where LD is length and T is time). This should be contrasted with a mechanism having a velocity (with dimensions LT7'-") as proposed, for example, by NATURE VOL. 225 JANUARY 31 1970 A simple order-of-magnitude calculation suggests that diffusion may be the underlying mec in embryonic development. anism in establishing morphogenetic gradients Goodwin and Cohen‘. Because in diffusion the length enters as the square, pure diffusion processes are very rapid over rather short distances (say, the size of a cell) and very slow over long distances (say, the size of an organ). . The concentration approaches its final value asymp- totically, so one must have some criterion for deciding whether the gradient at any time is sufficiently close to a straight line. I have arbitrarily taken the gradient to be effectively established when it is everywhere within . AC of the final value, and chosen AC as 1 per cent of C, (C, is the maximum value at the origin). It would make little difference to the argument if AC were considered to have half this value.. The gradient might be set up in various ways. The result in each case can be expressed as tm A (nl)* D where ¢=time in seconds to set up the gradient; n= number of cells between source and sink; /=Jength of each cell, in cm; and D=#diffusion constant, in cm! s-!, _A is a numerical constant, the exact value of which will depend on the way the gradient is developed. Mathematically the simplest way is to start with zero eoncentration of the morphogen everywhere at time zero, and thereafter to maintain the source at concentration C, and the sink at concentration zero. This gives a value of A of 0-42.. Biochemically more realistic models give values only a little larger than this, so a good general value for A would be 0-5. It was pointed out to me by © Dr Asron Klug that, if the initial concentration were uniformly C,/2, the time required is reduced to a little less than one-quarter, and A will have a value of about 0-098. More realistic models of this general sort give values of A of, say, 0-15. The calculations of A were carried out by Mrs Mary Munro. In what follows, I shall assume that A is 0-5 (the simple mechanism), but it should be remembered that the organ- ism might be able to reduce this to about a third of this value. The diffusion constant in water for all but the smallest molecules—-provided they are roughly spherical—is in- versely proportional to their mean radius, to a near approximation. Thus, increasing the molecular weight by & factor of 1,000, from, say, a small organic molecule such as ATP (mol. wt. 507) to a very large protein like polynucleotide polymerase (mol. wt. about 0-5 x 10*) reduces the diffusion constant only by a factor of 10. Now it is reasonable to expect that the morphogen will diffuse rather rapidly, and should be able to pass fairly efficiently from cell to cell. It is also likely to be a rather specific molecule. For these reasons I doubt if morphogens NATURE VOL. 225 JANUARY 31 1970 will turn out to be large proteins or common ions’ like K+ or Na*. An obvious choice would be an organic molecule of about the size of, say, cyclic AMP or a steroid. That is, with a molecular weight in the range 300 to 500. The diffusion constant® in water (at 20°C) for such a molecule is about 4 or 5x 10-* em* s-?, (The diffusion of salts Jike NaCl or KCI is about three or four times as fast as this.) ; The inside of a cell is very far from being made of water, and one must estimate the effective diffusion constant within a cell. This amounts to estimating the effective . viscosity. The cytoplasm being a concentrated mixturo of molecules having a large variety of sizes, the relative viscosity will be considerably higher than water at the same temperature. For a small molecule, which can, as it were, slip between many of the other molecules, the effective viscosity is unlikely to be as big as the bulk viscosity of the cytoplasm (wherover that may be). It is difficult to make any precise estimate of tho effective diffusion constant, which in any caso may vary consider- ably between different types of cell. Allowing a factor of increase of viscosity of x6 (corresponding to a sucrose solution 40 per cent by weight), which seoms not un- reasonablo*, would make the effective diffusion constant about 0-8 x 10-* em? 8-1, Source Concentration of morphosen Sink Distance occa. meter reece cece DCOCOOD ta Fig. 1. To filustrate how a source and a sink can produce a linear gradient of concentration. Each cell in the line has Jength 2 em. The distance between source and sink is Z cm. Because n is the number of cells in the line, Z= nl, How docs the morphogen get from ceil to cell ? It would seem inefficient to make all cell membranes easily per- meable to it. To do this it would in any casc have to be very small and rather hydrophobic?®, Moreover, a high gencral permeability would allow the molecule to escape too easily from the tissue and this would not only disturb the gradient but possibly interfere in other parts of the organism. One is therefore driven to postulate a special mechanism which allows a relatively quick passage of the morphogen from one cell to another in tho tissue of interest. Whether this mvolves tight junctions or other special structures remains to be seen. Such a mechanism could be facilitated diffusion. This is cha acterized by being independent of energy, specific 421 (usually stereospecific) and saturatable at high concentra. tions of the diffusing molecule. The passage of glucose through the membrane of a human red blood cell or an. ascites tumour cell is bolieved to have this character. At low concentrations (that is, far from saturation) the mechanism can be described by a permeability, P, meas- | ured in em s-!. It is easy to show that the effective diffu- sion constant, D’, for our problem is given by 1/D’ = 1/D+- 1/Pl, where 1 is the length of each cell in the direction of diffusion". Thus, if D is 0-8 x 10-* em* s-! and 7=10 pm’. (say) we see that if the ‘resistance”’ to flow of the morpho- gen because of permoability between colls was equal to that. due to diffusion within a cell, P would have to have the value 8 x 10-* em/second. This is a high value, but prob- ably not impossibly high". If wo arbitrarily take P aa about half this and D as before, we obtain D’=0-27 x 10-# em s-?, 7 We now need an experimental estimate of the ‘time needed to set up a gradient. This is not casy to obtain. Most embryologists would feel that o day is too Jong... A minute seems far too short. A few hours would seem about right—Wolpert has suggested that: between 5 to 10h is not unreasonable for many of the well-studied cases (ref. 1, page 41). I shall assume a figure of 10‘ s (approximately. 3h), because some time must be allowed for the changes which take place after the gradiont is set up. Combining our formulas we obtain rane Beh) and substituting the chosen values: £== 104 s, A=0°5, £=10 um, D=0-8x 10-* em? 5-1, P=4x 10-4 em s-! we. obtain n270 cella, If 2 were 30 zm, n would come to a- little over thirty cells. Even allowing for the ‘very approxi- mate nature of the calculations, the agreement with the figures given in the quotation at the start of the article is striking. In broad outline what the calculation shows is. that, for the times considered, distances or the order of a millimetre (or less) are possible, but distances of a centi- . Of course, for organisms which -: develop vory rapidly the distances would have to be smaller . metre are too great}, than a millimetre. We can take it, then, that assuming the effective © viscosity of cytoplasm has not been grossly underestimated, -; and provided there is a special mechanism to increase the rate of permeability of the morphogen between cells,th ere are many cases in embryology where the times .ond distances involved are quite compatible with a mechanism based on diffusion. It is important, however, to make two - reservations. There may be special cases, involving setting up gradients quickly over large distances (of the order of .. several centimetres) which may require other mechanisms, such as tho signalling devices suggested by Goodwin and Cohen®. Cases of “mushroom growth” (as, for example, the growth of mushroom) are unlikely to be due to diffusion alone. Secondly, in one’s enthusiasm for. diffusion, it is important to realize that the many other problems remain . to be tackled. Even when the gradiont has been set up the coll has to recognize it. Because at least in the insect integument the “gradient” appears to impose a polarity on those epidermal cells which becomo scales and bristles** the cells must have some additional mechanism (involving microtubules ?) to do this. In the case of the amphibian oye the retinal ganglion cells must not only recognize the presumed gradient (so that they know where they aro in the retina); cach cell must also convey this information to the far ond of its growing axon, so that it can make the connexion at tho appropriate place on the optic tectum’. Moreover, there arn likely to be subsidiary mechanisms to guide the growing bundlo of nerves along the right path to the tectum. In two brilliant articles published about ten years ago, Locke's showed that the pattern of wrinkles obtained on 422 tho adult cuticle of Rhodnius after operations on tho carlier larval stages (usually the last larval stage) can only be oxplained by a “gradient” of some sort, running from one intersegmental membrane to the noxt, and repeating in successive sogments.. He showed convincingly that neither mechanical expansion alone, nor polarity alono, could explain the results. Dr Locke kindly sent us the original photographs of some of his matorial. Mrs Mary Munro and I, together with Dr Poter Lawrence, have attempted to fit these patterns to a pure diffusion model. Although, following Locke’s arguments, the observed wrinkles have roughly the expected pattern, they differ in - detail from the computed pattern. Moreover, various estimates of the diffusion constant disagree drastically. We are thereforo currently exploring a model in which each cell in the epidermis attempts to maintain the con- centration of the morphogen within itself to a previously preset level, determined soon after tho gradient is first set up. This model, which has only one disposable parameter, -is a much better fit with Locko’s data. More elaborate hypotheses, within the basic diffusion framework, are also under consideration. It is important, therefore, not to approach these problems with too naive a model!*, Finally, one should emphasize that gradients are unlikely to command general acceptanco until their biochemical basis is discovered experimentally, and that this may not - prove an easy task. Mathematically minded biologists could well object that any theory which has the same . mathematical formalism would necessarily fit the observed patterns, and that the agreement between the calculated and observed distances (on the diffusion theory sketched -above) may only be coincidental. In spite of theso possible _ objections it is my belief that mechanisms based on diffusion are not only plausible but rather probable. Nature usually has such difficulty evolving elaborate biochemical mechanisms (for example, those used in protein synthesis) that the underlying processes are often rather simple. If this approach serves to make the idea _ of diffusion gradients respectable to embryologists it will ~ have served its purpose. I thank my wife for drawing tho figure, my colleagues, especially Dr Peter Lawrence and Mrs Mary Munro, for NATURE VOL. 225 JANUARY 31 1970 many helpful discussions, and Professors Lewis Wolpert and W. D. Stein for sending me information. Received January 2, 1970, ? Wolpert, L., J. Theoret. Biol., 25, 1 (1969). This article, entitled “Positional nformation and the Spatial Pattern of Cellular Differentiation’, should be consulted both for a modern statement of the problem and also for references to earller work. * The basic idea of this article was at the Fourth NATO Advance in July 1969 at Spetaal. *Child, C. M., Patterns and Problems of Development (Chicago University Press, Chicago, 1941), { “For a review sco Lawrence, P. A., Adv. Insect Physiol. (in the press). Sea especially Stumpf, H., Houx Arch. Ent, Mech. Org., 158, 315 (1967), * For a review sce Gaze, R. M., Growth and Differentiation, Ann. Rev. Physiot., 29, 50 (1967). * Goodwin, B., and Cohen, NM. H., J. Theoret. Biol,, 25, 49 (1860). ‘The mechanism for forming a source and o sink for fons also presents special problems, whereas for organic molecules rather simple enzymatic processes could do the trick, oo * The effect of temperature on the viscosity of water does not seriously affect the calculations. Taking the viscosity of water (in arbitrary units) a8 1-0 at 20° C, its value at §° C Is about 1} and at 30° C fs close to resonted ‘at a lecture given to students Study Institute of Molecular Biology * See, for example, a very ingenious fluorescent method used by Victor W, Burns (Biochem. Biophys. Res. Commun., 37, 1008; 1969) using a smali organic molecule. He obtained a factor of about x6 for Euglena. The figure for yeast was about twice this. ** Lieb, W. R., and Stein, W. D., Nature, 224, 240 (1969). ‘| This assumes that the permeability is fairly evenly distributed over the cell membrane. If it were concentrated in a small patch the effective diffusion would be slower. : ™* For examplo, P for glucose in ascites cells at 87° C in about 4-5 x 10¢ cm/s (Kolber, A. R., and LeFevre, P. G., J. Gen. Physiol., $0, 1907; 1967), or in human red cella at 37°C about 1x 10- cm/s (Millar, D. Bf, Biophys. J., 8, 407; 1965), Admittedly these are among the higher values of P known so far. See Stein, W. D., The Movement of Molecules Across Cell Membranes, chap. 4 (Academic Press, New York, 1967). It should be remembered that in going from one cell -to the next the morphogen may have to cross two cell membranes. 33 An upper limit can be calculated assuming that the morphogen Is an fon, that it diffuses (at 37° C) in the cytoplasm as freely as in water, that the permeability between cells is so high that it does not slow down the process at all, and that the most efficient method is used to sot up the ‘adient. The distance then comes to 1:5 cm, but I feel that this com- ination of assumptions is quito unrealistic. . od Piepho, H., Naturwissenschaften, 42, 22 (1955); Lawrence, P. A., J. Exp, tol., 44, 607 (1966). . 4 Locke, M., J. Ezp. Biol., 86, 459 (1959); J, Erp. Biol., 32, 308 (1960). 1* Tho present mode! could be elaborated by assuming two different morpho- gens, one having a gradient sloping m left to right, and the other with a ient from right to left: on this model the position of a cell . would characterized by the ratio of its concentration of the two morphogens, This particular elaboration, however, would not signi- ficantly alter any of the arguments given in this article.