420 NP,lURE VOL 225 JANUARY 31 1970 Diffusion in Etibryogenesis by FRANCIS CRICK Medlal Research Council Laboratory of Molecular Blolo~y, Hills Road, Cambridge -- A simple order-of-ma nitude calculation suggests that diffusion may be the underlying met anism In establishing morphogenetlc gradients f in embryonic development. It haa be-m a great ~urpties and of cot&k&e import- ance to find that moat embrpnti PJds seem to involve diatunees of lean than 100 ceh, and a* less than 60. Professor Lewis Wolpert' WHEN I read this sentence I wee delighted because it seemed to oonflrm some conclusions I h8d come to on purely theoreti& grounds*. It ie an old idecr th8t ~`gmdients" are inv&ved in embryological development -in fmt. C. M. Child' in 1941 wrote 8 whole book on tho subjeat. ' M8ny of the gradients to whioh Child referred seem more likely, in retrospeat, to be the reeulte of development &her than ite cause. An outsider to embryGlogy h8s the imprassion th8t in recent yeere ' gredhnta h8Ve beoome 8 diI%y Word. This is p8Idy baamae of the failure to isolete umunbiguously the m&mules involved, whose aonaentrafion is presumed to constitute the gmdient, snd pertly beceuse 8 feeling h8s grown up th8t2iffUf3ion is n& 8 ih8t enough xxmdi&m for establishing gradiente. In this 8rticle I 8im to show that this fear is unfounded, end, on the aontrsrg, thsf the known f8et.8, sparse 8s they 8re, fit rather well to 8 mech8nism based on dif%ion. The problem a8n be stated in this way: wh8t is the m8ximum distanae over which 8 steady conoentretion gr8dient could pleusibly be set up in the times 8VSil8bla during the development of the embryo 1 The obvious model for setting up 8 simple gradient is illustreted in Fig. 1. At one end of a line of cells one postulates a scwce-a cell which produces the chemical (which I shall call 8 morphogen) and m8inteins it at 8 constant level. At the other end the extreme oell 8ct.a as a sink: that is, it dastroys the moleaule, holding the aoncentration at that Doint'to 8 fkd low level. The morphogen can difbe from one cell to another along the line of aalls. @bea a time the system 8ppro8ahas 8 dynamii equilibrium, 8nd it is easy fo show thet if the effeative diEusion constant is everywhere the 8ame, the concentretion gradient will be lineer. Of course, real embryological struotures will have three dimensions, but if for convenience w-e restrict ourselves to sheets of cells such 88, for exBmple, the insect epi- dermis* or the developing Bmphibian retina6 the problem becomes two-dimensional. The source a8n be aonsidered to be 8 line of cells (the line being perpendiauler to the paper in Fig. 1) end simil8rly for the sink, thus reducing the problem to one dimension. It is not difbdt ta doulate how long it would teke ,to asat up suah 8 systam, eupposing that both the sourae and the sink are turned on at time xero. Diffusion is 8 random walk process, and the dimensions of the diffusion constant, D, 8re L'T-1 (where L in length and T is time). This should be contrasted with 8 meahanism h8ving 8 velocity (with dimensions LT-1) aa proposed, for example, by Goodwin and Cohen'. Because in diffusion the length anters sa tha square, pura diffusion proaeases 8ro very rapid over rrrther short distanoea (say, the size of 8 aell) 8nd very slow over long distancea (6ay, the eize of an organ). The concentrefion 8pproaehes its fin81 Value asyrnp- toticelly, so one must h8ve some criterion for deciding whether the gxudient et 8ny time is stioiently olose to 8 straight line. I heve 8rbitmrily taken the gradient to be e&atively established when it is everywhere within A0 of the M value, 8nd ahosen AC 8s 1 per oent of C,, (a, is the m8ximum value et the origin). It would make little difference to the argument if AC were considered to h8ve half thie v8lue. The gradient might be set up in various ways, The resultine8aheasec8nbeexpressedss where I = timo in seconds to set up the gradient; n= number of cells between source 8nd sink; I=length of eech cell, in cm; and D=difhion constant, in cm' s-1. i A is 8 numeric81 eon&& the BIBct velue of which will depend on the w8y the gredient is developed. hfathemtrfically the simplest way is to start with zero conoentretion of the morphogen everywhere at time zero, and thereah to m8int8in the source at concentration C?, and the sink 8t conoentr8tion zero. This gives e value of A of 0.42.. Biochemicellv more teelistia models nire values only 8 litfle hger &8n this, so a good gen&al value for A would be 04. It was uointed out to me by Dr Aaron Klug thet. if the St&l aonaentration wore rmiformlv a.12. the time reauireci is redUced to 8 little less thenl on&&ktm, end A- will have e value of about O-09. &fore raalistia models of thla general sort give values of A of, say, O-16. The calculstione of A were carried out by Mrs Bf8ry Munro. In wh& follows, I sh8ll assume th8t A is O-6 (the simple mechanism), but it should be remembered that the organ- ism might ba 8ble to raduae tbia to about a third of this V8h3. The diffusion constent in water for 811 but the smallest molecules-provided they are roughly spherical-is in- versely proportion81 to their mean radius, to 8 neer 8pproxim8tion. Thus, inofeeeing the moleoular weight by 8 fhUtOr Of 1,99@ from, Sey, 8 6UU811 OrgeniO IUOk'.?llle such 8s ATP (mol. wt.' 607) to 8 very lerge~ protein like polynucleotide polymerase (mol. wt. about 045 x 10') reducea the diffusion constent only by 8 f8otor of 10. Now it is raasoneble to expeat th8t the morphogan will diffuse rether wpidly, end should be 8ble to p8ss fairly ficiently from cell to cell. It is also likely to be 8 rather speaific molecule. For these reasons I doubt if morphogens NATURE VOL. 225 JANUARY 31 1970 421 wJI turn out to be large proteine or common iorul' like K+ or Se+. An obvious choice would be an organic ~olcc~~lc of about the size of, say, cyclic AMP or a steroid. That is, with a molecular w-eight in the rangc 300 to 500. The diffusion constant0 in water (at 20" C) for such a ~&m&3 is about 4 or 6 X lo-' cm* s-l. (The diffuRion of salts Jikc SaCI or KC1 is about three or four times as fast 6s this.) no infiide of a cell is VOW far from be&g mado of water, md one must estimate the effective diffusion constant &thin a cell. This amounts to estimating tho offectivo v&osity. The cytoplasm being a concentrated mixture of ,nolecules having a large variety of sizes, the relative viscosity will be considerably higher than water at the mme temperature. For a small molecule, which can, as it bare, slip between many of the other molecules, tho eficctivc viscosity is unlikely to be as big as the bulk viscosity of the c~tiplaem (wherever that may be). It is difficult to make any pFccise estimate of the effective diffusion constant, which in any case may vary considor- ably between different types of cell. Allowing a factor of incremo of viscosity of x 0 (corresponding to a sucrose solution 40 per cc& by weight), &hich &ms not nn- rcasonablog, would make the cffectivc diffusion constant about 098 x,10-6 cm* 0-I. (usually stcreospccific) and saturatablo 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 believed to have this character. At low concentrations (that is, far from saturation) the mechanism aan bo doearibod by a permeability, P, meas- ured in cm s-1. It is easy to show that the effective diffu- sion constant, D', for our problem is given by l/D'= l/D+- 1IPl, where E is the length of each cell in tho direction of &ffusion'l. Thus, if D is 0.8 x lo-* cm' s-l and I= 10 pm'. (sav) we sea that if tho "resistance" to flow'of the morpho- kc; because of permeability between cells was equal to-that duo to diffusion within a cell. P would have to have the value 8 x 10-a cm/second. This is a high value, but prob- ably not impossibly high". If we arbitrarily take P 88 about half this and D as before, we obtain D'fiO-27 x lo-` cm s-l. We now need an experimental estimate of the t&e needed to set up a gradient. This is not oasy to obtain. Most embryologists would feel that a day is too,long. I, A minute seems far too short. A few hours would seem ab&ut right-Wolpert has suggested that between 5 to 10 h is not unreasonabln for many of the well-studied cases (ref. 1, page 41). I shall assume a figure of 10's (approximately 3 h), because some time must be allowed for the changes which take place after the gradient is sot up. Combining our formulaL( wo obtain and uubstituting the chosen values: t = 10' s, -4=0*5, I= 10 pni, D = 0.8 x 10-e cm' s-1, P= 4 x lo-' cm s-1 we obtain 7~270 ~011s. If 2 were 30 pm, n would come to a little over thirty cells. Even allowing for tho'vcry approxi- mate nature of the calculations, the agrcemont with the figures given in the quotation at the F&I% of the article is strikinn. In broad outline what the calculation shows is. that, f& the times considered, distances or the order of a millimetre (or less) are possible, but distances of a centi- metre are too great". Of course, for organisms which . . develop very rapidly the distances tiould have to be smaller than a millimotre. We can t.ake it, then, that assuniing t.he effective viscosity of cytoplasm has not been grossly underestimated, : and provided there is a special mechanism to increase tho , I # : rato of permeability of the morphogen between cells,th ere 0 1 , are many c&808 in embryology where the times nnd I I I I distances involved are quit0 compatible with a mechanism &-me--m I bssod on diffusion. It is important, however, to make two _ --------- & r0.scrvation.s. There may be special'cases, involving setting up gradients quickly over large distances (of the order of several ocntimntres) which may require other mechanisms, 4 1. - nl a such ns tho signnlling devices suggested by Goodwin and Cohen'. Cases of "mushroom growth" (as, for example, the FIG 1. To iilrlstrnte how R nource ntid B Rink can produce n linear `Cdh~ Of concn~tmtlon. growth of mushroom) are unlikely to be due to diffusion Each cdl In the line has length 1 cm. The (lMnnCe bctwcn ROIWX nnd Rink is I, cm. Bccauw. n In the number of cells in the line, A=nZ. alone. fiecondly, in one's enthusiasmfor.diffusion, it is important to realize t,hat the many other problems remain to be tackled. Even whon the gradient has been set up the co11 has to mcornizo it. Because at least in the insect How does the molphogcn get from cell to cell ? It would scorn inefficient to mak; al7 cell membranes easily per- meable to it. To do this it vr~uld in anv case have to be verv Small and rather hydrophobiclo. MO&over, a high gene&l pcrmoability would all& the molecule to &cap