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<too' caRily
from the tissue and this wouldnot onlv disturb the nradient
hut possibly interfere in other parts6fthe organigm. One
is thercforo driven to postulate a special mechanism which
. .
~110~s a rolativcly qiick passage of the morphogon from
one cell to another in the tissue of interest.  Whether this

involves tight junctions or other special structures remains
to be seen.

Such a mechanism could be facilitated diffusion. This is
cha &crized by being independent of energy, specific

integument t,ho "gradient" appears to impose a polarity
on those opidermal cells which become scales and bristles14
tha cells must have some additional mechanism (involving
miorotubulos ?) t.o do this.

In the case of the amphibian eye the retinal ganglion
cells must not only recognize the presumed gradient (so that
they know where they ax-o in the retina); each cell must
also oonveg this information to the far end of its QXN<~
axon, so &at it can make the conncxion at tho appropriate
alaco on the ootic teotums. Moreover. there am likclr to be

kubsidiary m&haniams to guide the growing bu<dic of
nerves along the right path to the tectum.
In t\o brilliant articles published about ten year+ ago,
Lock+ showed that the pattern of wrinkles obtained on


422

tho adult cuticle of Rhodnius after operations on tho
earlier larval stages (usually the last larval stage) can only
be explained by a "gradient" of some sort, running from
one i+emegmental membrane to the next, and repoating
in successive sogments. He showed convincingly that
neither mechanical, oxpansion alone, nor polarity alone,
could explain the results. Dr Locke kindly sent us tho
original photographs of some of his material.  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 tho diffusion oonstant disagree drastically.
We are thereforo emntly exploring a model in whioh
eaah cell in the epidermis attempts to maintain the con-
centration of the morphogen within itself t,o A previously
preset level, determined soon after tho gradient is first set
up. This model, which has only ono disposable parameter,
is a much better fit with Locke's data. More elaborate
hypotheses, within the basic diffusion framework, are also
under consideration. It is important, themfore, not to
approach these problems with too naive a model**.
  Finally, one should emphasize that gradients are unlikely
to command general acceptance until their biochemical
basis is discovered experimentally, and that this may not
Provo an easy task: Mathematically minded biologists
could well obje& 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
,abo~e) may only be coinoidental. In spite of theso possiblo
objections it is my belief that mechanisms based on
diffusion are not only plausible but r&her probable.
Nature usually has such difficulty evolving elaborate
biochemical mechfmisms (for example, those used in
protein synthesis) that the underlying processes are ofton
rather simple. If this approaoh serves to make the idea
of diffusion gradients respeotablo to embryologists it will
have served its purpose.

I thank my wife for drawing the figuro, my colleagues,,
especi&y Dr Peter L awrenco and Mrs Mary Munro, for

NATURE VOL. 225 JANUARY 31 1970

many helpful disaussions, and Professors Lewis Wolpert
and W. D. Stein for sending me information.

ILccelved January 2, 1070.

I Wolpert, L., J. T&oret. Biol.,25.1 (1969). This article, entitled "PosItional
Information and the Spatinl Pattern of Cellular DMerentlation", shotid
be consulted both for a modem statement of tbc problem nnd also for
mfemnces to eadler work.

* Tbo basic idea of this Article WIM
at the Fourth NATO Advance s ?CeOntOd'nt II ktUM ivall lo students
in July 1060 at SpeWI.   Study Institute of 3~olccular I)iology
                ,

a Cblld, C. M., Pallerns and Pfoblen~u of DacIopmenl (Cl~icngo University
Press, Chleago, 1941).
4 For a revfcw seo Lawrcncc, P.`A.. Adv. Insect JWsiol. (In tbo press). See
espoclalty Stumpf, H., How drch. Ent. dlee~. Org.. lSS, 315 (1~67).
' For D revlew aoc Qnzc, L N.. Growth and DWerenlinlion, dnn. Rev. Phpiol.,
29, GO (lU67).

a Qoodwln, D., and Cohen, 11. II., J. Thuoruf. Viol., 26, 10 (IOOD).
7 Tho meehnnlsm for formlng a source and a sink for Ions also preaenls
special problems, whereas for organic molecules rather slmpte enzpmatlc
  DIQCCSW could do the trick.

`The effect of tern
at&t the talc up" rature on the viscosity of writer does not seriously
           atlons.  Taking the vlseosity of writer (in nrbiw
units) as 1.0 nt 20' C, its value at 6' C Is about It and at 38' C IS ~1~
to 2/s.

S See, for example. n very ingerdous fluorescent method used by Victor W.
  Buma (Biwhem. Biophus. RN. Commnan., S77,100S; 1960) using a small
orgmlc molecule. He obtained 8 f8ctOr of nbout x 3 for Englena. The
  5gum for yeast was about twice this.
I0 Ueb, W. It., and Stein, W. D.. Nafure,224, 240 (1969).
It This assumea thnt the permenbttIty is fairly evenly distributed over the
  cell membrane. If It wem concentmted In a small patch the effwtire
  diffusion wonld be slower.

** For exnmplo, P for glucose in a&tea cells nt 8;' C in nbout 4.5 x 10-t cr&
  (Kolbcr, A. R., and LeFevm, P. 0.. J. Ocn. Ph!laid., 60, 1007; lQ3i).
or In human md cells at 37' C about 1 x lo-' cm/s 0flllar, D. Al.,
Biephfl8. J., 5, 407: 1035). Admittedly these are among the higher
  valued of P known so far. See Stein, W. D., The JZowmcnt 01 Mde&t
Acrorr Cdl Mmbrane8. chap. 4 (Academic Press, Rew York, 1967). It
  should be remembered that in going from one cell. to the next the
  morpbogen may have to cross two cell membranes.
1' An upper limit can be calculated assuming that the morpbogen Is an Ion.
  that It dlm.tsea (at 37' C) In the cytoplasm as freely as In water. that the
  permeability between cells Is so hi
  process at all, and that tho meet e $r that it does not stow down the

   adlent.             clent method Is used to set up the
Q    The distance tben come8 to l-6 cm, but I feel that this com-
  in8t10n of asaumptlons 1s quit0 unreslistic.
aa Pio ho, R.. Nalurwimnachqflen,
&                 Q
   Id.. 44. 807 (1036).       , 22 (1055): Lawrence, P. A., J. Ezp.
1' Locke, N., J. Ezp. Bid., 36. 450 (lQ6Q); J. Erp. Bid.. 17. 303 (lD30).