THE TIME NEEDED TO SET UP A GRADIENT:
DETAILED CALCULATIONS

By MARY MUNRO anpF. H.C. CRICK

Medical Research Council Laboratory of Molecular Biology,
Hills Road, Cambridge CB22QH, England

INTRODUCTION

In embryological development there ate many cells which acquire ‘posi-

tional information’ (see, for example, Wolpert, 1969) and it is plausible

that this is obtained from the concentration of certain unknowti chemicals

(called here morphogens) which are diffusing down concentration gradients

set up by sources and sinks at special places in the embryo. It is thetefore

important to be able to calculate how much time is needed to set tip 4 steady

concentration gradient. This problem has been discussed in a recent paper

(Crick, 1970) which suggests that the times available are roughly what might

be expected on theoretical grounds. Here we give the details of various’
calculations quoted in that paper.

The physics and mathematics of diffusion are fairly straight-forward.
The classic references are two books: The Conduction of Heat in Solids,
by Carslaw and Jaeger, 2nd edition, 1959 (here referred to as C. and J.)
and The Mathematics of Diffusion, by J. Crank, 1956. Results which could
not be obtained easily in an algebraical form were calculated by computer.

Although a tissue consists of discrete cells, we have often found it conven-
ient to treat it as a continuous medium, in which the morphogen has a diffu-
sion constant D cm?/s. On the other hand, in calculations on the computer,
this continuous medium has usually been approximated by a series of dis-
crete points. Details of these computations will be given later. For reasons
explained in the earlier paper (Crick, 1970) it is reasonable to calculate
one-dimensional cases, at least in the first instance.

A SIMPLE LINEAR GRADIENT

Mathematically, the simplest model to consider is one having a source at
the origin, holding the concentration there to the value C,, and a sink at the
point x = L, holding the concentration there to zero. If the diffusion con-
stant, D, is everywhere the same, then after an infinite time the concentra-
tion will tend to the value

Cac!

- —x

LS
[ 439 J

(o> x > L). (1)
440 MARY MUNRO AND F. H. C. CRICK

(Note that this expression does not contain D. However, the flux per unit
area does depend on D, and is given by (DC,/L).)

The concentration will approach this final value asymptotically. To
calculate the amount by which the concentration at any time differs from its
final value we adapt the formula given in C. and J. P- 99, section 3-4, equa-
tion (1). The general value c for the concentration at the point x at time t
is given by

 

E-x_2Cy © =sin ("F* exp (—ntat)

where c=f(x) at t=0,
and thereafter c=Cy at x=0,
c=o at x=L,
We have expressed time in a convenient dimensionless form by putting
T = (Dt/L?).

In the first instance let us assume that the initial concentration is every-
where zero (i.e. f(x) = 0). Then at any given time the maximum value of
AC, the difference between the concentration and its final value, is at the
midpoint « = }L, because of the symmetry of the problem. For this special
case the value of AC is given by

AC = — 70'S Lsin™ exp (—ntat), (3)

If we only consider cases in which AC is small, we need take no more than
the first two terms, so that
2Cy ,

AC = —~* [exp (—7T')—Jexp(—97°7)...] (4)
and usually the first term alone will suffice. In Fig. 1 we plot the value of
|AC/C,| against T (for the middle point). For example, if |AC/C,| is taken
as 1%, then T has the value of 0-42.

We have also computed the whole course of the concentration curve
for certain selected values of 7, using equation (2).

Linear gradient with initial constant background

A smaller value of 7 (for a chosen value of AC/C)) can be obtained if
we allow the tissue to have a uniform concentration of the morphogen at time

THE TIME NEEDED TO SET UP A GRADIENT 441
zero. Let this be aC,. To be of any advantage « must be less than unity,
In this case equation (2) becomes

~4E 2 3 Ur—cosnm)—1 (“) exp(—n'nT), (5)

Cy math n
It is only necessary to take the first two terms of this series, since for the
particular times we are interested in, the third and higher terms are neglig-
ible (for T= 0-1, n = 3, exp(—n°7°T) = 2x 10-7). For the special case

 

 

0-00 ! ! l t !
0-20 0-25 0-30 0-35 0-40 0-45 0-50 0-55 0-60 0-65 0-70
Time T

Fig. 1. AC/C, for the mid-point of the gradient as it approaches its final value.

a=} the term having 2 = 1 is zero, and the maximum value of |AC}
occurs at the points x/L = } or }. It turns out that for values of a less than
unity but not too close to a = } (that is, when a is between o and O-4, or
0-6 and 3-0) we can neglect all but the first term. For |AC/C,| = o-o1 we
easily derive the explicit equations:

 

 

 

x I 200
at T= T = slog. aan),
x
at >=]. 1 100
L T= 73 !0Be “7 N2 (20 — 1)
or =}

where T is, in this case, the time at which |AC/C,| is 1% at the point under
consideration. By symmetry, T for x/L = t with initial background aC, will
be the same as T for x/L = } with initial background (1 —@) Cy. The values
thus obtained are set out in Table 1.
442 MARY MUNRO AND F. H. C. CRICK

 

0-35

Tt

O24

aoc
“me

0-1

 

 

} | 1 | ! J 1 I I
00 Of 02 03 04 O05 06 07 O8 OF 10
a

 

0-0

Fig. 2. Time taken for the concentration at all points to come within 1% of their
final values, for different initial backgrounds (a@C,).

Table x

Values of T for |AC/C,| equal to o-o1 for the point specified, for various values of
the initial background @C,, calculated using only a single (non-zero) term in the
expansion. The value in brackets is inaccurate.

At At x/L = } (foro < @ < a5)

a x/L = 4 or x/L = } (for o-5 < a@ < 1-0) a
0°05 O°410 0375 0°95
o'10 0°398 0°363 9°90
O15 0385 0°350 085
0:20 0°369 0°334 o'80
O25 O°351 O°315 o-75
0"30 0°328 0'293 0'70
0°35 0°299 0264 0°65
"40 0258 0'223 0-60
o-45 0188 (0-15) Oss

0°50 90°000 0088 0°50

THE TIME NEEDED TO SET UP A GRADIENT 443

For values of « outside the limits 0-45 to o- 55 the last point to fall within
the 1% limit is always the mid-point. However, this is not true when
a = 4, since AC at the mid-point is at all times zero (because of the sym-
metry), and the last point to fall within the 1 % limit is x/L = } or x/L = 3.
For values of a in between 0-45 and o-5 the last point will be somewhere
between x/L = } and x/L = 4, and can be roughly estimated. For example,
when a = 0-475 the last point is approximately x/Z = 0:35, when T ~ 0°126.
We then adopt as a criterion the condition that all points on the gradient
must be within 1% of the final concentration difference between the ends
of the gradient from their final values, and define 7” as the time taken fot this
to occur. Fig. 2 shows the approximate value of 7” for all values of a between
oand 1. The values near @ = }, shown dotted in the curve, have been roughly
estimated from computer calculations for a = 0-45 and & = 0-475,

It will be seen that although a rather small value of 7” can be obtained if
a is exactly 4, the value of 7” rapidly increases if « differs appreciably from
this special value. However, if the level of a were within + 10% of this
special value, 7’ would always be less than o-I9.

Other initial conditions

We have shown that if there is zero concentration initially, the time taken to
set up an almost linear gradient is longer than for any other initial UNIFORM
concentration less than C,. By symmetry this time is the same as the time
when the initial concentration is Cy throughout.

We now prove that if the initial concentration has any arbitrary form,
but is always less than C, (and always positive), it takes a shorter time to set
up a nearly linear gradient than the simple case with zero initial background.

The time taken to set up the simple case is approximately T = 0-4.

When T = 0-4, exp(—7°T) = 0-0193 and exp (—47°T) = 1-7 x 1077.
Looking at the concentration equation (2), the second term is

co
ae “sin (") exp (—n*7?T).
Whenn = 2and T ~ o-4 thisis always numerically less than (1-7 x 10-7) C,/7
and is clearly negligible.

Whenz > 3 and T' ~ 0-4 the terms are very much smaller. Looking at the

third term of the equation (2), since

0 < f(x) < Cy by definition

and ~—I <sin (“Z") < +1 forall x andn,

_ fe) dx < | fe) sin ("F") dx < i i (x) dx
444 MARY MUNRO AND F. H. C. CRICK

L
and I f(x) dx is clearly less than C,L,
0
7 (nix
therefore -Q)L< I, f(x) sin (*F") dx < CyL.
a _ are Ly. (nm\ ,,
So P| L ) exp( nn r)| J(*’) sin (7 ) ax

is also negligible for n > 2 when T is as great as 0-4.
So the concentration function ignoring 1 > 2 is

c= aes —2sin ey) exp(—7°T) [= _ i | . J(«')sin (*) a:'|

AC = asin (=) exp(—7°T) [s -; I . ‘A (sin 7) as'| . 6

When f(x) = 0, initially we get

thus

JAC| = 2 sin (=) exp(—7°T),

as before, but 0 < f(x) < Cy

. (1x .
and o <sin (F) <1 forallxintherange o<x<L

 

 

7. nN {me\ ,, LL (nx' 2C)L
therefore o < {, F(x’) sin ("*) dx’ < Col sin ( Z. ) ax <—.

Putting this in equation (6)

|AC| < 7 sin (7) exp (— 727)
and the equality is obviously the simple case f(x) = 0 initially. So no initial
conditions, subject to the restrictions o < f(x) < Cy for o < x < L, will pro-
duce longer times to set up a linear gradient than the case of initial uniform
concentration of zero, provided that our criterion for setting up the gradient
is such that |AC/C)| is a small percentage.

THE TIME NEEDED TO SET UP A GRADIENT 445

SOME SPECIAL CASES
Step function

For times shorter than 7 = 0-4 the terms in n = 2 will not be negligible.
In particular, the term when n = 2 is

sin (#) exp(—47°T) Ff, ne sin *F) dx’ —~ =| .

7

Clearly the largest possible value this can have at any time, within the con-
ditions imposed on f(x), will occur when

L ¢
z i} J(x’) sin (=) dx’
0

is as large, negatively as possible.

As the function sin (27x'/L) is negative only for x/L > $, arid f(x) is always
positive, the function which would give the largest possible value to this

term is the step function: .

f(x)=0 for o

J(x)=Cy for 4

This integrates to give

<x/L <4
<x/L <1.

LC,
oa (cos nm —cos 4n7).

As far as n = 2 we get from equation (2)
= 30559 (27) oxy (— an? T
AC= 7 sin (77) exp 47?T)

(the term in ” = 1 cancels to zero). When x/L = } clearly AC = 0 for all 7.
So we must examine the point x/L = }. When this is within 0-01 Cy of its
final value, we get
30, (7 aT) = &o
7 sin (7) exo 4rT) =

100’

x ~ log 3°°
T qn log, 7

& Ollie.

°
440 MARY MUNRO AND F. H. C. CRICK

Reverse gradient

If we start with f(x) = (x/L) C) (that is an exact reverse of the final gradient)

and calculate concentration function as far as n = 2, from equation (2)
%

|AC| = 2 Coen (7*) exp (—47°7)).

Again the centre point is always }C,, so we examine x/L =}:
Cy _ 2Cyexp(—47°T)
100 7 ,

I 200
Tx 4m log, 7

& O10.

oO

As we have shown, when T = 01, exp(—g7?T) = 2 x 10~7 so the third and
higher terms of equation (2) are negligible.

SUMMARY
Time to within
Initial concentration (C,/100)% of Point No. of
function f(x) final value considered terms

F(%) =o all x 0°42 x/L=%3 3
F(x) = $C, all x 009 x/L =} 2
S(x) = B5Cp all x O41 a/L = 4 I
Step function

f(x) =o,o<x/L <4 O'lls x/L =} 2

F(@®) = Cyt <x/L <1

Reverse step function

F(x) = Co < x/L <4 009 x/L =} 2
f(x) =0,4 <x/L <1

Reverse gradient
F(x) = Cy x/L 0°10 x/L =} 7

Flux required to maintain the gradient

This can be expressed in a very simple manner. Once the gradient has been
established the amount of the morphogen in the tissue, provided the sink
holds the morphogen concentration at zero, is clearly 4C,LA, where A is
the area of the tissue perpendicular to the gradient. The flux needed to
maintain it is(ADC,)/L. We now calculate the time, t,, needed for the source
to produce the amount of morphogen in the tissue at any moment. Since

ADC, 4C,LA

Lb

THE TIME NEEDED TO SET UP A GRADIENT 447

we obtain t, = $L7/D. Thus t, is roughly equal to the time needed to set up
the gradient from scratch to 1% Cy of its final value. In other words, if it
takes three hours to set up a linear gradient to 1 %, the source will need to
produce the amount of morphogen then in the tissue about every three hours.
This assumes that D does not change after the gradient has been set up.

MORE COMPLICATED MODELS

The models so far discussed although convenient for calculation, have the
disadvantage that the flux of the morphogen at the origin has to be very
high at small values of the time. It is therefore necessary to consider other
models in which the flux varies in a way which is biochemically more
realistic.

The obvious type of model to try has a ‘pump’ at the origin producing the
morphogen at a rate which varies with its concentration at that point. The
destruction of the morphogen at the sink is specified in a similar way.

We have tried two such models. In the first model the sink destroyed
the morphogen at a rate proportional to the concentration there, so that
the flux at sink was equal to 8C;, where C,, is the concentration at the point
x = Land f is a constant. The source produced it according to the formula

flux at source = A(C’ —cy),

where cy is the concentration (not necessarily constant) at x =o and C’
is a constant. It is easy to show that under these conditions the final gradient
will run from C’ —(f,/f) at x = 0 to the value f,/f at x = L, where Jo is the
final steady flux. If we call the total difference Cp, then

Cy 2h

We have carried out the calculations for various values of m, where m is
the maximum possible value of the ratio initial flux to final flux for the
flux at the origin, and expressed the result as the values of 7” for each of
three values of |AC/Cz|, namely 2, 1 and $°%. This is for the case of zero
initial background. We have also calculated 7” for the case with a uniform
background equal to the mid-point value of the final gradient and another
with go % of this value (« = 0-50 and 0-45). All these results are presented in
Table 2. As can be seen, the times are considerably increased over those for
our simple model. (In the simple model Cy, and C, are the same.) However,
we feel that Nature is quite likely to have evolved a more efficient pump,
with sigmoid rather than linear characteristics. This might be expected to
have the characteristics shown in Fig. 3. We have approximated to this by
448 MARY MUNRO AND F. H. C. CRICK

Table 2. Values of T’ for the simple pump model

Final values of No Background Background
conen (x C’) background o5 Cy 0-45 Cy
[ enema Y oc ™~ t ~’ t ~
m Source Sink 4% 1% 2% $% 1% 2% 4% 1% 2%
Io O90 O10 0°79, 0°68, 0°57 016 0135 O11 0°435 0°325 O21
25 0°96 0:04 0°58, O51 0°42, 0°13 O10, 0°08; O'31g O23 O15,
50 0°98 0°02 0°53 0°45, 0°38, OTT, 0°09s 0°075 O27, 0°20 O13
100 O99 GOI O'505 0°44 0-365 OII 0°09 0°07, 0°26 OIg O12

 

Table 3. Values of T' for the sigmoid pump model

 

 

Final values of No Background Background
concn (x C4) background o5 Cy o45 Cy
coos a ~~’ £ +

m Source Sink 4% 1% 2% 4% 1% 2% 4% 1% 2%
I's roO 00) (O'695 0°62 0°55 9 or16 O14, 0125 0°36, 0°29 0°22,
20 10) OO 058s O52 0°45 «= O'TZg O'ETS O10 0°35 0°25 O17
30 = 10) 800535 0°46 0°39 «2 rT 0'08, 0°28, 0:22 O'r4s
50 TO OO 0°505 0°44 = 0°363 OTE 0°90 0°075 0275 O20 0°13

In both Tables 2 and 3 the value of 7” is listed for four values of the pumping
constant, three different initial backgrounds and three values of the maximum
permitted percentage difference from the final value. (m is the maximum possible
value of the ratio of initial flux to final flux at the source.)

15

14

1:3

Flux/f

12

1-1

 

 

j 1 !
0-0 0-2 0-4 0-6 0-8 1-0

Concentration
C,
Fig. 3. The flux at the point x = o for sigmoid type of pump. (fy is the final flux
required to maintain a linear gradient from Cy to 0.) The dotted line shows the
approximation used.

THE TIME NEEDED TO SET UP A GRADIENT 449

mposing a maximum value of the flux, both at the source and the sink,
which operates from time zero onwards. When either the source or the sink
reaches the final concentration (Cy and o respectively), the concentration
there is held constant from then on. This is (roughly) equivalent to the dotted

10%
@ Simple pump model
© Sigmoid pump model

Flux/f,

 

 

I 1 I 1 1 !
0-0 0-1 0-2 03 0-4 0-5 0-6
Time T

Fig. 4. The flux at the point x = o for both pump models.

line shown in Fig. 3. As in the simple model Cp = Cy. The results for 7”,
where |AC/C;,| are 3,1 or 2% are shown in Table 3 for a = 0 (no back-
ground), & = o-50 and @ = o-45.

Fig. 4 shows the variation of the flux at the origin with time for both the
simple and sigmoid pump models. The cases shown are initial flux equal to
10 times the final flux for the simple pump (m = 10), and a maximum value
of 1-5 for the sigmoid pump. These take comparable times to set up a linear
gradient.

29 SEB 25
450 MARY MUNRO AND F. H. C. CRICK

A comparison of Tables 2 and 3 shows that, as expected, the sigmoid
pump is the more efficient, since it can reach a given value of T” for a lower
value of the initial flux. However, the important point to notice is that for all
the cases considered, the time J” is not increased grossly above the value for
the ‘mathematical’ model described first.

Table 4. Calculated concentrations for the simple model
At the point x/Z = 4

Algebraic Reiterative
method method
T = 005 0711384 (Co) 0711446 (C4)
or 0°26276 0°26288
Ors 0°35515 O35511
o2 O°41157 041148
o-25 0°44601 0°44591
03 0°46704 0°46696
0°35 0°47988 0°47981
o-4 0°48762 0°48766
O45 049250 0°49246
O"5 0°49542 0°49540

With initial background 0-5 Cy, at the point x/L = }

Algebraic Reiterative
method method
T = 0-05 0°70578 (Cy) 070560 (Cy)

o1 0°74386 0°74376
O15 0°74915 0°74912
o'2 0'74988 0:74988
O25 0°74988 0-74988
03 0°75000 "75000

The results presented in Tables 2 and 3 were calculated on a computer,
for a model with twenty cells between x = 0 and x = L. The diffusion
equation (@c/ét) = D(éc/@x*) (where c is concentration; D, diffusion con-
stant) was approximated as a finite difference equation:

Dot
c(x, t+ dt) = c(x, t)+ (éxy2 [e(x + dx, t) + c(x — dx, t) — 2¢(x, t)] (7)
where c(x, t) is the concentration at time ¢ and the point x, dt is the time
interval between each step, and dx the distance between each point. We
took D = o-o1, 6x = o-os5L and dt = 0-01 (the conversion to dimensionless
time gave T = (Dt/L*) = o-ort). .

The concentrations c(o, t) and c(Z, t) at the source and sink were changed

according to the specifications for the flux of each model. In the simple

THE TIME NEEDED TO SET UP A GRADIENT 451

diffusion model, where it was possible to calculate the concentration easily
algebraically, we compared the result with those obtained by the reiterative

" process using equation (7). These are shown in Table 4.

For the times we are interested in, the error is fairly small (although

105-

0-9

Concentration
° ° °
ro ~ oo
i t T

9g
wn
t

S
&
T

Ce

0-2

041+

 

0-0 \ t J ! t 1 t I
0-0 0-1 0-2 0-3 0-4 0-5 0-6 0:7 0-8 0-9 1-0

r (distance from centre of sphere)

 

Fig. 5. The final gradients for the 3p cases.

slightly larger for cases with initial background of 0-5 C,) where the con-
centrations reached the required values at small values of T.

So far our results have been for one-dimensional systems. However,
our general conclusion is likely to apply equally well to systems having two
or three dimensions. To illustrate this we have calculated the three-dimen-
sional case of a hollow sphere. The inner radius (r,) is held at concentration
Cyand the outer (r,) atconcentration zero. Computation was carried out forthe
ratio outer/inner radius ( = r,/r;) either 5:1 or 10:1. The gradients obtained
after infinite time (which naturally are not linear) are graphed in Fig. 5.

29-2
452 MARY MUNRO AND F., H. C. CRICK

The calculations (not reproduced here) show that by the time T = 0-45
the concentration is everywhere very close to the final values. In this case we
define T = Dt/(r,—1,)*. To compare our results with the linear one-
dimensional case we have arbitrarily considered the point midway between
the outer and inner radius, $(r,—1,), and have plotted the approach to
equilibrium with time in Fig. 6. The dotted line in each case shows the
point at which the concentration has the same value as the final value at
a point at a distance of one hundredth of the thickness away, i.e. at a distance

0-17 -

=

Concentration
°
—_
an

So
=
wu

 

1 1 1 1 L 4
5 0-40 0-45 = 0-50 055 060 0-65
Time T

3

030 «60.

~

0

F =5 Internal radius 0-2

°
=
°o

Concentration
So
°
oO

0-08

 

0:07 L L 1 1 i
0-30 0-35 0-40 0-45 0-50 055 0-60

r Time T
2 =10 Internal radius 0-1.

Fig. 6. The chosen point approaches its final value for both cases
of the 3p spherical model.

of (r,—7,)/100 from the chosen point, since this gives some measure of the
possible error of position which might be produced by an inaccurate
gradient, and corresponds to the criterion adopted for the one-dimensional
linear gradient. As can be seen, T, for this error is about 0-35 in both cases.

Although any comparison between the three-dimensional case and the one-
dimensional case is necessarily inexact, these figures show that it takes a

THE TIME NEEDED TO SET UP A GRADIENT 453

similar sort of time to set up a gradient whatever the number of dimensions
involved. This is, of course, exactly what one would expect from general
arguments based on the random nature of diffusion.

We thank the Computer Laboratory, Cambridge, for providing facilities
for us on their Titan computer.

REFERENCES

CarsLaw, H.S. & Jarczr, J. C. (1959). The Conduction of Heat in Solids, 2nd
edition. Oxford University Press.

Crank, J. (1956). The Mathematics of Diffusion. Oxford University Press.

Crick, F. H. C. (1970). Nature, Lond. 228, 420.

Worrent, L. (1969). ¥. Theoret. Biol. 25, 1.