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![MARY MCN'RO 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 ( DCJL).) The concentration will approach this final value asymptotically, lo 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 „ L -X 2 C„ ™ I . (tlTTx\ . „ £ - C -T—7«5 1 ; 8 1 vxr xp( ' r ' + j 2 sm j-pj expi - J (2) where c =f(x) at t — o, and thereafter c = C 0 at x = o, c — o at x = L. We have expressed time in a convenient dimensionless form by putting T = (DtjL 1 ). In the first instance let us assume that the initial concentration is even - where zero (i.e. f(x) = o). Then at any given time the maximum value of AC, the difference between the concentration and its final value, is at the midpoint x = iL, because of the symmetry of the problem. For this special case the value of AC is given by AC = Sjsin^exp(-hW). (3) If we only consider cases in which AC is small, we need take no more than the first two terms, so that AC = - :^ [exp (~7 t 2 T)- ^exp ( - 9 n-T).. .] (4) and usually the first term alone will suffice. In Fig. 1 we plot the value of |AC/C 0 | against T (for the middle point). For example, if |AC/C 0 | is taken as i % then T has the value of 0-42. We have also computed the whole course of the concentration curve for certain selected values of T, using equation (2). Linear gradient with initial constant background A smaller value of T (for a chosen value of AC C 0 ) can be obtained if we allow the tissue to have a uniform concentration of the morphogen at time](https://iiif.wellcomecollection.org/image/b18173020_PP_CRI_M_1_4_0034.jp2/full/800%2C/0/default.jpg)
