CONCEPT OF IMPOSED GRADIE ON EMBRYON MEN A.K. Saha Embryology Unit, Indian Statistical Institute, 203, Barrackpore Trunk Road, Calcutta 700 035, India ABSTRACT It is observed in Nature that the einhryonir developments of different kinds of species require different type of environments. It is ;iswmed ill this pitper that there niay exist some kind of imposed gradient or pre-as\igned gradient, generated as a resultant effect of the environnient, acting on the developing embryo. This system bas heen studied by constructing a siiitahle iiiatlieniatical rnodel of epigenetic mechiinisin. It has been observed that the iiitroduction of this: itriposetl gradient ran easily change any exiting pattern of the ein1)ryo. Modcl analysis has been pcrfornied in the line of Turing ;ipproach. Key words : Epigenetic tnech:inisin, 4111 coefficient, Bifurcation, imposed gradient. INTRODUCTION The aim of this work is to show how an existing pattern within an embryo generated by a reaction diffusion system can be deformed hy imposing a gradient. It is assumed that the gradient is generated in the extra cellular space of the emhryo via some unidirectional signals. As an application we have considered here a simple mat1iein:itical model proposed by Tapaswi and Saha (1986) on epigenetic mechanism which is essentially the case in inany different embryos. MA THEMATICAL MODEL ?I?e incrth~~iritrtic.cr1 irrodel, considered here, involves mRNA(X), rqulotor (y) (which itself is a protein, perhaps an enzyme) and rnoryliogcvi (Z) taking into account the self and cross-d@usion qf morphopt I (z) . with zero-jlux bounrlriry rotidition. We shall consider a qlindrirally slropd mhiyo as (I modcl. Tlilzerq%re A' denms the Laplace opmitnr in the cy1intlric.d coordinates. Here p arid yi > 0 (i = I ,23) are the systein paraineters ad p is the Hill coefficient, 7 is a real parairretc~r and 8 is the imposc.11 grudient. D, z 0 and aj3 = D, Z 0. D' = (b,) ij = 1,2,3 ; whmJ ull b, ure actp bJ3 = D, # 0. INHOMOGENEOUS CONCENTRATION PATIERNS IN CYLINDRICAL COORDINATES When the imposed gradient 6 = 0, (1) becomes with zero-flux boundary condition. The inhomogeneous concentration patterns emerge as a solution of GF(GQU + Y,~H(CQU + DA~U - o (3) Here wo = (%,yo,%) is the steady-state. (3) has a non-trivial solution U(r) # 0 when det [GF(od + y3 6H(o,) - ej D,,] - 0 which implies U - ll Y3 = Y3" = K&D33[ Y,(l+Y, Y2 P Y,"") where D, = -DZ3l as D,' > 0 and D:3 a-- D33 (4) (5) represents the line of bifurcation below which the system is stable and above unstable . Thus Turing (Turing, 1952) structure is possible when y3 2 yc. Solutions of (3) is given by 'UL - v nkj 6 cos(k~t/h)zJ, (x,r/R) cosn 4 (7) U* and ?iT can be found from < U,, U, > = 1 and [6F(wo) + y3 yGH(oo) - k,kj2D] 6 = 0. Here h and R are the height and radius of the embryo. For n = 0,k = 0 andj = 3, we have a real pattern of 3-germ layers ectoderm, mesoderm Proceedings RC-IEEE-EMBS & 14th BMESI - 1995