A Probabilistic Model for Morphogenesis 



365 



These configurations bearing llic same probability, while they are not isomorphic 

 in the sense mentioned earlier, have an important property in common. If 

 each configuration is represented by a graph (13), identifying the cells as nodes 



ir, 12 7r_ 12 TT-U ■"•-14 TT-M T'-M '"'U 



s s 



(2) te (') ft ") ^ (1) t# (1) E§° (2) cfe <') S 

 9.2642 6 SS30 4.4603 3.S67S 3.3181 3.1906 2.6296 



TT- 14 V 14 TT^U Tr=14 TT-H Tr^\t TT^IS 



(» #3 (')[ffl3^ Wdffi^ <5)cnrffl (3)%° (') ^ (') P^ 



25119 2.3261 18499 1.2956 0.7837 0.7730 0.5952 



7r= 16 77- 16 7r.i6 7r= 16 '"■=16 "■ 16 



'" ^ '" ^' '^''rrftTT. '" 4™' ""rftrrn "' rfWi ^ ''' ^n^ 



0.5357 0.5060 0.4067 0.3869 0.3075 0.2976 0.2679 



TT 16 V 16 TT 16 7r=16 TT; 16 IT 16 



S „ S rn rn S S S 



(') °#^ (i9) ftTTTn (ugi™ (I) H (luElnfl 



H I I I I II 

 0.2530 0.1587 0.1554 0.1488 0794 0.0397 



7- ARRAY 

 Fig. 2 



"■=12 rr=i4 ir=u ir=i4 v,u ir=u w-i4 in^u itm ir^n v^u 



(">& <'>^ ("^ <«^ 0)^ ">^ ("te^ 0)^ (»fe (l)ffiffl (4) rT4ffl 



5-12 3.50 3JS 3J4 3.01 2.61 2J5 2J« 2.19 1.12 



■ir=i4 77=14 Tr-14 ir=i4 vm ir=u 7r=i4 v-}i itM ■"■=16 



1.66 

 "■=16 



(I) 



(1) 



1.40 1.30 



(1) 



"'cfflD '" 



(2) 



rffFP 



(2) 



(2) 



(I) 



(2) 



(3) 



1.127 1.094 0.9462 0.6456 0.7292 0,7257 0.6841 



Wz. 16 7r=)6 ir=i6 Tii6 TTiii Tr=i6 ir=]i 



S 



<«a& '"^'"'d* '« cP "'-f "'c#^ 



(3) 



(5) 



16 _7r=i6 ir=i6 



$, 



(I) 



(IS) 



7r=i6 

 w>cflffl 



0.62719 0.6093 0.5960 0J77O 0.5280 OJ20i 0.4716 0.4340 0.3672 0.3480 0.3364 



IT: 16 Tr=u "=1« "^16 ir-l« "=i< "^16 ir=.M ir^M itm "=16 



s s „„ p n PS 



(2)rTffP (l)'#^ (lO)rrTfFP (1) ^ (2) rrFRn (1) #5^ 112) rrrrfB (2) F#P (4) 



0^040 0l2920 0J644 0JZ79 02170 Ol20K1 0.1707 0.1649 0.162S ai520 0.I4S8 



"=14 "^16 "=14 "=16 V^U ■"=16 "=16 ■"-16 '"=16 "=.16 "-16 



(1) ^^ (8) ^ (I) ^' (1) °f°'(13)nflg3 (3) ^ 0) !§: '(15)B^ (i) CdgJ (4) ^ (3) ^ ' 



ai42l 0.1190 0.1166 0.1116 0.1073 aiOtO 0.1642 0.0043 0.0613 0.0769 0.0744 



T»I6 ir=I6 77^16 7r=16 "=16 



osi 



TT-U ir^M -77=16 ir=16 '"=16 ■"=16 



rrrWl (SWrrftrfl (D "^ (2) S=™ (37) ^TTTf^ (3) rftrfll (5) Ftfffl 0) M (52) illB O §™^ 0) g^ 



aOMS 0.0546 0.0531 0.04216 0.0391 0.034« 0.0394 0.019* O.Om 0.0196 0.0192 



Tml6 "--16 



aoo** 0M496 8 — ARRAY 



Fig. 3 



and the covered edges as branches between nodes*, it will be seen that all 

 configurations represented by the same graph have the same probability. 

 Certain other properties are also suggested by consideration of these small 

 'organisms'. The configurations with the largest number of inner cornersf are 



* This representation is analogous to the graph obtained by identifying countries on a map 

 with nodes and common frontiers between countries with branches. 



t We can use the perimeter, n, i.e. the number of open edges, instead of the inner corner, C, 

 in describing the property in question since tt ^ 2{k i- 1 — C). 



