B U L L E T I N OF MATHEMATICAL BIOPHYSICS VOLUME I9, 1957

TOPOLOGICAL BIOLOGY: A NOTE ON RASHEVSKY'S TRANSFORMATION T. ERNESTO T R U C C O MENTAL HEALTH RESEARCH INSTITUTE U N I V E R S I T Y O F MICHIGAN In t h e b i o - t o p o l o g i c a l t r a n s f o r m a t i o n b e t w e e n g r a p h s d e n o t e d b y ( T ( I ) X) N. R a s h e v s k y (Bull. Math. Biophysics, 18, 1 7 3 - 8 8 , 1956) c o n s i d e r s t h e n u m b e r of f u n d a m e n t a l s e t s w h i c h (a) h a v e o n l y o n e s p e c i a l i z e d p o i n t a s s o u r c e ( a n d n o o t h e r s o u r c e s ) , (b) h a v e n o p o i n t s in c o m m o n ( a r e ~ d i s j o i n e d ~ ) ; h e p r o v e s t h a t t h i s n u m b e r i s a n i n v a r i a n t of t h e t r a n s f o r m a t i o n . In t h i s n o t e we s h o w t h a t R a s h e v s k y ' s T h e o r e m c a n b e e x t e n d e d a s f o l l o w s : The number of fundamental sets of the first category is an invariant of the transformation. We must~ h o w e v e r , c o u n t t h e s u b s i d i a r y p o i n t s of t h e t r a n s f o r m e d g r a p h a s s p e c i a l i z e d p o i n t s . We r e c a l l t h a t f u n d a m e n t a l s e t s of t h e f i r s t c a t e g o r y a r e t h o s e w h o s e s o u r c e s c o n s i s t of s p e c i a l i z e d p o i n t s o n l y ( T r u e c o , Bull. Math. Biophysics, 18, 6 5 - 8 5 , 1956). But in t h i s m o d i f i e d v e r s i o n of t h e T h e o r e m t h e f u n d a m e n t a l s e t s m a y h a v e more than one source and need not be disjoined.

In studying some properties of his bio-topological transformation T, N. R a s h e v s k y (1956, pp. 1185-87) has proved the following theorem: The number of fundamental sets (FS's) which (a) have only one s p e c i a l i z e d point (and no others) as source and (b) are disioined is an invariant of the transformation; i.e., it is the same for the transformed graph T(P) as for the primordial graph P. In this note we show that R a s h e v s k y ' s theorem can be generalized in two ways, namely: (1) the F S ' s in question may have more than one source, provided only all t h e s e sources are s p e c i a l i z e d points (or subsidiary points in the transformed graph), (2) the F S ' s need not be disjoined. In the transformed graph, T, we distinguish between specialized, subsidiary, and nonspecialized or residual points. If we u s e the term " n o n r e s i d u a l " to denote both s p e c i a l i z e d and subsidiary points, we may c l a s s i f y the F S ' s of T as follows: a) F S ' s of the first category; their sources c o n s i s t only of nonresidual points, 19

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b) FS's of the second category; all their sources are residual points, c) FS's of the third category; their sources include both nonresidual and nonspecialized points. The same classification applies to the primordial graph P, in which, however, " n o n r e s i d u a l " and " s p e c i a l i z e d " mean the same thing, since there are no subsidiary points. In a previous paper (Trucco, 1956, referred to as I in the following) we called p; p"~ and p"" the number of FS's of the first, second, and third category in P. With the above terminology our theorem may now be stated as follows: The number of FS's of the first category is an invariant of the transformation T. As in I we denote by Ps and P5 the graphs obtained after the steps T a and T 5 of the transformation. Also, we call s(P) the i specialized points of the primordial graph P, and s i (or Sjo ) the corresponding points of Pb; the subsidiary points attached to sj are called s i v ( v >1 1). The index j runs from 1 to n, and the index v from 0 to /1; the lj's are non-negative integers such that n

Eli

= n(m - 1).

j-1

The correspondence between the point-bases of P and P8 was made clear in I. Furthermore, each non-subsidiary point of P5 is a source if and only if it corresponds to a source of Ps" However, some of the subsidiary points in P5 may also become sources. In particular, if the specialized point s i is a source of the third category in P~ all its subsidiary points sir are also sources, whereas if s i is not a source none of its siv's can be a source (this follows easily from the rules of the transformation T and from Theorems 1 and 2 of I). There is thus a one-to-one correspondence between the FS's of the first category in P and in Pb" The same is true for the FS's of the third category; the number of FS's of'the second category, on the other hand, is multiplied by m in going from P to P~, so that the point-base number of P5 has the value p" + roD'" + p ' " . We still have to consider the last step of the transformation T, i.e., the addition of n(m - 1) residual graphs R'/~, one such graph being attached to each of the subsidiary points. First of all, since the residual graphs consist of nonspecialized

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points only, their presence cannot add any specialized sources to the final graph, and therefore the number of FS's of the first category, g'~ will not increase. Next, let II be a FS of the first category in P5 and B the pointset formed by its sources. All points of B are nonresidual, a n d any two of them are connected by at l e a s t ...sone way. Denote by Bin the point-bases of the residual graphs Riv. The FS II will c e a s e to exist and become part of some other FS of the secbnd or third category, if and only if there is a way going from at l e a s t one of the B iu ' s, say B-~i~, to B (i.e., from a point of B-~ to a point of B). Since the connection between B~i~ and P5 is made through a subsidiary point, say s ~ , t h i s implies that: (a) there is a way from -B;u to s;v, (b) the subsidiary point s;~ is a source of II (I, Theorem 2). This, in turn, means that the specialized point s~ must b01ong to II and also be a source of II. But then condi6on (a) is not satisfied (I, ,end of page 80), and hence the number of FS's of the first category remains constant throughout the transformation T, which is what the theorem states. Moreover, let HI(P) , Hz(P) be two FS's of the first category in P, H1, II 2 their corresponding FS's in T. Then it is e a s y to see that II 1 and II 2 will be disjoined if and only if the same holds for [I~(e) and H2(P) Furthermore, if a FS [I(P) of the first category has o~tly one specialized point (say si(P) ) as source, the same is true for the corresponding FS II in the transformed graph: none of the subsidiary points sir attached to sj can become a source (this does not hold for some other types of transformations considered by N. Rashevsky). In this way we obtain Rashevsky's special case of the theorem. LITERATURE Rashevsky, N. 1956. "What Type of Empirically Verifiable Predictions Can Topological Biology Make?" B u l l Math. B i o p h y s i c s , 18, 173-88. Trucco, E. 1956. " A Note on Rashevsky~s Theorem about P o i n t - B a s e s in Topological Biology." Bull. Math. B i o p h y s i c s t 18, 65-85. RECEIVED 8-16-56