BULLETIN OF MATHEMATICAL BIOPHYSICS VOLUME 19, 1957

A NOTE ON THE GEOMETRIZATION OF BIOLOGY N. RASHEVSKY COMMITTEE ON MATHEMATICAL BIOLOGY THE UNIVERSITY OF CHICAGO In c o n n e c t i o n w i t h a r e c e n t l y proposed t o p o l o g i c a l model of an organism

(Bull. Math. Biophysics, 18, 31-56, 1956), it is pointed out that the dep e n d e n c e of some p r o p e r t i e s of the s t u d i e d t o p o l o g i c a l s p a c e s on t h e type of s p a c e in w h i c h t h e y are imbedded r e f l e c t s some a s p e c t s of the d e p e n d e n c e of the organism on i ts environment.

In a recent paper (Rashevsky, 1956) we suggested a p o s s i b l e " g o o m e t r i z a t i o n " of biology, by studying geometrical, and in particular topological, structures, the topological relations within which are logically isomorphic to the biological relations within an organism. We pointed out that one of the basic phenomena in biology is the p r o c e s s of s e l e c t i o n of certain elements of the environment, followed by subsequent assimilation. This selection by the organism occurs, however, only when the organism simultaneously " l o s e s " some of its elements either through catabolic breakdown which produces the energy n e c e s s a r y for the s e l e c t i v e p r o c e s s e s or by secretion of enzymes which act to a large extent as " s e l e c t o r s " on a molecular level. A selection logically implies the division of a s e t of elements in two c l a s s e s : the selected ones and the rejected ones. It was pointed out that c a s e s of division of a s e t in two c l a s s e s by a geometric structure are quite common in mathematics. They are, however, i n s u f f i c i e n t for our purposes. What we need is a s u b s e t M of a s e t S, such that M divides S - ?4 in two c l a s s e s if and only if M l o s e s some s u b s e t s of its own. Since, to the b e s t of our knowledge, no such s y s t e m s have boon studied in topology, we invented, as an i l l u s t r a t i o n only, a few c a s e s of topological s p a c e s which have the following characteristic: 201

202

N. RASHEVSKY

If certain points of a subspace M of the space S are removed, then the points of S - i become divided in two classes. One class is characterized by the points acquiring some property P with re. spect to M; the other, by the absence of this property. A space i which is characterized by this peculiarity we shall call a A-space. In the examples which we studied, the property P consisted either in making M connected, or in reconstructing U by being added to the lost elements of M. This situation as has been discussed in detail in loc. cir. presents some basic logical analogies to the biological situations in organisms. It was also shown on an example how we may arrive at empirically verifiable biological statements by " t r a n s l a t i n g " the geometrical properties into corresponding biological ones. The purpose of this note is to point out that the spaces which we have studied in loc. cir. have another characteristic which offers a formal logical analogy to organisms. Namely, the characteristic property of those spaces M depends on the structure of the space S of which they are subspaces. As an example consider the c a s e discussed on pages 48-49 and illustrated on page 47 of loc. eit. The space M, considered as subspace of S-- E 2, shows the n e c e s s a r y characteristics. Upon removal of the points fll and f12 which may lie on any of the subspaces A'~ A'*, B'~ or B'" of ~d, any Jordan arc of S - M, which does not intersect M, which connects M'" and fd"" (see loc. cir.), and whose end-points are cut points of either M'" or M"'~ acquires the property of reconstructing a second space M~ which is homeomorph to ~/but distinct from it. Consider, however, the same space M as a subspace of the space S~ which is a strip in E 2 contained between the lines pq and p'q" (Figure 1). Let M be situated with respect to S I as shown in Figure 1. Now, in order to achieve the same result, the points fll and f~2 must necessarily lie either both in one of the subspaces A'~ A"~ B'~ B"~ or they must lie both in either A ' U B" or both in A " U B " , If for example /31 lies in A') while/3 2 lies in B"~ then A'

g'

p

q

S'

B" 9

p, FIGURE

Io

q/

GEOMETRIZATION OF BIOLOGY

203

there is no Jordan arc in S I - M, which does not intersect M and which connects A" and B", If we consider M as a subspace of $2, where S s is shown in Figure 2, then M no longer possesses the characteristic s~udied above. Thus a given space M is or is not a A-spacer depending on the space S of which it is a subspace. All the above is perfectly obvious and to some extent trivial from a mathematical point of view. It is, however~ much less so when we "translate" the geometric relations into biological ones. To the quality of being a A-space there corresponds the quality of an organism to select certain elements from its environment and to eventually reproduce itself. The space S - M corresponds to the environment of the organism. Translated into biological language~ our above remarks amount to the statement that the ability of an organism to select elements of its environment depends on that environment. This at first also sounds quite trivial and obvious. The obviousness of this, however, is due to our familiarity with this fact which is so generally true. Of courses if the elements which must be selected by the organism are absent from the environment, then it is obvious that the organism cannot select them. But it is not obvious or self-evident at all that certain plants can select nitrogen from nitrogen compounds, but not directly from air. If those plants are left in an environment which does not contain nitrogen salts, they will not grow in spite of the fact that nitrogen is present in air. If a general theory of A-spaces is developed, then the mathematical conditions which the space S (or S - M ) must satisfy in order that a given M be a A-spacer will give us, when translated into biological language, the biological or physico-chemical conditions which an environment must satisfy in order that a given organism may exist in it. But this will have definitely a predictive value.

l

I FIGURE 2e

204

N. RASHEVSKY

This work was aided by a grant from the Dr. Wallace C. and Clara A. Abbott Memorial Fund of The University of Chicago. L ITERATUR E R a s h e v s k y l N.

1956.

~ T h e Geometrization of Biology. t~

Bull. Math.

Biophysics~ lilt 31-56. RECEIVED 11-2-56