Some theorems in topology and a possible biological implication |
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Authors: | N Rashevsky |
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Institution: | (1) Committee on Mathematical Biology, The University of Chicago, Chicago, USA |
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Abstract: | In a previous paper (Bull. Math. Biophysics,16, 317–48, 1954) a transformationT of one graph into another was suggested, which may describe the relations between organisms of different complexity. In this
paper some topological properties of the transformationT are studied. It is shown that the fundamental group of the transformed graph is homomorph to the fundamental group of the
original graph. An expression is derived for the number of points in a point base of the transformed graph in terms of the
number of points of the point base of the original when the point base of the latter consists only of residual points, and
it is shown that the ratio of the number of points of the point base to the total number of points of the graph is in that
case greater in the transformed graph than in the original. A combinatorial problem arising in connection with the transformationT is solved by deriving the number of possible ways in whichn-n
i
indistinguishable elements may be arranged inn
i
classes, permitting some of then
i
classes to be empty.
The possible biological meaning of the increased ratio of the number of points of the point base to the total number of points
of the graph is discussed. It is suggested that it may be interpreted as a decrease of regenerating ability with increase
of differentiation of the organism. Those considerations suggest the possibility of deriving some general biological laws
from the consideration of the properties of the transformation only, regardless of the choice of the primordial graph. |
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Keywords: | |
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