Robert Rosen

A relational theory of biological systems II

The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.

THE REPRESENTATION OF BIOLOGICAL SYSTEMS FROM THE STANDPOINT OF THE THEORY OF CATEGORIES

A mathematical framework for a rigorous theory of general systems is constructed, using the notions of the theory of Categories and Functors introduced by Eilenberg and MacLane (1945,Trans. Am. Math. Soc.,58, 231–94). A short discussion of the basic ideas is given, and their possible application to the theory of biological systems is discussed. On the basis of these considerations, a number of results are proved, including the possibility of selecting a unique representative (a “canonical form”) from a family of mathematical objects, all of which represent the same system. As an example, the representation of the neural net and the finite automaton is constructed in terms of our general theory.

Recent Developments in the Theory of Control and Regulation of Cellular Processes

Publisher Summary This chapter reviews some of the theoretical aspects of regulation and control in cellular systems, and the implications which such theoretical work will have for the experimental cell biologist. The experimental biologist is familiar with the method of studying real systems of actual interest bymeans of model systems, i.e., systems which exhibit functional properties similar to those in which he is interested, but which differ more or less completely from them in physicochemical terms. Thus by the employment of enzyme models a biochemist can learn about enzymes by studying things which are not enzymes; a membrane physiologist can learn about biological membranes by studying collodion films and silica gels, which preserve basic functional properties of real membranes while differing radically from them in terms of physics and chemistry. The models of regulation and control are all relational models, and differ only in their level of generality; i.e., the degree to which their defining relations admit a variety of physicochemical realizations.

ABSTRACT BIOLOGICAL SYSTEMS AS SEQUENTIAL MACHINES.

It is shown that a rather close relationship exists between the (ℳ,ℛ)-systems, defined previously as prototypes of abstract biological systems, and the sequential machines which have been studied by various authors. The theory of sequential machines is reformulated in a way suitable for its application to the study of the intertransformability of (ℳ,ℛ)-systems as a result of environmental alteration. The important concept of strong connectedness is most useful in this direction, and is used to derive a number of results on intertransformability. Some suggestions are made for further studies along these lines.

A quantum-theoretic approach to genetic problems.

A quantum-theoretic picture of the transfer of genetic information is described. The advantage of such an approach is that a number of genetic effects appear to be explicable on the basis of general microphysical laws, independent of any specific model (such as DNA-protein coding) for the transmission of genetic information. It is assumed that the genetic information is carried by a family of numerical observables belonging to a specific microphysical system; it is shown that a single observable is theoretically sufficient to carry this information. The various types of structure that this observable can possess are then described in detail, and the possible genetic effects which can airse from each such structure are discussed. For example, it is shown how the assumption that the genetic observable possesses degenerate eigenvalues may lead to a theory of allelism. To keep the treatment self-contained, the basic quantum-theoretical principles to be used are discussed in some detail. Finally, the relation of the present approach to current biochemical ideas and to earlier quantum-theoretic treatments of genetic systems is discussed.

On a logical paradox implicit in the notion of a self-reproducing automation

The notion of automaton as used by J. von Neumann is formalized according to methods previously described (Rosen, 1958,Bull. Math. Biophysics 20, 245–60; 317–41). It is observed that a logical paradox arises when one attempts to describe the notion of self-reproducing automaton in this formalism. This paradox is discussed, together with some of the recent attempts to construct automata which exhibit self-reproduction. The relation of these results to biological problems is then investigated.

Church's thesis and its relation to the concept of realizability in biology and physics.

An attempt to characterize the physical realizability of an abstract mapping process in terms of the Turing computability of an associated numerical function is described. Such an approach rests heavily on the validity of Church’s Thesis for physical systems capable of computing numerical functions. This means in effect that one must investigate in what manner Church’s Thesis can be converted into an assertion concerning the nonexistence of a certain class of physical processes (namely, those processes which are capable of calculating the values of numerical functions which are not Turing-computable). A formulation which may be plausible is suggested, and it is then shown that the truth of Church’s Thesis in this, form is closely connected with the “effectiveness” of theoretical descriptions of physical systems. It is shown that the falsity of this form of Church’s Thesis is related to a fundamental incompleteness in the possibility of describing physical systems, much like the incompleteness which Godel showed to be inherent in axiomatizations of elementary arithmetic. Various implications of these matters are briefly discussed.

Two-factor models, neural nets and biochemical automata☆

Abstract It is shown that recent models for the regulation of macromolecular synthesis in cells can be described in terms of a two-factor theory. Models for differentiation can then be described as networks of two-factor elements. Thus these models are analogous to models which have for many years been used to describe properties of peripheral nerve and the central nervous system, in the sense that they are both realizations of the same mathematical formalism. The relationship of two-factor elements to McCulloch-Pitts neurons is reviewed, and in view of the resulting equivalence between two-factor nets and McCulloch-Pitts networks, the concept of “molecular automaton” in regulation and differentiation is precisely defined.

Abstract biological systems as sequential machines: III. Some algebraic aspects

Using the relationship between (M,R) and sequential machines developed in previous work, it is shown that the totality of (M,R) which can be formed over a given categoryA itself forms a category in a natural fashion.

A relational theory of the structural changes induced in biological systems by alterations in environment

It is shown that a wide variety of structural alterations in both the “metabolic” and “genetic” apparatus of (