molecular set theory

curve

Description: Molecular set theory (MST) is a mathematical formulation of the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets. In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets.

Molecular set theory was introduced by Anthony Bartholomay ([1, 2, 3]) and its applications were developed in Mathematical Biology and especially in Mathematical Medicine. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine.A precise mathematical presentation of the basic concepts in molecular set theory is as follows.

0.1 The Representation of Uni-Molecular Biochemical Reactions as Natural Transformations. Quantum Observables of a Molecular Class Variable

The uni-molecular chemical reaction is here represented by the natural transformations $\eta :h^A⟶h^B$, through the following commutative diagram:

(0.1)

with the states of the molecular sets $Au=a_1,\mathrm{\dots },a_n$ and$Bu=b_1,\mathrm{\dots }{b}_n$ being represented by certain endomorphismsin H(A,A) and H(B,B), respectively.

The observable of an $m\mathrm{.}c\mathrm{.}v$, $B$, characterizing the products “$B$” of a chemical reaction is defined as a morphism:

$$\gamma :H(B,B)⟶R,$$

where R is the set of real numbers. This mcv-observable is subjectto the following commutativity conditions:

(0.2)

with $c:A_u^{*}⟶B_u^{*}$, and $A_u^{*}$, $B_u^{*}$ beingspecially prepared fields of states, within a measurement uncertainty range, $\mathrm{\Delta }$.

0.2 An Example of an Emerging Super-Complex Organism as A Quantum–Enzymatic System.

Note that in the case of either uni-molecular or multi-molecular, reversible reactions one obtains a quantum-molecular groupoid, QG, defined as above in terms of the mcv-observables. In the case of an enzyme, E, with an activated complex, ${(ES)}^{*}$,a quantum biomolecuar groupoid can be uniquely defined in terms of mcv-observables for the enzyme, its activated complex ${(ES)}^{*}$ and the substrate, S. Quantum tunnelling in ${(ES)}^{*}$ then leads to the separation of the reaction product and the enzyme, E, which enters then a new reaction cycle with another substrate molecule S’, indistinguishable–or equivalent to–S. By considering a sequence of two such reactions coupled together,

$Q{G}_1\leftrightarrows Q{G}_2$,

corresponding to an enzyme f, coupled to a ribozyme $\varphi$, one obtains a quantum-molecular realization of the simplest (M,R)-system, $(f,\varphi )$

The non-reductionist caveat here is that the relational systems considered aboveare open ones, exchanging both energy and mass with the system’s environment in a mannerwhich is dependent on time, for example in cycles, as the system ‘divides’–reproducing itself; therefore, even though generalized quantum-molecular observables can be definedas specified above, neither a stationary nor a dynamic Schrödinger equation holdsfor such examples of ‘super-complex’ systems. Furthermore, instead of just energetic constraints–such as the standard quantum Hamiltonian–one has the constraints imposedby the diagram commutativity related to the mcv-observables, canonical functors andnatural transformations, as well as to the concentration gradients, diffusion processes,chemical potentials/activities (molecular Gibbs free energies), enzyme kinetics, and so on. Both the canonical functors and the natural transformations defined above for uni- ormulti- molecular reactions represent the relational increase in complexity of the emerging,super-complex dynamic system, such as, for example, the simplest (M,R)-system,$(f,\varphi )$.

Definition 0.1.

Multi-Molecular Reactions are defined by a canonical functor:

$$h:M⟶[M,\mathrm{𝖲𝖾𝗍}]$$

which assigns to each molecular set $A$the functor $h^A$, and to each chemical transformation $t:A⟶B$, the natural transformation ${\eta }^{AB}:h^A⟶h^B$.