0.1 Organismic Supercategories: I. Complex System Biodynamics
0.1.1 A Dynamic System Example: Connecting Topological and Algebraic Structures.
Let us consider an example of a system whose state space consists of a torus, T3, such
that the states of the system are contained inside the torus, and all transitions lead to
states inside the torus. The homology theory offers in this case two intuitive examples of
generators as the entire torus is generated by only two cycles. These two cycles generate two
homology groups H0(T) = Z and H1(T) = Z ⊗ Z, where Z is the group of integers, and ⊗
denotes the product operation. These homology groups give a characterization of the
topological space represented by the torus. In this way a connection is established between a
topological structure-that of the toroidal space, and algebraic structures, those of homology
groups, for example. Even more, we can assign two numbers to a given complex K
: the Betti number-which is the number of repetitions of Z in the homology group,
Hp(K) = Z ⊗Z ⊗...⊗Z ⊗GPT of a complex K-and a number p which is the number of elements
of a finite abelian group GPT . The Betti number gives the number of p-dimensional
holes of the complex K, and the number p gives the number of p-dimensional turns of
K. A natural physical interpretation that one can give to the holes inside of the state
space of a system will be that of instability fields of the system under consideration.
Consequently, the Betti number will give a coarse idea about the instability of the system,
being the number of instability fields of the state space. However, homological as well as
homotopy techniques would allow a much finer characterization of the local and global
properties of the dynamics of a system, being able, for example, to locate singularities in a
state space (Hwa and Teplitz, 1966). It must be mentioned here that the theory of
categories and functors provides a natural framework for homological techniques, mostly
for abelian structures in commutative Homological Algebra. In the example discussed
above, a number was assigned to a quality, that is, a Betti number was assigned to a
topological space. Another example of such an assignment is found in the theory of
elementary particles, where one associates a probability with a Feynman diagram that
qualitatively represents particle interactions via quantum fields or exchanged quanta.
However, the general procedure is not to assign a single number to a quality, but an
entire set of numbers or elements. In our second example the operation of addition
induced a corresponding operation on diagrams. This fact suggests that operations which
are used in metric, or quantitative, biology may induce corresponding operations in
relational biology. Conversely, one can think of significant relational operations with
notions which would permit us to obtain solutions of complicated problems of quantitative
biology.
0.2 Observables, Generators and Qualitative Dynamics
Observables of a biological system may be introduced as intensities of some (metabolic /
biochemical / enzymatic, or genetic, etc.) activities of the living system (as suggested for example
by Rashevsky and Rosen); alternatively, they may be considered either as parameters
characterizing specific or global processes inside the system, or as variables (e.g., molecular set
variables) that specify the quantities of certain bioproducts which are formed as a result of
the activities of the system. Among observables, structural parameters as defined in
(Rosen 1968a, b) and time observables play distinguished roles. Some observables are
“linked”, in the sense that, a change in one of the observables implies a corresponding
change of the others. Linked observables were represented as morphisms in a categorical
diagram. Such a diagram may represent, for example, the linkage group of observables
and is a part of the generating class of the system. Since diagrams can be regarded
as functors (Mitchell, 1965) one can represent the linked observables that specify the
generating classes of a system by ‘dynamic configuration’ functors between categories of
possible system configurations at different times; therefore, qualitative changes in system
dynamics may be represented by natural transformations of such dynamic configuration
functors, whereas transition-state morphisms assign numerical values to observables in
different system states. There is then a classification problem associated with defining, or
determining through measurements the biodynamics of super-complex systems as the latter are
endowed with variable state-space (or configuration space) topologies (Baianu et al, 2007a).
This leads unavoidably to the consideration of supercategories of variable topological
structures and their associated algebroids related to global stability and qualitative
dynamics.
Let us define a state of a system or organismic set, biomolecular set, (M,R)-system,..., any
theoretical model of a biosystem, at a given moment as an n-tuple of the values of essential
observables at that moment. In this representation, either a state, or system ‘configuration’, is then
defined by a dynamic functor from the category of generating classes of the system to
mathsfℛ–the set of real numbers regarded for example as a discrete category (or as a category
whose objects are real numbers, and whose morphisms are mappings; the operations with real
numbers in this category are induced by the structure of the category of generating
classes). Let us also consider here a specific example from microbial genetics. An operon as
defined by Jacob and Monod (1961) may be considered- in a very simple model- as
having two states: an active state and an inactive one. In its inactive state the operon
will not induce the synthesis of the corresponding enzyme, while in its active state, it
will induce the synthesis of a determined quantity C of synthesized enzyme per unit of
time. Now, if we consider a linkage group of operons O1,O2,...,On, which are all active
in the same time and, if the synthesized quantities of enzymes per unit of time are
respectively, C1,C2,...,Cn, a state of the linked operons may be defined by the n-tuple
(C1t,C
2t,...,C
nt) of the values of C
1,C2,...,Cn at a given instant t. However, suppose
that only C1,C2 and C3 are essential, and all other observables can be expressed in
terms of C1,C2 and C3. Moreover, let us suppose that we may find some operators such
that C2 = Y C1 and C3 = ZC2. In this case there exists a third operator X such that
C3 = XC1, which makes the three operon diagram commutative. This is the generating
diagram of the linkage group of operators. In this simple model, a genetic system, or
organsimic set (discrete organismic supercategory) of zero-th order, S0, will be then
represented by a generating class, whose objects are generating diagrams of the linkage
groups of operons, and whose morphisms are the functional connections among the
activities of the operons. Suppose that a mutation takes place in such a genetic system, so
that an operon will begin to induce the synthesis of an enzyme which was not induced
previously by the genetic system. The state corresponding to the very moment when the
change takes place will be considered as a singularity of the state space, because at
that moment when the genetic mutation occurred one cannot characterize the state
of the genetic system either by Ckt –the quantity of synthesized enzyme E
k per unit
of time, or by Ck the quantity of the ‘mutated’ enzyme E per unit of time (that is,
E is the enzyme which begins to be synthesized only after the mutation took place).
It may happen that a mutation produces effects such as the complete inactivation of
an essential operon. In this case, it is conceivable that the whole linkage group will
become inactive. If the inactivated operon is the replicon (Jacob et al., 1963), then
the cell will cease to divide, and eventually die. The singularity of the genomic state
space in the case of a mutation would last much less than other states of the dynamical
system, and may be therefore considered as an unstable state inside the state space of the
system.
Generally, if the unstable state leads only to other unstable states, it may result in the destruction
of the system-generating an unstable field outside the state space stable attractors of the system.
Consequently, states in growth processes should have to be considered as metastable, and cannot be
simply considered as unstable. The replacement of an observable by another, in the case of a
mutation discussed here, is in fact a change of the structure of the genetic system– a structure
which has both algebraic and topological representations in a supercatgeory, that may also be
viewed as a family or class of multiple, (variable) categories; the latter type of generalized
structure necessary for the representation of super-complex system biodynamics will be
defined more precisely in the next section as an interpretation of ETAS axioms, a natural
extension of Lawvere’s ETAC axioms for the foundation of the theory of categories and
functors.
1 Biodynamics in Organismic Supercategories
Axiomatic definitions of categories and supercategories based respectively on ETAC and ETAS
interprertations provide a framework for representations of Super- Complex Biological Systems and
also allow for dynamic computations of cell transformations that may lead to neoplasia, and in
certain intriguing cases to malignancy. The concepts of quantum automata and quantum
computation are applied in the context of quantum genetics and genetic networks to study their
nonlinear dynamics. In a previous publication (Baianu,1971a) the formal concept of quantum
automaton was introduced and its possible implications for genetic and metabolic activities in
living cells and organisms were considered. This was followed by a report on quantum and
abstract, symbolic computation based on the theory of categories, functors and natural
transformations (Baianu,1971b). The notions of topological semigroup, quantum automaton, or
quantum computer, were then suggested with a view to their potential applications to
the analogous simulation of biological systems, and especially genetic activities and
nonlinear dynamics in genetic networks. Further, detailed studies of nonlinear dynamics in
genetic networks were carried out in categories of n-valued, Łukasiewicz Logic Algebras
that showed significant dissimilarities (Baianu, 1977; Baianu et al., 2006-2008) from
the oversimplified Boolean models of human neural networks.A categorical and topos
framework for Lukasiewicz Algebraic Logic models of nonlinear dynamics in complex
functional genomes and cell interactomes is proposed. Lukasiewicz Algebraic Logic models of
genetic networks and signaling pathways in cells are formulated in terms of nonlinear
dynamic systems with n-state components that allow for the generalization of previous
logical models of both genetic activities and neural networks. An algebraic formulation of
variable ‘next-state functions’ is extended to a Łukasiewicz Topos endowed with a n-valued
Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and
nonlinear network activities as well as their transformations in developmental processes and
carcinogenesis. Novel results and specific applications concerning cell interactomics, dynamics of
genetic-proteomic networks and signaling pathways, development, regeneration, the control
mechanisms of cell dynamic programming in cells, neoplastic transformations and oncogenesis
are then derived on the basis of complex system modeling and biomolecular network
representations in categories of Lukasiewicz Logic Algebras and Lukasiewicz-Topos. Molecular
models in terms of categories, functors and natural transformations were then formulated
for unimolecular chemical transformations, as well as multi-molecular chemical and
biochemical transformations (Baianu, 1983,2004a). Previous applications of computer
modeling, classical automata theory, and relational biology to molecular biology, oncogenesis
and medicine were extensively reviewed (Baianu,1987). Novel approaches to solving
the realization problems of Relational Biology models in Complex System Biology are
introduced in terms of natural transformations between functors of such biomolecular
supercategories. Natural transformations of organismic superstructure were developed for
modelling protein biosynthesis, embryogenesis and nuclear transplant experiments. Other
possible realizations in Molecular Biology and Relational Biology of Organisms are here
suggested as a novel approach to Bioinformatics to Interactomics and Relational Quantum
Genetics.
