symmetry and groupoid representations in functional biology
Functional biology is mathematically represented through models of integrated biological functions
and activities that are expressed in terms of mathematical relations
between the metabolic and repair components (Rashevsky, 1962 [2]). Such representations
of complex biosystems, mappings/functions, as well as their super-complex dynamics
are important for understanding physiological dynamics and functional biology in terms of algebraic topology
concepts, concrete categories, and/or graphs; thus, they are describing or modeling theost important
inter-relations of biological functions in living organisms. This approach to biodynamics in terms of
category theory
representations of biological functions is part of the broader field of categorical dynamics.
In order to establish mathematical relations, or laws, in biology one needs to define the key concept of
mathematical representations. A general definition of such representations as utilized by mathematical or theoretical biologists, as well as mathematical physicists, is specified next together with well-established mathematical examples.
Definition 0.1
Mathematical representations are defined as
associations
between abstract structures
and classes
, or sets (
) of concrete structures
, often satisfying
several additional conditions, or axioms imposed by the mathematical context (or category) to whom
the abstract structures

belong. Thus, in
representation theory one is concerned with various collections
of quantities which are similar to the abstract structure in regard to one or several mathematical
operations.
Notes. Abstract structures are employed above in the sense defined by Bourbaki (1964) [4].
Unlike abstract categories that may have only morphisms
(or arrows) and `no objects' (or vertices), other abstract structures are simply defined as `pure' algebraic
objects with
no numerical content or direct physical interpretation, whereas the concrete structures do have
either a numerical content or a direct physical interpretation.
Examples
- An abstract symmetry group,
with multiplication “
" has mathematical representations by matrices, or numbers, that have the same multiplication table as the group (McWeeny, 2002 [1]). In this example, such similarity in structure is called a homomorphism. As a specific illustration consider the symmetry group
that admits a numerical representation by the sextet of numbers
(or line matrix)
for the group symmetry elements
, where the latter
five are rotations (or the generators of this symmetry group) and
is the unit element of the
group. Note that the symmetry group
has the obvious geometric interpretation as the
collection of symmetry operations of an equilateral triangle. Such symmetry operations are defined by
the abstract group elements, with the group unit element playing the role of the `identity
symmetry operation'
that leaves any physical object (or space on which it acts) unchanged, such as a
degree rotation in three-dimensional (real) space. Note that each such symmetry operation of the symmetry group has an inverse which `cancels out' exactly the action of its opposite symmetry operation (e.g.,
and
),
and of course, multiplication by
leaves all symmetry operations unchanged. (This is also true for non-Abelian, or noncommutative groups with
acting either on the left or on the right of all the other group operations).
- The previous example extends to abstract groupoids
whose representations are, however, defined as
morphisms (or functors), to either families or fiber bundles of spaces- such as Hilbert spaces
. Moreover, one notes that groupoids exhibit both internal and external symmetries
(viz. Weinstein, 1998). Whereas a group can be considered as a one object category with all invertible
morphisms, a groupoid can be defined as a category with all invertible morphisms but with many objects instead of just one. Therefore, the groupoid structure has a substantial advantage over the group structure
as it allows for the simultaneous representation of extended symmetries beyond the simpler symmetries represented by groups.
- The favorite family of group representations
in the current, Standard Model of physics
(called SUSY) is that of the
product of symmetry groups; this choice might explain some of the limitations encountered in High energy physics
using SUSY and the corresponding physical representations of the symmetry associated with this product of groups, rather than quantum groupoid-related symmetries. It is also interesting that noncommutative geometry models of quantum gravity
seem also to be `consistent with SUSY' (viz. A. Connes, 2004).
- The quantum treatment of gravitational fields leads to extended quantum symmetries
(called `supersymmetry') that require mathematical representations of superfields
in terms of
graded `Lie' algebras, or Lie superalgebras (Weinberg, 2004 [3]).
- Simplified mathematical models of networks of interacting living cells were recently formulated
in terms of symmetry groupoid representations, and several interesting theorems
were proven for such
topological structures
(Stewart, 2007) that are relevant to relational and functional biology.
Several areas of functional biology, such as: functional genomics, interactomics,
and computer
modeling of the physiological functions in living organisms, including humans
are now being developed very rapidly because of the huge impact of mathematical representations
and ultra-fast numerical computations
in medicine, biotechnology and all life sciences.
Thus, biomathematical and bioinformatics approaches to functional biology
utilize a wide range of mathematical concepts, theories and tools, from ODE's to biostatistics,
probability theory, graph theory, topology, abstract algebra, set theory, algebraic topology, categories,
many-valued logic
algebras, higher dimensional algebra
(HDA) and organismic supercategories.
Without such mathematical approaches and the use of ultra-fast computers, the recent completion of the first
Human genome projects would not have been possible, because it would have taken much longer and
would have been far more costly.
- 1
-
R. McWeeney. 2002. Symmetry : An Introduction to Group Theory and Its Applications.
Dover Publications Inc.: Mineola, New York, NY.
- 2
-
N. Rashevsky.1962. Mathematical Biology. Chicago University Press: Chicago.
- 3
-
S. Weinberg. 2004. Quantum Field Theory, vol.3. Cambridge University Press: Cambridge, UK.
- 4
-
N. Bourbaki. 1964. Algèbre commutative in Éléments de Mathématique, Chs. 1-6, Hermann: Paris.
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