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The Fundamental Theorem of Biomatics - Biomatics.org

The Fundamental Theorem of Biomatics

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If Biology is to reveal it's mathematical secrets it is likely to do so in the form of some fundamental concept. The interactions of two carbon atoms seems about as fundamental as can be.  A "fundamental theorem" may thus be that - two carbon atoms yield a "set" and an "operation" as well as possible "relations".  
 

In mathematics, a primitive notion is a concept not defined in terms of previously defined concepts, but only motivated informally, usually by an appeal to intuition and everyday experience. For example in naive set theory, the notion of an empty set is primitive. (That it exists is an implicit axiom.) For a more formal discussion of the foundations of mathematics see the axiomatic set theory article. In an axiomatic theory or formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be "defined" by the axioms, but this can be misleading.

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?
 

Consider two adjacent carbon atoms.  The hydrogen atoms will be in an energetically favorable position when they are in the staggered so called "trans" configuration.

image: trans&newman.jpg      

 Notice that each covalent bond could rotate one hundred and twenty degrees to the next favorable spot.  Each carbon thus has potentially three favorable states.

image: 3_states.jpg

We now add a few more lines to complete the cube.

image: cube.jpg


Consider a set of three carbon atoms that can each exist in one of two states. We therefore have 2**3 or eight possible states which can be labeled as 000, 001, 010, 011, 100, 101, 110, 111. To go from 000 to 001 requires the rotation of a single bond. To go from 000 to 110 requires rotation about two bonds etc. The following diagram, where the eight corners are the states and the twelve edges are the transitions between states, can depict this situation.

image: cube1.jpg


Label edges

E1 =  000,001
E2 =  001,011
E3 =  011,010
E4 =  010,000
E5 =  000,100
E6 =  100,110
E7 =  110,111
E8 =  111,011
E9 =  101,001
E10 = 111,101
E11 = 100,101
E12 = 010,110


Inverses
E1' = 001,000
E2' = 011,001
E3’ = 010,011
E4’ = 000,010
E5’ = 100,000
E6’ = 110,100
E7’ = 111,110
E8’ = 011,111
E9’ = 001,101
E10’= 101,111
E11’= 101,100
E12’= 110,010
 
If we consider all sequences of moves along the edges of the Cube, we notice the following:

Closure- One sequence followed by another sequence is a sequence.
Associativity- That is E1*(E2*E3) = (E1*E2)*E3.
Identity element- The do nothing sequence does nothing.
Inverses- Every sequence of moves can be done backwards and therefore undone, e.g. E1 and E1′.

Thus, the set of all sequences of moves on the above Cube is a group.

Coincidentally we could relabel the above cube as follows: 
 
image: cube2.jpg

to represent the relation “containment” of one subset in another for the partially ordered set of all subsets of the set (a,b,c).


In addition, represent the relation of divisibility for the partially ordered set (1,2,3,5,6,10,15,30) as follows:

image: cube3.jpg

 

Molecular Spin Groups

A Group is a set together with an operation on the members of that set such that the operation applied to any 2 members of the set yields another member of that set. The Integers and the operation of addition form a group, for example.

Given a carbon chain with N carbons and angle of  109.5 degrees between successive carbon atoms, and thus N-1 covalent bonds.

Initial state of the molecule is simply state at time zero. Each bond = zero degrees ie home

Assign spin (ratio) rate to each covalent bond eg radians per unit time:  R1,R2,...RN-1. The elements of the set are the possible transformations of the system, which will be ratio preserving multiples of R1...RN-1.

If Bond 1 and 2 have ratio values of 2 and 3 then a transformation in the set will preserve this ratio (as well as the ratios of all other respective covalent bonds). So for example, 20 degrees and 30 degrees will satisfy the required conditions. So will 30 and 45, as well as 1 and 1.5 etc. Thus we have an infinite set.

Note also that the transformations are additive. eg 20+2=22, 30+3=33 satisfy the required ratio. Thus any 2 members of the set added together yield another member of the set. And thus the requirements of a Group are satisfied.

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