Computational Structures in Non-Coding DNA and the Histone Code

From Biomatics.org

1 Computational Structures in Non-Coding DNA and The Histone Code: Molecular Algebra and Finite State Disease Models
2 Abstract
3 Carbon Atom
4 The Cube and it's properties
5 Genetic Control Group
- 5.1 Multiplication Table under binary operation *
- 5.2 Control Group acting on set of Gene Clusters
6 Hardware analogies
7 Pascal's Triangle Augmented
8 DNA Microarray Experiments

Computational Structures in Non-Coding DNA and The Histone Code: Molecular Algebra and Finite State Disease Models

Groups (in the common and strictly mathematical sense) of covalent bonds in the DNA molecule and associated histone proteins are rich in mathematical and logical structure, which must help to control transcription patterns of gene clusters.

Abstract

The human genome project has shown that protein-coding genes comprise only a small percentage of the DNA molecule. The functioning of the remaining portion and the histone proteins is thus far mostly a mystery. The observations contained here may help in unraveling this mystery. The dynamics of covalent bonding between tetrahedral atoms such as carbon and phosphorus exhibit rich mathematical and logical structure yielding possible patterns of gene transcription. Computer Science has a wealth of concepts that are relevant to this understanding, as a start similarity to concepts of logic-circuit design are observed, specifically n to 2ⁿConversion and multiplexing.

Much research is going into Genomics, Proteomics, and DNA microarray technology. This model of genetic control should help explain many results as well as suggest fertile areas of research especially related to genetic conditions and diseases. DNA microarray experiments related to cancer and asthma among other conditions have found many instances where clustering techniques have yielded precisely eight clusters of genes. These results may be related to the description that follows here.

Carbon Atom

DNA is composed of five basic elements: Carbon, Hydrogen, Nitrogen, Oxygen, and Phosphorus. A carbon atom needs four electrons to fill its outer shell; it thus can form four covalent bonds with other atoms, including other carbon atoms. These electrons behave according to principles of thermodynamics and seek to be as far apart as possible. The resulting configuration is Tetrahedral.

These are actually electron clouds, which seek to distance themselves from similar clouds on adjacent carbon or phosphorus atoms. Thus are formed long varying chains of organic molecules, such as drugs, sugars, proteins, fats, and deoxyribonucleic acids (DNA).

Consider DNA and proteins as a chain of carbon atoms although other atoms are also involved; Phosphorus for example also shares carbon’s tetrahedral geometry. The key characteristic of these covalent bonds is the relationship to the other three bonds of the neighboring carbon. Thermodynamic laws dictate that these three bonds of a carbon (or phosphorus) be as far away as possible from the three bonds of an adjacent carbon atom thus forming what is known as the trans conformation between two carbon atoms. For small organic molecules, Ethane, for example, the thermodynamic barrier to rotation is so small at room temperature that the carbon atoms spin freely. Much larger macromolecules such as DNA achieve relatively but not completely stable conformations.

Conformational variation in DNA arises from these restricted bond rotations, which occur at 120-degree intervals.

Conformers of butane. The gauche and anti forms are staggered conformations. The eclipsed conformation of butane is unstable because it is at an energy maximum.

Each individual Carbon atom potentially has three stable trans conformations corresponding to rotations of 120 degrees relative to adjacent carbon atoms. Some atoms may not be able to rotate at all due to steric or physical hindrance, while others may be limited to two possible states for the same reason. For the sake of simplicity and tractability, assume for the moment that each carbon can exist in one of two rotational states.

For the simplest case, assume that the functioning of some part of the genome is dependant to some extent on the state of this single carbon atom. This change of state could affect a single gene or a single cluster of genes. Depict this situation with a vertical line whose endpoints represent the two states. For the case where two carbon atoms are involved we have 2²=4 possible states. In general, for n carbon atoms we have 2ⁿ possible states.

http://mathworld.wolfram.com/Hypercube.html

The Cube and it's properties

Note that replicating the previuos diagram and then connecting the vertices constructs the above sequence of diagrams. The next object to follow the cube is the hypercube or tesseract.

To get a feel for the geometry involved here try viewing an ordinary die (cube) as it balances on a single corner. Note the four horizontal levels in the 1 3 3 1 configuration. To convince yourself of this connect the dots below to form a cube-

                             *

          *                 *                 *

         *                 *                 *

                             *

Note also that the cube has 6 sides 8 corners and 12 edges, (or 24 edges if we consider that direction matters).

Also, note in the above sequence of diagrams the occurrence of Pascal’s triangle if for each n we count how many states there are at each horizontal level.

1 (n=0)

1,1 (n=1)

1,2,1 (n=2)

1,3,3,1 (n=3)

This pattern is relevant to the topics of Algebraic expansions, combinatorial analysis, and many other mathematical patterns and curiosities.

For example: Coefficients of the Binomial expansion

(x+y)⁰ = 1

(x+y)¹ = 1x + 1y

(x+y)²= 1x² + 2xy + 1y²

(x+y)³= 1x³ + 3x²y + 3xy² + 1y³

Coincidentally we could relabel the above cube as follows 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 relationof divisibility for the partially ordered set (1,2,3,5,6,10,15,30) as follows:

Observe that since 15 is not divisible by 10 you must move upward along an edge to get from 15 to 10.

The following inductive construction shows how to calculate the number of vertices, edges, squares, and cubes contained in a hypercube.

The number of vertices doubles with every dimension:

The segment has 2 of them
The square 4
The cube 8
The tesseract 16.

In general, the n-dimensional hypercube has 2ⁿ vertices, built up of (n-1)-, (n-2)-… and 0-dimensional elements. For the case n=4 these appear as the coefficients of the expanded polynomial (2x + 1)ⁿ. For example,

(2x + 1)⁴ = 16x⁴ + 32x³ + 24x² + 8x + 1

This says that the tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes.

n	2ⁿ(corners)	Edges	Pascal Pattern
0	1	0	1
1	2	1	11
2	4	4	121
3	8	12	1331
4	16	32	14641

[edit]

Genetic Control Group

The situation where 3 two-state tetrahedral atoms (A, B, C) influence 8 (=2³) gene clusters may be depicted as follows (no significance to actual physical relationship):

G0—G1—G2—G3—A—B—C—G4—G5—G6—G7

Relabel the corners of the cube to correspond to clusters of genes G0 to G7 i.e. where 1,1,1 = 7. The three control atoms thus regulate a system of eight (possibly overlapping) gene clusters. Call this control group ABC.

AC for example would read “A and C,” similar to what logic-circuit designers refer to as a “minterm”.

Or alternatively,

If we now consider the changes of state (along edges of cube) and assume that for example BC*AB=AC indicates that we start with state BC and “multiply” (in abstract algebra sense) by state AB (i.e. atoms A and B change state) we arrive at state AC, we construct the following tables:

Multiplication Table under binary operation *

*	0	A	B	C	AB	AC	BC	ABC
0	0	A	B	C	AB	AC	BC	ABC
A	A	0	AB	AC	B	C	ABC	BC
B	B	AB	0	BC	A	ABC	C	AC
C	C	AC	BC	0	ABC	A	B	AB
AB	AB	B	A	ABC	0	BC	AC	C
AC	AC	C	ABC	A	BC	0	AB	B
BC	BC	ABC	C	B	AC	AB	0	A
ABC	ABC	BC	AC	AB	C	B	A	0

Control Group acting on set of Gene Clusters

*	0	A	B	C	AB	AC	BC	ABC
G0	G0	G1	G2	G3	G4	G5	G6	G7
G1	G1	G0	G4	G5	G2	G3	G7	G6
G2	G2	G4	G0	G6	G1	G7	G3	G5
G3	G3	G5	G6	G0	G7	G1	G2	G4
G4	G4	G2	G1	G7	G0	G6	G5	G3
G5	G5	G3	G7	G1	G6	G0	G4	G2
G6	G6	G7	G3	G2	G5	G4	G0	G1
G7	G7	G6	G5	G4	G3	G2	G1	G0

Here we have the specification of a finite state machine model- eight states with state transitions specified for each possible input.

Hardware analogies

Thus, three binary inputs control eight outputs. This is isomorphic to what circuit designers refer to as an: n to 2ⁿ decoder. The 3-to-8 converter is a chip with three logic inputs and eight logic outputs. Depending on which combination of inputs is on, a single of the eight outputs will be potentiated.

Decoders are one of the Standard Integrated circuits that in the 1960’s replaced the use of discrete gates as the main building blocks of logical circuits.

Decoders are used for the following purposes:

· Routing input data to a specified output line

· Code conversions

· Basic building blocks in implementing arbitrary switching functions

· Memory address decoding.

· A minterm generator to implement logic functions by combining outputs with logic gates (eg OR gates)

· Multiplexing

The multiplexer circuit combines two or more digital signals onto a single line, by placing them there at different times. This is known as time-division multiplexing.

If we use two addressing inputs, we can multiplex up to four data signals. With three addressing inputs, we can multiplex eight signals.

The circuit symbol for such a genetic multiplexer is as follows:

Pascal's Triangle Augmented

For the more general case, we need to augment Pascal’s triangle. We have n+m control atoms with n 2-state atoms and m 3-state atoms controlling (2ⁿ)(3^m) gene clusters. Each tetrahedral atom can have at most three rotational states. Some may only exist in one or two states based on the structure of the molecule.

Thus (2ⁿ)(3^m) (for n + m =3) has possible values as follows: 1,2,3,4,6,8,9,12,18, 27

1= (1)

2= (1 1)

3= (1 1 1)

4= (1 2 1)

6= (1 2 2 1)

8= (1 3 3 1)

9 = (1 2 3 2 1)

12 = (1 3 4 3 1)

18 = (1 3 5 5 3 1)

27= (1 3 6 7 6 3 1)

For the case 3*3*3=27 (1 3 6 7 6 3 1) the breakdown is as follows for example:

(abc)

(a’bc, ab’c, abc’)

(a’’bc, a’b’c, a’bc’, ab’c’, ab’’c, abc’’)

(a’’b’c, a’’bc’, ab’’c’, a’b’’c, a’bc’’, ab’c’’, a’b’c’)

(a’’b’’c, a’’b’c’, a’’bc’’, a’b’’c’, a’b’c’’, ab’’c’’)

(a’’b’’c’, a’’b’c’’, a’b’’c’’)

(a’’b’’c’’)

This model thus predicts that proteins function in matehmatically organized clustered systems. For example in the case with control group ABC, DNA microarray experiments may find 8 clusters of genes involved in certain disease states. These clusters may be further broken down into a 1 3 3 1 configuration. Elucidation of the above model may allow for targeted therapy interventions.

DNA Microarray Experiments

DNA microarray experiments have found asthma to be related to eight clusters of genes. According to the above finite state model, there may be three binary dynamics, A B and C. responsible for asthma. Three proteins (Topoisomerases?), methylation, acetylation, or ubiquitination reactions manipulated in a specific combination can adjust this system through above state transitions.

Consider the normal state of these eight clusters as a vector V1 as follows-

V1 =   |G0|
          |G1|
          |G2|
           |G3|
          |G4|
          |G5|
          |G7|

Similarly, define the cancerous state as V2-

V2 =   |G0|
          |G1|
          |G2|
         |G3|
           |G4|
         |G5|
           |G7|

The desired therapy result vector is then V3 = V2 - V1 to restore V2 to V1

*	0	A	B	C	AB	AC	BC	ABC
G0	G0	G1	G2	G3	G4	G5	G6	G7
G1	G1	G0	G4	G5	G2	G3	G7	G6
G2	G2	G4	G0	G6	G1	G7	G3	G5
G3	G3	G5	G6	G0	G7	G1	G2	G4
G4	G4	G2	G1	G7	G0	G6	G5	G3
G5	G5	G3	G7	G1	G6	G0	G4	G2
G6	G6	G7	G3	G2	G5	G4	G0	G1
G7	G7	G6	G5	G4	G3	G2	G1	G0

So for example if GN is over expressed and GM is under expressed- locate GM on the left hand side of the table and apply the associated dynamic combination to transition to state GM. E.g. applying AB to G4 yields G0.

Computational Structures in Non-Coding DNA and the Histone Code

From Biomatics.org

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