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Enzymes - Coenzymes
Some elementary views:

1. How regard protein enzymes in relation to the dimension model?
They are obviously acting as "forces", binding and breaking (~ polarizing) in relation to substrata, however, per definition, not transforming into other molecules, not involved in that sense.

Hence, the 0- and 00-poles defining Direction as d-degree 4, a vector geometry, which according to hypotheses in physics have been assumed transforming (possibly through inversions) to mass and particle structures (fermiones), is not the case here.

Instead we have to regard enzymes as expressions for these poles when they meet in last step 1 → 0/00 of motions, the enzymes here as "oscillating" between binding and breaking, expressions for poles 1a ("motions towards each other", defining a new centre) and 1b ("motions from each other", defining a new anticentre).

Chains of enzymes illustrate metabolic pathways (as such primitively seen as "linear"). There is however only one enzyme for each step on the way. Individual enzymes represent only the very jumps between stages (the metabolites): happenings in their active centra with just a few amino acids involved.
   One example: the enzyme which breaks fructose dividing it in two parts, where His and Cys co-operate.

Last step 1→ 0/00 is represented in every step of decreasing d-degree in the dimension chain (or inwards increasing (1←0/00).
   Hence, it would be possible to imagine that protein enzymes (Ep) originated from development of secondary dimension chains in each step of underlying chains, as the level development have been assumed in this model. In similarity with how new bubbles of economic activity and money circulation are developed through history and modern societies from more direct ways of "metabolism".

An immediate objection against these sketched views is of course that steps between metabolites or substratum/product(s) seldom appear as changes in d-degrees in any identifiable way. Why should they? The assumption here is that the enormous complexity in a cell and hierarchies of newer and newer levels out of earlier steps hide the origin in elementary dimension chains. Levels and distribution of charge and of energy may at higher levels be the only observable expressions for the different d-degrees.

2. Structure:
Keeping to the view on protein enzymes (Ep) as representing new levels developed out of d-degree steps, it could explain why enzymes geometrically enclose substrata as anti-centra in relation to centra. In the same snse that lower d-degrees represent manifolds and anticentra (00) in relation to higher d-degrees, superposed levels become anticentra in relation to an underlying level.

Geometrically the relation is a typically complementary one, earlier called "the lock and key" model, later modified to "the induced fit model". It's a matrix relation and one expression for the complementarity between poles in the dimension model here.
   Cavities in enzymes with hundreds of amino acids are to a certain degree adapting to the small molecules of substrata, which also may undergo slightly changes in structure to fit in.
   This main centre of the reaction could be described as positive/negative forms, similar to the opposition Mass/Space;

In terms of energy, this relation could be described as partly inversed:
   If the needed activating energy for a reaction without enzymes is regarded as an energy wall, similar to the potential wall (+) around an atomic nucleus, the complementary principle would appear in negative energy (-) of space, as a hole.

The observations concerning activation energy levels (just decreased by enzymes) and speed of reactions should perhaps be regarded as only secondary results of the decisive complementarities in geometrical forms, in charge and hydrophobic / hydrophilic polarities.

Both the long chains of amino acids in the proteins, the "global structure"or complicated (stepwise) folding where each part has a special function - and the specificity of the enzymes, generally catalysing only one special reaction in their active centra, seems to agree with the view on enzymes as developed from the steps in the dimension chain.

Protein enzymes (Ep) with coenzymes (CoEp), including "prosthetic groups", appear as a new level with the same anticentre - centre relation as protein enzymes to substrata.
   On this level the coenzymes as a kind of "substrata" are carrying mass, very small molecules beginning with H+ and e- as primary "carriers of forces" on the biochemical level. (About the coenzymes, see below.)

The coenzymes are mostly more or less closed ring-formed molecules and many are derived from the codon bases, the nucleic acids. If we regard the ring-forms from the aspect of sp(d)x-hybridizations, they imply steps in the 2x2-chain for electron orbitals from the "end" towards the middle of the number chain. We find similarities between the d-orbital shape in some illustrations and the ac/c-relation of proteins to coenzymes, which also include several metals in the d-orbital of the periodic system.
   We get enzymes illustrating anticentra on different levels and in relation to different dimension steps.

RNA has also been found to have a certain catalytic capability, and a central enzymatic function is performed by the ribosomes at the protein synthesis (step 3 - 2 in our earlier interpretation). The complementary polarity between the base pairs in the relation codons — anticodons at sites of ribosomes could perhaps be regarded as the first example of an enzymatic principle, generalized into all enzymatic activities by pure proteins?
   In that context one could ask if there is a deeper connection between the
few amino acids chosen as active centra of Ep and the substratum, e.g. between His and Cys and fructose? Or if they just happen to fit the needed operation?

Mass numbers of side chains of His and Cys is 81 and 47 (uncharged). These numbers reappear in the distribution of 128 C-atoms to codon grouped amino acids as well as approximately between C-atoms of side chains and backbone parts of 20 + 4 double-coded ams (82 - 48). First two bases in their codons are anti-codons to each other when read in opposite directions. Could this have a deeper sense, a connection with the division of fructose in 2 C3-molecules in step 3 - 2 ?
   With such facts and others it seems as the whole processes of glycolysis and citrate cycle should be regarded as in some way "created"  by the proteins, developed from peptide relations or follow from analogous, connected schemes, not only the other way around. (Cf. e.g. that the sum of the 10 stations in citrate cycle + 8H from it amounts to the same sum as the side chains of the 20 + 4 double-coded amino acids.)

It's said that there are about 4000 catalytic reactions in a cell. It would be interesting to know how many of these that are performed by pure Ep-enzymes, how many through coenzymes. (An eventual number relation?) Another question concerns the relation between structure proteins and protein enzymes, if a border is possible to draw. The "walking" of molecules on microtubules in cilia for instance (and in ion canals?) have also been described as enzymatic. Correct? If so, we have an illustration of the last step in our dimension model from linear d-degree 1 to motions to and from as steps in walking.

reflecting complementary features between poles in the dimension model, a summary:
a. Protein part of enzymes versus coenzymes
b. Enzymes in relation to Substrata
c. Binding versus breaking enzymes
d. Activating versus blocking function (stimulating - inhibiting) in regulating systems

3. Origin of protein enzymes?
It would be much easier to believe that substrata or substrata relations create their own enzymes or in any case decide the folding of these long protein chains, than that they are synthesized independently and just happen to fit a certain catalysis. The "induced" adaptation possibility of the folding of enzymes seems to reflect a little of such a thought.
   However, it should demand that proteins or chains of amino acids in some way could create their own codons for storage in DNA. Is there anything pointing to such a possibility in the earlier evolution. If possible, it seems left to the future to detect. (?)
   More points to a similar but parallel evolution of amino acids and nucleic acids, following from related schemes. (Cf. in documents about the genetic code mass sums of amino acids, 2 x 544 + 2 x 208, derived from the "exponent series", and mass of codon bases in the same chain when transformed to number-base 8.)

We may remind of the arithmetical relations between sums of amino acids and bases in the figure here and ask if a similar relation could exist between more or less virtual chains of substrata or metabolites as pathways and the protein enzymes catalysing the steps?
   (In a complex enzyme as amino acyl tRNA synthetase, binding both tRNA, ATP(AMP) and the single, specific amino acid for this tRNA, it has been shown that also all kind of organic bonds are included. Bonds that are possible to interpret as steps in a dimension chain. )

The figure referred to above with arrow bows for multiplication could lead to another speculation:
   Frameshifts in decoding of a ring-formed RNA should lead to totally different proteins. Could such frameshifts be connected with the difference between protein enzymes and other structure building proteins?
   Frameshifts may illustrate the brick wall principle, however not applicable to the relations between enzymes (Ep) and substrata, only to hierarchies of proteins.

One more question concerns the number of enzymes involved in the transformation of a molecule as substratum to the "end" product; for instance 5 enzymes for transformation of the amino acid Thr to Ileu, 10 for synthesis of the amino acid His, 8 for synthesis of Arg, according to a forgotten source.
   Could these numbers of enzymes in their turn have a deeper connection with d-degrees?

One example of enzymes transforming the amino acid Thr to Ileu in 5 steps *:

Intervals as expressions for underlying or superposed level - or secondary developed dimension chains in the steps? Representing other coordinate axes?
   The process here in mass numbers: -2, +44, -32 as -16, -16, + 2.

4. Control enzymes:
They illustrate the hierarchy of proteins (as of genes) and represent still another level, a bureaucratization.
   The brick wall principle, following from displacements in half steps between levels appears as the most natural for this relation between enzymes and control enzymes. (Possible to identify as such?) If they control not only individual steps but the order of steps, they represent changes of second order and may be interpreted as substantiations of the very process as such.

In the dimension model what is motion in one d-degree derives from a d-degree step and is structure in higher d-degrees. A process as a series of changes may be regarded as a chain of motions. The dimension chain of motions in the model here get the opposite direction to the chain of structure. The motional patterns, the changes themselves, may be substantiated during evolution, saturated to structures.

5. Activating enzymes and product inhibition of two types:
Inhibitors as products, or substances often similar in structure, act as a feed back mechanism. In the dimension model poles of each d-degree have character inherited from the 0- and 00-pole respectively as 4a ~ inward direction, 4b ~ outward direction.
   Hence, the "product inhibition" could represent this inward pole in each step, affecting the active site of the enzyme. The complementary pole with feature from the 0-pole should represent the "feed forward" mechanism.
    At the other kind of inhibition, the "allosteric" one, an enzyme in the start of the chain is affected and at another location than in the active site of this. It means that the enzymes also have the centre - anticenter polarity in themselves, centres as active sites, the anticentre expressed in the distant other location. With the suggested illustration below it could express a) the close connection between step 5 - 4 and 1 - 0/00 in the loop model of a dimension chain (the polarization steps 5 → 4/1 → 3/2), and b) the connection between anti-center at end of the chain (in d-degree 0/00 of Motions*) with the pole 00 of d-degree 4 (00 — 4 — 0).
    * (The "d-degree 0/00" substantiated during evolution toward higher levels.)

Regulating mechanisms of the types "product inhibition" and "allosteric" inhibition":

It seems as a congruent, similar pattern of inhibition is repeated on the higher electromagnetic level, in a developed nervous system where a cell nearby may inhibit a cell from activation through signals "- ½". Another system is the "lateral inhibition" which blocks incoming signals further away from the cell?

A blocking synapse on another location than in the pathway of the signal.

A corresponding regulating structure seems to appear between genes in DNA:

Operator gene and Regulator gene as a primary polarization in complementary poles:

With the aspect from the dimension model in mind:
-    From 0-pole: quantified steps, structure building from inside outwards.
Structure genes connected with the 0-pole. (Cf. "feed forward" mechanism ~ outward direction.)
-   From 00-pole, the pole representing continuum, repetition and multitude:
00 = anticentre, regulating gene in another location in the chromosome, reminding of the "allosteric inhibition" among enzymes. (Cf. feed back" mechanisms ~ inward direction.)


Forms of the coenzymes?

An attempt to range the forms in a scheme of polarizing steps and in a certain degree according to heaviness is made below: Only giving a hint of eventually possible aspects.
   Since most data indicate that nucleic acids and amino acids is coupled with d-degree step 4-3 as the step for "A-Z-numbers" of N, nitrogen, this position is kept in the sketch.


Calculations with mass numbers:

In the primary reference here (P. Karlson:Biokemi 1976) 25 coenzymes are mentioned, that's 52.
   The simple number 25 became a temptation to study the coenzymes a bit closer to see if they in some sense could be interpreted as representing a dimension chain in agreement with this model. Such a task is of course a very suspicious one:
  Firstly, the number of coenzymes may be just a selection out of many more. Secondly, regarding mass analysis, many of the coenzymes appear in different forms, for instance with or without carried groups or in -TP-, -DP-, -MP-forms etc. Following choices were made in the table below.
- Codon base enzymes in their -TP-form.
- THF, tetrahydrofolacin as formyl-THF
- PMP, pyridoxaminphosphate (PLP-form = -1 A)
- TPP, thiaminpyrophosphate
- Ubiquinone (said to include "6-10" isoprene molecules), here 8 supposed.
In mass numbers +1 are included for all negative charges and for bonds to other molecules, without regard to some (balancing?) positive charges.

5 H+                         2865
5 e-                           3272
5 codon bases           2495
3+2 C1+C2               2310
3 other groups           2334
2 quinones                   1478..... Sum 14754,
                                                -1 if PLP = 14753.

An alternative form of the dimension chain has been tested before (not published) in connection with amino acids. Here d-degree 0/00 as equivalent with 5' is the start, hence with outer poles 1a —1b. The superposed chain gets altered to the number chain


- Triplet 975 from ordinary superposed chain (9-7-5-3-1): 97,52 = 2 x 4753,1 25.

- 2 x 753 = 1475 + 31 = 1506 = 24 amino acids (R-chains) with +2H. (31 = Serin)
   (1475 + 4753 = 3278 = 24 amino acids, R+B-chains, with + 2H.)

- Quinones: 1-4-7-5 + 3 = 1478 A

- √543210    x 2 = 1474. = sum of quinones without 2 x 2H

- Quinones + e-group = 1478 + 3272 = 4750 A

Mean value of a coenzyme = 14753/25 = 590,12.
590 is the sum of one of the "factor chains"

The coenzymes in 5 groups - as in 5 steps - in another way, where those transporting other molecules are split on the C2- and C1-groups.
   Dividing the mass sums with factors 54, 43, 32 etc., gives a sum which is an inversion of the real total:

* Explanations:
3-2: C2-group HS-CoA + TPP = 1192, + Active sulphate 507 = 1699.
+ Quinones as 0/00-centres meeting in step 3-2 = 1478. Sum 3177.
2-1: C1-group 1118, + PLP + B12 = 2944.



Mass sum of all 25 coenzymes with PMP: 14754.

Porphyrines, the e--transporting coenzymes: 5 cytochrome and chlorophylls = 3272,
+ B12 (1579) = 4851. 4851 = 21 x 231 or 11 x 21 squared..
   Without +1 for bonds and negative charges in cytochromes and chlorophylls (=-6) the sum becomes 4845.

Without adding +1 for negative charges in the rest of coenzymes and only 2 H rolled in quinones and H+-group (= -12 H) we get the sum 9876 for the rest.

A dimension chain with poles:

Divisions within the porphyrine group:
- The potentials of the dimension chain x 3:

(Here porphyrines as charged and in bonds, a part only from uncharged P-group in B12. B12 connected with activity of haemoglobin?)

Last potential 101 should here be imagined as appearing within the 3-2-step. There we get the border then between cytochromes and chlorophylls, a border between plant and animal life too.

If the thought above should have some sense, it could throw light upon why porphyrines are stored as piles (or columns), discs upon one another just inside cell membranes, d-degree 2 in form. (Cf. total number 4851 = number 21 for step 2-1 squared, x 11.)

And why the "virtual" displacements of the metal atoms (here illustrated as polarized in directions on level 4-3-4; poles 4a-4b representing opposite directions)? Illustrating some transition functions in a development towards higher d-degrees??

Sum of 5 code base enzymes in -TP-, -DP, -MP-form, uncharged:

(See further about these coenzymes here, part 2, Transformations between number-base systems.)

NADP - ATP, Z-numbers 256-260, 381-387:

If these arithmetical relations should have some hidden sense, it could imply that there existed some "dynamic balance" between charges of these molecules.
   Or some principle between different parts of the molecules at the construction?

(Both these coenzymes are involved in the glycolysis and citrate cycle, where amino acids have their origins - but these divisions in N-Z-numbers and mixed sums seem as a cruel violation of even the slightest possibility of a scientific theory?)



© Åsa Wohlin
Free to distribute if the source is mentioned.
Texts are mostly extractions from a booklet series in Swedish, made publicly available in 2000.