The seriousness of homochirality
Taking the chemical homochirality of life seriously can provide incredible clarification in our understanding of the most famous complex unsolved problems of modern mathematics
Homochirality: a group of molecules that possess the same chirality
Chirality: Objects that are not superposable with their mirror images. For instance, a person’s hands.
Today pharmaceutical industry is connected with production of chirally pure biomolecules.
Some pathologies can be caused by the use of a sedative hypnotic drug thalidomide containing a mixture of enantiomers of a synthetic derivative of glutamic acid ( alpha - phthalimido - glutarimide ).
Whereas, D (right) - isomer is safe, the L (left) - isomer provokes malformation in the form of phocomelia or a congenial malformation. This is characterized by agenesis of the long bones in the arms (or legs) and resulting in limbs that look like the flippers of a seal.
This sort of pathology demonstrates the importance of homochirality for natural systems in some deepest fundamental sense.
Life exists in merely one chirality - sugars are exclusively right - handed, whereas amino acids are left - handed.
This kind of single handedness is usually biologically defined as "homochirality".
Such asymmetry may seem mathematically trivial at the first glance, however, life scientists still do not understand why do so many biological molecules exist in just merely one chirality and how it emerged on the Earth at all. ( Miller,1953; Frank-Soai,1995; Joyce,1987; Steendam,2014; Powner,2009 and Hein,Tse, Blackmond, 2011).
Following Pasteur's pioneering experiments, many organic compounds having identical sets of atoms (but different mirror symmetry) nevertheless are able to demonstrate different biochemistry (different taste and smell).
As is known today the functional proteins of all organisms contain only L - amino acids which represent merely structural proteins but not signal. From another side, there are short polypeptides with D - amino acids as well.
Such neuropeptides, however, are not structural, but are merely signal proteins. It is remarkable that enzymes are completed of more than 100 L - amino acids and living organisms use only D - amino acids for information transfer.
Monosaccharide L- arabinose is found in plant biopolymers such as pectin and hemicellulose from plant cell walls. Single - celled archaea use L - glycerol to make their phospholipid membrane, etc (Powner,2009)
If we make experiments with spontaneous decomposition of the symmetric mixture resulting from the formation of L- and D - crystals , we may also assume an existence of some kind of memory which provides the effect that the average number of homochiral L- and D - crystals will be the same.
In other words, chirality always is restored in a large samples. (Miller,1953)
Such observation suggests that there exists an analogy between self - organization processes of spontaneous decomposition of the symmetric mixture into L- and D - molecules and chemical self - organization of the Belousov - Zhabotinsky type.
Indeed, in chemical self - organization auto waves break up as they encounter an obstacle, giving rise to pairs of L- and D - chemical structures.(Ivanitsky ,2010)
Perfect Mathematical Homochirality
There is remarkable analogy of biological homochirality in the theory of perfect numbers of today's mathematics.
Perfect numbers were established by Ancient Greeks (Plato, Euclid) and they are represented by following even natural numbers : 6, 28, 496, etc.
A perfect number is equal to the sum of its own divisors. For example , 6 =1+2+3, 28 = 1+2+4+7+14.
Perfect numbers are very rare. Mathematicians have only discovered 47 perfect numbers.
There are even and odd integers, it is easy to suppose an existence of the odd perfect numbers.
However, it is clear even for non-mathematicians that odd perfect numbers cannot exist at all.
Indeed, simple algebra proves that If perfect numbers are defined as
6 = 2^3 - 2^1 ( because 2^1(2^2 - 1 ) = 2^3 - 2^1)
28 = 2^5 - 2 ^2
496 = 2^9 - 2^4, and
2^86225217 - 2^43112609 is perfect number, etc.
Then an odd perfect number cannot exist in general, because any 2^n ( where n = 1,2,3,4,5,... ) is always Even ( correspondingly, Even - Even = Even ). ( Popov, 1999).
The Millennium Riemann problem
Another analogy with L - and D - homochirality could be found in modern theory of zeta function, where for negative meanings of (Euler's) zeta function - only negative odd meanings have a nonzero meaning :
zeta (-3) ≠ 0
zeta (-5) ≠ 0
zeta (-7) ≠ 0
zeta (-9) ≠ 0
zeta (-11) ≠ 0
Whereas negative even meanings equal trivial zero:
zeta (-2) = 0
zeta (-4) = 0
zeta (-6) = 0
zeta (-8) = 0
zeta (-10) = 0
In this context, famous Riemann Hypothesis suggests an existence of biological - like homochirality for all complex meanings of zeta function which are given by
zeta(1/2 ± 14.135i) = 0
zeta(1/2 ± 21.022i) = 0
zeta(1/2 ± 25.011i)= 0, ...
Thus generally Riemann hypothesis is false if All nontrivial zeros do not lie on the critical line 1/2 + it.
In other words, Riemann Hypothesis assumes an existence of some unknown complex homochirality connected with distribution of the prime numbers in number theory.
Similar with homochirality by biologists, such sort of asymmetry is not trivial and mathematicians still do not understand why do certain meanings of zeta function must exist in just merely one chirality (homochirality ).
Correspondingly, how and what for did it emerge in mathematics and in Nature indeed. Some physicists consider even an existence of new kind of physics of Riemann Hypothesis.
This problem is called “Millennium Riemann problem " which is recognized today as the most famous fundamental unsolved problem of contemporary mathematics.
Catastrophe produced homochirality
Let us suppose that both biological and mathematical homochiralities have to have the same mathematical logic of emergency.
Correspondingly, there is some initial event (not only formal mathematical proof ) which we would like to call " Catastrophe " that is producing unified homochirality.
Probably, Vladimir Arnold was the first mathematician who considered such sort of theory of catastrophe - inspired homochirality in the 20 century
Arnold showed that initial non-systematic attempts were made already by A. Poincare and A.A. Andronov (in 1920s).
Following Arnold some equations of the type Uxxxx + 2Uxx + U^2x + U = 0 have periodic solutions and special bifurcations ( mathematical catastrophes) connected with a symmetry in their roots : + x ~> - x ( and so-called " hamiltonianness" of the equation of the problem ).
In other words, symmetry and hamiltonianness are produced mathematical catastrophes.
Hence, If and only If there is asymmetry ( homochirality ) such symmetric and hamiltonian system can become more Stable and " Catastrophe - lessnessful". ( Arnold,1983 ).
Of course, Arnold's result needs new mathematical generalizations, however, it is clear now that in order to explain homochirality of natural and symbolical systems we are needed more than pure formal proofs of Riemann - like problems.