A Sketch Of Test-Tube Evolution In A Primeval Number Soup
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~activweb/
Related URL-sites: http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication released by the author on: 25/11/97. Recovered: 30/11)
Abstract
The closed form sequence algebraic expression for Nat(z) is defined as z/(z-1). The
expression is by mathematical definition capable of infinite expansions as a Laurent series
expansion by long divisions. A sequence is uniquely identified by two properites, viz., its
order given by the indicial values of the order variable z in the denominators and its
duplicities given by the numerator coefficients. If these properties represent genetic
information, then offsprings with new genetic properties can be generated by sequence
multiplications. On the other hand, sequence addition can be considered as symbiogensis by
the merging of two species and subtraction as viral infection which represents death if the
resultant sequence contains purely negative signed terms. The intelligence of a species is
measured by its information content, i.e., the number of alphanumeric characters needed to
encode it in closed form. Only one species, viz., Nat(z) is capable of spontaneous
generation. Once started other species could follow by ways of sequence arithmetic. Thus
evolution is a boostrapped process. A virtual primeval number soup is created in which test-
tube evolution of species could be played out. Competitions can be simulated by species
fighting over limiting memory storage. Without closed forms as shortcuts, the probability of
spontaneous generation from the primeval number soup is far too low. This might suggest
that Nature might have reserved shortcut functional descriptions for RNA, and DNA.
However, if Nature's mathematic is totally different from the human kind, then it is very
difficult to unravel.
1. Introduction
This paper give a sketch of a computer simulated test-tube evolution by mathematic
sequences. The outline of the game is described in sufficient detail so that those who are
interested could develop the simulation software. A simpler version simulating the initial
stage where Nat(z) is spontaneously generated and detected by string pattern recognition
has already been developed and is available from this website [1]. You will need knowledge
of ActiveX and VBScript to develop the present software using the previous version as a
base. You will also need some knowledge of sequence algebra to understand the
mathematical basis of the evolution game [4,5].
2. The Primeval Number Soup
The primeval number soup is simply a universe containing random distributions of the 128
ASCII decimal codes from 0 to 127. This range is sufficient to generate all the published
scientific and mathematical literatures on records. The statistical probability of a monkey
typing randomly on a 128-keyed typewriter to generate an English sentence string with
meanings can be accurately predicted. An infinite 1-dimensional alphanumeric string can be
generated by a random number generator from which a pattern matcher could be developed
to detect spontaneous generations of specific string patterns representing chemical formulas
or algebraic equations. In the present simulation, only the string expression for Nat(z) given
by z/(z-1) is assumed capable of spontaneous generation and is to be detected. At least in
this primeval number soup, evolution can be boostrapped from a single primitve species.
Life science biologist like Lynn Margulis postulated that every living creature carries within its
cell the history of metabolic pathway back into the murky past of 3600 million years ago [2].
Thus modern living beings are windows through which one could backtrack in time to view
early evolutionary history. A minimal living unit is a one celled bacteria with 2000 to 5000
genes wrapped within a membrane capable of generating as many types of proteins in its
bag of metabolic tricks. It is already so complex that the probability of spontaneous
generation of this individual is an impossibility. Neither DNA nor RNA alone is enough to
form life but it is postulated that the latter is a prime candidate for early life's supermolecule
[3]. One attractive feature is that it supports autopoeisis which is absolutely required at all
times for any life form. It only needs to occur once and it will never be completely lost. RNA
is also too complicated and is unlikely a candidate for spontaneous generation. Many
workers have already demonstrated spontaneous generation of simpler chemical
components in laboratories but these experiments have been conducted with artificially
raised probabilities. Imagine someone shooting a bullet into a long pipe with the other end
directed at the bull's eye! The odd of spontaneous generation of even the simplest life
supporting molecule is so very low that one can rule out its happening in real life let alone the
co-presence of the supermolecule together with the membrane ready to wrap it up at that
instant of a thunder flash. This is the stuff of Hollywood science fantasy!
In this paper primordial chemical soup is reserved for life science and primeval number soup
for mathematic simulation. The primeval number soup to be studied is modelled by an
infinite 1-dimensional random number string. Viable building blocks are primitive number
sequences which have closed form expressions with all terms positive. Primitve number
sequences such as Nat(z), Even(z) and Odd(z) have closed form representations. Since
Even(z) and Odd(z) can be generated from Nat(z), only the spontaneous generation of Nat(z)
is necessary to kickstart the evolutionary process.
Nat(z) = z/(z-1): ASCII decimal code of the RHS is given by "122 47 28 122 45 49 29".
Since each character has 1/128 chance of occurring, therefore the probability of generating
the RHS expression for Nat(z) is one in 562949953421312. However the chance of two
copies being spontaneously generated is one in 316912650057057350374175801344. It is
defined that the process can only be initiated if two copies of Nat(z) enter into sequence
multiplication.
If there are no closed forms, the probability of generating a 100 digit natural number
sequence such as {0,1,2,3,.....................98,99} would be one in 24008890760512425
330814857516899576105018151691160832802708344378095698991146369156964318347
742989995443107492151430854636339662327138670341097333788761016363476555073
70061701735589111476563621882077520149696001 which is a 211 digit numbers.
Viewing it this way, the probability of spontaneous generation of even the shortest RNA is
practically nil. There must be a missing clue somewhere not discovered. This clue is found
in the primeval number soup, i.e., sequence expressions in closed forms. The author
speculates that there must be closed form functions for RNA and DNA sequences hitherto
undiscovered. If these can be found, it would prove that the probability of spontaneous
generation of suitable chemical components is much higher than current assumptions. There
were just too many coincidences of biochemical events with extremely low probabilities to
make the evolutionary process viable.
Given below in table 1 are samples of higher ordered sequences which can be built from
Nat(z) by sequence arithmetic. Table 1 gives an inkling of an infinite pool of higher ordered
number sequences which can be built from the primitive sequence of Nat(z).
Table 1 - Some Primitive And Compound Sequence Expressions
----------------------------------------------------------------------------------
Even(z), and Odd(z)
Comp(z), Oddcomp(z) and Prime(z)
Nat(z)^n, Even(z)^n, Odd(z)^n
Nat(z)^p*Even(z)^q, Nat(z)^p*Odd(z)^q, Even(z)^p*Odd(z)^q
Nat(z)^p*Even(z)^q*Odd(z)^r
Nat(z)*Comp(z), Nat(z)*Oddcomp(z), Nat(z)*Prime(z)
Comp(z)^n, Oddcomp(z)^n, Prime(z)^n
........................
----------------------------------------------------------------------------------
3. Sequence Operations
Three mathematical sequence operations are defined with a short example given for each.
(i) Sequence Multiplication
.................z
Nat(z) := ----- ......................................(1).
...............z - 1
On series expansion by Maple syntax: series(z/(z-1),z=infinity,10), we get
...............................1........1........1.........1........1.........1.........1.......1............1
Nat(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) ........(2).
.................................2........3.........4.........5........6.........7.........8.......9...........10
...............................z........z.........z.........z.........z.........z.........z.........z............z
The growth of the series can be controlled by defined time intervals and even availability of
food sources. Theoretically, it can grow infinitely but along the away it could be destroyed by
diseases or transformed by mutations.
If two copies of Nat(z) are available, sequence multiplication could increase variabilities
within a genera. A genera is defined by the order of the sequence which is governed by the
powers of the z-order variables. Variable properties which distinguish one species from
another within a genera are accounted for by the numerator coefficients. Thus Nat(z)^2 still
retains the order of Nat(z) but the numerator coefficients have changed. Mutation occurs if
the original order is disturbed. Sequence multiplication may or may not create mutations but
sequence subtraction often does, especially causing changes of arithmetic signs to the
terms.
....................................3........4........5........6........7.........8........9.......10...........1
Nat(z)^2 := 1 + 2/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) .........(3).
......................................2........3........4........5........6.........7........8.........9...........10
....................................z........z.........z........z.........z.........z........z..........z.............z
(ii) Sequence Summation And Subtraction
It is impossible to generate Even(z) directly from Nat(z) without summation. It is impossible
to generate Odd(z) from Nat(z) without subtraction. Both summation and subtraction lead to
mutations. Subtraction sometimes lead to auto-destruction of the particpating sequences
since the resultant sequence contain purely negative terms.
In equation (4), first we need a mutation in Nat(z) in the form of Natm(z). All that is needed
is the change of minus sign to plus sign in the denomator expression.
Natm(z):= z/(z+1) ..................................................................(4).
Series expansions of Natm(z) gives equation (5) alternative signs amongst the terms.
.................................1.......1.........1.......1.........1........1........1.......1............1
Natm(z) := 1 - 1/z + ---- - ---- + ---- - ---- + ---- - ---- + ---- - ---- + O(---) ..........(5).
...................................2.......3.........4.......5.........6........7........8.......9...........10
.................................z.......z..........z.......z.........z..........z........z.......z............z
Note that by defintioin if a sequence contain purely negative term, it leads to auto-destruction
as if killed by a disease or virus. Equation (5) can still survive as it contains some positive
terms but it is considered not in the peak of health and will affect the health of subsequent
generations.
Then Even(z) can be generated by summation of Nat(z) and Natm(z) as follows:
...............................................................1........1.........1.........1..........1
Even(z):= (Nat(z) + Natm(z))/2 := 1 + ---- + ---- + ---- + ---- + O(---) ..................(6).
.................................................................2........4.........6.........8...........10
...............................................................z........z.........z..........z............z
Even(z) contains only positive signed terms and is considered a viable species. Thus
sometimes less healthy species could be restored to health by mutations.
Once Nat(z) and Even(z) are present, we can get Odd(z) by subtraction as shown in equation
(7):
..........................................................1........1.........1........1
Odd(z):= (Nat(z) - Even(z) := 1/z + ---- + ---- + ---- + ---- .................................(7).
............................................................3........5.........7.......9
..........................................................z........z.........z........z
Odd(z) contains only positive signed terms and is considered a viable species.
(iii) Auto-Destruction
Sequence operations which result in purely negative signed terms lead to auto-destruction.
Here is an example arising from sequence subtraction.
....................................3........4........5........6........7........8........9........10...........1
Nat(z)^2 := 1 + 2/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) .........(8).
.......................................2.......3........4........5........6........7........8.........9............10
....................................z........z.........z.........z.........z........z........z.........z.............z
................................1.......1.........1........1........1........1........1.........1............1
Nat(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) ........(9).
...................................2.......3.........4........5........6........7........8.........9...........10
................................z........z.........z........z.........z.........z........z.........z.............z
....................................................2.......3........4.......5.......6.......7........8........9
Thus Nat(z)-Nat(z)^2 = - 1/z - ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- ...........(10).
......................................................2.......3........4.......5.......6.......7........8........9
....................................................z.......z.........z........z........z.......z.........z........z
which leads to auto-destruction. Sequence summation seldom lead to auto-destruction but
this can occur with sequence subtraction. Summation and subtraction can be viewed as the
merging of two species somewhat like symbiogenesis. Very often such combinations are
lethal but sometimes beneficial results could arise. Auto-destruction can also be viewed as
an attack by a viral sequence.
Thus in general sequence multiplications create variability within a genera whereas sequence
summation and subtraction could create mutations. Subtraction could sometimes cause
auto-destruction.
(iv) Mutation and Replication
Replication is a form of sequence multiplication where one participating sequence has only a
constant term. For example 2*Nat(z) generates two identical copies of Nat(z). If included in
simulation, replication should be dictated by availability of food supplies. Since all offsprings
are identical, these are nearer to cloning. Sequence multiplication described under heading
(i) are nearer to sexual reproduction where the offspring is not identical to any of the two
parents provided the order is conserved. If order is changed, then, it is a mutation.
4. The Family Tree Of Sequences
Sequences can be classifed by their orders. There are basically six types of orders:
Natural order
Even order
Odd order
Composite order
Odd composite order
Prime order
Each order can be likened to a genera in biological species. Changes in order causes
mutations.
5. Summary
There is potential in simulating test-tube evolution in a primeval number soup based on
sequence algebraic manipulations. This once again shows the versatility of this branch of
mathematics. This is in the realm of applied mathmeatics where one will need to force fit
sequence algebraic operations to the processes under study. Pure algebraic symbols and
arithmetic operations do not carry any real world meanings unless we assign these to them.
The fact that the whole evolution could theoretically starts from a single species is attractive.
The author speculates that there must be closed form functions describing lengthy RNA and
DNA sequences. Most probably each function describes the expressions of genes.
6. Reference:
1. Huen Y.K.: Spontaneous Gneartion Of Number Sequences In A Primeval Number Soup,
URL site: http://web.singnet.com.sg/~huens/
2. Margulis L. and Sagan D. (1995): What is Life? The Orion Publishing Group, London,
Printed in Italy. Chapter 4, pp 68 to 78.
3. Weaver R.F. and Hedrick P.W. (1995): Basic Genetics, WCB Publishers, Second Edition,
pp 223-224.
4. Huen Y.K.: A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994,
Vol.25, No.6, pp 913 - 920.
5. Huen Y.K.: Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997,
Vol.28, No.3, 333-344.
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