A Sequence Algebraist's Attempts To Learn From Life Sciences
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 14/1/98)
Abstract
For brevity in this paper, life sciences are defined to encompass conventional biology,
molecular biology, and metaphysical papers by life scientists. There is a gulf of difference
between mathematics and biology. Mathematics is neat and deterministic but biology
at the macro and molecular level is tedious and messy. A few years back
the author's interest was drawn to life sciences based on his conjecture that Nature
is more a mathematician than one suspects. The topic of most interest is Symbiosis
which covers various modes by which two types of organisms have become interlocked
by evolution where in extreme cases, these have lost their individual identities and
become a new being. This is synergy where entities behave as more than the sum
of their parts. It is discovered that sequence algebra could benefit from
an analogous process although it is not called symbiosis. The author has tried
merging different generating functions and found that sometimes the results are
quite unexpected. Nature's mathematics is quite
different from those devised by Man and will not yield its secret if one does not
observe Nature's paradigms. Mathematics is inadequate in biology unless
mathematicians are prepared to devise now operators and functions to handle
new problems.
1. Introduction
The contents of this paper are drawn mostly from Chapter 5 of Margulis and
Sagan's book entitled "What is Life?" [1]. This is not a book on mathematical biology.
The chapter on "Permanent Mergers" speculates that biological evolution is more
the product of symbiosis than mutation. Where mutations proceed in small steps
symbiosis is abrupt and spectacular. These authors conjectured that the
evolution of the kingdom of animals, plants, and fungi could not have come about
by mutations but by symbiosis and that all higher animals are multicellular composites.
The present author believes that if such complex organisms as bacteria could evolve by
symbiosis, so can mathematical processes. When disparagingly different mathematical
functions are merged, the results can be very bizaare but occassionally
something spectacular happens giving rise to synergistic forms with totally unexpected
properties some of which lead to new insights. The main theme of this paper is that
in sequence algebra there is a process analogous to symbiosis when different generating
functions are merged. Examples will be cited to support the author's conjectures.
2. Biological Symbiosis Explained
A Narrow Definition of Symbiosis: Symbiosis is a process of
permanent merger between different types of living beings which could lead
to new life forms ... a process which is very unlikely if Darwinian's theory of evolution
by genetic mutations and survival of the fittest are the only forces in play.
There are many forms of Symbiosis from the loose but beneficial relations
between plovers and crocodiles called behavioral symbionts to parasitism which
does not appear to benefit the hosts. The most extreme form is
endosymbiosis where one symbiont lives inside the larger or host symbiont
in a mutually beneficial relation leading to loss of individuality for both and the
result is a new living being. Thousands of different fungal species combine with
a few photosynthetic partners to produce a great variety of lichens. The result
of symbiosis can be synergistic and specutacular as it is not predictable by
simple addition of the constituent properties of the symbiotic partners. Most
life scientists believe that biological evolution is more symbiosis driven than mutation
driven.
3. Functional Symbiosis In Sequence Algebra
The author's original interest in life sciences came from the fact that there are
some similarities between algebraic sequences and DNA sequences. He thought
that maybe sequence algebra could be used to model replication processes in
DNA sequences. This was found to be not that simple as many operations in
DNA sequences have no algebraic counterparts. Let us cite the process of
DNA synthesis at the fork [1]. A double helix can open up at one end into a Y-shaped
fork so that DNA replication can proceed continuously, semidiscontinuously, or
discontinuously. To add to the problem, replication can only start in the presence
of primers. The nearest analogy to a primer is the Normc( ) function defined in
sequence algebra. Chronological developments of mathematics were well ahead of
life sciences which might explain why past mathematicians seldom look to
biology for inspiration.
Some processes in Nature
are so complex and yet evolve with such high precision that the mathematical angle
cannot be ignored. Many operations are found missing in traditonal mathematics
which need to be defined in order that progress can be made in applying sequence
algebra to life sciences. The content of this paper is tentative and probably
represents the tip of the iceberg in terms of the amount of efforts needed
to secure future progress.
Some life scientists believe that all members of the three kindgoms of animals,
plants, and fungi are composite beings. Put it rather simplistically, a Man is a
composite being formed by multicellular eukaryotic cells. But eukaryotic cells
are themselves composites being formed by the merger of simple prokaryotic cells.
Prokaryotic cells are bacteria cells which have no nucleus. The simplest
autopoietic unit of life on planet earth at this moment are bacterial cells. Anything
less than this such as DNA or RNA sequences or virus are not alive. Eukaryotes
are made of nucleated cells. All eukaryotes come from protoctists or multicellular
cells; bacteria are prokaryotes which are unicellular protists.
Prokayrotes and eukayrotes form the two "supergroups" on life on Earth. Eukayrotes,
being nucleated cells,
are composite beings since these have two sets of DNA which do not mix and
replicate at different time. It is suspected that these two sets of DNA came from
two different bacterial beings when one ingested the other. Somehow the ingested
did not get disgested and if this is a photosynethic being, it could supply food to the
predator provided the latter is able move rapidly into sunlit regions. This mutually
beneficial relation eventually evolved into a permanent merger where the original parties
lost their identities resulting in a new being.
Man is the spectacular example of a similar merger without involving photosynthetic
partners. Man is not simply the arithmetic sum of all his body cells. If that is true
whales should be more intelligent than Man by simply having more cells. Drawing
lessons from symbiosis, examples are given whch demonstrate that an analogous
process exists in sequence algebra. Most of the examples shown are not simply
functional embedments since some of the mergers were formed at random with
no rational basis at all.
Example 1: In sequence algebra, we can
regard Nat(z), Even(z), Odd(z), Prime(z) and so on as individual species. We can pull
together any two species and define arbitrarily a sequence algebraic function or
fomulation for them very often for no good reasons other than curiosities.
A simple yet important example is that of the defintion of Goldbach's sequence given
by equation (1):
.............Goldbach(z) := Normc( Prime(z)^2); ..............................(1).
There was probably no instance of any function in conventional number theory which form
functions out of the square of primes, much less prime sequences. As it stands,
Prime(z)^2 generates only even integers which is an entirely new sequence as it
is not exactly Even(z). This
draws attention to Goldbach's Conjecture where it is defined that every positive
integer greater than 3 is the sum of two primes not necessarily distinct. That was
why the resultant sequence is called Goldbach's sequence since it gives a lot of
insights on Goldbach's Conjecture. If we draw analogy from biology, then
Prime(z)^2 represents algebraic symbiosis. The Normc( ) operation is analogous
to a primer in a DNA replication since Goldbach(z) cannot be further processed
unless it is normalised.
Example 2: The equation of divisibles (see equation (2)) is
regarded by the author as an anchor in sequence algebra [4]. From this
basic formulation, several new variations with useful properties can be derived
as shown in equations (3) and (4).
..................................................ub
................................................-----
.................................................\ ..........1
................................Comp(z) := )...--------- .................................(2)
................................................./.........i....i
................................................----- z (z - 1)
................................................i = 2
Equation (2) represents an infinite summation of self-similar generating functions
of the general form of 1/(z^i*(z^i-1)). These surely can be described as algebraic fractals.
If each of these is regarded as a unicellular entity,
then this equation can be treated as a multicellular composite, an algebraic symbiosis.
When this equation is
series expanded we get the global composite number sequence. Since this
equation involves infinite summation, it is not something which one could
derive in total with pencil and paper. One could guess the first few terms but
the proof of its validity is asymptotic. It is almost like symbiosis in biology when
the end result is unpredictable until it is completed. In fact 2000 million years
after the first symbiosis between single-celled bacteria, Man is still evolving. The
process will never be completed.
Prior to this
there was no determinstic formulation for composite numbers.
This formulation has synergy in number theory since we
can now also develope global formulations for Prime(z), Oddcomp(z), and Perfect_num(z)
amongst others. Are sequence algebraists reaping windfalls resulting from
some symbiotic mergers?
Example 3: Perfect Numbers and Amicable Pairs
It is generally accepted by mathematicians that number theory is the purest
branch of mathematics. Number theorists seldom even bother to find real world
applicaitons for their discoveries. An exception in recent time is that number
theory has found an important niche in crytography. But so far perfect numbers
and amicable pairs belong to the realm of mathematical curiosities. Only after
the development of sequence algebra has there been some progress made in
their predictions [4]. One curious fact about sequence algebraic functions is that
these can return multiple answers, not only because one can control the number
of terms in the expansions but also the number of variables within each term.
Thus sequence algebraic functions deviate from the standard definition of
a mathematical function as one which returns a single output value.
Furthermore, within limits, one could control whether variables should be
multiplied or added. Furthermore one could introduce special filters to
suppress terms which do not satisfy a predefined criterion such as whether
the resultant value is a prime or not a prime, is even or odd and so on.
In generating functions, the variables themselves are global composites.
By a minor modification to the equation of divisibles one gets a generating
function capable of predicting the occurrences of both perfect numbers and amicable
pairs as shown in equation (3). This was surprising to the author who did not plan for
the double outcomes.
From this formulation, the author was able to develop search algorithms
for both of these numbers. Generating functions are algorithmic and are thus
easily programmed using symbolic packages.
Case (i): By introducting an "i" in the numerator as shown in equation (3), the sum of divisors for each integer
number will be displayed. Perfect numbers are identified by terms of the form k/z^k where
both the numerator coefficient and the denominator power index have the same value.
Amicable pairs are identified by two terms exhibiting reciprocated relations between
the numerator coefficients and the denominator power indices, i.e., giving rise to
a pair of the form m/z^n and n/z^m. Picking up amicable pairs are more difficult than
picking up perfect numbers since these pairs could occur at large intervals apart. The
search algorithm developed for amicable pairs is made easy by the use of a symbolic
package called Macsyma 2.2 [4]. The search algorithm would be difficult to uncover
using conventional algebra.
..................................................ub
................................................-----
.................................................\ ..........i
...................................Perf_z := )...--------- .................................(3)
................................................./.........i....i
................................................----- z (z - 1)
................................................i = 2
Case (ii):By introducing an unassigned order variable of x in the numerator as shown in
equation (4), the individual
divisors of each integer number will be displayed [4]. Are these functional symbiosis
at work?
..................................................ub
................................................-----........i
.................................................\ ..........x
..........................Perfactor_z := )...--------- .................................(4)
................................................./.........i....i
................................................----- z (z - 1)
................................................i = 2
Example 4: Functional Evaluations Along A Sequence
Suppose you have a finite sequence which have identical functional expressions
in the numerators but each function will take the input from the power index of its
respective denominator. How will you implement this in sequence algebra?
The answer is quite simple -- we show how to compute the factorial function using
equations (5) and (6):
F(z):=sum(G(i)/z^i,i=0..5);
................................................G(1)....G(2)...G(3)...G(4)...G(5)
..........................F(z) := G(0) + ---- + ---- + ---- + ---- + ---- ......................(5).
...................................................z...........2........3........4.........5
............................................................z..........z........z.........z
A numeric case is given in equation (6) where G(i) = i! (6):
F(z):=sum(i!/z^i,i=0..5);
........................................................2........6.......24.....120
...........................F(z) := 1 + 1/z + ---- + ---- + ---- + --- ...........................(6).
..........................................................2........3........4........5
........................................................z........z.........z........z
Suppose each numerator expression is a generating function and the number of
terms to be generated by each function is determined by the denominator power
index i. We repeat the factorial example given in equation (6).
F(z):=sum(sum(i!/x^i,i=0..j)/z^j,j=0..5);
...............................................2..........................2.........6.........................2........6......24
...............................1 + 1/x + ----......1 + 1/x + ---- + ---- ....1 + 1/x + ---- + ---- + ----
.................................................2..........................2........3..........................2........3.......4
...................1 + 1/x.................x..........................x.........x.........................x.........x.......x
F(z) := 1 + ------- + -------------- + --------------------- + ----------------------------
.....................z....................2.............................3........................................4
..........................................z.............................z........................................z
.................2.........6.......24.....120
1 + 1/x + ---- + ---- + ---- + ---
...................2.........3.......4.........5
.................x.........x........x.........x
+ ---------------------------------- .....................................................(7).
.............................5
...........................z
The output of equation (7) is interesting since it can be viewed as a main spine
sequence with side-chains emanating from each term or node like a feather. This
feathery structure is like a protein-coated DNA actively being transcribed into
messenger RNA molecules. Sequence algebra would have to do more than
the above in order to describe biochemical processes realistically.
Example 5: Order and Chaos
Order and chaos have appeared in various theories such as information theory,
chaos theory, and and life sciences. Life scientists believe that life process
itself represents attempts by matters to postpone indefinitely the eventual outcome
of the Second Law of thermodynamics, i.e., heat equilibrium or heat death [1]. Whilst
this view is now not as popular amongst scientists, informaton and biological
evolution
have an inescapable common basis in this law. It is an established scientific fact that
DNA sequences encode information. In a completely random
system, there is no information. To build up order, one must build up information.
but there is no free lunch in evolution as this is done at the expense
of generating waste as heat so that in a closed system there is no nett gain. However
life on the Third Rock is based on solar economy which is not exactly a closed
system. The eventual supernova explosion of the Sun which spells the doom of
the Solar System is another story altogether.
Chaos and order are strange bedfellows. Here is an example which will
demonstrate this point effectively. Let us start with a binary ascii string which is
entirely random, i.e., without any information. Since there are only two symbols,
we choose an ordering paradigm by which the populaton of one symbol is
arranged in periodic order by the Theorem of Pseudo-Periodicity [4].
This is based on the assumption that a string of random bits has maximum entropy
but a perfect periodic bit sequence has zero entropy. The interval
can be of any finite value since all periodic sequences have zero entropy. One
will immediately discover that whilst it is possible to order one subset , the
other subset will defy
ordering. Thus given a random string with two distinct subsets, it is only possible
to order one of these at the expense of the other. We can call the remaining
unordered subset waste, chaos, or entropy. A perfectly random binary string
should have exactly half of each type of symbol. This
means that no matter how one tries, order can only be put into half of the content
of the string. Of course if one pupulation is larger than the other we can go
beyond 50% but then the string is less than random [9,10].
This finding is not surprising as there are previous workers who have
arrived at the same conclusion via different pathways such as Godel's
Theorem, Chaitin's algorithmic information theory on a computer, and
Turing's halting problem. The problems with previous discoveries are to
find convincing examples to demonstrate validity. The advantage of the
present approach is that
one could always start with a finite random string to demonstrate the
futility of 100% ordering. Any forced ordering by a recursive process on the remaining
symbol will require the string to be lengthed until eventually it reaches
infinite length.
The above finding has important implications in information theory and also
life sciences. Unlike Chaitin, the author proposes that number systems have zero
entropy, i.e., there is no ambiguity about them so long as these are written in
as sequence algebraic expressions which are periodic. Human languages always
contain
redundancies. The point is that Life is the
process of increasing information in DNA-sequences and this is done by putting
order into matter which is initially in a random state. This implies that the DNA
sequences can never be like the natural number system and that at least 50% of its
content is meaningless. Life scientists already know that there is a lot of
meaningless repeats in DNA sequences. Sequence algebra says as much
from the Second Law of thermodynamics.
Example 6: Mendelian Genetics [2]
If one wishes to use sequence algebra to predict the two principles in Mandelian
genetics, i.e., the principle of segregation and the principle of independent
assortments, one will discover that without defining new operators, sequence
algebra is inadequate to the task. In other words, it is impossible to
simulate Mandelian genetics purely using conventional arithmetic operations.
In the two cases below, the new operator to be defined is Slice( ) which will
split a double helix into two separate strands in which the relative positions
of nucleotide pairs must be preserved. A nucleotide pair in the double helix is
represented by the term A*B/M or A*B/F where M and F in the denominators
represent male and female respectively and A and B in the numerators represent
the n.p. pairs. A will be kept in the A-strand and B will be kept in the B strand
after slicing operations.
Case(i) Mendelian Principle Of Segregation [2]. The textbook example
is based on figure 2.5 on page 28 of reference [2].
.......................................................2
........................................Dominant
................Parent1(z) := ----------- ......................................(8)
..............................................M
.......................................................2
.........................................recessive
................Parent2(z) := ------------ ......................................(9)
..............................................F
...................................................................2
....................................................Dominant
................Gamete1(z) := Slice(------------- )
..........................................................M
...........................................Dominant
.....................................=-------------- .......................(10).
.................................................M
2
..............................................................recessive
..........................Gamete2(z) := Slice(------------- )
....................................................................F
.............................................recessive
....................................= --------- ..........................(11).
...................................................F
Dominant * recessive
..........F1(z) :=Gamete1(z)*Gamete2(z):= --------------------------- ...............(12).
...............................................................................M * F
........................................................Dominant recessive
.................Gamete11(z) := Slice(---------------------- )
...................................................................M F
..............................................Dominant..........recessive
..........................................= --------.........+ --------- ...............................(13).
..................................................M.....................M
...................................................Dominant recessive
....................Gamete22(z) := Slice(--------------------- )
..................................................................M F
..............................................Dominant..........recessive
.........................................:= ------------ + --------- ............................(14).
......................................................F...................F
................F2(z):=Gamete11(z)*Gamete22(z):=
..................................2...............................................................2
...................Dominant...............Dominant recessive........recessive
...............= -----------........+ 2 -------------------- + ---------- .............(15).
........................M F.............................M F.......................M F
We can define Dominant^2 = Dominant
......................Dominant * recessive = Dominant
and..................recessive ^2 = recessive. ........................................(16).
So that
........................................Dominant.............recessive
.......................F2(z) := 3 ------------ .....+ --------- ............................(17).
...........................................M F....................M F
which is Mendelian's 3:1 principle. The use of M*F in the denominator is optional
but it serves tracing of parentage.
Case(ii) Mendelian Principle Of Independent Assortment [2]. The textbook
example is based on figure 2.7 of page 30 in reference [2].
.....................................................................2.................2
.......................................................D_round........r_green
.................................Parent1(z) := ---------- + --------.................(18).
.............................................................M...................2
................................................................................M
....................................................................2
...................................................r_wrinkled.......D_yellow d_yellow
.............................Parent2(z) := ------------- + -----------------........(19).
............................................................F.........................2
.....................................................................................F
...............................................................................2.................2
.................................................................D_round.......r_green
...............................Gamete1(z) := Slice(------------ + -----------)
.......................................................................M..................2
.........................................................................................M
..................................................D_round.........r_green
............................................ := ----------- + -----------.........................(20).
.........................................................M....................2
.............................................................................M
.............................................................................2
...........................................................r_wrinkled........D_yellow d_yellow
.........................Gamete2(z) := Slice(-------------- + --------------------)
.....................................................................F...........................2
................................................................................................F
................................................. r_wrinkled.....D_yellow
............................................. := -------------+ -----------..........................(21).
...........................................................F...................2
..............................................................................F
F1(z):=inner_prod(Gamete1(z)*Gamete2(z))
................................/D_round.......r_green \ /r_wrinkled........D_yellow\
........ := inner_prod( |---------- + --------- | |-------------+ -----------| )
................................|...M...................2......| |.....F.......................2.......|
................................\.......................M......./ \............................F......../
..........................................D_round r_wrinkled.....r_green D_yellow
..................................... := --------------------- + ----------------.................(22).
........................................................M F........................... 2 2
.......................................................................................M F
....................D_round r_green...D_round D_yellow...r_wrinkled r_green...r_wrinkled D_yellow
Gamete11(z) := --------------- + ------------------. + ------------------ + -------------------
.....................................2...........................2............................2...............................2
..................................M..........................M..........................M.............................M....(23).
.......................D_round r_green...D_round D_yellow...r_wrinkled r_green...r_wrinkled D_yellow
Gamete22(z) := ----------------- + ------------------ + ------------------ + -------------------
........................................2..........................2...............................2...............................2
.....................................F..........................F...............................F...............................F....(24).
..........................2..........2...................2..............................................2
............D_round r_green.......D_round D_yellow r_green.....D_round r_green r_wrinkled
F2(z) := ----------------- + 2 -------------------------- + 2 ---------------------------
..........................2 2.................................2 2...........................................2 2
........................M F...............................M F..........................................M F
..................................................................................2..............2.............................2
......D_round D_yellow r_wrinkled r_green...D_round D_yellow....D_round D_yellow r_wrinkled
+ 4 --------------------------------------- + ------------------+ 2 ----------------------------
..................................2 2............................................2 2......................................2 2
...............................M F..........................................M F....................................M F
....................2..........2.....................2.................................................2..............2
...r_wrinkled r_green.........r_wrinkled D_yellow r_green.....r_wrinkled D_yellow
+ -------------------- + 2 ---------------------------- + ---------------------......(25).
................2 2......................................2 2........................................2 2
.............M F.....................................M F......................................M F
There are a total of 16 cells in the 4x4 matrix in equation (25) but due to diagonal
symmetries, F2(z) phenotypes observe the 9:3:3:1 ratio as predicted by Mendalian Law of
independent assortment. Unlike in case(i), the denominators in the final equation
are of higher order but these do not affect final outcome. These are retained as they
are useful for tracing parentage history.
4.Conclusions
This paper shows that some processes in biology can only be analysed algebraically
if new operators are defined. More new operators may be necessary as we go deeper
into this subject. Mathematics has seldom drawn analogies from biology in the past
but with
rapid advance in life sciences, it is now opportunistic to put analyses on a mathematical
basis. This is especially relevent in sequence algebra since the building blocks of
life are DNA sequences. In fact, sequence algebra itself is benefitting from
the analogy of symbiosis borrowed from life sciences.
5. References
Comments: Not all references in this list are directly referred in the main paper.
Most are provided for readers not familiar with sequence algebra. These papers
can be easily hyperlinked whilst you are in the web. Most html files are quite short
and can be download quite fast with the exception of those published in journals.
1.Margulis L. and Sagan D.:What is Life? Chapter 5, Permanent
Mergers, The Orion Publishing Group, Printed in Italy, 1995, pp 90 to 117,
2. Weaver R.F. and Hedrick P.W.:Basic Genetics, WCB Publishers,
2nd Edition, 1995, Printed in Dubuque, pp 154-159.
3. Burton M. Burton: Elementary Number Theory, Third Edition, WCB publishers, 1994, 17 to
48 to 50.
4.
Search Algorithms For Odd Perfect Numbers
- Huen Y.K. (Date Released 7/1/98, 23 Kbytes)
5. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci.
Technol., 1994, VOL. 25, NO.6, pp 913-920.
6. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
7. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1996,
VOL.??, NO.?, ???-??? (In the press as proof paper mes 100421).
8. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
========================================================
9. The Canonical Generating Function
or CGF(z) ... - by Huen Y.K.
(date released : 27.5..97) (24 KBytes, 7*A4s).
========================================================
10. Visual Solutions Of Number Theoretic
Problems ..... - by Huen Y.K. (date released : 3.6.97) (38.3 KBytes, 10*A4s).
=====================END OF PAPER ======================