Complexity of the DNA double helix -- -- introduction -- -- September 2006

A Mathematical Analysis of the DNA Double Helix

  Thomas J McCabe

  Copyright c, September 2006

How to read this

Don't worry about not understanding all of it.  See the direction, the motivation, and the spirit.  There is enough here to get mathematicians interested in genetics, and there's plenty to get geneticists interested in the possibilities of applied 
mathematics.  There is more than enough to get the general reader interested in the new breakthroughs in genetics and excited by the possibility that the underlying genetic machinery may be mathematically tractable.

Introduction

There is a seduction here.  It seduced me.  I hope it seduces you.

This is a story that interrelates two branches of study -- -- molecular biology and pure mathematics.  Molecular biology is about genes and the biological structure that holds them -- -- the DNA double helix.  The new science of molecular genetics has had momentous breakthroughs in the last several years: sequencing the human genome, identifying and locating over 30,000 human genes, identifying various morbid genes that lead to fatal diseases.  The breakthroughs in molecular genetics are happening every day; for example, today October 30, 2006 there was an announcement in the news that genetic have identified a 100 million year old honeybee and have sequenced the genome of current honeybees to understand their social 

behavior.  The breakthroughs in molecular genetics concerning our DNA and genes will be the defining event of the 21st century.  Many diseases presently fatal will have genetic prescriptions, lifespan will be extended, and a host of chronic disabling diseases will be both prevented and have more effective palliative treatments. More broadly the striking genetic similarly of many species is being identified, our evolutionary ancestry is being clarified, and the underlying genetic machinery common to all life will be fully understood and replicated.

The universality of DNA and genes, the common machinery of its cell division and protein synthesis, is stunning.  Every living species has such DNA and genes.  Likewise, every extinct species had its own DNA and genes.  The breakthroughs being made in genetics, as often as not, are pertinent to several species.  In fact, the similarities across species are striking -- -- the underlying mechanism for the splitting of cells and the protein synthesis of genes within each of many varied creatures is almost identical. The genome of a mouse is 99 per cent similar to a human; our genome is only 15 per cent larger than a mustard plant’s.

The structure of the DNA is intriguing; besides it’s universality as the code book for all living species it is also mathematically symmetrical and regular. There are two north-south backbones; one male, oriented north to south, one female oriented south to north. The horizontal latter has the four letter alphabet of the genes; which are encoded as bases, 

made up of four nucleotide chemicals: adenine, cytosine, thymine and guanine, usually denoted by the letters A,C,T, and G. The letters form base pairs that link together to form the rungs on the ladder of the DNA double helix. Genes are finite sequences of bases along each of the vertical backbones.

The other thread binding this story is of applied mathematics. Mathematics is called the purest of the sciences.  Mathematics requires no experimentation, or physical validation. Mathematics is not dependent upon a contemporary view of physics or chemistry, or astronomy. To mathematicians the classical life sciences and biology had seemed intractable; their processes seemed way to randomly chaotic and loose to admit of mathematical analysis.

Molecular biology has changed all that.  The DNA double helix is such a regular, stable, and rich structure that it can be thought of as mathematical object. Also the recent discoveries of the detailed process of cell replication and gene synthesis is so uniform, even across different species, and meticulous, that we can now apply mathematical concepts. The incredible recent genetic breakthroughs have demonstrated a universal regularity; the molecular biologists have opened the door for mathematicians.

This paper will present mathematical insights that systematize and simplify the way genes are produced from DNA. The mathematical concept we will use is a Vector Space -- a rigorous concept which has been developed over the course of many centuries. This paper will attempt to model the DNA double helix as such a mathematical Vector Space; if successful there will be many direct results. One of which will be a classification of genes into two camps -- -- basis genes and composite genes.  Vector Space analysis will provide a way to generate composite genes without the usual DNA protein synthesis -- -- composite genes will be shown to be linear combination of basis genes. Vector Spaces can potentially lead to alternate ways to generate genes and proteins; it could enable man made genetic drugs and medicines.

Partly because mathematics is pure and not dependent on physical science it’s subjects tend to be abstract, obscure, and lifeless. Not here, the target of our mathematical analysis, DNA and gene generation, is life itself.  Every living thing, present or past, has DNA and genes -- -- so our subject is life universal. My finger tactile genes are being generated faster than I can type about the mechanism my finger DNA cells are using.  Your DNA has produced genes for your optic nerve faster than you can read about how it does this. This  is a mathematicians delight; our mathematical target is the most universal of life forms… the mathematical analysis of the DNA double helix … I hope you’re getting seduced.

The physical characteristics of DNA and genes are indeed mystical and seductive.

Size

Each Homo Sapiens has 500 billion cells.  Each cell has a nucleus containing 24 chromosomes that contain the DNA double helix.  Within each cell the information along the male and female strands of the double helix is rich enough to produce a complete clone --- not just of the cell itself, but a clone of the complete person.  If we spliced a human's DNA together and stood it end-on-end, it would stand about a six story-building high.

Speed

Each cell generates a new gene about every four seconds.  In the time it takes you to read this page your cells would have produced about 2 trillion new genes.

Omnipresence

Every living thing has DNA and genes.  Whales do. Microscopic life forms which could have an 8 million metropolis located on the head of a pin, indeed have DNA and genes.  Chimpanzees have 99.9% of the same genetic makeup that we do; we share 75 percent of our genetic makeup with a mouse.  For that matter the sea squirt, which Aristotle thought was a plant, has embryos which are strikingly similar to those of humans.

Symmetry

to be continued ……………..