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Cell Division and the Pascal Triangle
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"Hooke, Robert , 1635-1703, English physicist, mathematician, and inventor. He became curator of experiments for the Royal Society (1662), professor of geometry at Gresham College (1665), and city surveyor of London (1667). Considered the greatest mechanic of his age, he made many improvements in astronomical instruments and in watches and clocks, was the first to formulate the theory of planetary movements as a mechanical problem, and anticipated universal gravitation. In 1684 he devised a practicable system of telegraphy. He invented the spiral spring in watches and the first screw-divided quadrant and constructed the first arithmetical machine and Gregorian telescope. He stated Hooke's law . In his Micrographia (1665) he described his microscopic observations of plant tissues. Hooke coined the term cell".
The Columbia Electronic Encyclopedia Copyright 1994, 2000, Columbia University Press. |
The cell is in biology, the unit of structure and function of which all plants and animals are composed. The cell is the smallest unit in the living organism that is capable of integrating the essential life processes. There are many unicellular organisms, e.g., bacteria, in which the single cell performs all life functions. In higher organisms, a division of labor has evolved in which groups of cells have differentiated into specialized tissues , which in turn are grouped into organs and organ systems.
All cells share a number of common properties; they store information in genes made of DNA; they use proteins as their main structural material; they synthesize proteins in the cell's ribosomes using the information encoded in the DNA and mobilized by means of RNA; they use adenosine triphosphate as the means of transferring energy for the cell's internal processes; and they are enclosed by a cell membrane, composed of proteins and a double layer of lipid molecules, that controls the flow of materials into and out of the cell.
The cell cycle and cell division lie at the
heart of cell
biology. During the development of any multicellular organism, as well as
during adult life,cell multiplication
and division
contribute to the generation of specialized cells, which are necessary to
form tissues and carry out particular functions.Cell division is also
central to the replenishment of cells with a limited life span, such as
blood and skin cells.Cell division has been studied since the 19th century, when Rudolf Virchow`s aphorism "omnis cellula e cellula" encapsulated the concept that all cells are derived from pre-existing cells.
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Virchow, Rudolf. Born in 1821. Leading Prussian scientist, the founder of cellular pathology and a leader of German anthropology. Virchow was a major advocate of the cell theory and published an influential theory that cells arose from each other in a continuous series of generations . He opposed spontaneous generation and carried out experiments in the 1850's to show that nematodes do not arise spontaneously. A moderate leftist and a materialist, Virchow opposed evolution on the grounds that it was speculative, dependent upon spontaneous generation, and a thinly-veiled justification for revolutionary politics. A speech on this topic at an 1877 scientific meeting in Munich had international repercussions. He argued that all variation was pathological, and therefore that apparently distinct fossil humans (including the original Neandertal skeleton) were merely diseased individuals. These anti-evolutionary views on anthropology were presented at an 1887 meeting in Vienna. Virchow's greatest opponent was his former student Haeckel. Virchow's evolutionary and anthropological views had an indirect influence on Boas and his school.
Died 1902. |
"Where a cell exists, there must have been a preexisting cell, just as the animal arises only from an animal and a plant only from a plant." "Omnis cellula e cellula" or "All cells from cells." - Rudolf Virchow (1855).
The ability to reproduce distinguishes living organisms from nonliving entities and this perpetuation of life is based on the reproduction of cells or cell division. In unicellular organisms, the division of one cell to form two reproduces an entire organism. In multicellular organisms, cell division allows growth and development plus replacement of damaged or dead cells. Cell division is not just simple pinching in half, but a complex process that passes along the genome from one generation to the next. Cell division involves: precise replication of DNA, allocation of DNA to opposite ends of cells, and separation into two identical daughter cells.
During the mitosis a living cell duplicates into two daughter cells. So the living beings survive. The amount of the genetic material in daughter cells is the same as in the mother cell.
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In cycle 1, there is a cell-creator: 1 A0
In cycle2, our mother cell A0 during the mitosis duplicates into two daughter cells: 2 A1
So in cycle 3, the two mother cells, 2 A1, duplicate into four daughter cells: 4 A2
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In cycle 4 the four mother cells, 4 A2, during the mitosis duplicate into eight daughter cells: 8 A3;
In cycle n, the 2n-2 An-2 mother cells, duplicate into 2n-1 daughter cells: 2n-1 An-1 .
The number sequence which represents the cell division is a geometrical series:
|
1, |
2, |
4, |
8, |
16, |
32, |
64, |
128, |
256, |
512, |
As we have seen, by geometrical model of cell division, each cycle leads to the new cells of the same generation without surviving of the old cell. The cell A0 becomes older by the law
t=n*T
where n is the number of the cycle andT is the period of of the cycle.But, by the previous model the age of the cells is always T. That is in contradaction with the natural law and with the reproduction of the more - cellular organisms.
When we have reproduction of the more-cellular organisms the parents kept their generation and the children represent a new generation. The reproduction of the more- cellular organisms is universal law of the cell division:
Cell division results into two cells of different age!
We recognise the mother cell and the daughter cell.
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1.So in cycle 2, our young cell becomes a mother for the first time and produces her first daughter cell: A0 +
A1
2.In cycle 3, the mother cell A0 reproduces into A0 + A1, as well as cell-daughter reproduces into A1 + A2 . Now, three generations are present: A0 + 2 A1 + A2. |
3. In cycle 4, the original mother cell produces another daughter cell. Two mother cells A1 reproduce into 2 A1 + 2 A2. The mother cell A2 also produces its own daughter cell. Now four generations are present:A0 + 3 A1 + 3 A2 + A3 ;
4. In cycle 5, there are: A0 + 4 A1 + 6 A2 + 4 A3 + A4;
5. In cycle 6, there are: A0 + 5 A1 + 10 A2 + 10 A3 + 5 A4 +
A5 ;
etc...
The numbers of cells in each cycle produce the rows of the Pascal`s triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1
5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56
28 8 1
1 9 36 84 126 126 84 36 9 1
Pascal triangle is the Model of the cell division. According to the Pascal Triangle and Fibonacci numbers, the planets of the Solar
system are arranged.Also Pascal Triangle is the universal law of the atomic
structure.There is the connection between numbers of electrons and protons on one side and triangular and tetrahedral numbers, on the other. So, the Pascal Triangle is the basic number`s expression in Nature.
By adding diagonal numbers of the Pascal Triangle Fibonacci sequence can be obtained.
In the 13 th century Fibonacci investigated the numerous
rules of the cell division. The original problem that Fibonacci investigated ( in the year 1202 ) was about how fast rabbits could breed in ideal circumstances.
Suppose a newly - born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the femele always produces one new pair ( one male, one female ) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs of rabbits will there be in one year?
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At the end of the first month, they mate, but there is still only one pair At the end of the second month the female produce a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. |
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. The number of pairs of rabbits in the field at the start of each month is the Fibonacci number :
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1, |
1, |
2, |
3, |
5, |
8, |
13, |
21, |
34, |
55, |
The Fibonacci numbers play a significant role in Nature. Many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. The leaves are often arranged so that leaves above do not hide leaves below. This means that each one gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.If we count in the other direction we get a different number of turns for the same number of leaves. The number of turns in each direction and the number of leves met are three consecutive Fibonacci numbers!
Let`s now have a look at the leaves (1,4 and 9) on the plant in the next picture:
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We have 2 anti- clock wise rotations before we meet a leaf directly above the first ( 1-4 ), passing 3 leaves on the way. The Fibonacci fractal is 2/3.
We have 5 anti- clock wise rotations before we meet a leaf directly above the first ( 1-9 ), passing 8 leaves on the way. The Fibonacci fractal is 5/8.
We have 3 anti- clock wise rotations before we meet a leaf directly above the first ( 4-9 ), passing 5 leaves on the way. The Fibonacci fractal is
3/5.
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The above is computer-generated "plant" but you can see the same thing on real plants. One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers, Some common trees with their Fibonacci leaf arrangement numbers are:
1/2 elm, linden, lime, grasses 1/3 beech, grasses, blackberry 2/5 oak, cheery, apple, holly, plum 3/8 poplar, rose, pear, willow 5/13 pussy willow, almond
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The DNA molecule, the program for all life, is based
on the Golden section. It measures 34 angstroms long by 21
angstroms wide for each full cycle of its double
helix spiral.34 and 21, of course, are numbers in the Fibonacci
series and their ratio, 1.6190476 closely approximates Phi,
1.6180339. |
The Pascal Triangle is the ideal law of the cell
division. All cells have the infinite division possibilities and the cells never die. However, that is no case in Nature.
Real Model of the cell division must assume that a cell dies after a few cycles of the division.
In cycle 1, we have cell- mother A00. There is 1 cell present.
In cycle 2 cell- mother produces cell - mother A01, cell- first daughter A10 and cell- second daughter A-1 which doesn`t create a cell. There are 3 cells present.
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In cycle 3, cell A01 becomes a mother and produces her daughter : A02 + A10. Cell A10 becomes a mother for the first time and produces its own daughter: A11 + A20. Cell- second daughter A-1 doesn`t have its own production. There are :A02+A10+A11+A20 . There
are 4 cells present.
In cycle 4 there are : A03+A11+A20+
A12+A20+A21+A30 . The cell A02 has no production
and becomes A03. There are 7 cells present.
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In cycle 5 the cell A03 passes away and the cell A12 becomes A13. There are: A12+A20+A21+
A30+A13+A21+A30+
A22+A30+A31+A40. .There are 11 cells present.
In cycle 6 there are : A13+A21
+A30+A22+A30+A31+
A40+A22+A30+A31+
A40+A23+A31+A40+
A32+A40+A41+ A50.
There are 18 cells present.
In cycle 7 there are 29 cells and etc..
The number of cells in each cycle of cell division is :
| 1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |
199 |
322 |
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That is Lucas series of numbers. This series has the characteristic that each term
is the sum of the two previous terms. Lucas numbers Model of Cell Division is Real Model of Cell Division. The connection between Cell Division on one side and Pascal Triangle, Fibonacci numbers and Lucas numbers, on the other side, is shown.
Model Mitosis and Meiosis
- Cell
Division Construction Kit (CDCK)—public version available to everyone
-
NOTE: CDCK requires that the
Macromedia Flash™ 6 plugin is installed in your browser. Download the
Flash 6 player below.
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