Wednesday, April 23, 2008

Exponents



The way probability relates to biology and evolution can be controversial and yet is important. I'd like to talk about the simple mathematics of exponents for those who don't know much about it. Origin of life (OOL) and evolution are both important subjects in biology, and it helps to understand exponents in both cases.

The picture at the top shows 10 to the first, second and third powers. In my text, I use the ^ sign for the power after 10, such as 10^1, 10^2, 10^3. The exponent is the power to which the base, in this case ten, is multiplied by itself (e.g. 10^2=10x10). If the base is 10, the exponent, or power, can also stand for the number of zeroes after the 1. For instance, 10^1=10, 10^2=100 and 10^3=1000. You can see that the numbers increase dramatically as the exponents increase. If the line on your screen is about 3 1/2 inches as it is on mine, the next number, 10,000 would be at the end of a yard and the next, 10^5, at 30 feet.

The animation from Harvard about ATP Synthase (at right in Links) shows the cellular mechanism for construction of molecules which deliver energy to the rest of the cell (ATP). The mechanism would have had to be present in some of (all?) the earliest life on earth. The Synthase machinery has several components, each made of several sub-units. One of the proteins, ATP synthase subunit a, in the fruitfly has 224 amino acids in specific order. These are (from Uniprot, P00851):

10 20 30 40 50 60
MMTNLFSVFD PSAIFNLSLN WLSTFLGLLM IPSIYWLMPS RYNIFWNSIL LTLHKEFKTL

70 80 90 100 110 120
LGPSGHNGST FIFISLFSLI LFNNFMGLFP YIFTSTSHLT LTLSLALPLW LCFMLYGWIN

130 140 150 160 170 180
HTQHMFAHLV PQGTPAILMP FMVCIETISN IIRPGTLAVR LTANMIAGHL LLTLLGNTGP

190 200 210 220
SMSYLLVTFL LVAQIALLVL ESAVTMIQSY VFAVLSTLYS SEVN

Remember--the letters stand for amino acid abbreviations. For example, the A is for Alanine, D for Aspartic Acid, etc. (see Amino Acids--Wikipedia link at right).

The machinery would have had to be in working order from the days of the first bacteria. In fact, the bacteria E. Coli has more amino acids in this subunit than the fruitfly--271 (Uniprot, P0AB98). {By the way--scientists talk about viruses being simpler organisms which evolve, but viruses are not able to reproduce by themselves. They need other cells for the reproductive machinery in order to replicate and therefore could not have come earlier than those cells.} The bacteria, so far the simplest life forms which replicate, have over 1000 proteins in working order.

The probability atoms and molecules would have arranged themselves by chance into working machinery is astoundingly low. There are ways to figure the numbers, such as the probabilities of getting started with the strictly left "handedness" of amino acids as they are found in functional, living proteins (see Amino Acid Mirror Images link on right). Let's use the bacteria number of 271 amino acids for this sub-unit. For the ATP subunit shown above, just for the mirror-images to be only the L-(Levo for left) orientation by chance, we'd use 2 for the base because the outcome could either be left or right as it is in non-living nature. In cases of independent probability, you get the total probability by multiplying together each possible outcome. So that would be 1 in 2 for each amino acid to be L-oriented and 1 in 2^271 for the whole subunit to be L, which converts to about 1 in 10^81. The estimated number of particles in the universe is 10^80. Probabilities for the whole ATP Synthase molecule forming in the L-orientation by chance would be much smaller. William Demski defines what he calls the universal probability bound in his book, The Design Inference. He shows that all events in the universe, including chemical reactions, are within the number 10^150. That is a probability boundary for our universe. It's the multiplication of the number of particles in the universe, 10^80, times the maximum amount of events per second including chemical reactions, 10^45, times the number of seconds in an old universe of about 14 billion years, 10^25. If there is less than 1 chance in all the events of the universe for something to happen, we can be assured it didn't happen. (See The Design Inference, Cambridge UP, for a more precise definition of Universal Probability Bound.)

These numbers have caused some scientists to speculate on "Infinite Universe" theories. In that scenario, at least one of an infinite set of universes can by chance form life. Articles about infinite universes and life from the production of infinite universes have appeared in scientific journals. One interesting problem of the infinite universe (inflationary) theory that proponents have stated and now must consider: how can we know the probability of the formation of infinite universes?

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