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Author
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Topic: Is 2nd Law a special case of 4th Law?
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Poul Willy Eriksen
Member
Member # 1976
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posted 06. December 2006 08:33
Melvin wrote:
quote:
I wonder if it would help if we could show each of theses states to be measurable.
I would say 'yes'; at least trying to make them measurable might clean up some confusion about, what each of these states really are made up of ![[Smile]](smile.gif)
And how well does the thermodynamic laws carry into other areas?
In Shannon information theory we are dealing with two Markov processes (a message process and a noise process) acting on each other; but how well would that model biological inheritance?
- pwe
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Allen Lints
Member
Member # 1453
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posted 07. December 2006 04:36
Poul,
Disclaimer: I fully stipulate that I am strictly an amateur in these matters. I have been study these issues for some fifteen years and do have some opinions so I will put in my two cent worth.
I hope this doesn't increase your confusion. If it does perhaps I can decrease it by giving you some good references.
When Shannon proposed his theory he restricted his informational sources to Ergodic Markov Process. This is important because it not only guarantees an equilibrium but a special kind of equilibrium. The long term average probability distribution will equal the average global probability distribution. (i.e. Ergodic)
A Mathematical Theory of Communciation
Ergodic Markov Theory
I am not suggesting that Shannon theory can not be extend to other distributions, but it is important to note when Darwinist suggest that life is at nonequilibrium, that they never specific where life equilibrium is, hence how can they know how far away from equilibrium life has evolved. It seems we need to baseline the genetic process before we can proceed to other aspects of it.
Quine McClusky Simplification for System 18
Shannon Information and Kolmogorov Complexity
I will try and go in to more depth this weekend but it is late tonight and I need my rest.
ADL
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Poul Willy Eriksen
Member
Member # 1976
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posted 07. December 2006 08:36
Hi Allen;
You write:
quote:
I am not suggesting that Shannon theory can not be extend to other distributions, but it is important to note when Darwinist suggest that life is at nonequilibrium, that they never specific where life equilibrium is, hence how can they know how far away from equilibrium life has evolved. It seems we need to baseline the genetic process before we can proceed to other aspects of it.
I cannot say for sure; but the nonequilibrium would appear to be environmental changes. If the environment changes, the relative fitness of alleles may change as well, which over time should change their frequences.
Oh, and thanks for the links It's some time ago that I last read Shannon's paper - maybe I should give it a reread.
quote:
I will try and go in to more depth this weekend but it is late tonight and I need my rest.
Ok, I wish you a good sleep And I'm sure you'll come up with something usefull, when you have the time for it.
- pwe
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Allen Lints
Member
Member # 1453
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posted 11. December 2006 01:42
Allen;
Perhaps we can go into the Darwin’s issue a little later in the conversation. Let us start with the generic entropy function.
Poul:
quote:
H = − k ∑ i pi log pi
This gives the entropy for defining a minimal binary representation for an alphabet associated with an ergodic equilibrium probability distribution for the frequency of an occurrence of letter in the language. In bits provided k is ln (2). This is quite a mouthful to say. What does it mean? Let us say we have a language A with a 21 letter alphabet and we know is ergodic equilibrium distribution and Shannon entropy of 4.01. Likewise, we have a language B with a 21 letters alphabet and Shannon entropy of 4.99. Now precisely what does it mean for language A to have 4.01 entropy. What exactly does one bit of information correspond to in terms of the probabilities? It is easy to determine a solution to w*ln(w) =1. w is 1.763222834351896710225201776952… but this isn’t a probability, and there is no solution for -w*ln(w)=1. Does this mean that in effect that five letters of the alphabet are uses so infrequently that we can in effect discount them and use only sixteen letters most of the time? Obviously, in both languages, we need five bits to express all letters, but in language B do we have to use all 32 5-bit representations or can we use some letters more frequently and still achieve entropy of 4.99?
Kolmogorov complexity works only a little bit better. It maps the binary representation to the natural numbers 0,1,2,3 … n. However, there is no general purposes algorithm to accomplish this. In fact, for some problems no solution may exist.
Fortunately, I was not attempting to solve the general problem but a specific problem the standard genetic code for life. There is only 2.04e+42 possible binary representations of the genetic code. So I can use a Karnaugh map after the manner of Dr. LeMay.
Dr. Lemay Doctoral Thesis
Standard Genetic Code
Measuring the simplicity as the ratio of the number of 1’s in the Karnaugh map divide by the number of adjacent 1’s. The lower the simplicity ratio, then the simpler the solution will be with the Quine-McClusky algorithm. I set a genetic algorithm to test various Karnaugh maps. It found a plateau at .200000 fitness and the solutions were dominant by sequence numbers 11 to 31 with the proline amino acid being 31.
U was assigned to 00, C was assigned 01, A was 10 and G was 11. A quick application of DeMorgan Theorem and Nagle Algorithm converted the amino acids into sequences 0 to 20. It seems the algorithm solves the problem for the genetic code not only for Shannon but also for Kolmogorov. However, there was not one solution but 178 solutions for the standard genetic code. Naturally, I extend my search to twelve other genetic codes and the results can be summarizes below.
Genetic Code Number of System Minimum Simplicity
Ascidian Mitochondrial 76 0.199234
Blepharisma Nuclear 75 0.199433
Ciliate Nuclear 621 0.198502
Echinoderm Mitochondrial 804 0.198502
Euplotid Nuclear 272 0.198682
Flatworm Mitochondrial 714 0.198308
Invertebrate Mitochondrial 808 0.198488
Standard Nuclear 178 0.199248
Vertebrate Mitochondrial 58 0.199241
Chlorophycean Mitochondrial 44 0.199431
The minimal representation does not depend on the frequency the amino acids have in the language.
So I was lead to find a ergodic Markov process which would give me an equilibrium frequency for amino acids with each of the genetic codes. Subsequently I could uses this to determine the relative entropy for each distribution.
There were some eighty-eight Markov matrices know. Starting with Kimura, Dayhoff, Blotsum etc. These were all specific to some set of genes or proteins. I want matrices specific to genetic code and I wanted it to be ergodic, I go in to why a little latter in the discussion.
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Allen Lints
Member
Member # 1453
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posted 11. December 2006 01:57
Eighty-three Markov Matrices
I counted the number of codon in this manner. If X was a codon that did not change and Y was the codon that does change, then for single codon changes in the triplet the possibilities were XXX, XXY, XYX, and YXX. Since Y can have three values, the total number of single mutation for each codon is ten. What effect this has on the amino acid distribution can easily be calculated for the initial probability distribution.
Simply matrix multiplication and solve the higher number of mutations. P^n being the n mutation matrix.
Inital Probability Matrix
First Passage Markov Matrices
Variance Matrix
Covariance Matrices
EigenSpace
Covariance Eigenspace
These matrices are all ergodic, aperiodic, Perron-Frobenius irreducible, time reversible. Time reversible in Markov terminology means it is a Markov chain going forward in time and it is a Markov chain going back in time and since it aperiodic it is the same Markov Chain. Perron-Frobenius irreducible means that all states can communicate with all other states in a single ergodic set or that in a finite amount of time any triplet codon can mutated into any other triplet codon.
How does this related to what the members have been talking about the Omega space.
Dr. Motoo Kimura formulated a neutral theory of evolution. He started with Diffusion Equation.
While this is a general-purpose equation, it has a special solution if the D function is constant or proportional to the local allele density. If these conditions are met, then equation takes the form of the Heat Equation.
This is easily seen by deriving the Heat Equation from the Continuity Equation (conservation of charge) and Frick’s First Law.
By a simple substitution, you can derive the heat equation. Note that the left hand side of the heat equation is solely dependent on time, while the right hand is solely dependent on space. The only way this can happen is by the separation of variables and the left hand side of the equation must be a constant. So also, must the right hand side and it must be the same constant. Therefore, the solution of the problem must be ergodic. The fact that the densities are a constant matrix changes nothing of the fundamental nature of the equation. After all the genetic code is a constant as far as we know.
Well this is enough for a start, perhaps next time we can discuss Zipf’s Law.
Wishing you well,
Allen Lints
PS. Sorry my message was divded into two posts, I not entirely used to this form.
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Allen Lints
Member
Member # 1453
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posted 13. December 2006 04:04
Poul,
Let me try to answer an objection before you raise it. How long time to reach equilibrium?
Forunately, Grinstead and Snell has answer this question concerning Kemeny's Constant. If we don't know the inital distributions what is an reasonable expection on average for the mutation to achieve equilibrium.
There is a couple of ways to calculated this. One is to take the trace of the fundamental matrix TR(Z)-1 where Z=(I-T)^-1.
Fundamental Matrix for the Genetic Codes
Or altenatively to take the sum of 1/(1-lambda) of all the subdominant eigenvalues.
- Genetic Code Kemeny Constant
- Alt 28.92413034
- Chl 29.00888464
- Yea 29.20704982
- Ble 29.20776845
- Std 29.31287183
- Eup 29.58547845
- Asc 29.63147812
- Ver 29.64486555
- Flt 29.75874552
- Ech 29.80822176
- Inv 29.88207374
Considerable research has been undertaken in the Kemeny's constant area. Establishing it as intimately related to potential theory.
The negative eigenvalues just demonstrated the eveness of the language see Wentian Li page 9.
Grinstead and Snell Reply to Peter Doyle
Catral Research on Kemeny's Constant
Doyle's Research on Green Function and Kemeny's Constant
Wentian Li Negative Eigenvalues Power Spectra
I contact Dr. Christian Schwabe, he work on a monograph about a Genomic Potential and I ask him if this was the potential he was referring. He send me a polite e-mail back saying it wasn't.
Surely all of the Darwinist using ergodic Markov chains can tells us how long it will be to expect equilibrium to be achieved for their particular distribution.
Regards,
Allen Lints
[ 13. December 2006, 22:18: Message edited by: Allen Lints ]
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