The Ghost of Paley
Posts: 1703
Joined: Oct. 2005
|
By the way, I keep hearing about how Mystery, a joke, etc. Well, if this is true, can someone give a specific example?
Here's where they discuss configurational entropy:
| Quote | 3. Configurational Entropy Work
Finally, we need to quantify the configurational entropy change (Sc) that accompanies the formation of DNA and protein. Here we will not get much help from standard experiments in which the equilibrium constants are determined for a polymerization reaction at various temperatures. Such experiments do not consider whether a specific sequence is achieved in the resultant polymers, but only the concentrations of randomly sequenced polymers (i.e., polypeptides) formed. Consequently, they do not measure the configurational entropy (Sc) contribution to the total entropy change (S). However, the magnitude of the configurational entropy change associated with sequencing the polymers can be calculated.
Using the definition for configurational "coding" entropy given in eq. 8-2c, it is quite straightforward to calculate the configurational entropy change for a given polymer. The number of ways the mass of the linear system may be arranged © can be calculated using statistics. Brillouin20 has shown that the number of distinct sequences one can make using N different symbols and Fermi-Dirac statistics is given by
= N! (8-6)
If some of these symbols are redundant (or identical), then the number of unique or distinguishable sequences that can be made is reduced to
c = N! / n1!n2!n2!...ni! (8-7)
where n1 + n2 + ... + ni = N and i defines the number of distinct symbols. For a protein, it is i =20, since a subset of twenty distinctive types of amino acids is found in living things, while in DNA it is i = 4 for the subset of four distinctive nucleotides. A typical protein would have 100 to 300 amino acids in a specific sequence, or N = 100 to 300. For DNA of the bacterium E. coli, N = 4,000,000. In Appendix 1, alternative approaches to calculating c are considered and eq. 8-7 is shown to be a lower bound to the actual value.
For a random polypeptide of 100 amino acids, the configurational entropy, Scr, may be calculated using eq. 8-2c and eq. 8-7 as follows:
Scr = k lncr
since cr = N! / n1!n2!...n20! = 100! / 5!5!....5! = 100! / (5! 20
= 1.28 x 10115 (8-8)
The calculation of equation 8-8 assumes that an equal number of each type of amino acid, namely 5, are contained in the polypeptide. Since k, or Boltzmann's constant, equals 1.38 x 10-16 erg/deg, and ln [1.28 x 10115] = 265,
Scr = 1.38 x 10-16 x 265 = 3.66 x 10-14 erg/deg-polypeptide
If only one specific sequence of amino acids could give the proper function, then the configurational entropy for the protein or specified, aperiodic polypeptide would be given by
Scm = k lncm
= k ln 1
= 0
(8-9)
Determining scin Going from a Random Polymer to an Informed Polymer
The change in configurational entropy, Sc, as one goes from a random polypeptide of 100 amino acids with an equal number of each amino acid type to a polypeptide with a specific message or sequence is:
Sc = Scm - Scr
= 0 - 3.66 x 10-14 erg/deg-polypeptide
= -3.66 x 10-14 erg/deg-polypeptide (8-10)
The configurational entropy work (-T Sc) at ambient temperatures is given by
-T Sc = - (298oK) x (-3.66 x 10-14) erg/deg-polypeptide
= 1.1 x 10-11 erg/polypeptide
= 1.1 x 10-11 erg/polypeptide x [6.023 x 1023 molecules/mole] / [10,000 gms/mole] x [1 cal] / 4.184 x 107 ergs
= 15.8 cal/gm (8-11)
where the protein mass of 10,000 amu was estimated by assuming an average amino acid weight of 100 amu after the removal of the water molecule. Determination of the configurational entropy work for a protein containing 300 amino acids equally divided among the twenty types gives a similar result of 16.8 cal/gm.
In like manner the configurational entropy work for a DNA molecule such as for E. coli bacterium may be calculated assuming 4 x 106 nucleotides in the chain with 1 x 106 each of the four distinctive nucleotides, each distinguished by the type of base attached, and each nucleotide assumed to have an average mass of 339 amu. At 298oK:
-T Sc = -T (Scm - Scr)
= T ( Scr - Scm)
= kT ln (cr - lncm)
= kT ln [(4 x 106)! / (106)!(106)!(106)!(106)!] - kT ln 1
= 2.26 x 10-7 erg/polynucleotide
= 2.39 cal/gm 8-12
It is interesting to note that, while the work to code the DNA molecule with 4 million nucleotides is much greater than the work required to code a protein of 100 amino acids (2.26 x 10-7 erg/DNA vs. 1.10 x 10-11 erg/protein), the work per gram to code such molecules is actually less in DNA. There are two reasons for this perhaps unexpected result: first, the nucleotide is more massive than the amino acid (339 amu vs. 100 amu); and second, the alphabet is more limited, with only four useful nucleotide "letters" as compared to twenty useful amino acid letters. Nevertheless, it is the total work that is important, which means that synthesizing DNA is much more difficult than synthesizing protein.
It should be emphasized that these estimates of the magnitude of the configurational entropy work required are conservatively small. As a practical matter, our calculations have ignored the configurational entropy work involved in the selection of monomers. Thus, we have assumed that only the proper subset of 20 biologically significant amino acids was available in a prebiotic oceanic soup to form a biofunctional protein. The same is true of DNA. We have assumed that in the soup only the proper subset of 4 nucleotides was present and that these nucleotides do not interact with amino acids or other soup ingredients. As we discussed in Chapter 4, many varieties of amino acids and nucleotides would have been present in a real ocean---varieties which have been ignored in our calculations of configurational entropy work. In addition, the soup would have contained many other kinds of molecules which could have reacted with amino acids and nucleotides. The problem of using only the appropriate optical isomer has also been ignored. A random chemical soup would have contained a 50-50 mixture of D- and L-amino acids, from which a true protein could incorporate only the Lenantiomer. Similarly, DNA uses exclusively the optically active sugar D-deoxyribose. Finally, we have ignored the problem of forming unnatural links, assuming for the calculations that only CL-links occurred between amino acids in making polypeptides, and that only correct linking at the 3', 5'-position of sugar occurred in forming polynucleotides. A quantification of these problems of specificity has recently been made by Yockey.21
|
Now, the assumption of an 100 amino acid polypeptide might be too restrictive, because I believe that there is some evidence that a smaller polypeptide might have been able to kick things off. If so, the Scr component of the configurational entropy would be decreased, thus reducing the work involved. But to what extent? Or is this calculation relevant? If not, why not?
--------------
Dey can't 'andle my riddim.
|
?? 
(to keep the biologists happy) 
