Combinatorial Calculations and the Value of Fi



Abstract

Pseudoscientific disproofs of abiogenesis rooted in mistaken use of combinatorial mathematics have recently been adopted by some scientists opposed to the idea of ETI (extra- terrestrial intelligence). Two recent major arguments of this sort are reviewed and found to contain serious flaws. However, an approach of similar type revised to eliminate fallacious reasoning can be used to estimate the chance of life arising on the early Earth. A valid consideration of conditions on the primitive Earth seem to indicate that the fL term in the equation of Drake (1965) is equal to unity; that is, that given liquid water, organic chemicals, shock waves from micrometeorite impacts and time, life will always arise on a planet.

Introduction

"Creationists" believe that Earth's biosphere arose through divine fiat in ways incompatible with abiogenesis. Some Creationists have used combinatorial arguments to "disprove Darwin;" that is, to show that the chance of some complicated molecule coming into being at random is too small to have happened in the history of the Universe. Increasingly sophisticated arguments of this type have been presented by du Noy (1947), Yockey (1977), Hart (1980) and Thaxton et al. (1984).

The first two of these have been refuted elsewhere (Asimov 1957, Doolittle 1983, Freske 1983, Jukes 1983). They contain two fatal flaws. First, randomness is invoked where it is not appropriate (du Noy actually implied that atoms combine at random). Second, an unnecessarily narrow range of targets is chosen: only one combination of monomers or class of polymers is the "right" one and the rest are useless. Similar flaws exist in Hart and Thaxton et al. However, when similar methods are revised to be consistent with biochemical reality, they can be used to reliably predict the probability that life will arise in a given environment.

Hart's Argument

Hart (1980) argues as follows. The advent of life on Earth required a "genesis DNA" at least 600 nucleotide residues long. An unknown process "favors the formation of... strands of just that length." The early Earth had about 2 x 10^44 nitrogen atoms at or near its surface, and there are 2,000 such atoms in a 600-n DNA strand. Therefore, 10^41 600-n strands could have existed at once. Each strand fragments and recombines in a new pattern three times per second, so almost 10^58 unique strands could arise in a billion years. There are four types of nucleotides in DNA, so the chance of a given combination existing at a given place and time is one in 4^600, or about 10^360. The chance of a particular combination arising in a billion years, then, is one in 10^302.

Hart is aware that nucleic acids and proteins have substitution points where a different monomer has no effect on chemical function. He allows 100 points on the genesis DNA where either of two nucleotides are allowed and 400 where any nucleotide is allowed. This gives a chance of one in 10^90 (4^100 x 2^100) that a given combination will "work," and a net chance of one in 10^32 that a self-replicating molecule will arise in a billion years. He concludes by asserting that even this is too generous: real DNA can only direct protein synthesis if transfer RNA is present, and then only in a ribosome.

The argument has the following flaws. First, nucleotide sequence is not perfectly random.

1a. Hart assumes all nucleotides in the primeval seas occur in equal quantities. Some evidence (see 1b below) indicates that this is not the case.

For equal quantities, the number of possible combinations Nc in a polymer of length L is

            L
    Nc  =  a                                (1)

with a the number of monomer types. But for unequal quantities

            H L
    Nc  =  a                                (2)

where

            a
    H  =  -SUM f  log  f                    (3)
           i=1  i    a  i

in which fi is the fraction made up by the ith monomer (Yockey 1977). Equation (2) reduces to equation (1) for the equal- quantities case.

Assume that cytosine and guanine made up 30% each of the nucleotides in the primeval sea while adenine and thymine (for DNA, uracil for RNA) only made up 20% each. The number of possible combinations is then 4^591 10^356, not 4^360 10^361. The difference is only five orders of magnitude, but it shows how rapidly combinatorial values can change with a small change in input figures.

1b. Diener (1981) found an excess of cytosine-guanine over adenine-uracil links to be characteristic of viroid RNA, and Eigen et al. (1981) found that C-G links had ten times the "stickiness" of A-U links, which "offered special advantages in early evolution." Being harder to break, such bonds would be likelier to persist.

1c. "The possibility of rapid transpeptidation at elevated temperatures conceptually allows selection of a small number of most stable sequences" (Fox et al. 1971). In fact it now seems probable that natural selection applied to prebiotic molecules as well as to organisms -- "Selection in the Darwinian sense can be found even at the molecular level" (Kppers 1983; see also Eigen et al. 1981, Biebricher 1983). This is a fatal blow to the assumption of randomness.

Hart's second mistake is that the range of target combinations is too narrowly defined.

2a. No argument was advanced for why one combination would be self-replicating and all the rest would not. If the protein coded by the genesis DNA was an autoreplication enzyme, it is even possible that "...any sequence would replicate itself" (Futuyma 1982, p. 135).

2b. The assumption of a "genesis DNA" is not well founded. "RNA is generally believed to be the original information carrier in evolution, replaced much later by DNA" (Biebricher 1983). Rich (1981) suggested that transfer RNA is four billion years old, predating the earliest microfossils. The genesis DNA idea slants the argument against the likelihood of life, for two reasons. First, only two types of DNA are known, while over a thousand forms of RNA have been found (Kppers 1983). Second, DNA tends to be much longer than RNA. This makes it less likely a combinatorial calculation will hit a predefined range of targets.

Some examples: Potato Spindle Tuber viroid RNA is 359 nucleotides long (Diener 1981). Viroids in general tend to be 270-380-n long (Lewin 1983). Transfer RNA is typically 50-100-n long (Eigen et al. 1981).

2c. Hart's assumption that 2/3 of chain length might be substitution points is unrealistically low. As a counterexample, only seven of the 140 amino acids in the protein hemoglobin are always the same from one animal to another (Freske 1983). Thus the genes for hemoglobin must be 95% substitution points. And since the genetic code is highly redundant (64 codons for 20 amino acids and "stop"), even the codons for the crucial seven aminos may differ greatly from one animal to another.

Ohno (1972) suggested that 94-98% of human DNA is "nonsense;" introns and meaningless repetitions of the same protein-coding instructions. Lowenstein (1986) goes further, suggesting that 99% of all eukaryote DNA is "junk" or even "parasitic," and that the "vast majority" of amino-acid substitutions in proteins "make little or no difference to the function of the protein or the survival of the organism."

2d. DNA-tRNA protein synthesis happens in ribosomes because of evolution which favored that location. Self- replicating molecules (probably some form of tRNA by itself) came before DNA-tRNA protein synthesis, not the other way around. MacElroy et al. (1978) found that "polynucleotides can make limited base pair copies in the absence of enzymes." Price and Cech (1986) add that "RNA self-splicing can occur at a rate sufficient to support gene expression in a prokaryote, despite the likely presence of ribosomes on the nascent RNA."

2e. The possibility of self-replicating molecules other than DNA and RNA was not considered by Hart. Schwartz and Orgel (1986) found that new nucleic-acid-like chemicals formed in one advanced type of Urey-Miller experiment.

Thaxton et al.'s argument

The major conclusions of the book by Thaxton et al. (1984) are roughly as follows:

1) Amino acids created by one energy source in the primeval sea would have been destroyed by others. No organic sediments of great age have been found. Therefore, the Urey- Miller experiment and its successors support only "the myth of the prebiotic soup."

2) Nonbiological proteins, abnormal nucleotides and miscellaneous other monomers would have joined in the creation of most polymers, and even if selective sorting mechanisms excluded all but a few monomers, the bonds need not have been at the locations in present biochemicals. Polymerization experiments show a preponderance of abnormal linkages. Combinatorial mathematics thus imply that the likelihood of forming a recognizeable biochemical is small.

3) Divine special creation should be treated as a serious alternate hypothesis to abiogenesis. (It can be seen that this argument depends on the strength of the previous two.)

The first argument has some strength. Thaxton et al. make a good case that ultraviolet synthesis experiments, by using 'constructive' short-wave UV and not 'destructive' long- wave UV, overestimate the rate of amino acid synthesis as opposed to fragmentation. They cite Miller et al. (1976) on the relative flux strengths of various energy sources on the early Earth. 'Destructive' UV light was the major energy source. Shock waves, mostly from meteorite impacts, are much less strong (1.1 cal/cm/year versus 3,359, respectively).

However, shock waves are some one million times more efficient than short-wave UV at promoting amino acid synthesis (Bar-Nun et al. 1970). Thaxton et al. note this, yet here is their conclusion on shock waves (p. 47):

"Such optimism regarding possible shock-wave synthesis should be tempered by what we shall call the 'Concerto Effect' ... Amino acids produced in the atmosphere by electrical discharges or shock waves, for example, would be vulnerable to long-wavelength (> 2,000 ) ultraviolet photodissociation..."

If we assign relative efficiencies of effect of 1 to ultraviolet light and 10^6 to shock waves, synthesis should dominate over dissociation by a factor of 327. Thaxton et al. should have made some quantitative estimate of the relative rates of synthesis and dissociation, yet they rely on rhetoric about the "Concerto Effect" and the fact that organic-soup sediments have not been found. They ignore evidence which does suggest widespread amino acid synthesis in the early Solar system, such as the presence of amino acids in meteorites (Engel and Nagy 1982).

Their contention (p. 65) that "organic remains should be literally all over the earth [sic] in deep sediments of great age" ignores two problems: 1) Once life arose, it would have eaten the organic material, which would then be in the biosphere, not the sediments. 2) There are no sediments of sufficient age for analysis. Microfossils occur in rocks 3.8 billion years old (e.g. the Issua fragments of Greenland), but the oldest sediments are only 3.5 billion years old (Walker 1982). Thaxton et al. ignore the existence of continental drift, and thus of the likelihood that any organic-soup sediments would long since have been subducted under the Earth's crust.

Their second argument is probably their strongest. Protein and polynucleotide synthesis do indeed show a preponderance of abnormal monomers and linkages. Thaxton et al. show how this affects the probability of forming a given biochemical sequence (p. 157). They define the number of unique sequences:

                     N!
    _O_    =  ---------------               (4)
       cr     n1! n2! ... nk!

where N is the size of a polymer and ni the number of ith monomers in it. For a 100-mer evenly divided between 20 amino acids, this gives _O_cr 1.28 x 10^115 possible configurations. This is a more restricted definition than equation (1), which would give 20^100 1.27 x 10^130.

They then assume 50% of the links would be alpha-type peptide bonds, noting (p. 157) that "some studies indicate less than 50% alpha-links" (they only cite one, Temussi et al. 1976). They note that any monomer could have either dextro- or levulo-rotatory optical activity. The probability of hitting the "right" combination then becomes one in 1.28 x 10^115 x 2^99 x 2^100, or 1 in 10^175. Clearly this will never happen.

A similar calculation is made on pp. 145-146. The number of ways to arrange a 101-mer, the number of polypeptides in a one-meter thick layer covering the Earth (10^41), a reshuffling rate of 10^14 Hertz and a time lapse of five billion years are combined to yield a probability of only one in 10^45 that the target polypeptide would form. Thaxton et al. have missed the point that although their particular 101-mer was unlikely to form, some 101-mer would have. They never explain why only one combination would work.

Modeling the prebiotic Earth

Following Hart, I assume nitrogen was the element in shortest supply for building organic compounds. I then assume a discrete distribution of monomers based on the number of nitrogen atoms (Nn) each monomer contains:

             k
    Nm  =  ------                           (5)
           Nn + 1

where Nm is the number of monomers having Nn nitrogen atoms. Nn ranges from 0 to 5, with anything larger arbitrarily classed as a polymer. It seems realistic to have lighter molecules more common. The 1 in the denominator is there simply and solely to prevent Np = infinity at Nn = 0.

The total number of monomers of all nitrogen contents is then:

            5     k
    Tm  =  SUM  ------  =  2.45 k           (6)
          Nn=0  Nn + 1

The total of nitrogen atoms occurring in all monomers of a given size Nn is

                k
    Sp  =  Nn ------                        (7)
              Nn + 1

therefore the grand total number of nitrogen atoms present is

            5        k
    Sn  =  SUM  Nn ------  =  3.55 k        (8)
          Nn=0     Nn + 1

Sn is approximately 2 x 10^44 (Hart 1980), so the constant k must be Sn / 3.55, or about 5.63 x 10^43, and the total number of individual monomers Tm is 2.45 k or about 1.38 x 10^44.

I allow a reasonable number of different types of monomer at each size level:

    ty  =  5 (4 Nn^2 + 1)                   (9)

The 1, of course, prevents having no monomer types when Nn is zero. The total number of monomer types is therefore

5 Sy = SUM 5 (4 Nn^2 + 1) = 1,130 (10) Nn=0

In other words, we are considering a primeval sea which is a mixture of over a thousand simple types of chemical. Lastly, the number of monomers present of a given specific type is

    Ny  =  Nm / ty                          (11)

Polymer distribution

The assumed distribution of polymers is simply

    Npo  =  k2 / L                          (12)

Length L ranges from 1 to 1,000, which should not be too unrealistic. The number of monomers in a polymer of a given size is obviously L, so the number in all polymers of that size must be

    NL  =  L (k2 / L)  =  k2                (13)

The total number of monomers is

          1000
   Tm  =  SUM  k2  =  1,000 k2              (14)
          L=1

We know from equation (6) that Tm is about 1.38 x 10^44, thus k2 is Tm / 1,000 or about 1.38 x 10^41.

Target distribution

The number of "target" polymers present (i.e. those with exactly the qualities sought for) is given by

    Nt  =  Npo fm fl                        (15)

where fm is the fraction of polymers made only of monomers from the target set and fl is the fraction of those with all correct linkages. The first fraction is

             L
    fm  =  Pm                               (16)

where L is polymer length and Pm is the probability that a given single monomer is of the target set:

           a   Ny
    Pm  = SUM ----                          (17)
          i=1  Tm

Here a is the number of monomers in the target set. For nucleotides, a = 4; for amino acids, a = 20.

The second fraction is

             L - 1
    fl  =  Pl                               (18)

where Pl is the probability that a single linkage is of the correct type. (The L-1 comes about because a 2-mer has one link between its two parts, a 3-mer has two links, etc.)

For a range of possible targets of length from A to B, the cumulative Nt (equation (15)) becomes

           B
    St  = SUM Nt                            (19)
          L=A

Finally, the number of times the target species actually arose on Earth is

    X  =  St R t                            (20)

where R is the recombination rate of the average polymer in the primeval sea; I hope R = 10 per year is not unrealistic. I conservatively estimate t, the time available for abiogenesis, at 10^8 years.

Note that this assumes only one type of polymer is the "correct" one. In order to find the probability that any self-replicating molecule will arise, another factor, Na, is needed -- the number of possible functional analogues to the one that arose on Earth. With this in place, we can write:

    f   =  Na St R t                        (21)
     L

where I have deliberately used Drake's "fraction of planets with life" term (Drake 1965), since the target set I intend to calculate for is the set of self-replicating molecules. Equation (21) calculates the probability that life will arise on Earth.

The range of targets

But how to calculate the number of possible functional analogues? We have no experience of other self-replicating molecules than RNA and DNA, though Schwartz and Orgel's work and the large number of transfer RNAs indicates such molecules will be plentiful. A very crude attempt to provide an estimate follows.

A self-replicating polymer is composed of a certain set of monomers. The number of possible sets, given the 1,130 molecules in the model primeval sea, is

                1,130!
    Ns  =  ---------------                  (22)
           m! (1,130 - m)!

where m is the number of monomers in the set.

This is, of course, merely the combinations equation -- how many sets of n things are possible taken m at a time.

Estimating m is a problem. DNA combines four nucleotides at a time in 3-n codons. RNA does the same, save that one of the nucleotides is different (uracil instead of thymine). Both code for amino acids, of which there are 20 in our biology. But these arrangements are not universal even on Earth:

1. According to Dickerson (1978), "Studies with molecular models show that it is possible to construct a double-strand DNA helix with paired bases and a 5',2' connection, but the helix appears to be less stable than one with a 5',3' structure." The numbers, of course, refer to the location at which links occur.

Anyone familiar with DNA knows that it is a stable molecule indeed, stable almost to the boiling point of water. A 5',2' DNA might still be quite stable. A target species should have all links at the same location, but there is no reason to think ours the only one possible.

2. Our amino acids all have levulo-rotatory optical activity, but Urey-Miller type experiments produce half levulo- and half dextro- molecules. Our arrangement may simply be the chance result of the first successful replicant having had that optical activity; there is no reason dextro-rotatory amino acids would not have worked just as well (and levulo- rotatory sugars in place of our dextro-rotatory ones). As it happens, a substance related to L-glucose is used by Streptomycin, and D amino acids are found in the cell walls of some bacteria (Asimov, 1972).

3. Rich and Kim (1978) found more than 50 types of nucleotide in transfer RNA. Schwartz and Orgel (1986), as mentioned earlier, found novel species similar to nucleic acids. The stop code UGA ("umber") in our genetic code produces tryptophan in mitochondria and yeasts, and a eukaryote codon for leucine yields threonine in mitochondria.

So what is the proper value for m? Clearly, any value that might yield a consistent coding set. One obviously would not work, as only one substance could be coded for, but any greater value could be used depending on how long a codon was. Arbitrarily restricting m to the range 2 to 50 (the upper limit after Rich and Kim), the number of functional analogues to our first self-replicating molecule is:

             50       1,130!
    Na  =  F SUM  ---------------           (23)
             m=2  m! (1,130 - m)!

where F is the fraction of such sets that will actually prove self-replicating in a primeval sea.

What is the proper value of F? How many polymer sets are self-replicating compared to those that are not?

The fact is that there is no way to estimate F at this time. For the moment I will assume an unrealistically high value, F = 1 (100%), so that this factor can be ignored while the rest of the estimate is calculated. We will return to F later.

I wrote the program which appears as Appendix A to run through the calculations specified so far. I take a generalized RNA as the target polymer with N = 4 (the classic four nucleotides only, ignoring the rich variety found in transfer RNA). L is computed from over the range 22 to 1000 (since Eigen et al., 1981, produced self-replicating RNA variants in their Q- phage experiments as small as 22 nucleotide residues long). In fact it turns out that Nt declines rapidly with increasing length, so an accurate value can be produced in less than two dozen terms (computationallly convenient given the limitations of present-day computers). St turns out to be approximately 1.57 x 10^-29. Na is about 5.17 x 10^87 (greater than the estimated number of particles in the observable Universe, by the way). Thus fL = about 8.14 x 10^67.

Now F becomes important. Clearly all types of polymer are not self-replicating. But in order for fL to be less than 1 (i.e. less than a 100% chance of life arising from abiogenesis), F must be less than about 1.23 x 10^-88. Knowing that DNA, the 1,000-plus varieties of RNA and the new nucleic- acid-like chemicals found by Schwartz and Orgel exist, a value that low seems less than a prima facie sure thing.

References

Asimov, Isaac 1957. "The Unblind Workings of Chance." In ONLY A TRILLION. NY: Abelard-Schuman, Inc.

Asimov, Isaac 1972. "The Asymmetry of Life." 68-80 in THE LEFT HAND OF THE ELECTRON. NY: Dell Publishing Co., Inc.

Bar-Nun, A. et al. 1970. "Shock Synthesis of Amino Acids in Simulated Primitive Environments." Sci. 168:470-473.

Biebricher, C.K. 1983. "Darwinian Selection of Self- Replicating RNA Molecules." 1-52 in EVOLUTIONARY BIOLOGY 16, Ed. Hecht, M.K. et al. NY: Plenum Press.

Dickerson, R.E. 1978. "Chemical Evolution and the Origin of Life." Sci. Am. 239:70-85.

Diener, T.O. 1981. "Viroids." Sci. Am. 244:66-73.

Doolittle, R.F. 1983. "Probability and the Origin of Life." 85-97 in SCIENTISTS CONFRONT CREATIONISM, Ed. Godfrey, L.R. NY: W.W. Norton and Co.

Drake, Frank 1965. "The Radio Search for Intelligent Extraterrestrial Life." 323-345 in CURRENT ASPECTS OF EXOBIOLOGY, Ed. Mamikumian, G. and Briggs, M.H. Pasadena: Pergamon Press.

du Noy, L. 1947. HUMAN DESTINY. Cited in Asimov, 1957.

Eigen, Manfred et al. 1981. "The Origin of Genetic Information." Sci. Am. 244:88-118.

Engel, M. and Nagy, B. 1982. "Distribution and Enantiometric Composition of Amino Acids in the Murchison Meteorite." Nature 296:837-840.

Fox, S.W. et al. 1971. "The Primordial Sequence, Ribosomes and the Genetic Code." 252-262 in MOLECULAR EVOLUTION I. CHEMICAL EVOLUTION AND THE ORIGIN OF LIFE, Ed. Buvet, R. and Ponnamperuma, C. NY: American Elsevier Publishing Co., Inc.

Freske, S. 1983. "Creationist Misunderstanding, Misrepresentation and Misuse of the Second Law of Thermodynamics." 285-295 in EVOLUTION VS. CREATIONISM: THE PUBLIC EDUCATION CONTROVERSY, Ed. J.F. Zetterburg. Phoenix: The Oryx Press.

Futuyma, D.J. 1982. SCIENCE ON TRIAL: THE CASE FOR EVOLUTION. NY: Pantheon Books.

Hart, Michael H. 1980. "N is Very Small." 19-25 in STRATEGIES FOR THE SEARCH FOR LIFE IN THE UNIVERSE, Ed. Papagiannis, M.G. Boston: D. Reidel Publishing Co.

Jukes, T.H. 1983. "Molecular Evidence for Evolution." 117- 138 in SCIENTISTS CONFRONT CREATIONISM, Ed. Godfrey, L.R. NY: W.W. Norton and Co.

Kppers, B.-O. 1983. MOLECULAR THEORY OF EVOLUTION. NY: Springer-Verlag.

Lewin, R. 1983. "The Birth of Recombinant RNA Technology." Sci. 222:1313-1315.

Lowenstein, J.M. 1986. "Molecular Phylogenetics." Ann. Rev. Earth Planet. Sci. 14:71-83.

MacElroy, R.D. et al. 1978. "An Approach to the Origin of Self-Replicating Systems. I. Intermolecular Attractions." 249-254 in ORIGIN OF LIFE, Ed. Noda, H. Tokyo: Center for Academic Publications Japan.

Miller, Stanley L. et al. 1976. J. Mol. Evol. 9:59. Cited in Thaxton et al. 1984.

Ohno, S. 1972. "An Argument for the Genetic Simplicity of Man and Other Mammals." J. Human Evol. 1:651-662.

Price, J.V. and Cech, T.R. 1986. "Coupling of Tetrahymena Ribosomal RNA Splicing to Beta-Galactosidase Expression in Escherichia Coli." Sci. 228:719-722.

Rich, A. 1981. "Transfer RNA and the Origin of Protein Synthesis." 211-228 in LIFE IN THE UNIVERSE, Ed. Billingham, J. Cambridge: The M.I.T. Press.

Rich, A. and Kim, S.H. 1978. "The Three-Dimensional Structure of Transfer RNA." Sci. Am. 238:52-62.

Schwartz, A.W. and Orgel, Leslie E. 1986. "Template-Directed Synthesis of Novel, Nucleic Acid-Like Structures." Sci. 228: 585-587.

Temussi et al. 1976. J. Mol. Evol. p. 105. Cited in Thaxton et al., 1984.

Thaxton, C.B. et al. 1984. THE MYSTERY OF LIFE'S ORIGIN: REASSESSING CURRENT THEORIES. NY: Philosophical Library.

Walker, J.C.G. 1982. "Climatic Factors on the Archaean Earth." Paleogeog. Paleoclimatol. Paleoecol. 40:1.

Yockey, H.P. 1977. "A Calculation of the Probability of Spontaneous Biogenesis by Information Theory." J. Theoret. Biol. 67:377-398.

Appendix A. Simulation Code (Borland Turbo Pascal 5.0)
{--------------------------------------------------------------------}
{ Sea models the primeval sea to estimate the probability of }
{ abiogenesis. }
{--------------------------------------------------------------------}
{$N+}
PROGRAM Sea;
  
USES
      CRT;
CONST
      ty:     array [0..5] of extended = (5, 25, 85, 185, 325, 505);
  
VAR
      DfL:            extended;       { Drake's f-sub-L. }
      DiagFile:       text;           { Diagnostic file. }
      F:              extended;       { Fraction of monomer sets useful. }
      Fact1130:       extended;       { 1,130! }
      fl:             extended;       { Fraction with links okay. }
  
      fm:             extended;       { Fraction with monomers okay. }
      i:              integer;        { Loop counter. }
      k2:             extended;       { Polymer constant. }
      L:              extended;       { Length of polymer. }
      Na:             extended;       { Number of functional analogues. }
  
      Npo:            extended;       { Number of polymers of length L. }
      Ns:             extended;       { Number of possible monomer sets. }
      Nt:             extended;       { Number of polymers just right type. }
      Pl:             extended;       { Probability 1 link is okay. }
      Pm:             extended;       { Probability 1 monomer is okay. }
  
      R:              extended;       { Recombinations per year. }
      St:             extended;       { Cumulative Nt. }
      t:              extended;       { Years available for abiogenesis. }

{--------------------------------------------------------------------}
{ Fact returns the factorial of the (integer) argument. }
{--------------------------------------------------------------------}
FUNCTION Fact (iarg: integer): extended;
CONST
      small: array [1..20] of extended = (
  
      1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
      3628800.0, 39916800.0, 479001600.0, 6227020800.0,
      87178291200.0, 1307674368000.0, 20922789888000.0,
      355687428096000.0, 6402373705728000.0, 121645100408832000.0,
      2432902008176640000.0);
VAR
      f:              extended;       { Factorial. }
      i:              integer;        { Loop counter. }
      n:              extended;       { Current multiplier. }
BEGIN
      if iarg <= 20 then
                Fact := small[iarg]
      else
       begin
              f := small[20];
              n := 20.0;
  
              for i := 21 to iarg do
               begin
                      n := n + 1.0;
                      f := f * n;
               end;
  
              Fact := f;
       end;
END;  { Fact }
  
{--------------------------------------------------------------------}
{ Pow raises "base" to the power "exponent." }
{--------------------------------------------------------------------}
FUNCTION Pow (base: extended; exponent: extended): extended;
BEGIN
      Pow := exp (exponent * ln (base));
END;  { Pow }

{====================================================================}
{ Main routine starts here. }
{====================================================================}
BEGIN
      Assign (DiagFile, 'DIAG.DAT');
      rewrite (DiagFile);
  
      F := 1.0;                       { Fraction of monomer sets useful. }
      k2 := 1.38E+41;                 { Polymer constant. }
      R := 10.0;                      { Recombinations per year. }
      t := 1.0E+08;                   { Years available for abiogenesis. }

{ Single potentially useful set -- RNA!: }
  
      Pm := 4 / 1130;                 { Or 1 in 282.5. }
      Pl := 0.2;                      { 1 in 5. }
      St := 0.0;                      { But not for long. }
        
      for i := 22 to 251 do
       begin
              writeln (i, Nt, St);
              L := i;                         { Float. }
              Npo := k2 / L;                  { This many are this long. }
              fm := Pow (Pm, L);              { Pm ^ L. }
  
              fl := Pow (Pl, (L - 1));        { Pl ^ {L - 1). }
              Nt := Npo * fm * fl;            { Number perfectly in set. }
              St := St + Nt;                  { Sum over all lengths. }
       end;
  
      writeln ('St = ', St);
      writeln (DiagFile, 'St = ', St);
  
{ Find Na: }
  
      Fact1130 := Fact(1130);         { By definition. }
      Na := 0.0;                      { So far. }
  
      for i := 2 to 50 do
       begin
              writeln (i);
              Ns := F * Fact1130 / (Fact (i) * Fact (1130 - i));
              Na := Na + Ns;
       end;
  
      writeln ('Na = ', Na);
      writeln (DiagFile, 'Na = ', Na);

{ Find St: }
  
      DfL := Na * St * R * t;
 
      writeln ('fL = ', DfL);
      writeln (DiagFile, 'fL = ', DfL);
      close (DiagFile);
END.  { Sea }


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