a good article on randomly assembling proteins that is very detailed
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Probabilities of randomly assembling a primitive cell on Earth
Summary
We evaluate the probability Pr that the RNA of the first cell was
assembled randomly in the time available (1.11 billion years
[b.y.]). To do this calculation, we first set a strict upper limit
on the number of chemical reactions nr which could have occurred
before the first cell appeared.
In order to illustrate the consequences of the finite value of nr,
we make some extremely minimalist assumptions about cells. We
consider a cell composed of Np = 12 proteins, each containing Na =
14 amino acids. We refer to the minimum (Np , Na) set as a (12-14)
cell. Such a cell is smaller than some modern viruses.
The ability to perform any of the basic tasks of the cell is not
necessarily limited to a single protein. Many different proteins
among all those which were available in the primeval soup may have
been able to perform (say) waste disposal. In order to allow for
this in estimating Pr, we include a factor Q to describe how many
different proteins in the primeval soup could have performed each
of the basic tasks of cell operation. The larger Q is, the easier
it is to assemble a functional cell by random processes. However,
there is a maximum value Qmax that is set by phase space arguments.
The hypothesis that life originated by random processes requires
that Pr be of order unity. We estimate how large Q must be (Qra :
subscript “ra” denotes “random assembly”) in order to ensure Pr = 1
in the time that is available (1.11 b.y.). We find that Qra must be
so large as to exceed the maximum permissible value Qmax in the
phase space of proteins comprised of a set of 14 distinct amino
acids. Such a large value of Qra would have serious consequences
for biology: if Qra were truly as large as Qmax in the primeval
soup, then essentially all 14-acid proteins must have possessed
the ability to perform each of the fundamental tasks in the cell.
That is, there was no task specificity among the proteins: a
protein which was able (say) to maintain the membrane in a cell
would also have been able to control (say) the replication
process.
In such a situation, the very concept of a cell, as a wellorganized
factory in which the task of each department is
regulated, and each department must coordinate dependably with all
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others, would no longer be valid. A cell would quickly be reduced
to an unpredictable entity which lacked robust properties.
In the “real world”, where a cell must be able to preserve itself
and replicate faithfully from generation to generation, it seems
inevitable that the various proteins must be prevented (by nature)
from performing multiple tasks. That is, there must be a certain
amount of specificity to the task that any given protein can
perform: of all available proteins, only a fraction F should be
capable of performing the task of (say) membrane repair. In a cell
where the number of proteins is Np, the restraints of specificity
require that the value of F can certainly not exceed 1/Np. But F
might be much smaller than this upper limit. This leads us to
introduce a “protein specificity index” m such that the actual
value of F in the primeval soup is usefully written as (1/Np)m. In
the modern world, the value of m ranges from 1 to a maximum value
between about 10 and 20.
We find that, even assigning the minimum possible specificity (m =
1), the probability Pr of assembling the RNA of a (12-14) cell by
random processes in 1.11 billion years using triplet codons is no
more than one in 1079. And if the protein tasks are even marginally
specific (with m = 2-3, say), the chances of random assembly of
RNA for the first cell decreases to less than one in 10100.
In order to improve the chances of random assembly of the first
cell, we consider a situation which might have existed in the
young Earth. We suppose that proteins could be constructed using a
smaller set (numbering Naa) of distinct amino acids: we consider
the case of Naa = 5 (instead of the modern 20). If, in these
conditions, the number of bases in DNA remained as large as 4,
then doublet codons sufficed to encode protein production with the
same amount of error protection as occurs in the modern (triplet)
genetic code. In such conditions, the probability of randomly
assembling the RNA for the first cell in 1.11 b.y. improves.
However, it is still small: the optimal probability is no more
than one in 1063.
To improve the probability even further, it is tempting to
consider the possibility of singlet codons. But we point out that
these are not relevant in a realistic biology.
In the context of doublet codons, we can improve the probability Pr
of random assembly by considering a larger set of distinct amino
acids. The number of distinct amino acids for which doublet-codons
can encode ranges from 5 to 14 (allowing for start and stop
codons). As Naa increases above 5, there is a marked improvement in
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Pr for a (12-14) cell: in fact, Pr may approach a value of order
unity when Naa = 11 provided that the specificity index m is
smaller than 1.3. (This is far below the average value of m, and
represents very marginal specificity.) And Pr formally exceeds
unity for Naa in the range from 12 to 14, provided that m does not
exceed 2.5. This value of specificity is still well below the
average value. It is not clear that a functioning cell could
survive for long with such low protein specificities.
Nevertheless, the fact that Pr formally reaches a value as large as
unity suggests that we may have found a window of opportunity for
random assembly of the first (12-14) cell.
However, these cells face a potentially fatal problem: even with
11 amino acids to be encoded by 16 codons in the RNA, there is
little redundancy in the genetic code. And for Naa = 14, the
redundancy vanishes altogether. As a result, there is a much
reduced error protection in the code which translates the
information in RNA to proteins. In the limit Naa = 14, there is no
error protection at all: transcription from RNA to protein then
has no immunity against noise. Moreover, in the limit Naa = 14
(plus a start and stop), proteins would be equally able to encode
for RNA, in violation of the Central Dogma of biology. Therefore,
although the probability of randomly assembling the RNA for a (12-
14) cell in such a world may approach unity in a mathematical
sense, it is not clear how useful such a cell would be for
biology.
We stress that our assumptions about a (12-14) cell are minimalist
in the extreme. In the “real world”, it is not obvious that a
protein containing only 14 peptides will be able to fold into a
stable 3-dimensional shape at the temperatures where water is
liquid. And in the “real world”, a cell probably requires as many
as 250 proteins to function. In such case, even if Naa = 14, Pr
approaches unity only if the specificity index m lies in the very
restricted range between 1.0 and 1.17. We identify this as a
narrow window of opportunity for random assembly of primitive
cells. But even this narrow window closes altogether if our
estimate of the number of chemical reactions is too large by
several orders of magnitude (as it may well be).
Our calculations refer only to the assembling of a cell in which
the genetic code is already at work. We do not address the origin
of the genetic code itself.
We conclude that, even if we assume that the genetic code was
already in existence (by some unspecified mechanism), conditions
in the early Earth must have been “finely tuned” in order to
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“squeeze through” the narrow window of opportunity and assemble
the first cell on Earth in a truly random manner.
1. Introduction
Evolution theory claims that all species of animals and plants
that now exist on Earth came into existence as a result of random
variations in pre-existing species. It is presumed that life on
Earth began as a single cell. An essential aspect of evolution
theory is that the first living cell originated in the early Earth
also as a result of random processes.
When Darwin proposed his theory of evolution, he did not know the
chemical make-up of a cell. Therefore, when he appealed to random
processes at work in nature, he could be excused for not knowing
what exactly was entailed in such processes. But in our day and
age, advances in microbiology and biochemistry have opened up to
us the molecular details of the processes that occur in living
cells. For example, we now know the make-up of proteins and DNA.
In fact, we will need to describe these in some detail in order to
proceed with our discussion of the probability of random
formation. (We will return to these details below.)
We are now in a position to spell out the chemical processes that
must have occurred if the first cell was indeed put together by
chance.
2. The challenge of creating the first cell
The question we wish to examine here is the following. If the
process of assembling the first cell occurred in a truly random
manner in the early Earth, what conditions would be needed?
To address this question, we need to answer two more basic
questions: (a) how much time was available before the first cell
appeared? And (b) how many chemical reactions of the correct type
could have occurred in the time available? The aim here is to
answer these questions as quantitatively as possible.
The answer to question (b) will set a limit on the properties of
the first cell that would have been created by random processes in
the early Earth.
We turn first to the question of how much time was available for
the development of the first living cell.
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3. The earliest life forms on Earth
The fossil record indicates that the first life forms to appear on
Earth existed some 3.45 billion years ago. These are cyanobacteria
(formerly called blue-green algae) which are found in rocks from
Apex Chert, Australia. ). The first life forms on Earth were
single-cell organisms. (See
http://www.unimuenster.
de/GeoPalaeontologie/Palaeo/Palbot/seite1.html
It is hardly surprising from an evolutionary standpoint that the
earliest forms of life on Earth were single-cell organisms.
Presumably it is easier for random processes to give rise to a
single cell first, before bringing forth a multi-cell organism.
4. How much time elapsed before the first cell appeared on Earth?
The age of the Earth, based on radioactive dating of rocks, is
estimated to be 4.56 billion years old. Comparing this with the
cyanobacteria ages, we see that the first living cells appeared
within a time interval of 1.11 billion years of the formation of
the Earth.
Therefore, the time tfc required for the development of the first
cell on the Earth is certainly no longer than 1.11 billion years.
Actually, the value of tfc might be much shorter than this.
Astronomers who calculate the internal structure of the Sun find
that the Sun has not always been as luminous as it is today: the
young Sun is calculated to have had a luminosity that is some 20-
30 percent fainter than it is today. Therefore, the mean
temperature on the early Earth might have been considerably colder
than it is today, so cold that the water on Earth’s surface was
frozen. (This is the “faint young Sun problem”: Sagan and Mullen,
1972, Science vol. 177, 52).
It is likely that the development of life requires water to be in
liquid form. The solar structure calculations suggest that the
energy provided by the Sun to the Earth might not have become
sufficient to melt the ice until the Sun was about 700 hundred
million years old. This means that the first living cell appeared
no more than about 400 million years after liquid water became
available.
Moreover, the early Earth would have been subject to a more or
less heavy bombardment by the debris of the proto-planetary disk
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before the latter was finally cleared out. The impacts of
planetesimals (such as that which destroyed the dinosaurs some 60
Myr ago) would have interrupted the processes which were “trying
to form” the first cell. Large impacts might have reduced the
interval for assembling the first cell to even less than 400
million years.
However, in order to improve the chances of evolution, let us
grant a full 1.11 billion years and ask the question: could the
first cell have developed by random processes in 1.11 billion
years?
The number of seconds of time in 1.11 billion years is 3.5X1016. We
will need this number in what follows.
5. Some essential constituents of cells
Now that we know how much time is available, we move on to the
main question that we wish to address: how was the first living
cell formed? Evolution theory asserts that it was formed by random
processes. We wish to assess the probability of such processes.
To assess realistically the chances of assembling the first cell
by chance, we need to know certain fundamental properties of the
components that go to make up a cell. Let us first summarize
these.
5.1. What do we need to know about proteins?
There are three levels of structure within a protein which are
relevant to us here.
(a) Primary Structure
A protein consists of a series of amino acids that are linked (by
peptide bonds) into a chain in a specific order. The change of
even a single amino acid in a chain of dozens or hundreds of amino
acids may in certain cases disrupt the functioning of the protein.
(b) Secondary structure
In order that proteins may function, the primary structure (i.e. a
chain of amino acids) is not sufficient. Certain segments of the
amino acids in the chain group themselves together into sub-units
known as alpha-helixes, beta-sheets, and beta-turns. For example,
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an alpha-helix consists of a chain of consecutive amino acids
arranged in a twisted three-dimensional structure (including 3.6
acids per turn of the helix) with well-defined angles between
neighboring acids in the chain.
These well-defined sub-units form the secondary structure of the
protein: they are stable and rigid, like “lego” blocks which can
be “fitted together” into a larger structure.
(c) Tertiary structure
Once the “lego” blocks are available, the stage is set for the
protein to go beyond the secondary structure: using available
thermal energy, the protein twists and folds itself into a certain
3-dimensional structure with specific bumps and hollows. These
bumps and hollows, which are referred to as the tertiary structure
of the protein, determine where electric charge builds up, and
these localized charges control the protein’s function, including
the reactions that it can catalyze (if it is an enzyme). For
example, insulin (one of the shortest proteins in the human body,
with 51 amino acids) folds itself naturally into a wedge-like
shape which enables groups of six insulin molecules to pack
themselves tightly into spherical clusters.
The sequence of amino acids in a particular protein may be highly
specific at certain locations. There are certain sites in the
protein (“invariant sites”) where even a single alteration in the
sequence can lead to drastic changes in the shape of the folded
protein, thereby disabling the protein. For example, human
hemoglobin, the protein that carries oxygen through the blood,
contains Na = 574 amino acids arranged in four secondary sub-units,
with an overall spherical tertiary structure. Two of the invariant
sites in hemoglobin have attracted widespread attention because of
the drastic consequences they may have in a certain segment of the
population. If one of the amino acids (glutamic acid) in a certain
position in two of the sub-units of the hemoglobin molecule is
replaced by another amino acid (valine), the result is the painful
and deadly disease known as sickle cell anemia. Although it would
seem that switching only 2 out of 574 amino acids ought to have an
insignificant effect, this is not the case for these two
particular sites. Just by changing 2 amino acids and leaving all
the remaining 572 as before, the process of folding the molecule
is altered so much that the 3-dimensional shape of the hemoglobin
changes is no longer spherical. Instead the molecule takes on an
elongated structure resembling a sickle.
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There are some proteins in which essentially all sites are
invariant. For example, histones which have at least 125 amino
acids in the peptide chain, have 122 invariant sites. Such
proteins are therefore exceedingly specific in the arrangement of
amino acids.
However, not all sites in all proteins are invariant. In many
proteins, there are sites where the amino acid can be replaced by
a number of other amino acids without affecting the functioning of
the protein. Yockey (Information Theory and Molecular Biology,
1992, Cambridge Univ. press, 408 pp; Table 6.3) discusses the
example of a particular protein (iso-1-cytochrome c, with 110
amino acids), with a list of all amino acids which are
functionally equivalent at each site. Some sites can have up to 13
different amino acids and still the protein retains functionality,
whereas others (the invariant sites) must contain one and only
particular amino acid in order to protect against protein
dysfunction.
At the primary level, the linear sequence of amino acids in a
protein is important to the proper operation of a living cell. But
in order to reach the final operating stage (which is fully threedimensional),
the creation of the “lego” blocks (i.e. stable and
reproducible secondary structure) is an essential intermediate
stage.
(d) How long are the secondary structures?
A central question in the present context is: what is the minimum
requirement for the “lego” blocks to be formed? What does it take
to be able to create the rigid sub-units which are used in making
the final protein? The answer is found in the quantum chemistry of
an alpha-helix and a beta-sheet: in principle, a sequence of at
least 4 amino acids is required in order to make the smallest
alpha helix (this allows for one complete turn of the helix). The
minimum size of a beta-sheet may be comparable.
However, the minimum size is not the only factor that is at work
in creating the “lego” blocks in proteins: the question of
stability also enters, because it is a fundamental requirement for
living cells that the secondary structures must be rigid.
Otherwise, the shapes of proteins in a cell would be subject to
chaotic fluctuations. Studies of reproducible structure of subsequences
in proteins suggests that chains of at least 7 amino
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acids are required in order to create a stable and reproducible
“lego” (Sudarsanam and Srinivasan, 1996, abstract E0274, IUCR
Seattle meeting). It therefore seems unlikely that stable “lego”
blocks can be constructed with a chain that is less than 7 amino
acids long.
Now, the tertiary structure of a protein comes into existence only
if at least two stable “lego” blocks are joined together in a
reproducible 3-dimensional structure. (Many proteins require more
than 2 secondary structures: e.g. hemoglobin contains 4.) Thus,
the bare minimum requirement for a protein is that Na should be at
least twice the bare minimum needed for rigid and stable secondary
structure. According to the estimates of Sudarsanam and
Sreenivasan, this requires Na = Nmin = 14.
We emphasize that this assumption of a mere 14 amino acids in a
functioning protein is extreme. A protein with only 14 amino acids
is very short in terms of the proteins that exist either in the
modern world (e.g. insulin, with its 51 amino acids, and
hemoglobin, with its 574 amino acids), or even in ancient
proteins. For example, bacterial ferrodoxins, with at last 56
amino acids, “are believed to date nearly to the time of the
origin of life, and the histones which are also believed to be
ancient and have at least 125 amino acids” (Yockey, p. 143). Even
in the earliest stages of life on the planet, before the so-called
“breakthrough organism” had appeared, the proteins that might have
been operational back then have earned the title of “miniproteins
” because the number of amino acids they contained was
“perhaps 20 or shorter” (Maniloff, Proc. Natl. Acad. Sci. USA,
vol. 93, p. 10004, 1996).
Computational attempts to “construct” proteins which are capable
of folding into a certain unique and stable tertiary structure
have been made by several groups. Dahiyat and Mayo (Science vol.
278, p. 82, 1997) found that, using only amino acids which occur
in modern nature, the shortest protein without sulfides or metals
that folds into a stable tertiary structure contains 25 amino
acids. An earlier computation (Struthers et al., Science vol. 271,
p. 342, 1996) had obtained a stable tertiary structure with a
chain of only 23 amino acids: however, one of the 23 was a nonnatural
amino acid. It seems that polypeptide chains with fewer
than 23-25 amino acids can probably not create the tertiary
structure which is key to protein function unless they are
assisted by sulfides or metals.
How far below the 23-25 limit can a functional protein go when
assisted by sulfides and metals? The answer is not clear. However,
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it seems unlikely that the limit will be reduced below 14, which
is our limit based on the stability properties of at least two
“lego” pieces (alpha-helices and beta-sheets). In fact, in terms
of the thermal energy which is available, it is not clear that a
protein as short as 14 amino acids will be “foldable” or
“bendable” at temperatures where water is liquid.
Nevertheless, in the spirit of optimizing probabilities, we assume
that polypeptides in the primeval soup could indeed function as
proteins while containing no more than 14 amino acids.
5.2. What do we need to know about DNA?
DNA is a molecule that has the shape of a long twisted ladder (the
"double helix"). In this ladder, there are "rungs" connecting the
long "sidepieces". The "sidepieces" are long linear chains of
sugars and phosphates, while each "rung" is composed of two
interlocking bases. The four bases consist of two purines and two
pyrimidines. The bases in the ladder are arranged in a definite
order, just as amino acids are arranged in a definite order in a
protein.
When a cell wishes to reproduce a certain protein, the section (or
"gene") of DNA that is responsible for that protein must undergo a
well-defined process. First, the two bases that are interlocked in
each rung of the ladder in that section must be "unzipped" so as
to expose a sequence of bases. The exposed sequence then creates a
strip of RNA whose task is to assemble amino acids from the cell
medium in the correct order.
The order of the bases along the DNA “ladder” (or along the RNA
strip) is highly specific, just as the order of acids in the
protein is crucial for protein function. The change of even a
single base inside a gene may result in the creation of the wrong
protein, and the organism may die as a result. This indicates the
need for serious error-protection in the process of replication of
a cell.
6. Cell structure: high information content
Even a “simple” cell is a complicated system where chemicals of
various kinds operate in a synergistic way to provide various
functions that are essential to cell viability.
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The outer wall (or membrane) provides the cell with its own
identity, and separates it from the rest of the world. Apart from
the membrane, i.e. inside the body of the cell itself, there are a
number of sub-systems that must run cooperatively in order to keep
the cell in operation. The most important chemicals are proteins
and the DNA that has the capacity to reproduce those proteins.
Some proteins provide the structural characteristics of the
different components of the cell. Some proteins serve as catalysts
in the various chemical reactions that keep the cell running.
There are also regulatory proteins which ensure that each protein
performs its function only in its proper location within the cell:
it would not do, e.g., to have energy generation occurring in the
cell membrane. In a multi-cell organism, these regulatory proteins
ensure that (e.g.) kidney cells do not grow in (say) the eye.
It is amazing that there is enough information in a linear object
(a DNA strand) to determine a three-dimensional object (a
protein). How is it that the sequence of bases in DNA instructs
the cell to make proteins, each of which is a “sentence” composed
of a specific sequence of various choices from a “vocabulary” of
the 20 (or so) amino acids which occur in modern proteins? (There
are many more amino acids in nature, but they are non-proteinous,
and we do not consider them here.) The beginnings of an answer
were first proposed by Gamow (1954: Nature 173, 318): there exists
a code which translates the information in the bases in DNA into
the amino acids in protein. This was an amazing insight on Gamow’s
part. As Yockey says (p. 4): “The idea…of a code is so
unconventional that had Gamow’s paper been submitted by almost
anyone else, it would most certainly have been rejected”.
The eventual identification of the code at the heart of biology is
a triumph of human ingenuity. The bases in DNA are now known to be
grouped into 64 “code words”, and the sequence of these words
contain the information which is eventually translated into the
20-letter vocabulary of proteinous amino acids.
A more difficult question to answer is: how do the amino acids
“understand” the “language” of the “words of information” that are
contained in the DNA? (For example, a string of letters may mean
one thing to a Frenchman, something else to a German, and nothing
at all to an Englishman.) It is not obvious that an answer has yet
been given to this question. It may in fact be the most difficult
question of all to answer. For example, Yockey (2000: Computers
and Chemistry 24, 105) argues that the answer may simply be beyond
the powers of human reasoning. In the present calculation, we do
not address the issue of the origin of the code. We merely assume
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that the code is already in existence as a result of unspecified
processes in the early Earth.
Returning to a question about the links between DNA and protein
that can be answered, the distinction between 64 and 20 is
noteworthy and essential for living cells. In terms of coding
theory, the fact that 64 greatly exceeds 20 means that DNA code
has a lot of built-in redundancy: there are more code words (or
symbols) at the source (DNA) than at the destination (protein).
Coding theory proves that this redundancy of source relative to
destination is an essential feature of a code in order to protect
from errors in transmission. One of the theorems of coding theory
(Shannon’s channel capacity theorem) makes a strong statement
which at first sight appears counterintuitive (Yockey, p. 8): even
if there is noise in a message, the proper use of redundancy
allows one to extract the original message “with as small a
probability of error as we please”.
Therefore, if we were to attempt to construct a biological system
based on a code where redundancy is absent (and we shall mention
one such attempt in Section 19 below), the process of cell
replication would inevitably be prone to errors in transmission.
Since even a single error may prove to have mortal consequences
for a protein (and its host organism), it is hard to see how cells
that are subject to serious errors in replication could be
regarded as “living” in any meaningful sense.
The code words in DNA in the modern world consist of a series of
triplets of bases. Each triplet (written as ACG, or UGA, etc,
where each of the letters A, C, G, and U is the initial letter of
one of the 4 bases) encodes for a particular amino acid. There are
64 such triplets available as a source code. (We will consider
below the possibility that triplet codons were not necessary in
the primeval soup, but that doublet codons might have sufficed
then.)
If a cell contains a particular protein that is a chain of Na amino
acids in a certain sequence, then the DNA of that cell contains a
corresponding segment containing 3Na bases also arranged in a
sequence that exactly parallels the Na acids in the protein.
However, this is not all that is required for a gene. Since the
DNA consists of a long chain of bases, we need to ask: how does
the RNA know where to start “reading” the code for a particular
protein? The answer is: in the DNA itself, associated with each
gene, there must be a “start code” and a “stop code”. In fact, a
triplet of bases serves to encode START and another triplet to
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encode STOP. (E.g., in modern cells, the triplet AUG encodes for
start, while stop has three possible codons: UAA, UAG, UGA.).
Therefore, although a strip of RNA needs to have 3Na bases in a
particular order, the gene (i.e. the corresponding piece of the
DNA) must have 3Na+6 bases in a particular order.
As an example, we note that among the shortest proteins that exist
in human beings, insulin contains 51 amino acids in a particular
order. Such a protein requires a sequence of 153 bases in human
DNA in a specific order, plus 6 bases for start and stop.
7. What does a cell need in order to function?
To determine the probability that the first cell was assembled
randomly, we first need to answer the following general question:
what is required in order to make a functional living cell?
In other words, what is the bare minimum number of proteins for a
cell to function at all? If we can answer this, it should help us
determine what the very first cell might have looked like.
As a first step in answering this, it is worthwhile to consider
the simplest known cell that exists in the world today. This is an
organism called "Mycoplasma genitalium" (MG) whose genetic
information is many times smaller than the information in the
human genome: the number of genes required for the functioning of
MG in its natural state is only 517. (Humans have tens of
thousands of genes.)
Recently, researchers have raised the interesting issue: are all
517 of these genes really necessary for MG to function properly?
The answer is No. By removing genes one at a time, researchers
have been able to show that the cell continues to function with
fewer than the total complement of 517. By eliminating more and
more of the genes, it has emerged that MG continues to function
normally as long as there are between 265 and 350 protein-coding
genes (see Hutchison et al., Science vol. 286, p. 2165, 1999). An
earlier estimate of the minimum cell size in nature had suggested
that the minimum number of proteins for cell operation might
indeed be about 250 (J. Maniloff, Proc. Natl. Acad. Sci. USA Vol.
93, p, 10004, 1996).
It appears, then, that the simplest cell in the modern world
requires at least 250 proteins in order to survive in viable form.
Many of the 250 (or so) essential proteins in MG have identifiable
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functions. Hutchison et al. list 13 categories of identified
functions in the MG genome: (1) cell envelope, (2) cellular
processes, (3) central intermediary metabolism, (4) co-factors and
carriers, (5) DNA metabolism, (6) energy metabolism, (7) fatty
acid metabolism, (8) nucleotides, (9) protein fate, (10) protein
synthesis, (11) regulatory functions, (12) transport/binding
proteins, and (13) transcription. Each of these 13 categories
contains multiple genes, so that (e.g.) protein synthesis does not
depend solely on a single protein for its operation: there are
backups and multiple redundancies in each category. For example,
some 19 proteins are used for membrane maintenance (category (1)).
About 150 of the MG proteins can be assigned with some confidence
to one of the 13 categories.
However, more than 100 of the MG genes perform functions that are
currently unidentified. Nevertheless, the cell certainly requires
them: without them, there is empirical proof that the cell fails
to function.
8. The first cells to appear on Earth: reducing the requirements
to an absolute minimum
It might be argued that the first cells to appear on Earth were
smaller than the simplest cells (such as MG) that exist in the
world today. Those primitive cells might have been able to operate
with many fewer proteins than the 265 needed by MG.
Although we will use this argument below, it is actually difficult
to substantiate. The mathematician John Von Neumann estimated the
bare necessities which are necessary in order to construct what he
referred to as “a self-replicating machine” (Theory of Self-
Reproducing Automata: Univ. of Illinois press, 1966). It has been
a popular exercise among science fiction writers to use this idea
in connection with how a civilization might colonize a galaxy by
sending out machines. Von Neumann concluded that the number of
parts in one such machine must be in the millions. Other authors
have reduced this estimate somewhat, but even according to the
most optimistic estimate, the numbers remain very large: the best
estimates suggest that there must be between 105 and 106 parts in a
self-replicating machine. This means that the genome needs at
least 105 bits in order to metabolize and replicate (Yockey, p.
334). Using the information content in a typical modern protein,
Yockey concludes that the original genome must have been able to
specify at least 267 proteins. The fact that this is close to the
minimum number required for a modern cell (such as MG) suggests
that one is not necessarily permitted to assume that the original
15
cell contained significantly fewer proteins than the smallest
modern cell.
Nevertheless, other authors have argued that the Von Neumann
approach is overly restrictive. E.g., Niesert (1987, origins of
Life 17, 155)) estimates that the first cell might have been able
to operate with as few as 300-400 amino acids.
Which of these various estimates of minimum requirements for the
first cell should we consider? There must be some absolute minimum
requirements for making even the simplest cell. For example, one
might argue that, among the 12 non-regulatory categories of gene
functions listed by Hutchison et al., one representative protein
should be present in the first cell. And each of these 12 proteins
should have an accompanying protein to serve in a regulatory role.
This line of reasoning would suggest that 24 proteins are a
minimum for cell operation.
Can we reduce this to an even barer minimum? Examples of minimum
cell requirements have been summarized by the paleontologist
George Gaylord Simpson. Of the 13 categories listed by Hutchison
et al, Simpson narrows down the bare minimum to the following: (i)
energy generation, (ii) storing information; (iii) replicating
information; (iv) an enclosure to prevent dispersal of the
interacting sub-structures; (v) digestion of food; (vi) waste
product ejection (Science vol. 143, p. 771, 1964).
In view of these bare-bones requirements, it is hard to imagine
how any cell could function without at least the following six
types of proteins: (i) those that help to digest food, (ii) those
that generate energy for cell operations, (iii) those that carry
away waste products, (iv) those that preserve and repair the cell
membrane, (v) those that determine when reproduction is to occur,
and (vi) those which actually catalyze the tasks of reproduction.
Corresponding to each of these six, there must be a regulatory
protein which ensures that the corresponding protein does not
“express itself” in the wrong location in the cell.
It is hard to imagine how a living cell would exist at all if it
failed to contain at least these 12 proteins.
The fact that the simplest cell in the modern world (MG) requires
265 proteins as a bare minimum in order to function makes our
estimate of 12 proteins look ridiculously small. But since it is
possible that the first living cells may have been much simpler
than those we find in the world today, let us make the (perhaps
16
absurdly reductionist) assumption that the first cells in fact
were able to operate on the basis of the bare minimum 12 proteins.
As an illustration of how reductionist our assumption is, we note
that in the first cell, we are assuming that a single protein is
responsible for ensuring proper functioning of the lipid membrane
of that cell. In contrast, the smallest known cell in the modern
world (MG) uses 19 genes to encode for lipoproteins (Hutchison et
al. Science vol. 286, p. 2166). The use of 19 genes in the modern
cell is an example of the large amount of redundancy that nature
uses to ensure that the membrane survives. But the first cell may
not have had the luxury of redundancy: it may have been forced to
survive using only one gene for its membrane. It would have been a
precarious existence.
We have argued that each protein must contain at least 14 amino
acids: thus our bare minimum cell, with 12 proteins and 14 amino
acids in each, contains 168 amino acids. This is even smaller than
the bare minimum of 300-400 amino acids described by Niesert
(1987, Origins of Life, 17, 155). The DNA of our minimal (12-14)
cell would contain only about 500 bases. This is 10 times shorter
than the genome of a certain virus (PHI-X 174) which transmits 9
proteins. It is widely believed that a virus cannot be regarded as
a “living cell” (it has no self-contained replication system), so
this again indicates the extreme nature of our assumption that the
first cell could have as few as 12 proteins. But let us proceed in
the spirit of optimizing the probability that the first cell
appeared by chance.
8.1. The first cell: putting the proteins together by chance
In the early Earth, the commonest concept of conditions back then
is that the primeval "soup" consisted of various chemicals that
were stirred up and forced into contact with one another as a
result of the forces of nature (including rain, ocean currents,
lightning). Simple chemical reactions in the soup were easily able
to create amino acids: these molecules are so small (containing no
more than 10-30 atoms each) that random processes can put them
together quickly from the abundant C, O, N, and H atoms in the
soup. As a result, we expect to find in the primeval soup, in
abundant supply, all of the 22 amino acids that occur in modern
life forms. (For the number 22, see Nature vol. 417, 478, 2002).
In fact, there are more than 100 amino acids in modern nature, but
only 22 are used in proteins. And of those 22, numbers 21 and 22
are rare. Most living material relies on only 20 of these amino
acids, and we will use that number here.
17
To be sure, the “primeval soup” hypothesis is not without its
opponents (e.g. Yockey, pp. 235-241). Laboratory experiments which
claim to replicate conditions in the primeval Earth generate not
only amino acids but also a tarry substance (as the principal
product). This substance should have survived as a non-biological
kerogen in ancient sedimentary rocks, but no evidence for this has
been found. It should not be surprising that, in the primeval
soup, other amino acids, not currently used in life forms, could
have been formed. (This would include the acids that are used in
nylon.) And each of the amino acids which are created randomly in
the primeval soup would be created in two forms: the D-variety and
the L-variety. (These varieties refer to the ability of the
molecule to rotate the polarization of light either right or left:
this ability depends on the chirality of the molecule, i.e. on the
handedness of its 3-dimensional structure.) For reasons that are
not yet obvious, only one of these varieties (the L-variety) is
actually used in present-day life forms. However, the basic
property of amino acids, that they polymerize, operates only
between L alone or D alone: when an L and a D amino acid combine,
their opposite chirality has the effect of locking out any
possibility of further polymerization.
Another difficulty of a very different nature has to do with
reactions in an aqueous solution. The very process of assembling
amino acids into a polypeptide chain (so as to make a protein)
requires the removal of H from the amino radical and the removal
of OH from the acid radical: it is not obvious how these
constituents of a water molecule can be removed in an aqueous
solution.
Despite these difficulties with the primeval soup hypothesis, the
idea of the soup is so widespread in textbooks that it is a
natural starting point for an optimized estimate of probabilities.
In the spirit of the present approach (where we do whatever we can
to optimize the chances of assembling the first cell randomly), we
will simply go along with the textbooks. We shall assume that the
formation of the first cell in the early Earth began in liquid
water where only 20 L-amino acids need to be taken into account.
Other simple chemical reactions in the soup also give rise more or
less quickly to the four bases (two purines and two pyrimidines)
that form the "rungs" of DNA. Why are these formed relatively
readily? Because each base consists on no more than 13-16 atoms,
random processes can also assemble these bases rapidly from the
abundant C, O, N, and H atoms. It was probably more difficult to
form pyrimidines than purines, but the principle is robust:
18
formation of small molecules is essentially inevitable in the
early Earth.
In order for the first cell to come into existence, at least 12
proteins, each with Na amino acids in a specific order, had to
emerge in the same patch of the "primeval soup". To be sure,
individual proteins were probably emerging at random at many
places around the world. But if our aim is to form a complete
living cell, it will not help if the membrane protein emerged (at
random) in China, the energy protein in Russia, and the
replication protein in South America. That will not do: the only
way to have the first cell develop is if all 12 proteins emerge in
close enough proximity to one another to be contained within a
single membrane.
How might this have happened in random processes? By way of
example, let us consider one particular protein, in which the
chain of amino acids happens to be denoted by the series of
letters ABCDEFGHIJKLMN. In order that this protein be made by
chance, amino acid E (say) (one of the 20 commonest in nature)
might have started off by entering into a chemical reaction with
amino acid F (another of the 20), such that the two found it
possible to become connected by a peptide bond. Then amino acid D
might have had a chemical reaction so as to join onto the EF pair
at the left end, forming DEF by means of a new peptide bond. Note
that it is important to form DEF rather than EFD, which would be a
very different protein. This process presumably continued until
the entire 14-unit protein chain ABCDEFGHIJKLMN was complete.
8.2. The first cell: putting the DNA/RNA together by chance
It is not enough to assemble 12 proteins to have a functional
living cell: the cell must be able to reproduce, and for that
the cell needs DNA (or at least RNA). In order to ensure
reproduction of the cell, there had to be (also in the same patch
of the primeval soup) at least 12 genes on an RNA strand, each
containing 3Na+6 bases in a specific order.
Thus, in the very same patch of "soup" where the protein
ABCDEFGHIJ formed by chance, a strand of RNA must have been formed
where the three bases that encode for amino acid A were joined in
a specific order along the RNA strip by a series of chemical
reactions. Then the three bases that encode for amino acid B had
to be added in a specific order to the sidepieces, right next to
the three bases that encode for A. This process must have
continued until the triplets of bases that encode for each of C,
D, E, F, G, H, I, J, K, L, M, and N respectively were assembled in
19
a specific order into a chain of 30 bases. There would also be one
triplet at each end of the 30-base sequence to serve as markers
for start and stop. This 36-base sequence would then form the gene
for the first protein in the first cell.
Now that we know how the first proteins and RNA/DNA were put
together, we are in a position to estimate the probability that
this will occur by random processes.
9. Probability of protein formation at random
In the example given above, we recall that amino acid (say) E is
only one of 20 amino acids that exist in living matter. Amino acid
F is also one of 20. Therefore, a process that successfully forms
the sequence EF at random out of a soup where all amino acids are
present in equal abundances, has a probability p2 which is roughly
equal to (1/20) times (1/20) = 1/400.
Actually, however, pre-living matter contains not only the Lvariety
of each amino acid, but also the D-variety. Therefore, a
better estimate of the probability p2 that the correct pair of Lamino
acids be formed is (1/40) times (1/40), i.e. p2 = 1/1600.
However, once an L-acid unites with a D-acid, the opposite nature
of their chiralities leads to a “lock-out”: no further
polymerization is possible. So we will optimize probability by
assuming that only the L-variety is present. We therefore take p2 =
1/400.
Another way to state this result is that if we wish to create the
combination EF (both L-variety) by chance, the number of chemical
reactions that must first occur between amino acids in the
primeval soup is about 1/p2, or about 400. That is, if we allow so
much time to elapse that 400 reactions can occur in the primeval
soup, then there is a high probability (close to a certainty) that
the combination EF will appear simply at random.
This argument assumes that the only amino acids in the primeval
soup are the 20 which occur in modern living organism. However,
there were certainly other non-biological amino acids available.
As a result, many more than 400 reactions was almost certainly
required before the combination EF appeared at random. However, we
will optimize the chances for random assembly of the first cell by
ignoring the non-biological amino acids.
After creating EF by random processes, the next step is to have
the next amino acid to join the chain be the L-variety of (say) G,
i.e. only 1 out of the 20 types available. Then the probability
20
that the three amino acids EFG will be assembled in the correct
order is about (1/20)3.
Continuing this all the way through a sequence of Na amino acids
in a protein, the chance f1 of correctly picking (at random) all
the necessary amino acids to create one particular protein is
roughly equal to (1/20) raised to the power Na. This corresponds to
f1 = (1/10)x where x = 1.3Na. Actually, to the extent that some
amino acids may be replaced by others without affecting the
functionality of the protein, the above expression for f1 is a
lower limit. (We will allow for this later in this section.)
Yockey (p. 73) shows that instead of 20N for the value of 1/f1, a
more accurate estimate is 2NH where H is the mean value of a
quantity known as the Shannon entropy of the 20-acid set (see
below). In the limit where all amino acids have equal probability
of being encoded, and are equally probable at all sites in the
protein, 2NH turns out (from the definition of H) to be equal to
20N . In all other cases, 2NH is less than 20N. This returns us to
the previous conclusion: the above expression for f1 is a lower
limit on the true value.
Suppose that the particular protein with probability f1 has been
formed in a particular patch of the primeval soup. Then in order
to form a single cell (with at least 12 proteins as a bare minimum
to function), eleven more proteins must also be formed in the same
patch of soup, in close enough proximity to one another to be
contained within a single membrane. Each of these proteins also
has a certain number of amino acids: for simplicity let us assume
that all have length Na.
The overall probability f12 that all twelve proteins arise as a
result of random processes is the product of the probability for
the twelve separate proteins. That is, f12 is roughly equal to f112,
i.e. f12 is roughly (1/10)y where y = 15.6Na.
We can now quantify the claim that the first cell was assembled by
random processes. If the first cell consisted of only the bare
minimum 12 proteins, and if each of these proteins was uniquely
suited to its own task, the probability that these particular 12
proteins will be formed by random processes in a given patch of
primeval soup is f12.
Now let us turn to the fact that a protein may remain functional
even if a certain amino acid is replaced with another one.
(Obviously, we are not referring to invariant sites here.) For
example, it may be that the protein which we have specified as the
one that is responsible for (say) energy generation in the cell is
21
not unique. There may exist other groupings of amino acids which
also have the shape and properties that enable the task of energy
production for the cell. Maybe the others are not as efficient as
the first one, but let us suppose that they have enough efficiency
to be considered as possible candidates for energy production in
the first cell. Then we need to ask: how many energy-producing
proteins might there be in the primeval soup?
It is difficult to tell: in principle, if Na has the value 14
(say), then one could examine the molecular structure of all 14-
amino acid proteins (of which there are some 2014 , i.e. 1018.2 if
all amino acids are equally probable) and identify which ones
would be suitable for performing the energy task. Presumably there
must be some specificity to the task of energy production:
otherwise, a protein which is supposed to perform the task of
(say) waste removal might suddenly start to perform the task of
(say) membrane production in the wrong part of the cell.
Therefore, it is essential for stable life-forms that not all
available proteins can perform all of the individual tasks.
Suppose the number of alternate energy-producers Q is written as
10q. In a world where all proteins have Na = 14, the absolute
maximum value that q can have is qmax = 18.2. This is the total
number of discrete locations in the “14-amino acid phase space”.
In the real world, a more realistic estimate of qmax would be
smaller than the above estimate. First, not all amino acids have
equal probability of being encoded: there are more codons in the
modern genetic code for some amino acids than for others. (E.g.,
Leu, Val, and Ser have 6 codons each, whereas 10 others have only
2 codons each.) When these are allowed for in the probability
distribution, it is found that the “effective number” of amino
acids in the modern world is not 20 but 17.621 (Yockey, p. 258).
Thus, with Na = 14, a more accurate estimate of qmax(eff) is 17.4
(rather than 18.2).
As a result, in the real world, qmax(eff) may be considerably
smaller than 18.2. However, in the spirit of optimizing
probabilities, let us continue to use the value 18.2.
The requirement that some specificity of task persists among
proteins requires that the value of q must certainly not exceed
qmax. At the other extreme, in a situation where each protein is
uniquely specified, q would have the value qmin = 0 (so that one
and only one protein could perform the task of energy production).
22
Now we can see that our estimate of f12 needs to be altered. We
were too pessimistic in estimating f12 above. Each factor f1 needs
to be multiplied by 10q. For simplicity, let us assume that q has
the same value for each of the 12 proteins in the cell. Then the
revised value of f12 is 1/10z where
z = 15.6Na - 12q . (eq. 1)
This result applies to a cell with 12 proteins, each composed of
amino acids chosen from a set of 20 distinct entries.
10. Random formation of DNA/RNA
The first cell could NOT have functioned if it consisted only of
proteins. In order to merit the description living, the cell must
also have had the ability to reproduce. That is, it must also have
had the correct DNA to allow all 12 proteins to be reproduced by
the cell.
In order to estimate the probability of assembling a piece of DNA
by random processes, we can follow the same argument as for
proteins, except that now we must pick from the available set of 4
bases.
Repeating the arguments given above, we see that for each protein
which contains Na amino acids in a certain sequence (plus one start
and one stop), there must exist in the DNA a strip of B = 3Na+6
bases in a corresponding sequence. If we pick bases at random from
a set of 4 possibilities, the probability of selecting the correct
sequence for a particular protein is (1/4)B. Therefore, the
probability of selecting the correct sequences for all twelve
proteins, if each protein is unique, is (1/4)D where D = 36Na + 72.
Writing this with the symbol fRNA, we see that fRNA is equal to
(1/10)E where
E = 21.7Na + 43.3. (eq. 2)
Again, however, if instead of unique proteins for each task, there
are 10q proteins available to perform each task in the cell, then
we must increase the above value of fRNA to 10-G where
G = 21.7Na + 43.3 - 12q. (eq. 3)
23
11. Probability of random formation of a complete cell
Since both the RNA and all 12 proteins have to be formed in the
same patch of primeval soup in order to form a viable cell, the
probability fcell that random processes will perform both tasks in
the same patch of soup will be the combination of the separate
probabilities. That is, fcell is roughly equal to fp X fRNA, i.e.
about 10-J where
J = 37.3Na + 43.3 - 24q. (eq. 4)
Therefore, once enough time elapsed in the primeval soup to
allow the chemicals there to undergo a certain number of
reactions, R12p = 1/fcell, there would be a high probability (in
fact, a near certainty) that the proteins and the requisite DNA
for a (12-14) cell could indeed have been assembled by chance in
the primeval soup.
In order to optimize the chances of forming the first cell, we
ask: is it possible to find ways to make R12p smaller than the
above estimate? The answer depends on the theory that one adopts
for the development of the first cell.
Suppose one were to theorize that the only thing one would have to
provide to get the first cell going was the RNA containing the
genetic code for the 12 proteins. (It might be beneficial if the
RNA could catalyze its own replication: however, this is not
altogether desirable, since it leads to possibilities of ‘‘errorcatastrophes
” [Niesert et al. 1987, J. Mol. Evol., 17, 348].)
According to the "RNA-first theory", one would not have to "wait
around" for proteins to be constructed by random reactions in the
primeval soup. Instead, once strips of RNA were formed (as a
result of random processes), DNA could be assembled from the RNA
strips. At that point, proteins should be reproduced more or less
automatically, apart from the necessity of certain enzymes
(proteins) to catalyze the "unzipping" of the DNA itself, and to
catalyze the collection and assemblage of the amino acids.
In order to optimize the chances of cell formation at random, let
us assume that the unzipping can be done with the help of a single
protein, and that the collection and assemblage of amino acids can
also be done with a single protein. (This is a far cry from the
modern world, where multiple proteins exist in even the simplest
cell to perform each task.) Then the first cell will require the
RNA to be assembled by chance (with probability fRNA, as given
above) plus just two proteins (with probability f2) also assembled
24
by chance. If this theory is correct, then R12p(RNA-first) would be
equal to 10K where
K = 24.3Na + 43.3 – 14q. (eq. 5)
This may provide a substantial reduction below the original
estimate of R12p.
Should we also consider the obvious alternative to the RNA-first
theory? That is, should we also consider the “protein-first”
theory? The answer is no, provided that the modern genetic code is
at work. The structure of the modern genetic code is such that,
according to the Central Dogma, proteins do not pass on
information to DNA: the flow of information goes only from DNA (or
RNA) to protein, and not the reverse. As Yockey (2000) puts it,
“The origin of life [as we currently know it] cannot be based on
‘protein-first’.” However, the “protein-first” theory may need to
be considered when we consider a certain “window of opportunity”
in the early Earth (see Section 19).
Because we now know how many reactions are required in order to
create the first simplest possible cell, we are in a position to
test the evolutionary claim that the first cell was assembled
randomly. To do this, we proceed to the crucial question that is
at the heart of the present argument. This question, and its
detailed answer, is the subject of the next section.
12. How many reactions occurred in the primeval soup?
Is random assembly of the first cell possible? To address this, we
need to answer the following question: How many chemical reactions
(of the sort we are interested in) actually occurred in the
primeval soup during the first 1.11 billion years?
We will not be surprised to find that the number of reactions nr
is a "large" number (in some sense). Nevertheless, nr is a finite
number.
Once we obtain nr, we can then estimate how large the value of q
must be in order that the probability of randomly assembling the
first cell of order unity. That is, we will equate nr to 10J (or to
10K, if we accept the "RNA-first theory"), and solve for q,
assuming that Na is at least as large as 14. The value of q which
we obtain from this estimate will be labelled qra to denote that
this is how large q must be in order that random assembly of the
first cell in the primeval soup becomes essentially certain.
25
We are interested in chemical reactions involving amino acids or
bases. To proceed with this discussion, we need to consider in
detail what happens during such a reaction. The most basic
requirement of a chemical reaction is the following: the two
reacting molecules must at the very least come close enough to
each other to have a collision. However, the very fact that two
molecules collide does not guarantee that a reaction will occur.
The reaction is controlled by many factors, e.g. the energy
involved, the angle of the encounter, the removal of by-products,
etc. As a result of these factors, many collisions may occur
before even a single reaction occurs. This explains why it is so
difficult to manufacture (e.g.) nylon: the creation of the peptide
bonds that hold nylon together (exactly equivalent to those which
hold proteins together) requires careful quality control. The
quality control which the DuPont engineers are forced to impose in
order to create nylon was certainly not available in the primeval
soup: therefore, the efficiency of the reactions which led to
peptide bonds (i.e. proteins) in the primeval soup was almost
certainly very small.
In view of this, we can derive an absolutely firm (and probably
very generous) upper limit on the number of two-body reactions n2
that occurred between two amino acids during any time interval by
calculating the number of collisions ncoll that occurred between
those two amino acids during that interval. In practice, n2 is
probably orders of magnitude smaller than ncoll. The purpose of a
catalyst is of course to increase n2 as much as possible: however,
even with a “perfect” catalyst, n2 can never exceed ncoll .
So let us turn to estimating ncoll. This number, which is “large”
but finite, will provide us with a firm piece of quantitative
evidence that will allow us to test the assertion that the first
cell was assembled randomly.
13. Collisions between amino acids in the primeval soup
We begin the calculation of ncoll by estimating the mean time tc
that elapses between successive collisions of molecule A with
molecule B. The general formula for tc is straight-forward. Let us
consider molecule A as the projectile, and molecule B as the
target. If projectile A moves with mean speed v cm/sec through an
ambient medium where there are nt target molecules per cubic
26
centimeter, then tc equals 1/(v nt A) seconds. Here, A is the area
(in square centimeters) presented by the target molecule.
13.1 Mean time interval between collisions
Let us now estimate the three quantities that enter into tc.
First, the area A. Amino acids and bases in nature have linear
dimensions of a few Angstroms (where 1 Angstrom = 10-8 cm).
Therefore, a typical amino acid or base molecule has A equal to
about 10-15 sq. cm.
Second, as regards v, there is a standard formula for the mean
speed of the molecules in a medium at temperature T: v2 = RgT/m
where Rg is the gas constant (= 8.3 X 107 ergs/degree/gram) and m
is the molecular weight. Amino acids and bases have m = 100 or so.
Moreover, living cells require liquid water in order to survive:
this means that T must be in the range 273-373 degrees Kelvin.
Taking an average value for T of about 300 K, we find that v for
the molecules in which we are interested here is about 104 cm/sec.
Even if we consider the extremely hot conditions at the ocean
bottom, near the hot thermal vents, where temperatures may be as
large as 1000 K, this will increase our estimate of v by a factor
of no more than 2. This will have no significant effect on our
conclusions below.
Third, as regards nt, we note that at the present time, the total
mass of living organisms on Earth is Mliving = 3.6 X 1017 grams (see
http://www.ursa.fi/mpi/earth/index.html). In the early Earth,
before the first cell appeared, the mass of living material was by
definition zero. But there were amino acids and bases present in
the primeval soup. So in order to optimize the chances of cell
formation, let us make a second gross assumption: let us assume
that all of the mass that is now in living organisms was already
present in the primeval soup in the form of amino acids (if we
wish to assemble proteins) or bases (if we wish to assemble RNA).
With a molecular weight of about 100, each amino acid (or base)
has a mass maa of about 1.7 10-22 grams. Therefore, the total number
ntotal of amino acids (or bases) in the primeval soup was of order
Mliving/maa. With this assumption, we find ntotal = 2 X 1039.
Naturally, this estimate is quite uncertain. Other estimates of
this number are larger. E.g. Bar-Nun and Shaviv (Icarus 24, 197,
1975) estimate 5.4 X 1041, while Shklovskii and Sagan (1966
Intelligent Life in the Universe) estimated 1044. We shall see
that our results are only slightly affected by these
uncertainties.
27
Finally, to derive nt in the primeval soup, we need to divide ntotal
by the volume of the material where living material existed on the
early Earth. In the present Earth, the volume of the biosphere is
of order 1019-20 cubic cm. However, life probably started in
particular locations, and so the relevant volume of the primeval
soup was probably much smaller. Let us suppose that the early
Earth had a biosphere with a volume that was 10-100 times smaller
than it is at present. (This putative decrease in volume will help
to speed up reactions.) That is, let us suppose that all of the
amino acids which now are present in living matter on Earth were
concentrated in the primeval soup into a favored volume of only
1018 cubic cm. Combining this with our estimate of ntotal, we see
that the mean density of amino acids in the favored volume of the
primeval soup nt could have been about 2 X 1021 per cubic cm.
Is this a reasonable value? To answer this, we note that this
value of nt corresponds to a mean mass density of 0.34 gram/cubic
cm for the amino acids in the primeval soup. This density is very
high (the molar concentration is about 0.1): it is questionable
whether such a high density of amino acids could ever have been
dissolved in water. This estimate of mass density is certainly
close to the upper limit possible: it could hardly have been any
higher. In order to remain dissolved in water (with mean density 1
gram/cubic cm), the mass density of amino acids can certainly not
exceed the density of water. Therefore, our estimate of the upper
limit on nt is not unreasonable as we try to optimize the chances
of randomly assembling a cell. (If we were to use Bar-Nun and
Shaviv’s estimate of the total number of amino acids, we would
need to dilute them by dissolving them in at least 100 times more
volume than we used above in order to keep the mean density less
than that of water. With Shklovskii and Sagan’s estimate, the
volume must be larger still by a further factor of 200.) The
actual value of nt in the primeval soup was probably orders of
magnitude less than the estimate given above. Maximum molar
concentrations of amino acids in the primeval soup have been
estimated to be as low as 10-7 or 10-8 (Hulett 1969 J. Theor. Biol.
24 56; Dose, 1975, Biosystems 6, 224). Thus, our estimates of nt
are probably too large by 6 or 7 orders of magnitude. However, in
the spirit of optimizing the chances of making a cell, let us use
the above upper limit as the value of nt.
Now we have all of the ingredients we need to calculate tc, the
mean time between collisions in the primeval soup. We find tc = 5 X
10-11 seconds.
13.2. Number of collisions by a single amino acid in 1.11 b.y.
28
Now that we know the mean interval between collisions, we see that
in the primeval soup, a given amino acid experienced 2 X 1010
collisions every second as an upper limit. Therefore, each amino
acid experienced no more than 2 X 1010 reactions every second with
other amino acids.
How many collisions did an amino acid experience in the primeval
soup in the course of a time interval of 1.11 billion years, i.e.
in the 3.5 X 1016 seconds before the first cell appeared on Earth?
The answer is straightforward. Multiplying the above reaction rate
by the number of seconds available, we find that each amino acid
in the primeval soup experienced at most nr(1) = 7 X 1026 reactions
with other amino acids before the first cell appeared on Earth.
13.3. Total number of collisions between amino acids in 1.11 b.y.
Finally, we ask: what was the total number of reactions between
amino acids that occurred in the primeval soup before the first
cell appeared? The answer is again straightforward: since each
amino acid experienced nr(1) in that time, and since there were
ntotal amino acids in the primeval soup, the total number of
reactions nr between amino acids was about 1065 before the first
cell appeared.
This is a "large" number. But it is finite.
Moreover, we have artificially forced nr to be as large as possible
by making four extreme assumptions. (i) Every collision produces a
peptide-bonding reaction. (ii) The mass of pre-biotic material was
as large in the primeval soup as it is in today’s biomass. (iii)
The entire biomass in the primeval soup was in the form of amino
acids (or bases). (iv) All amino acids were concentrated in pools
where their mass density could build up to the maximum permissible
value. In the real primeval soup, conditions might have been such
that any or all of these assumptions could have failed by several
orders of magnitude. (In particular, (iv) almost certainly failed
by 6-7 orders of magnitude, and (i) almost certainly failed by
several orders of magnitude because of reaction kinetics.)
Therefore, it is highly likely that the actual total number of
collisions which occurred in the primeval soup before the first
cell appeared could have been 10 or more orders of magnitude less
than 1065.
Of course, our estimates refer to our estimates of the biomass
only, and also to binary collisions only. If we were to use the
estimates of Bar-Nun and Shaviv or of Shlokskii and Sagan, the
number densities per unit volume nt cannot exceed the value we have
29
already used above. Therefore, there will be no change in the
number of collisions per second. But the total number of
collisions would increase by 2-5 orders of magnitude above our
estimate.
For the sake of argument, let us assume that these other processes
compensated for orders of magnitude deficits associated with the
extreme assumptions (i)-(iv) above. That is, we will assume in
what follows that nr was indeed of order 1065. This appears to be a
very generous estimate of the total number of reactions in the
primeval soup.
14. Random production of the first cell
We are now in a position to estimate probabilities for randomly
assembling the first cell.
Let us return to our estimate of the number of reactions that were
necessary to create the first cell by random processes. In order
to create a cell containing 12 proteins with chains of N = Na amino
acids each, we recall that R12p was required to be 10J (where J is
given in eq. (4) above) if proteins and RNA were both assembled at
random.
However, if we accept the "RNA-first theory", we recall that the
number of reactions R12p(RNA-first) was "only" 10K (where K is given
in eq. (5) above).
Now that we know how many reactions actually did occur in the
primeval soup before the first cell appeared, we can equate nr
to the above values of R12p in order to determine how large qra must
have been in order to have reasonable probability of assembling
the first cell at random.
Setting R12p equal to nr, we find that the value of qra required for
random assembly of the first cell must satisfy the equation
37.3Na +43.3 -24qra = 65 (eq. 6)
if proteins and RNA were assembled together. On the other hand, if
we accept the RNA-first theory, then we find
24.3Na +43.3 -14q(RNA)ra = 65. (eq. 7)
30
As mentioned above, the value of Na is no less than 14. Inserting
Na = 14 in eq. (6) and (7) leads to qra = 20.8 or q(RNA)ra = 22.8.
The numerical value of qra increases linearly with the value of Na,
increasing by 1.7 for each unit increase in Na. However, qra is not
sensitive to the number of proteins in the cell. Moreover, qra is
not sensitive to errors in our estimates of the number of
collisions in the primeval soup: even if our estimated number of
collisions is wrong by factors of (say) one million times too
large or too small, our estimates of qra would change by only plus
or minus 0.4.
The above estimates of qra emerge from the two basic points of our
argument: (i) a finite time was available for chemical reactions
to operate, and (ii) a cell cannot function as a truly living
organism with less than the bare minimum of 12 proteins.
However, as we saw in Section 9 above, the total number of all
available proteins in the Na = 14 world is such that q has
certainly a maximum value qmax = 18.2. (The actual maximum would be
smaller than this for the reasons discussed in Section 9 above,
but let us continue to optimize the case for random assembly and
retain qmax = 18.2.) We see that the value of qra that is required
to ensure random assembly of the first cell is larger than qmax.
However, it is formally impossible for q to have a value in excess
of qmax: qra cannot exceed qmax even in optimal conditions. If qra is
equal to, or larger than, qmax it implies that every available
protein in the primeval soup must have been capable of performing
the task of every other protein. This indicates a serious lack of
specificity of tasks in the cell.
This conclusion does not depend sensitively on the choice of Na. If
functioning proteins actually require Na to be as large as (say) 20
(such as the mini-proteins referred to by Maniloff), we would find
q(RNA)ra = 33. However, the total number of proteins in an Na = 20
world would be of order 2020 , i.e. qmax = 26. The value of q(RNA)ra
again exceeds qmax, and so the conclusion about non-specificity
still applies.
15. Do proteins in the primeval soup have specific tasks?
The result that qra has a value in excess of qmax has significant
implications. It implies that there are no distinguishing
properties between proteins: each protein would have had the
31
ability to perform the task of all the other functional proteins
in the first cell. If that were to be the case, then there would
be no way to regulate the various distinct groups of cell
operations: replication could occur in the membrane, or membrane
generation could occur in the energy generation sites.
However, the nature of a cell requires that proteins have clearly
defined and distinctly specific functions. That is, not all
proteins must be capable of (say) membrane production: only a
fraction F (<1) of the proteins must have this capability.
What is a likely value for F? At one extreme, the smallest value F
can have is Fmin = 1/Qmax. Writing F = 10-f, this means that the
maximum possible value of f is fmax = qmax. In this limit, protein
specificity would be maximized: there would then be one and only
one protein out of the Qmax distinct proteins which could perform
any one of the basic tasks of the cell. In such a case, all 14
amino acids in each protein would be an invariant site, forbidding
any substitutions.
This extreme specificity is not true of most modern proteins:
typically, only a subset of sites are invariant. E.g., Yockey
(Table 6.3) discusses a 110-acid protein in which only 14 sites
are invariant. At the remaining 96 sites, a number of other amino
acids (from 2 to 19) may be substituted without degrading
significantly the functioning of the protein. The amino acids
which are functionally acceptable at a site are those which do not
impede the folding process or the biochemical requirements of the
protein. Because of these possibilities for substitution, the
probability of randomly “finding” a functional protein in “aminoacid
phase space” may be much improved over what one might expect
on the basis of the value of Qmax alone. Yockey (p. 254) describes
in detail how to compute the probability factor 2HN when one knows
how many different amino acids can be substituted at each site.
For the 110-acid protein discussed by Yockey, the improvement in
probability is enormous (from 1 in 10137 to 1 in 1093). It is not
clear how much improvement will occur in a small protein, where
there are only 14 amino acids. For the latter, the phase space is
limited to 1018.2. The 3-dimensional folding of such a small protein
might be quite sensitive to amino acid substitutions, more so than
for a larger protein. If this is true, then the improvement factor
might be quite small.
At the opposite extreme, F can certainly not exceed Fmax = 1/12 if
we are to preserve the distinction of 12 separate proteins for
each of the cell’s tasks. The limit F = 1/12 represents the
32
minimum possible protein specificity. This means that f cannot
have a value less than 1.08 in a cell with Np = 12 proteins.
In fact, it is probable that F is much smaller than 1/12. If F
were as large as 1/12, the prognosis for cell survival would be
slim: a single point mutation could convert (say) a membraneproducer
in any particular cell into (say) a waste management
protein. If this were to happen, the cell and its progeny could
hardly expect to survive for long.
This suggests that, in order to ensure long life for the cell, the
value of F should be much smaller than 1/12. How small might F be?
Let us introduce a “protein specificity index” m such that
F=(1/12)m, i.e. f = 1.08m. With this definition, the minimum value
that m can have is mmin = 1 (the minimum permissible specificity).
Values of m in the range (say) m = 3-4 represent conditions where
protein functions are only marginally specific. The maximum value
that m can have is mmax = qmax/1.08: in the example given above where
qmax = 18.2, mmax would have a value of about 16.9. In the limit m =
mmax, every protein performs a unique task.
With this well-defined range of the m parameter, we may usefully
refer to an “average specificity index” mav = (mmin + mmax)/2. With
the values just cited, we find mav is about 9. High specificities
can be considered as those with m values in excess of mav. Low
specificities are those with m values less than mav.
16. What are the chances of creating the first functioning cell
randomly?
The fact that the factor F departs from unity has the effect that
the Q factor which we used above in estimating the probability of
random formation of the first cell must be replaced by the product
FQmax. The quantity q in our earlier estimates must be replaced by
qmax-f where f cannot be less than 1.08.
In view of this, if we adopt the “RNA-first” theory, the necessary
number of reactions for random assembly of the first cell is 10L
where
L = 24.3Na + 43.3 –14(qmax - f). (eq. 8)
Setting Na = 14, the chance Pr of random assembly of the first cell
in the first 1.11 billion years of Earth’s existence (during which
time there were at most 1065 reactions) is one in 10b where
b = 14(f - qmax + qra ). (eq. 9)
33
With f=1.08m, and qra – qmax = 4.6, the chance Pr is about one in
1015m+64.4. Since m cannot be less than 1, Pr is certainly less than
one in 1079. If m takes on its average value mav = 9, Pr decreases
to 1 in 10200. Even if m takes on values that are much smaller than
mav (say 2-3), the probability Pr amounts to only one in 1094-109.
Note that the exponent b increases rapidly as Na increases: both
qra and qmax are proportional to Na. As a result, if we increase Na
to (say) 21, we would find that qra – qmax would increase from 4.6
to 6.9. Then even with m = 1 (its lowest value), exponent b
exceeds 100.
Even if we were to allow for a much older Earth, with an age of
(say) 100 billion years, the number 65 in our formula for qra would
increase only to 67. This would lead to a reduction of only 0.14
in qra in the “RNA-first scenario”. This would increase the chance
of random cell assembly, but even in the best possible case (m=1),
Pr would still be no better than one part in 1077.
The result Pr < 10-79 applies to a cell consisting of only the
absolute minimum set of Np = 12 proteins. Such a cell is extremely
small compared to the smallest known cell in the modern world
(where Np = 250). What if the minimum number of proteins in a
functional cell is 30 or 50 or 100? In such cases, the requirement
of specificity of protein function has the effect that the factor
F must be smaller than 1/Np , i.e. the exponent f must exceed
log(Np). In terms of the protein specificity index m,
f = m log(Np), (eq. 10)
where m cannot be less than 1. In view of this, the probability of
random assembly of the first cell is one in 10b where
b = (Np+2)[mlog(Np) – qmax + qra ]. (eq. 11)
Therefore, if the first cell required (say) 30 proteins to become
operational, the chance of assembling its RNA at random in the
primeval soup after 1065 collisions is less than one in 1047m+147.
The exponent in this result rapidly becomes large even if we allow
for only marginal specificity. For example, if m has a value of 2,
Pr is less than one in 10240. And if m is set equal to its average
value mav = 9, Pr falls to less than one in 10570.
If the modern genetic code was operative in the first primitive
cell (much smaller than the smallest cell in today’s world), the
above numbers are mathematical statements of the chances that the
34
RNA for the first cell was assembled by random processes. It is
clear that the probabilities are extremely small. We stress that
we have optimized a number of parameters in estimating the above
probabilities.
17. What about doublet-codons?
We can improve the situation for random assembly of the first cell
by considering the following possibility: suppose that, by some
means, the proteins in the first cell were assembled from a
smaller set of distinct amino acids than the Naa = 20 which exist
in nature today.
To be specific, let us suppose that the number of distinct amino
acids which were used in the first cell was as small as Naa = 5. It
is not obvious that functional proteins could actually exist with
such a small “vocabulary” of amino acids. However, it has been
claimed that protein folding is possible with as few as 5 distinct
amino acids (Riddle et al. 1997). Therefore, consideration of this
case probably does not violate any of the constraints of physical
chemistry. It also does not violate any of the limitations of
information theory: the quaternary genetic code might have begun
as a “first extension” using doublet codons (Yockey, p. 188).
(Vestiges of this early code might still exist in modern
mitochondria.) Doublet codons might have encoded for as few as 4-5
proteins (see Yockey, Table 7.1).
The major change in our calculation in this case is that the
codons in the RNA would no longer need to consist of triplets of
bases. Assuming that there are still 4 bases to use for RNA
coding, doublets would suffice to provide unique encoding for all
5 amino acids (plus a start and a stop code). Of course, one might
suspect that in a world where the number of useful amino acids has
been reduced from 20 to 5, there might also be a reduction in the
number of useful bases. For example, if there were only 2 useful
bases (i.e. if the genetic code were ever binary consisting of one
purine and one pyrimidine, a possibility discussed by Yockey (p.
184), then triplet codons would still be needed even to encode for
Naa = 5. In this case, we would return to the estimates derived
above for the triplet codon world. If there were 3 useful bases
available, doublet codons would suffice to encode for up to Naa = 7
(plus a start and stop code).
However, to optimize chances for random assembly, let us assume
that all 4 of the modern bases are available so that we can
exploit the possibility of doublet codons for the case Naa = 5.
35
In this case, the probability of assembling the RNA for a cell
consisting of 12 proteins, each with Na amino acids, would be fRNA =
(1/10)M where
M = 14.4Na +28.9 – 12q. (eq. 12)
We still need two proteins to allow DNA to do its work: with only
5 different amino acids to choose from, the chances of assembling
these two proteins at random are (1/5)P X 10-2q where P = 2Na.
Therefore fRNA in the 2-codon world would be equal to (1/10)R where
R = 15.8Na + 28.9 – 14q. (eq. 13)
In order that RNA for the first doublet-codon cell could have been
assembled at random in the first 1.11 billion years of Earth’s
existence, we must satisfy the equation
15.8Na +28.9 –14qd = 65 (eq. 14)
where subscript d denotes that we are dealing with doublet codons.
What is the minimum size of a protein in a world with Naa = 5? In
our previous discussion of our modern world where Naa = 20, we have
argued that proteins with Na = 14 are the smallest functional
units. Does this argument remain valid when Naa is reduced to a
value as small as 5? The answer is not obvious. For lack of
alternatives, we will assume that Na cannot be less than 14 in a
functional protein in the Naa = 5 world.
With this assumption, we find that qd cannot be less than 13.2.
This is many orders of magnitude smaller than the value of qra
which is required in the three-codon world. At first sight, this
might appear to represent a large increase in protein specificity.
However, results from the three-codon world are not relevant here.
Instead, we need to compare the new estimate of qd with the total
number of distinct proteins that are possible in the primeval
soup. With 5 distinct amino acids in the soup, and with each
protein containing 14 amino acids, we see that there are some 514
distinct possible proteins. Therefore, in this case, Qmax = 109.8 ,
i.e. qmax = 9.8. In view of the requirement that Q be at least as
large as 109.8, we see that the qd required for random assembly of
the RNA for the first cell again exceeds its maximum permissible
value, this time by 3.4. That is, once again essentially all
proteins are required to perform the task of all other proteins.
We are faced once again with the problem of lack of protein
specificity.
36
To satisfy the demands of specificity, we again introduce the
fraction F = 10-f of all available proteins which are able to
perform the task of (say) energy production. As before, we write f
= m log(Np) where m lies between 1 and qmax/log(Np). (With the above
numbers, mmax = 9.1, and the average specificity mav takes on a
value of about 5.) In view of this, we see that the probability of
assembling RNA for the first cell by chance in the 2-codon world
becomes one in 10c where
c = (Np+2)[mlog(Np) – qmax + qd ]. (eq. 15)
Since the difference qd - qmax is now “only” 3.4 (as opposed to 4.6
for the 3-codon case), we see that the probability of random
assembly of the RNA for a (12-14) cell has increased in the 2-
codon case by at least 16-17 in the exponent. This is a great
improvement indeed relative to the 3-codon case.
However, even with absolutely marginal specificity of protein
tasks, i.e. m = 1, the probability Pr of assembling RNA randomly in
the primeval soup for a (12-14) cell which uses only Naa = 5
distinct amino acids is no better than one in about 1063. If the
specificity has its average value mav = 5, then Pr = 10-123. Even if
the value of m is much smaller than mav (say m = 2-3), and with
more realistic numbers of proteins in the cell (say Np = 30), the
chances of randomly assembling the RNA for the first cell in the
primeval soup using doublet codons is no better than one in 10200.
18. What about singlet codons?
We might (in principle) improve the chances of randomly assembling
the first cell if the genetic code were able to operate with
singlet codons (instead of doublets or triplets). However, it
seems unlikely that such a world can exist. It is known that
folding of a protein simply cannot be achieved using an amino acid
set that is as small as 3 (Riddle et al. 1997): on the other hand,
folding can be achieved if the set of amino acids is as large as
5. For the sake of argument, let us make the extreme assumption
that folding CAN occur with an amino acid set consisting of only 4
species in the primeval soup. In this case, a singlet codon (one
of the four bases for each amino acid) would in principle suffice
for the RNA to encode for the amino acids, although with zero
redundancy (and therefore no error protection).
37
However, in order to assemble an accompanying DNA molecule, we
also need to have start and stop codons. That is, we must encode
not merely for the 4 amino acids, but also for the start/stop
codons. This means that the DNA is required to encode for at least
6 elements. This cannot be done with singlet codons (if only four
bases are available.)
We conclude that the doublet-codon world is as simple as we can go
and still have access to the flexibility of the genetic code.
19. A window of opportunity
When we considered what was probably the simplest example of a
doublet-codon world, with Naa = 5, we found that random assembly of
the first cell turned out to be more probable than in the triplet
codon case with Naa = 20. But still, the probability Pr is very
small.
However, this is not the only example we might consider. Doublet
codons with 4 useful bases can in principle encode for a
“vocabulary” of proteins made with Naa in the range from 5 to 14
(allowing for start and stop codes). And if proteins still consist
of Na = 14 amino acids, then the maximum available number of
proteins Qmax increases from 514 to 1414 as Naa increases from 5 to
14. That is, qmax increases from 9.8 to 16.0. The corresponding
values of mmax in a 12-protein cell are 9.1-14.8 (with mav = 5.05-
7.9).
Returning to the expression we obtained for the probability Pr of
random assembly of RNA for the first cell in a doublet codon
world, 1 in 10c, we recall from eq. (15) that
c = (Np+2)[m log(Np) + qd – qmax]
where qd = 13.2 (for proteins with 14 amino acids each) and m has a
value of at least 1. Inserting qmax values in the range from 9.8 to
16.0, we see that the difference qd-qmax is no longer in all cases
positive definite. In fact, when Naa grows to a value as large as
9, the value of qd-qmax becomes for the first time negative (-0.2).
This will certainly improve the probability of random assembly.
However, if we insert numerical values, and set the specificity to
its average value (mav = 7.2), we find that in a (12-14) cell, the
value of the exponent c for the case Naa = 9 becomes 106. If we
allow the protein specificity to fall to a very small value, say m
= 2, then c becomes 28. That is, the probability that the RNA of
38
the first cell with Naa = 9 was assembled by chance in the first
billion years of the primeval soup might be as large as 1 in 1028.
These represent large improvements over the probabilities we have
considered above.
Moving on to even larger values of Naa, the formal probabilities of
random RNA assembly become even larger. In fact, with Naa = 11, the
probability Pr approaches unity if m has a value less than
1.4/log(Np). Thus, in a (12-14) cell, a value of m less than 1.3
would ensure that Pr could have a value of order unity if Naa = 11.
Such a cell could have had its RNA assembled randomly with high
probability in the primeval soup in an interval of 1.11 billion
years.
In the limiting case Naa = 14 in the doublet codon world, a (12-14)
cell could be assembled randomly with high probability (in fact,
with near certainty) in 1.11 billion years as long as mlog(Np) does
not exceed the numerical difference between qmax and qd (i.e. 16.0-
13.2 = 2.8), i.e. as long as m does not exceed 2.5. This
represents the widest opening of the window of opportunity for the
random assembly of the RNA for a (12-14) cell.
We note that a specificity of less than 2.5 is much smaller than
the average mav: for the case Naa = 14, mav has the value 7.9. If
the protein specificity index in the primeval soup was indeed as
large as the value mav, the probability Pr of assembling the first
(12-14) cell randomly in a doublet codon world is no more than one
in 1080.
The window of opportunity in the doublet-codon world has an
interesting property that is relevant to the modern world. For a
14-acid cell where the number of proteins is as large as in the
smallest known modern cell (Np = 250), the probability of random
assembly Pr could have approached unity as long as m is in the
range 1.0-1.17. This is a very restricted window: but it is a bona
fide window. It indicates that, provided all of the various
optimized conditions are satisfied, random assembly of a (250-14)
cell might have occurred with high probability in the young Earth
with Naa = 14.
However, the restricted window for the Np = 250 cell closes
altogether if we have overestimated by too much the number of
collisions in the primeval soup. As was mentioned in Section 13.3,
our choice of 1065 for the value of nr (the total number of
reactions experienced by bases or amino acids in the primeval
soup) may be too large by 10 or more orders of magnitude. If nr is
in fact equal to 1058 (or less), then qd increases to 13.7 (or
39
more). In this case, the probability Pr (= 1 chance in 10c) falls
far below unity even if m has its minimum possible value (m=1):
the exponent c takes on the value 24.7 (or more).
Values of m as small as 1.17 or 1.3 (or even 2.5) represent
marginal specificities; they are far below the average
specificities, and are close to the absolute minimum value of m
(=1). Whether living cells could in fact survive (and replicate
faithfully) in the present of such marginal specificities is not
known. At the very least, it is a cause for concern in the context
of cell robustness.
The above calculation suggests formally that random assembly of
the first cell could have been achieved in the primeval soup if
certain conditions were satisfied. The requirements are: (i) at
least 11 distinct amino acids were available for use in the making
of proteins; (ii) 4 distinct bases were available for the DNA;
(iii) the protein specificity index m did not exceed 2.5 (for a
cell with 12 proteins); (iv) the number of amino acids in the
polypeptide chain of each protein equals 14; (v) the total number
of reactions between bases or amino acids in the primeval soup was
1065 ; (vi) we accept the RNA-first theory of cell assembly.
If any of these conditions was violated in the young Earth, the
probability of random assembly quickly falls to very low values.
20. Entropy constraints on the window of opportunity
At this point in the argument, we need to ask: is the mathematical
scenario described in Section 19 relevant in a robust biological
world?
In order to address this, we need to consider a certain aspect of
coding theory (Yockey, p. 5). The Central Dogma of biology states
that DNA encodes for protein assembly but proteins do not encode
for DNA assembly. To ensure this, coding theory states that the
“vocabulary” at the source (e.g. DNA) must have significantly more
symbols than the “vocabulary” at the receiver (amino acids).
In the modern world, there is no problem with this requirement.
With 64 codons in the DNA, and only 20 amino acids in (most)
proteins, there is a large excess in the “mutual information
entropy” of DNA compared to amino acids. The maximum information
content of a DNA sequence is 5.931 bits per codon, whereas the
information content of an average protein sequence is 4.139 bits
per amino acid (Yockey, p. 175). (These numbers are close to the
40
definition of Shannon entropy for the source log2(64) and receiver
log2(20) respectively: the slight differences arise because not all
modern amino acids are encoded with equal probability.) The
difference dH between 5.931 and 4.139 (dH = 1.792 bits per codon)
is (in the language of coding theory) a measure of the difference
in Shannon entropy between source (DNA) and receiver (proteins).
(Shannon entropy has nothing to do with the Maxwell-Boltzmann-
Gibbs entropy of thermodynamics). Because of this difference in
entropy, DNA can communicate information to amino acids, whereas
amino acids cannot communicate information back to the DNA.
The large amount of redundancy (represented by the ratio of 64 to
20) in the modern DNA “vocabulary” relative to the amino acid
“vocabulary” allows for error checking in the course of cell
replication. With the proper use of redundancy, the channel
capacity theorem (Yockey, p. 115) indicates that the error rate in
a code can be kept below any specified level. This is essential
for cells to ensure reliable and consistent replication in the
course of many generations.
As one possible measure of the level of error protection in a
code, we may refer to some results obtained by Yockey (p. 73). It
turns out that in a protein with N amino acids, the number of
high-probability states N(h) in parameter space is 2NH where H is
the Shannon entropy per amino acid. In the event that all sites
have equal probability of occupation by each and all of the Naa
distinct amino acids, the value of N(h) becomes equal to NaaN, as
expected from the probability arguments we have used in this
paper. In view of the formula for N(h), it seems reasonable to
use, as a measure of error protection in the translation from DNA
to proteins, the number E = 2NxdH. In the case of a modern protein
such as insulin (with N=51), E has a value of 3 x 1027, and we
interpret this to mean that insulin is extremely well protected in
the modern world from errors in transcription.
Now let us return to the doublet codon option in the primeval
soup. A world containing 14 distinct amino acids in the proteins
(plus one start and one stop code) would correspond to a doublet
code in which the source has 16 symbols but the receiver also
contains 16 symbols. In this situation, where dH = log2(16/16) = 0,
there is zero entropy difference between source and receiver. As a
result, E = 1, and the measure of error protection for (say)
insulin would be some 27 orders of magnitude smaller than it is in
the modern world. Replication of insulin in such a situation would
be subject to intolerable uncertainty.
41
Moreover, the Central Dogma of biology would break down: a protein
(such as insulin) would be able to control DNA just as much as DNA
controls proteins. This hardly seems like a prescription for hardy
life forms: there are too many options for lack of
reproducibility.
However, the break-down of the Central Dogma in the Naa = 14 world
suggests that in such a world, one might consider not only the
RNA-first theory, but also a “protein-first” theory. The numerical
factors entering into our estimates of the probability of random
assembly would then change. Thus, the value we have used above for
qd (=13.2) (obtained from eq. (14)) would have to be changed to a
value determined from a modification of the expression for z in
eq. (1). We recall that eq. (1) refers to the case where the set
of distinct proteinous amino acids contains 20 entries. Here, we
have only 14 entries in the set, and as a result, z changes to
13.8Na – 12q. Setting z equal to 65 and Na = 14, we find qd = 10.7.
The window of opportunity now widens somewhat: for the case Naa =
14, the value of Pr approaches unity as long as the specificity
index m does not exceed 4.9. This is still well below the average
value mav (= 7.9). Thus, we are still forced to confront the
requirement that protein specificities are quite small.
A doublet codon world, if it is to be of interest to biology in
the context of error-free replication, must certainly contain less
than 14 distinct amino acids. How much less than 14 should we
consider? We have seen that there is a good probability that RNA
can be assembled randomly as long as Naa has a value of 11 or more.
Including a start and a stop codon, this means that the genetic
code must use 16 symbols at the source to encode for 13 (or more)
amino acids. The difference in Shannon entropy between source and
receiver for this case is log2(16/13), i.e. dH = 0.3. With such a
value of dH, the error protection E of insulin would fall to 4 x
104, i.e. some 23 orders of magnitude weaker than the protection
which exists in the modern genetic code. And for the cases Naa = 12
and 13, the values of dH are 0.19 and 0.09 respectively. The
corresponding values of E for insulin would be 826 and 24, i.e. up
to 26 orders of magnitude less protection than in the modern
world.
Although it is sometimes claimed that error protection “must have
been” less in the early genetic codes than in the modern world,
this is not necessarily true. On the contrary, to ensure that
reliable replication occurs among millions of cells of even a
single species, it appears that the earliest genetic codes “must
have been nearly as accurate as those of today, otherwise even
short proteins could not have been transmitted in sufficient
42
numbers” (Yockey, p. 338). In other words, if the earliest genetic
codes were error prone, biology would not have been possible.
In order to ensure the same error protection between source and
receiver which exists in the modern world, there should be similar
redundancy to what exists in the modern world. That is, the ratio
of the number of codons in the DNA to the number of symbols in the
amino acids should be comparable to the modern value (64/20 =
3.2). This suggests that, at an epoch when there were 16 codons in
the DNA code (if there was indeed such a “doublet-codon epoch” in
the early Earth), the value of Naa should have been 5. This is
precisely the case we considered in the Doublet Codon section. The
Central Dogma would be just as robustly valid in such a world as
it is in today’s world. However, the chances of randomly
assembling such a cell is (as we have seen) only 1 in 1063.
21. Window of opportunity? or bottleneck?
There is a further constraint on the world of doublet codons in
which Naa lies in the range from 11 to 14. This has to do with how
well protected the genetic code is from noise-induced mutations.
Cullmann and Labouygues (1983, BioSystems 16, 9: hereafter C&L)
have discussed this issue in numerical detail.
In order to understand the results of C&L, a brief summary of
their terminology is necessary. In a doublet code, with 4 bases,
there are 16 possible codons. Of these, only a certain number (the
“sense codons”) are used to encode for proteinous amino acids. The
remainder are “non-sense codons” which serve to terminate the
translation. Mutations of various types can occur as a result of
noise. There is one class of mutations which causes a sense codon
to switch to a non-sense codon. In a second class of mutations, a
single mutation causes a sense codon to switch to another sense
codon. In the latter case, the protein may still function if there
are synonymous code entries. But if we dealing with an invariant
site, then the protein function is disabled, and C&L refer to a
“mis-sense” codon.
C&L have systematically analyzed all possible doublet codons in a
world where the number of amino acids being encoded varies from Naa
= 0 to 16 (thus including all numbers of interest to us here). In
each case, they count up how many single mutations N lead to nonsense
codons, and how many single mutations D(1) belong to
synonymous and mis-sense codons. C&L point out that the optimal
code (as far as immunization from noise is concerned) is one which
43
minimizes N and which simultaneously maximizes D(1). Codes which
have N not too far from its minimum value also possess significant
immunization against noise. C&L find that, starting with Naa = 0
and increasing Naa in steps of unity, there is at first a growing
number of doublet codes which satisfy the optimal condition.
In the present context, it is important to note that this growth
in available codes continues up to Naa = 8, at which point there
are thousands of codes which are not far from optimal. But for Naa
= 9 and larger, the number of available codes begins to diminish
rapidly. For Naa = 12, the number of codes has decreased to the
hundreds, and as Naa approaches 16, the numbers drop off towards a
value of 1. Thus, as a doublet-codon system attempts to encode for
more and more amino acids, there are less and less options the
closer Naa approaches 16.
Yockey (p. 190) refers to this as a “bottleneck” which has
evolutionary significance. He suggests that doublet codons might
have been successful in operating biology as long as Naa was
smaller than 16. But as more and more amino acids became available
for inclusion into proteins, and Naa eventually increased above 16,
it eventually became necessary to go to triplet codons. However,
before this happened, and as Naa increased upward through values of
9, 10,…16, the shrinking size of parameter space in which noiseimmunized
codes can exist would have exposed the organisms of that
time to an increasing lack of immunization against genetic noise.
Now, we recall that, in our discussion above, the probability of
randomly assembling the RNA for the initial (12-14) cell first
rises to large values when Naa is as large as 11. Using the results
of C&L, we now see that this value of Naa has a significant
property: it is already past the peak in available numbers of
doublet codes. Thus, we are already approaching the vicinity of
Yockey’s “bottleneck”. This makes it increasingly difficult for an
immunized genetic code to handle the large variety of proteins
which one might expect to find in a flourishing biosphere.
22. Overview on the window of opportunity
Let us now take an overall look at the window of opportunity in
the light of our discussions of the “bottleneck” (Section 21), the
entropy (Section 20), and the requisite marginal specificities of
proteins (Section 19). Taken in combination, these discussions
suggest that what appears as a window of opportunity for random
assembly of the first cell (in a formal mathematical sense) may be
44
subject to several classes of difficulties in the biological
context.
It is true that a scenario in which the doublet-codon window opens
up to its widest extent describes a system which is interesting
from a mathematical perspective. But from a biological
perspective, this system suffers from three serious drawbacks.
First, in the encoding process between DNA and proteins, error
protection is many orders of magnitude weaker than it is in modern
organisms. Second, the phase space of permissible genetic codes
shrinks to smaller and smaller volumes. Third, a huge number of
the available proteins must be able to perform each and every task
in the cell: the number is so large that there would have been
almost no specificity in protein tasks within a cell. That is,
there is a good chance that a protein which is supposed to be used
for (say) membrane repair, may switch to one whose function is
(say) enabling reproduction.
Any one of these features could be considered as posing
significant difficulties for cell survivability. The combination
of all three exacerbates the problem. It is difficult to see how a
cell (even of the primitive kind we consider here, no bigger than
a modern virus) could have survived. For the first robust cell to
have developed randomly in the doublet-codon phase of the
primitive Earth, conditions must have been “just right” to allow
survival in the presence of the above serious drawbacks.
23. Conclusion
We have numerically evaluated the probability Pr that, in the first
1.11 billion years of Earth’s existence, random processes were
successful in putting together the RNA for the first cell. In
estimating Pr, we initially assumed that the first cell follows the
rules which guide modern life-forms. That is, we assume there are
Naa = 20 distinct amino acids in proteins, and triplet codons in
the genetic code.
In calculating Pr, we consider only the random assembly of RNA: we
assume that once the RNA is present, it will generate the proteins
for the cell. (Thus, we are not requiring that the proteins be
assembled randomly: if we were to impose such a requirement, the
probabilities of random assembly of the first cell would be even
smaller than the results we obtain here.) Furthermore, we consider
45
a cell which is much smaller than those which exist in the modern
world. The latter contain at least 250 proteins. By contrast, we
have reduced the requirements of the first living cell to a bare
minimum: we assume that that cell was able to function with only
12 proteins. Compared to the smallest known living cell, our
choice of 12 proteins seems almost absurdly reductionist. Our
“cell” looks more like a modern virus (which cannot reproduce
itself) than a bona fide cell. But we proceed anyway.
Moreover we also assume that each protein consists of a chain of
no more than 14 amino acids. We refer to this as a (12-14) cell.
Again, a chain with only 14 amino acids is considerably shorter
than the smallest known protein in the modern world (which
contains a few dozen amino acids). It is not clear that a protein
with only 14 acids would be subject to the 3-dimensional folding
which is essential to protein functioning. Nevertheless, we make
these reductionist assumptions about a cell with the aim of
optimizing the probability of assembling the first cell.
In this spirit, we start with the assumption that the only amino
acids which existed in the primitive Earth were the 20 (or so)
distinct types of amino acids which occur in the proteins of
modern living cells. Also in the spirit of optimization, we assume
that the entire pre-biomass of the Earth was in the form of
proteinous amino acids. We specifically exclude the non-biological
amino acids (numbering more than one hundred) which may have been
produced in the primitive Earth. Moreover, we also assume that all
20 of the proteinous amino acids were present solely in the Lisomer
form so that the growth of a protein chain is not ended
prematurely by unintentional inclusion of a D-isomer. Furthermore,
we assume that the initial cell occurred in the physical
conditions which are most commonly cited in textbooks, i.e. in a
“primeval soup”. This allows us to obtain a firm (and generous)
upper limit on the number of chemical reactions which could have
occurred before the first cell appeared on Earth.
With all of these assumptions, we find that the probability of
assembling the RNA required for even the most primitive (12-14)
cell by random processes in the time available is no more than one
in 1079.
In order to improve on the probability that random processes
assembled the RNA for the first cell, we make the (unproven but
likely) assumption that proteins in the earliest cells were
constructed from a smaller set of distinct amino acids than those
which occur in modern cells. In order to ensure that the primitive
life forms had a similar level of error protection in their
46
genetic code as that which exists in the modern world, we consider
a case in which the early proteins consisted of only Naa = 5
distinct amino acids. For these, the genetic code can operate with
doublet codons. In such a world, the probability of randomly
assembling the RNA for the first cell in the time available is
certainly larger than in our modern (triplet codon) world. But the
probability is still small, no more than one part in about 1063.
We have identified a region in parameter space where, once the
genetic code exists, the probability of random assembly of the
first cell could have reached formally large values in optimal
conditions. These conditions include the following: (i) the first
cell contained 12 proteins; (ii) each protein in the cell
contained 14 amino acids; (iii) there were 4 bases in DNA; (iv)
the protein specificity index was no larger than 2.5 (far below
its average value); and (v) conditions in the primitive prebiosphere
were such that chemical reactions occurred at their
maximum possible rates. (The last of these conditions almost
certainly involves an optimization which is unrealistic by as much
as 10 orders of magnitude.)
(Note that we have said nothing about how the genetic code came
into existence. We merely assume that it is already in operation.
The origin of the code is a more formidable problem than the one
we have addressed here.)
If mathematics were the only consideration, our conclusions would
suggest that the RNA for the first cell could have been assembled
randomly in the primeval soup in 1.11 b.y. once there was a code
and abundant supplies of between 11 and 14 distinct proteinous
amino acids. However, when we take into account considerations of
coding theory (especially the necessity to protect the proteins
from errors of transcription), it appears that this region of
parameter space is hostile to protein production. And the genetic
code has to pass through a “bottleneck” in order to enter into the
modern world, with its 20 proteinous amino acids. As a result, the
first cell might have had serious difficulties surviving as an
autonomous biological system.
Finally, the extreme nature of our assumptions regarding the first
cell (12 proteins, each containing 14 amino acids) can hardly be
overstated. If a cell is to fulfil even the minimum requirements
of a Von Neumann self-replicating machine, it probably needs at
least 250 proteins. Even with multiple optimizations in our
assumptions about the primeval soup, the window of opportunity for
creating such a cell in 1.11 b.y. narrows down to a very
restricted region in phase space: (I) there must have been exactly
47
14 distinct amino acids in the cell proteins, (II) the protein
specificity index must have been between 1.0 and 1.17, and (III)
at least 1058 chemical reactions must have occurred between the
bases (or amino acids) in 1.11 b.y. The “fine tuning” of such
conditions presents a problem. However, there are more serious
problems than fine tuning: error protection in the genetic code
fails altogether in these conditions. Even the Central Dogma of
biology breaks down. A cell formed under these conditions would
truly be subject to serious uncertainties not only during day-today
existence but especially during replication. The cell could
hardly be considered robust.
Nevertheless, as Yockey (p. 203) points out, the possibility that
an organism from the doublet-codon world might have survived the
“bottleneck” may have some empirical support. According to the
endosymbiotic theory (L. Margulis 1970, Origin of Eukaryotic
Cells, Yale Univ. Press, New Haven CT), mitochondria might have
been at one time free-living bacteria which now survive in a
symbiotic relationship with the cytoplasma of other cells. In
mitochondria, the genetic code differs somewhat from the code in
other cells. Perhaps mitochondria are representative of organisms
which originated in the doublet-codon world, but which could not
survive on their own because of the difficulties associated with
the hostile zone of parameter space where they originated.
In summary, if the first cell actually originated by random
processes, the genetic code must already have existed, and
conditions must have been “finely tuned” in order to trace a path
through a narrow (and hostile) region of parameter space. The idea
that some of the constants of the physical world have been subject
to “fine tuning” in order to allow life to emerge, has been widely
discussed in recent years (e.g. in the book by J. D. Barrow and F.
J. Tipler, The Anthropic Cosmological Principle, Oxford University
Press, 1994, 706 pp). If we are correct in concluding that “fine
tuning” is also required in order to assemble the first cell, we
might regard this conclusion as a biological example of the
Anthropic Principle.