The theory of evolution depends entirely on the veracity of a self-replicating molecule arising on purely stochastic grounds.   One would expect this to involve a modest stretch of time.    

How much time?    In the "Blind Watch Maker" Richard Dawkins suggests that attaining such a probability is simply a matter of having the correct perspective - for example, if we lived for 100 million years and happened to play bridge we would not be surprised to see something as improbable as a perfect bridge hand - where each player was dealt the same suite - turning up from time to time.    So it would not be so improbable after all.

How accurate is this assumption?   Calculating the chance of one perfect bridge hand where we play bridge 100 times a day for 100 million years comes out at a paltry 1.63x10^-15  - the same degree of delight one would associate with winning a lottery event twice.  [r36], [r42] Which is not quite "from time to time".   To improve this to a chance of say one in a million we would have to keep playing a hundred games a day for  61,238,285,120,420,996 years.   By contrast, the best current estimate for the age of the earth weighs in at a mere 4,567,000,000 years.

Even mathematicians occasionally underestimate probability - in "A Brief History of Time" Stephen Hawking mentions that monkeys pounding away on keyboards will "very occasionally" by pure chance type out one of Shakespeare's sonnets.    The calculation for the sonnet "Shall I compare thee to a summer's day" shows that the chance is about 1 in 10⁶⁹⁰ i.e. 10 followed by 690 zeros. [r78]   As there have only been 10¹⁸ seconds since the Big Bang and there are about 10⁸⁰ atoms in the visible universe it is difficult to see where 10⁶⁹⁰ fits in comfortably.    Physical limits on monkeys and keyboards means we would have to cycle through the heat death or final collapse of the universe in excess of 10⁶⁰⁰ times to obtain a single sonnet.  

Could the universe have cycled through this many Big Bangs?   Thermodynamics dictates that the number of photons will increase relative to other particles with each cycle, thus given a finite number of particles, the entire universe would eventually be reduced to photons.   Our ability to read tells us the universe is not based on infinite cycles.

Of course, this is not saying that an event of 1 in 10⁶⁹⁰ could not take place - rather it is an objective measure of how unexpected life is given the assumption of a reductionist universe.   

The informational complexity inherent in a sonnet parallels the informational complexity that defines a protein.  Just as a sonnet is assembled from 26 letters (ignoring punctuation, spaces and capitals), proteins are assembled (by previously assembled proteins) from strings of 20 distinct amino acids ranging in length from 20 (TRP-Cage) to 26,926  (Titin).   Proteins are the "work-horse" of biological life - each cell in the human body produces about 2,000 per second, and as each protein is produced it is folded by other proteins or self-folds into a complex three-dimensional shape required to activate it's chemical function.

The most abundant proteins associated with the DNA of eukaryotes are the Histones.   As they are essential for maintaining the structural integrity of DNA and have a role in the transcription process they are structurally intolerant to change.   Histone H4 contains about 104 amino acids and differs in two or three places across a wide range of species.    The high level of invariance of Histone H4 with respect to cellular replication suggests it is a candidate for examining probabilities associated with the formation of an equivalent protein in a primeval cell on the basis of chance.

With 104 amino acids, there are 20¹⁰⁴ ways a primeval equivalent of Histone H4 could have been arranged through chance.      For convenience, we approximate 20¹⁰⁴ by 2 x 10¹³⁵

If we assume that the entire observable universe - approximately 10⁸⁰ atoms - was available to manufacture the very first Histone H4 equivalent protein - at an average of 10 atoms per amino acid - we would have 10⁷⁷ amino acids available.   If we spent all 10¹⁸ seconds since the Big Bang cycling through all possible proteins using all the available resources of the universe once every second we would have generated a maximum of 10⁹⁵ proteins.     Thus the chance of obtaining one Histone H4 equivalent protein using all the resources in the universe for a workable primeval cell would be 1 in 10⁴⁰.   

To put things in perspective, a perfect bridge hand would be 4,473,877,774,353 times more likely.

But this is only step one.   As noted a newly manufactured protein has to be correctly folded into its specific three-dimensional shape to become chemically active - if this step depended entirely on random processes it would take a small 100 amino acid protein in the region of 10⁸⁷ seconds to work through all possible configurations - 10⁶⁹ times longer than the entire age of the universe.      Amino acids have hydrophilic and phobic facets that are assisted into the correct shape by accessory proteins - some of which are synthesised along with the main protein itself.   [r37]

Replication of the DNA template proteins are derived from is carried out by proteins.  In prokaryotic cells DNA polymerase III is a complex of seven subunits ranging from 300 to 1,100 amino acids in length, one does the copying and the remainder are involved in accessory functions including error correcting.      In E. coli DNA repair processes are managed by approximately 100 different genes.    Just how does one "evolve" an error correcting process distributed across 100 genes on the basis of chance?

As a general rule about one third of the amino acids in a protein are directly involved in providing structural and chemical function and are therefore invariant while the remaining amino acids are drawn from a pool of about three or four types.    Given that the simplest bacteria requires some 2,000 enzymes (protein catalysts), and assuming an average enzyme chain of 300 links we have

    2000! x [ 1/( 20¹⁰⁰ x 4²⁰⁰)] ²⁰⁰⁰  => a probability of about 1 part in 10^500,000         [r18]

The evolutionary view is this calculation would be more realistic if performed using the intracellular parasite Mycoplasma genitalium as a model.  The disadvantage of using Mycoplasma genitalium is that it depends on the existence of the cellular machinery whose probability we are trying to estimate.    Another problem is an analysis of the genetic characteristics of the mycoplasma class suggests it arose through a loss of genetic material.  [r11]

Either way the probability for the random generation of the enzymes for this parasitic organism work out at about 1 part in 10^6,393.   In a purely reductionist universe, one Shakespearean sonnet would be more likely by a factor of 10^5,703.

There is also the matter of the coordination of enzyme timing.

Consider:

• When catalyzed by the enzyme uroporphyrinogen decarboxylase, the biosynthesis of chlorophyll and haemoglobin is increased by a rate equivalent to the difference between the diameter of a bacterial cell and the distance from the earth to the sun.  The reaction half-life of the chemical process is reduced from 2.3 billion years - half the age of the earth - to milliseconds.  [r83]

• Uridine 5'-phosphate is an essential precursor of RNA and DNA. In neutral solution, orotidine 5'-monophosphate undergoes spontaneous decarboxylation to uridine 5'-phosphate with a half-time of 78 million years. At the orotidine 5'-phosphate decarboxylase active site, the same reaction proceeds with a half-time of 18 milliseconds. [r84]

• The half-time for attack by water on alkyl phosphate dianions is 1.1 trillion years at 25°C, the phosphatase enzyme speeds this reaction up by 21 orders of magnitude to complete within 10ms. Every aspect of cell signaling follows the action of the type of phosphatase enzyme that breaks down phosphate monoesters. They also help mobilize carbohydrates from animal starch and play a role in the transmission of hormonal signals.
   
  One trillion years is estimated to be around one hundred times the age of the universe. [r85]

• Other examples of the half times of biological reactions proceeding in water at 25°C without the presence of catalysts include the following from [r85]:
   
  1.1x10⁶ years for alpha-O-glysocide hydrolysis
  1.4x10⁵ years for phosphodiester anion hydrolysis (C/O)
  9.8x10⁴ years for mandelate racemization
  6000 years for amino acid racemization
  450 years for peptide hydrolysis s
  73 years for cytidine deamination
  2 days for triosephosphate isomerization
  7 hours for chorismate mutation
  23 seconds for peptide cis-transisomerization
  5 seconds for CO₂ hydration.

Envisaging how the evolution of enzymes that would have initially been dependent on non-catalyzed processes taking 1 trillion, 2.3 billion and 78 million years to complete and then ensuring these catalyzed reaction times all lined up with an accuracy of tens of milliseconds on the basis of random chemical determination appears to be one of the more interesting mathematical challenges in this paper.

If one allows for the approximately 50,000 catalysts which need to line up for the process to work, this is analogous to fitting a key with 50,000 bittings into a lock, all of which need to line up with sufficient precision for the lock to open, or in this case for the cell to function and produce more "keys".

If the catalyzed time range is for the sake of discussion never more than 10 units - clearly not the case from the above - we would need to explore 10^50,000 different keys which once again is beyond the computational capacity of the universe.

Even following the standard evolutionary explanation and suggesting the key is produced gradually we are faced with the essential difficulty of a single evolutionary iteration for one enzyme falling exponentially beyond the maximum timeframe available for the entire evolutionary discourse.
  
At this point the primeval cell is still a long way off.   Other requirements for a cellular release candidate include:

• Creating a base template from which we can derive copies of the required primeval proteins at any stage in the future

• A mechanism to read the base template - to identify where instructions for a specific protein begins and ends.

• A mechanism for duplicating just the protein specific portions of the base template for manufacturing purposes as required.

• Proof reading and error correcting of the temporary template

• Transporting the proofed template to the production zone.

• A molecule capable of reading the temporary template and assembling amino acids in accordance with the template instructions.

• Identification and transportation services to move proteins to appropriate locations.

• A process to assist with the folding of the new protein into its particular three-dimensional configuration.

• Regulatory systems to ensure we never have too little or too much of a specific protein.

• Energy acquisition / management - each step requires that the right levels of energy to be available at just the right time in order to function.

• House keeping - a mechanism to disassemble and recycle protein components past their use-by date.

• The entire "factory" must be capable of replicating itself to facilitate "natural selection".

As noted in the prologue Kurt Gödel once observed that "The formation within geological time of a human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field, is as unlikely as the separation by chance of the atmosphere into its components."    With customary precision Gödel elaborated that unless the complexity of living bodies was innate to the material they were derived from, or present in the laws governing their formation, the laws of physics were fundamentally too simple to account for biological complexity within available geological time.  [r12]

Biologists pay tribute to Gödel the mathematician, but believe it unnecessary to pay much attention to Gödel's comments on evolution.  After all, they point out, Gödel was not trained as a biologist.     

Whether Kurt Gödel ever reciprocated this insight is not recorded.

An early "anthropic coincidence" was the observation that humans find themselves located more or less in the centre of the measurable universe.   More precisely, the measurable scale of things ranges from about 10^-26 at the bottom end of the subatomic scale to 10²⁷ metres across for the estimated width of the universe.   Human beings are measured in metres at the 10⁰ position.    So by a curious coincidence - in terms of scale - the universe extends as far down beneath us as it extends above us.

To gain an appreciation for the complexity represented by the human body imagine shrinking ourselves down to the smallest end of the measurable spectrum and looking back at the human body.    Focusing on the orders of magnitude, we are made up of approximately 100,000,000,000,000 cells of about 100,000,000,000,000 atoms each.

Thus our molecular makeup comes in at 10,000,000,000,000,000,000,000,000,000 - a total which exceeds the estimated number of stars in the universe roughly 100,000 times.  [r13]

Given that the human body is derived from a single cell, it can be reasonably argued that our cells are the most informationally rich objects in the universe. Just how complex is a cell in reality?   Follow the suggestion of molecular biologist Dr Michael Denton [r15] and imagine a model of a cell scaled up to the point at which each atom is the size of a tennis ball.  Such a model would be 20km in diameter, completely eclipse most city centers, and reach to more than double the cruising altitude of modern jetliners.     A construction worker welding in a representative atom per minute would take fifty million years to complete a single cell - and 50,000 million million million years to complete one human.

Expand all the cells in the human body to the same representative size, we would be able to cover the surface of the earth - both land and oceans - over 50,000,000,000 times.    Try building - and running - a factory of that size using goalless stochastic methods.

As atoms are clustered into amino acids and nucleotides within a cell, construction time at this level of representation reduces to under five million years, or by using mass production methods for the parts of the cell that are themselves products of the cell's mass production systems, less than one million years.

Yet a typical cell replicates itself within half an hour - 17.5 billion times faster.     Computer buffs would be interested to learn that it has been estimated that in the first year of life the number of chromosomal copy operations amounts to the transfer of over 40 Terabytes of information per second.   Come across any backup systems that match that?

A space measuring just tens of micrometers across contains sufficient information to control the placement of 6,000,000 skin cells per cm², maintain a red blood cell density of approximately 5,000,000 cells per mm³, develop a brain with 100,000,000,000 cells - deployed at approximately a quarter of a million per minute - establish an estimated 10,000,000,000,000,000 neural interconnections with the aid of 500,000 to 1,000,000km of neural wiring - and then for good measure wire the brain to the rest of the body through an additional 380,000km of nerve fibres - the latter matching the distance to the moon.

From information held at a density of 1.88x10²³ bits / cm³ we derive a retina containing 400,000 optical sensors per mm² with a total count of 110 million rods and 6 million cones - rendering the equivalent of an "ultra-high definition" picture via a 100 to 1 real-time lossless compressed signal sent to the brain through a 2mm thick nerve bundle containing one million nerve fibres - and from which the brain resolves real-time stereoscopic depth information to complete the picture.   

From one cell comes the totality of our sense of hearing - developed with the assistance of 12,000 sensory cells arranged on a 32mm long lamina in four parallel rows with a width of 0.05mm and a geometrical distribution similar to that of a piano - tuned to 20khz at one end and up to 30Hz at the other.     The hearing apparatus handles a range of 12 orders of magnitude while being able to discern the lateral displacement of a source within 3° - on the basis of a time difference of 0.00003 seconds.
    
The inner ear also contributes to the sense of balance through three semi-circular, fluid filled canals at approximately right angles to each other, containing fine hairs topped with a small node of calcium carbonate to enhance inertial sensitivity.  The deflection of these hairs across three spatial dimensions is translated into electrical signals measuring rotational and translational movement.    The balance mechanism of the inner ear is connected directly to the eyes through one of the fastest reflexive circuits in the body - enabling our eyes to track a fixed point while our heads turn.   [r27]

Sounds complex?   Here's a key question: how much more complex do you think the human genome is when compared to - say - a computer product like Microsoft Office or Ubuntu Linux?   Hundreds of thousands of times?   Millions?   

The human genome consists of approximately 3 billion DNA base pairs (150 billion atoms).    Every three nucleotides in the string of 3 billion codes for one of twenty amino acids.   Assuming equal frequencies, this equates to 4.32 bits per amino acid, which when apportioned back to the nucleotide level, means each base pair encodes for 1.44 bits of information.   The human genome thus contains 515MB (megabytes) of encoded information.

Molecular biologists assure us that only 1.1% to 1.5% of the genome actually codes for function [r19], so assuming the central dogma of molecular biology to be correct - the human phenotype is defined by 5 to 7MB of information - 200 times smaller than a typical Microsoft Office installation, and leaving us only 60% ahead in the race against the fruit fly!

Even allowing for the fact that the component parts of our 22,287 genes [r20] can be recombined in a multitude of different ways, the crunch question is whether a total of 5-7MB of information is sufficient to run a "universe sized" chunk of molecular machinery.

Perhaps it comes as no surprise to hear that Dr Craig Venter, whose company Celera Genomics produced the first complete sequence of the human genome, commented in 2001:  "We simply do not have enough genes for this idea of biological determinism to be right. The wonderful diversity of the human species is not hard-wired in our genetic code.  Our environments are critical."  [r21]    

Similarly the late biologist Stephen Jay Gould wrote in the New York Times in 2001: "The collapse of the one gene for one protein, and one direction for causal flow from basic codes to elaborate totality, marks the failure of [genetic] reductionism for the complex system we call cell biology."     [r23]

Fifty years of faith and billions of dollars research grants in deference to the single gene / single phenotype thesis failed to demonstrate appropriate dividends in medical and behavioural genetics: the tale of the "selfish gene" is catalogued under fictional history.     

As a replacement, Richard Strohman proposes that "In order to assemble a meaningful story, a living cell uses a second informational system. .... Let's say you have 100 genes related to a heart disease or cancer. These genes code for at least 100 proteins, some of which are enzymes, so you have a dynamic-epigenetic network, consisting of 100-plus proteins, their many biochemical reactions and reaction products. It is 'dynamic' because it regulates changes in products over time, and it is 'epigenetic' because it is above genetics in level of organization. And some of these changed products feed back to DNA to regulate gene expression. The key concept here is that these dynamic-epigenetic networks have a life of their own-they have network rules-not specified by DNA. And we do not fully understand these rules.

"In short, genetics alone does not tell us who we are, or who we can be. While, as Gould says, the reductionist theory of genetics has collapsed, the dynamic-epigenetic point of view retains genetics as part of a new paradigm for life, one that has striking implications for the future of the life sciences."   [r22]

The new vista is visible on the educational front:   "Genomes are the handmaidens, not the blueprints ... not 'selfish genes' but self organization, self-assembly, and emergent traits that are adaptive, purposive, taking advantage of genes to instantiate themselves.   Organisms create themselves, bottom up.   There's no blueprint for a human in the human genome.    Rather, regulated expression of genes in human embryos generates proteins that interact to bring about the adjacent possible - the next trait - which in turn makes possible the next trait."    [r24]

At least this generous explanation hints at an extraordinary degree of foreknowledge about the environment, and a complexity across all genomes capable of actively facilitating adaptation when catalyzed by external pressure.    

Unfortunately, the key question - how the original informational content was generated that facilitates such highly sophisticated recombinational techniques under control of genes that preceded particular selection pressures is not touched on.   The explanation is once again implicitly attributed to evolution and pushed back in time under the assumption that evolution is already known to be true.

 
The differences between humans and chimps have been found to be four times more likely to be accounted for in regulatory rather than protein coding genes.   [r25] [r26].   Apart from why up to 200 regulatory genes would (in the evolutionary scale of things) undergo near simultaneous changes, a wider question is how one genotype is able to pre-specify a range of different phenotypes before evolutionary pressure is brought to bear.

In the literature, we are told "Humans and chimps diverged from a common ancestor only about 5 million years ago, too little time for much genetic differentiation to evolve between the two species"  [r28].   This makes it reasonable to ask when language, musical and mathematical skills evolved - given that from an information-theoretic standpoint the genetic code shows no significant differentiation between chimps and humans.

The official response is that these capabilities were already integral to the common genome, and it was just a matter of "switching on" the appropriate combination of genes at the right time to express the relevant diversity.   By implication, chimpanzees (and mice, who are only 300 genes away from humans) share a heritage that "arose" yet remained entirely unexpressed format before the time of Mus musculus.   Was Douglas Adams right about mice after all?

So how long does it take a human brain to evolve?   If it took the Nautilus over 140 million years to add a handful of floatation chambers to give rise to the Ammonites, can we account for the most complex object in the universe arising in the space of only five million years?  What is the maximum rate of information gain that can be attributed to evolution?   This will be considered in some detail in the next section.

Curiously, other parts of the Nautilus failed to get any grip on macro evolution, even after 500 million years   Take its pin-hole camera eye - routinely touted in evolutionary literature as the archetypical candidate of eye evolution.    

Richard Dawkins puts it well:  "The [Nautilus] eye is basically the same shape as ours, but there is no lens and the pupil is just a hole that lets in seawater into the hollow interior of the eye.  Actually, Nautilus is a bit of a puzzle...Why, in all the hundreds of millions of years since its ancestors first evolved a pinhole eye, did it never discover the principle of the lens?... The system is crying out for a particular simple change...Is it that the necessary mutations cannot arise...I don't want to believe it."    [r29]

The challenge, however, is a wider one.    At a top level symposium convened to examine the problems many mathematical practitioners had run into attempting to come up with a viable neo-Darwinian mathematical framework, Dr Stanislaw Ulam (a co-inventor of both Monte Carlo statistical techniques and the hydrogen bomb) highlighted specific limitations with respect to eye evolution.    The chairman of the symposium countered: "This is a curious inversion of what would normally be a scientific process of reasoning. It is, indeed, a fact that the eye has evolved. ... The fact that it has done so shows that [your] formulation is, I think, a mistaken one." [r55] 

Either way a cogent proof of the principle of evolution would outclass the theory of relativity in terms of its importance as a contribution to the field of science.    

• The rate of evolution of any trait depends on the strength of selection on that trait, where a weak selection pressure results in negligible penetration of a population by a change in the phenotype.

• A species can only sustain a limited selection pressure for the obvious reason that when it is too high, the species simply dies out

Worden calculated a speed limit in terms of a measure called the Genetic Information in the Phenotype (GIP), a property of the population measured in bits.    So the more precisely defined a trait was in a population, the higher the GIP.   Halving the width of the spread of a trait increased the GIP by one bit.     GIP would thus be indirectly related to the information content of the genome, but always less than it.

Worden found the calculated speed limit to be inconsistent with some versions of Eldridge and Goulds' theory (such as pure species selection) but compatible with others.

Under the section "Human intelligence and learning" Worden noted that Chomsky had argued that the capacity for language arose de novo in mankind, and in this was supported by Pinker.   Pinker drew attention to the fact that although the human and chimp genotype differed by only about 1%, the difference of about 10MB of information was in his view sufficient to specify a complete language engine.

Worden estimated a total of 350,000 generations since the assumed split 5-7m years ago.   Working on an average number of three children per couple the calculated practical limit on total GIP growth for the human phenotype worked out at most 100,000 bits in total - less than the 10MB quoted by Pinker.

The question Worden considered next was how much of total GIP contributed to the evolution of intelligence, working from the basis that in a typical hominid group the difference in fitness between the least and most intelligent - in terms of survival to successful reproduction - could be assumed to be no larger than 10%.    Worden calculated that a 10% variation in survival probability contributed a GIP of at most 1/8 bit per generation. So the useful genetic information in the human brain, beyond that in the chimp brain, is limited to 40,000 bits, a shade short of 5KB.

Worden compared the 5KB extra design information in the human brain with a previous estimate of 100KB total GIP in the mammalian brain and concluded that the design information which distinguishes our brains from those of chimps is around 5% of the total.

Reviewing these results Worden noted that 5KB is equivalent to a computer program of around 300 lines:  "Experience suggests that the functionality one can express in 300 lines of program code is very limited; certainly not enough to design a complete facility for language learning and use, unless the underlying computing engine is already very well-adapted for the language task.

"We are led to the conclusion that the main difference between the human brain and other primate brains is one of capacity and power rather than of design complexity; and that our language ability is largely built on some pre-existing mental capacity which other primates have, although they do not use it for language."

Using an information gain of 20,000 bits per million years for a single species as a crude baseline estimate and assuming unchanging reproduction and selection pressures - the total amount of information gain with respect to a specific species since the Cambrian period under the standard model of evolution would be:

Thus evolutionary derived GIP available in our era is five times smaller than the theoretical information content represented by the protein encoding parts of the human genome - or 400 times smaller than the information content of the complete genome.     This factor is a further order of magnitude too small if information gain is considered only from the Cretaceous.    Another anomaly is highlighted by the fact that approximately 7,000 genes of the (959 celled) nematode C. elegans are shared by humans [r32]  - given the origin of C. elegans dates back from in excess of 600m years whereas human evolution has taken place in the last 5m years. [r34].  

If the difference between a preference for living in the soil, the trees or a castle is derivable by adjusting a set of pre-existing switches in the genotype, then the genotype either displays a considerable amount of really old fashioned luck or a modicum of forethought.

In the absence of any known rigorous selection process in our past how is a genotype generously shared between mice, chimps, humans and even dogs able to provide the capacity for the human brain in a time-frame that in evolutionary terms is the click of a mouse button?  

The rate at which proteins are expressed by genes doesn't help either as the "variation in gene expression between individuals within the species is substantial, relative to the differences between humans and chimpanzees.  For example, one human brain sample differs more from other human samples than the latter differ from the chimpanzee samples."   [r51]

The late Marcel-Paul Schützenberger commented: "Gradualists and saltationists alike are completely incapable of giving a convincing explanation of the quasi-simultaneous emergence of a number of biological systems that distinguish human beings from the higher primates: bipedalism, with the concomitant modification of the pelvis, and, without a doubt, the cerebellum, a much more dexterous hand, with fingerprints conferring an especially fine tactile sense; the modifications of the pharynx which permits phonation; the modification of the central nervous system, notably at the level of the temporal lobes, permitting the specific recognition of speech.

"From the point of view of embryogenesis, these anatomical systems are completely different from one another. Each modification constitutes a gift, a bequest from a primate family to its descendants. It is astonishing that these gifts should have developed simultaneously. Some biologists speak of a predisposition of the genome. Can anyone actually recover the predisposition, supposing that it actually existed? Was it present in the first of the fish? The reality is that we are confronted with total conceptual bankruptcy."  [r56]

Thus both gene mutation and combinatorics offer little theoretical insight into the problem of the information gain necessary to support correlated adaptation.    

The statistical event Spetner is interested in is the emergence of a new species. 
The four "input parameters" are:

Spetner makes the simplifying assumption that each evolutionary step is the result of the mutation of a single nucleotide.      Given that the chance of a specific nucleotide mutating in one birth is 1 in 10^-10, the chance of this happening over 50 million births is 1 in 200.   Assuming an approximately equal chance that the nucleotide will change to any one of the other three bases (from the set of adenine, guanine, thymine and cytosine), the odds of a specific change for a specific nucleotide is 1 in 600.  

Sir Ronald Fisher, a world expert on the mathematics of evolution, has shown that the odds of the survival of a single mutation with a survival benefit of 0.1% greater than the rest of the population is 500 to 1 against - because the majority of mutants are eradicated by random effects.  In other words, only 1 in 500 mutants with a positive benefit of 0.1% will end up taking over the entire population.

The chance that a specific change to a specific nucleotide will occur during a step is thus 1/600, and the odds that it will also take over the population is 1/500.   The total odds are thus 1/600 * 1/500 or 1/300,000.     This needs to happen 500 times in a row (the number of steps required to arrive at a new species).    We thus need to multiply 1/300,000 by itself 500 times.  The odds against this happening are approximately 3.6 x 10²⁷³⁸ to 1, or viewed the other way round, the chance of this happening is 2.7 x 10^-2739.  

Of course, one cannot simply assume that only one mutation is available at every step.  How many positive mutations are available?   Nobody knows the answer to this.   So Spetner turns the question around: for evolution to have a reasonable chance of working, how many positive mutations must be available at each step for the model to deliver a new species?   

What constitutes a "reasonable chance"?   A chance of one in a thousand could reflect the observation that for every species alive today approximately 1,000 have gone extinct.  However, as some species go for a very long time without changing - the well recorded phenomenon of stasis - Spetner chooses a chance of 1 in 1,000,000. 

The chance of a single step succeeding must be large - because we need to multiply it by itself 500 times (for the 500 steps) so that it comes out as close to 1/1,000,000 as possible (i.e. the chance of 1 in a million).    The smallest number that will do this is 0.9727 as

1 - (1 - 1/300,000 ) ^1,080,000   = 0.9727

So if the odds that a specific nucleotide will mutate and take over a population are to be 0.9727 for each step, there must be 1,080,000 potential positive adaptive copying errors for each of the 500 steps to arrive at a 1 in a 1,000,000 chance for the development of a new species.

The high number of adaptive paths has important consequences for the hypothesis of convconvergent (parallel) evolution.   Palaeontologists concluded that the mammalian brain evolved in parallel in the different orders and families of mammals and the eye evolved separately at least 40 to 60 times (e.g. Dawkins, "Climbing Mount Improbable", 1996, pg127).

But if there need to be over a million potential adaptive pathways at each step, this means it is impossible for precisely the same trait to evolve independently in two different species as the amount of selective freedom at each step would be too great.

Allowing for redundancy, even if only 100 of the 500 choices needed to be the same, the odds against this happening would be 1 in 10⁶⁰⁰ for convergence in a single species.    For the convergence of complex organs such as wings, kidneys or eyes the probability would be much smaller because one would need to allow for many species and thousands of steps.

Evolutionists believe this is still possible on the basis that many different changes to the underlying genotype could result in the same phenotype.    But given that genotype determines phenotype, freedom in genotype still translates into some level of freedom in the phenotype.   Even if one million genotype choices at each of the 500 steps of transition to a new species equate to only 10,000 phenotype choices, the total number of pathways in the 500 steps is still 10^2,000. 

In 1994 the gene that controls eye development in insects and vertebrates was found to be 94% identical, leading the discoverers to suggest - without having to resort to any calculation - that "the traditional view that the vertebrate eye and the compound eye of insects evolved independently has to be reconsidered."   

This is precisely what Dr Spetner's mathematical analysis suggests.

This "eye gene" thesis has been subsequently confirmed by experiments done by Professor Walter Gehring and others in which Drosophila, mouse and squid genes were successfully spliced into Drosophila, resulting in the targeted expression of ectopic compound eye structures on the wings, legs and antennae, and successfully demonstrating the non-random nature of the control genes across three evolutionary branches once considered independent.   [r39], [r40], [r41]

In a 1998 blog, Stephen Jones [r38] writes "That this is a real difficulty for the Darwinists is evident from [Richard] Dawkins reaction to it.  He admits that his "message...that eyes evolve easily and fast...might seem challenged by an intriguing set of experimental results, recently reported by a group of workers in Switzerland associated with Professor Walter Gehring."  (Dawkins R., "Climbing Mount Improbable", 1996, pp174-175).   Dawkins is surprised by it: "Amazingly, the treated adult flies grew up with fully formed compound eyes on their wings, legs, antennae and elsewhere" and "They even work....." (pp175-176), describing it twice as "remarkable" (p176) and "almost too startling" (p176). Rather limply he says he doesn't think that he was "wrong to think that eyes have developed forty times independently" and that "At least the spirit of the statement that eyes evolve easily and at the drop of a hat remains unscathed." (p176).

David Berlinski, with customary verve, noted "The ... group's more recent paper, "Induction of Ectopic Eyes by Targeted Expression of the Eyeless Gene in Drosophila" (Science 267, 1988) is among the most remarkable in the history of biology, demonstrating as it does that the ey gene is related closely to the equivalent eye gene in Sea squirts (Ascidians), Cephalopods, and Nemerteans. This strongly suggests (the inference is almost irresistible) that ey function is universal (universal!) among multicellular organisms, the basic design of the eye having been their common property for over a half-billion years. The ey gene clearly is a master control mechanism, one capable of giving general instructions to very different organisms.   No one in possession of these facts can imagine that they support the Darwinian theory.

"How could the mechanism of random variation and natural selection have produced an instrument capable of anticipating the course of morphological development and controlling its expression in widely different organisms?" [r52]  

From a purely mathematical analysis the answer is unambiguous.

Writing to the American botanist Asa Gray in 1860, Darwin wrote "The eye to this day gives me a cold shudder, but when I think of the fine known gradation my reason tells me I ought to conquer the odd shudder."   [r43]   To Darwin, the eye's ability to focus for different distances, admit varying amounts of light, and correct for both spherical and chromatic aberration made the concept of formation by natural selection appear "absurd in the highest degree".   However, in convincing himself that all that was required was for nature to undergo a series of small, easily imagined morphological changes, he saw it as correspondingly simple for nature to follow suit - in modern parlance, mutations and natural selection were all that it required.     And in accordance with Darwin's original view, biologists assure us the eye evolved not just once but independently between 40 and 60 times.   Albeit with some specific experimental limits.

Everyone is aware that the optic nerve runs from the eye to the back of the brain.   This raises a question: if processing happens at the back of the brain (distributed over a number of areas), why does the picture we see appear to emanate from the retina?  

Yet the actual image on the retina is inverted - highlighting that what we think we are seeing doesn't even match the reality of the light that falls on the retina.    Clearly, the brain is providing the original "virtual reality" show.

The reality of course is that photons do not intrinsically "shine" at any wavelength - the assumption that the universe is bathed in a benevolent glow of light is a perception that exists only in our brains.   What we "see" is a genetically determined biochemical reaction to electromagnetic energy in the range of 400 to 700 nanometers - coinciding with the property that water is 10⁷ times more transparent to radiation in this narrow window than it is to ultraviolet and infrared radiation and matched by the radiation from the sun.     With the exception of infrared, the rest of the electromagnetic spectrum passes undetected.  

Naturally it would be appropriate to demonstrate that the principle of Darwin's fine, gradual changes can be achieved at the microbiological level.   For a moment, let us step into a purely molecular, naturalistic universe, at a time corresponding to millions of years preceding the existence of vision, and task ourselves with creating vision for the very first time at the microbiological level.    Sight has never existed, the notion and memory of light a total unknown.   How to proceed?   

First, which part of the electromagnetic spectrum would we select?   But perhaps not too fast on this point - given that the concept of light is still to come into existence we would presumably have no notion that an electromagnetic spectrum exists.   We would have to infer it from molecular entanglements, the rise and fall of energy levels, the thermodynamic interplay of biochemical events.  

Perhaps we reason there is a chemical that changes shape in response to the event we are groping towards.   Which chemical structure - out of several million - should we select?     How do we know we have found the right one?    Can our hypothetical primeval being synthesize it?  If not, where would we get it from - and how?   How would we select, transport, and store it?   Does it have an optimal operating temperature or concentration?   How will it interact with the chemical structures in the rest of our primeval being?   If the - so far - unknown event causes the chemical to change shape in a picosecond, how would we know when to expect such a change, be aware of or be able to measure the change with any accuracy?    In particular, would we be able to differentiate this change from changes due to other biochemical phenomena?  

Let's say that we positively identified a chemical structure that changes shape for an as yet unknown reason.   As a first step, how would we be able to infer the important concept of a directional component with respect to the changes happening around us - given that we have no notion of direction, distance or dimensionality outside our microbiological universe - and photons would be impacting us from all directions?    Also, there is a practical consideration - what if our entire stock of chemical were to change shape in response to this unknown factor?   How would we know it was used up?   Would we be able turn it back into the original shape?    Would we require other chemicals to assist with this process?   Which ones?   How many, what quantities and concentrations?    Again, could we synthesize them - and if so, what precursors would we require in order to perform the synthesis?   How to identify the precursors?   Transport them?

How would we know that photons - which we haven't seen so far - are in fact directionally orientated?   If we could figure that out, would we be able to infer the concept of focus on the basis of photons impacting our chemical substrate and the way shapes change at an atomic level?   Could we infer the rules of optics without any prior optical knowledge?    If so, how?  How would we differentiate graduated shading from an out of focus area?     What microbiological activities would allow us to infer the necessary procedure to create a lens?    How would we persuade our molecular cohorts to (gradually perhaps) make a hollow cavity in an area reserved for the brain so we could set up a light detecting chamber?    How big should the hollow be?   What shape?     Should it be done over multiple generations?   Could we ensure the shape persisted - or could we leave this step to chance?    If by suitable diligence we are able to create an "early" protein for a lens - one that allows photons to pass through it, how would we know that we had the optimal protein construction?    Practical thought - how would we clean the lens?   

Would we understand - at a molecular level - that proteins denature and that we would need to find a way to stop this particular protein from denaturing?   Do we do this by continually replacing the protein or should we construct another protein to prevent the first protein from denaturing but in such a way that light could still be transmitted through it?    What length of time would be appropriate to guard against denaturing?  How do we determine or measure an appropriate timeframe at a microbiological level?   If we create an eye, can we figure out how to make it move?    If so how far and in what directions?     How do we control the eye's speed in any particular direction?  How do we supply the eye with nutrients without affecting vision?

How could we differentiate between photons of different wavelengths?   How do we distinguish wavelength at a molecular level?   Would we have any idea of the relevance of wavelength if we had never seen it?   Are there subtle variations to the chemical structure that would make it possible to distinguish between photons of different wavelengths? 

Making a suitably large leap - assuming we have figured out a number of the items above - what do we do with a molecule that changes shape?  How do we pass knowledge about a change in shape to the brain?    How do we know where the brain is located, or what signals it is expecting?  Should we do this mechanically or electrically?    If electrically, how do we convert the change in shape to an electric charge?   Chances are that molecules with electric charges simply repel or attract one another.   If we are going to use electrical charges, how do we ensure an electrical charge is effectively used as a communication medium?   How do we make a charge move form one cell to another, and in the right direction?   Where is the right direction?    How do we organize for a hole to grow in the skull to accommodate the optic nerve?   Where should it be located?    Can we stop a moving charge from dissipating as it travels to the brain?   What is going to interpret the information? 

Not least, how do we get one million nerve cells to grow back into the brain and ensure they link up to the appropriate target points - and validate these connections before using them?

How do we perform a 100 to 1 lossless real-time compression of visual data if we don't know what the signals represent?    How do we know that "10111001" coming out of a bundle of nerves means life whereas "10111011" signals an incoming predator?   Which muscles should contract in response if the signal is changing by one bit per millisecond from the right, and by how much?   How do we measure the duration of this millisecond?    Does this signal warrant a wider adrenaline response?    

How do we infer a projected, three-dimensional picture "out there" from varying charges emerging from millions of identical looking nerve endings?   How do we add a new part to the brain to handle the three-dimensional geometry that allows us to seamlessly stitch together two overlapping pictures in "real time"?      Do we develop this processing capability before or after we develop the eyes?   Do we need to start with one eye?   Can we resolve scenes we have never seen before?    How will we recognize, store and recall images even if they are presented substantially differently?   Will the visual field of our primeval being be able to compensate if we choose to invert our vision millions of years later?

Looking back, how would you rate your chances of providing the appropriate answer to all the above bio-chemical and mechanical questions without any prior knowledge or awareness of the existence of light?   If a sighted person struggles with the details, the chance of a molecular melange resolving these problems on the basis of a goalless stochastic process is going to be vanishingly smaller - no matter how many times the exercise is repeated.     With no preconception of light and millions of interdependent parts, we are presented with what is known as a tall order. 

The main point of this exercise is to highlight, in non-mathematical terms, the sheer scale of the problem domain that "natural selection" really faces.    Details, like gears, need to be specifically engaged.

Incidentally, until May 2007 evolutionists routinely drew attention to "profound optical imperfections" in the vertebrate eye as evidence that it could not have been designed: the vertebrate eye is wired up backwards with the nerves in front of the rods and cones which "interfere with the images" (e.g. PBS-TV series, episode 1).    

A 2007 paper in the Proceedings of the National Academy of Sciences modified that view by showing that Müller cells that traverse the retina - once thought to provide only structural support and nourishment to retinal neurons - provide an optimized optical fibre bridge through the nerve bundle.    The parallel array in the retina suggests a fibreoptic plate capable of low-distortion image transfer and solves a long-standing problem in our understanding of the inverted retina.    [r67], [r68]     The limiting factor is the diffraction of light waves at the pupil - the eye is quite capable of detecting single photons and has a dynamic range of one million million to one.   [r57]

There is in fact an excellent reason why the nerves are in front - the retina is easily damaged by light-induced thermal changes and requires cooling.   So the space behind the eye is taken up by the choroid, which provides the blood supply for the retinal pigment epithelium and regeneration of photoreceptors, reduces internal reflections and manages the required heat dissipation.   The blood flow through the choroid is the highest per gram of tissue of all tissues in the body and is - unsurprisingly - regulated in part by the brain in response to changes in illumination.   Thus the retina of each eye has an integrated, optically regulated liquid-cooling system to ensure long term visual acuity.    

By contrast, cephalopods - routinely quoted as being wired up the "right" way round - are exposed to a much lower light intensity so thermally-induced damage is less of an issue.    [r50]

Most discussions on the veracity of the evolutionary model focus on the probability of a one dimensional target "phrase" of amino acids assembling by chance within a particular timeframe.     

What is almost always overlooked is that DNA is without any biological function whatsoever - unless the cell already exists.  DNA comes with no inherent energy sources, proteins to assist it with translation or error correction and there is nowhere it can store its own products.  Without a cell, protein products would have simply been washed away into the primeval sea.     But the instructions for constructing the mechanism that decodes the DNA is held in the DNA itself.    DNA code cannot be translated - unless the cell it is going to create already exists.    It is decoded by the product of its own translation.

• In E. coli DNA enzymatic repair mechanisms and cellular responses to DNA damage involve approximately 100 genes.   Without the delta sub-unit of DNA polymerase III the number of errors that creeps in during transcription increases by a factor of 10⁵.   [r46]

• Proteins targeted for a mitochondrion are given a temporary polypeptide "label" and encapsulated by chaperone proteins designed to prevent the premature folding of the newly manufactured protein - but in such a way that leaves the label free.   The label is identified by a receptor protein in the outer mitochondrial membrane and the protein is then guided through a channel (made of other proteins) straddling the inner and outer membranes.    The unfolded state is necessary for the newly manufactured protein to be able to fit through the channel.  Inside the mitochondrial matrix another set of chaperone proteins prevents premature folding until a third group of chaperone proteins takes over and catalyses correct folding.  The label is then removed.   [r6]
   

• In the cell, the oxidization of glucose is mediated by about 30 different enzymes, each designed to fit a specific substrate at precisely the right step.  Energy is taken off these reactions in small doses and packaged into energetic molecules of adenosine triphosphate - ATP.   Thirty-eight packages are formed during  the conversion of each glucose molecule, two are used as part of the conversion process giving a net yield of thirty six small energy packets for the cell.    Outside the body, glucose burns at a single high temperature, inside the body, the available energy is dissected and apportioned with meticulous precision.   [r45]

• The fifth step in the eight step heme production chain - integral to the production of hemoglobin - has a reaction half-life of 2.3 billion years unless acted on by the enzyme uroporphyrinogen decarboxylase.  This enzyme increases the reaction rate by a factor that is equivalent to the difference between the diameter of a bacterial cell and the distance from the Earth to the sun.  [r82]

 
If one were to use the above as a model for cellular functioning in part of the brain or retina, with each "mini cube" representing a uniquely orientated, non-homogenous entity with a specific local function, the probability of correctly arranging 2,168 such cells in a 20 x 20 x 20 cluster reduces to the smaller probability of one in 3.351x10^2,123.  

To appreciate the magnitude of the 1,478 digit number shown in the above table - associated with the homogenous clustering of only six amino acid types - consider that the number of atoms in a "sun sized" star as about 8.4x10⁵⁶.   To set up a statistical experiment that exactly matches the likelihood of solving a 20 x 20 x 20 Rubik cube, we would require a blind-folded person to pick out just one pre-selected atom from a total of 3.98x10^1,430 stars.     The problem of course is the total number of atoms in the visible universe is estimated at around only 10⁸⁰.    

Time is similarly limited - the best Hubble estimates suggest we are 13.7 billion years or 4.323x10¹⁷ seconds old, with the universe expected to fizzle out (one way or another) in the next 15 billion or so years.    Even if we were to repeat the statistical experiment as fast as the laws of nature would allow us, we would be almost guaranteed to encounter the heat death of the universe first.

The scaling factor associated with the probability space goalless natural selection has to deal with to attain a specific molecular or cellular configuration can be gauged from the following:

These calculations are estimates in that they assume a simple three-dimensional arrangement of six distinct entities - whereas a realistic calculation would take into account 20 distinct amino acids or 200 cell types while clustering at multiple, overlapping levels.     The point of the exercise is primarily to highlight the order of magnitude entailed by a "three dimensional" probability space.    As the human genome does not vary by more than 0.1%, the overall probability measure lies well within these bounds.

By contrast, the probability tools at the disposal of natural selection are:

The number 10¹⁴⁰ is thus the limit on the total number of calculations, chemical reactions or distinct operations of any sort that could have been performed in the history of the universe.    As the limiting bounds of the universe are also evolution's terminus, if natural selection cannot produce an organism incrementally in 10¹⁴⁰ distinct steps it is not supportable as an explanation for the origin of life.    (See e.g. the total number of calculations performed by all the computers on the earth [r62b])

The central claim is that evolution traversed to the solution point of a domain too big to be incrementally traversed - with a resolution of 1 in 10^{64,600,000,000,000} - by flipping the biological coin no more than 10¹⁴⁰ times - in a goalless process - is mathematically unusual.  

A valid statistical explanation would need to take us along a demonstrably smooth path from the first molecule to the final aggregation, within the time available, subject to the molecular and chemical constraints of the earth - and with change to spare.

We know that the optimal number of steps to solve a 3x3x3 Rubik cube is 20.  Compare this with the potential number of steps - 43,252,003,274,489,856,000.   If the ratio between 20 and 43,252,003,274,489,856,000 is sufficient to demonstrate the presence of human intelligence, what does the ratio between 10¹⁴⁰ and 10^{64,600,000,000,000} suggest?