Shouldn't be a problem. Math is math.
"Remember that "random" means utterly discontinuous functions of time."
That's *one* kind of randomness, sure, but hardly an all-encompassing definition. But since the distinction is not key to your argument, I won't quibble.
A random process produces a time series output that is a function of time. The output is a vector, (x1,x2,x3...x(n-1),xn. Read, "sub." A new vector is generated every clock pulse, or over some other interval. Each vector can have any possible value, in the n-space involved, with equal probability. F of t at every interval has any value in the n-valued "domain and range", and is therefore discontinuous at every point except for the chance of a seeming correlation caused by any point in the n-space being a possible output value.
Accepting that as a given, your mathematical analysis is good, but the problem is that this is *not* how evolution actually proceeds. You're modeling the wrong problem, as it were.
This is why correlation coefficients have a probabilistic output. Now let us talk about evolution as a random process.
Okay.
Say the Earth has had life for 3,000,000,000 years, or use your own number. Figure the Earth's biomass, say surface area is 200,000,000 square miles, and say biomass is five feet deep over entire sphere. Say mass is 64 pounds per cubic foot. Say 2.8x10 to the 16th pounds.
Only if you forget to multiply by 64lb/ft3... 2.8x1016 is the number of cubic feet, not the mass. Not that a factor of 64 makes a big difference to to the combinatorial argument, though, so no need to rework the rest of the numbers.
Going to grams, say 1.3 x 10 to the 19th grams. Say a cell masses 1x10 to the -5th grams.
*Way* too massive. E. coli cells, for example, mass around 1x10-12g, and an average human body cell (like a liver cell) masses around 8x10-9g. This causes you to undercalculate the number of cells by a factor of about 10,000. Again, though, not a major problem for your estimate, since cells would not be packed shoulder-to-shoulder anyway, there would be a lot of "filler" in the biosphere.
Therefore about 1x10 to the 24th cells.
Close enough for the kind of "feasibility" estimate you're doing.
As a working hypothesis say that the cell is the evolutionary unit, and that an evolutionary random process occurs once a second in each cell, and has done so for 3 billion years, then over that 1x10 to the 17th seconds roughly 1x10 to the 41st evolutionary events have occurred.
Here it starts to get iffy. One thing you're not taking into account is that biochemical processes can occur at mind-boggling rates. Chemical reactions in general often occur in times measured in femtoseconds (1x10-15). Even something as complex as DNA replication, which is an extremely intricate process involving the precision coordination of dozens of coordinating molecular "machines" (enzymes, etc.) and must wait until a matching amino nucleotide comes within "reach", etc., can typically take place at speeds of over a thousand nucleotide additions per second.
And needless to say "random assembly" could take place much faster than the "exact reproduction" requirements of DNA replication.
The other side of the same coin is that vast amount of molecular components in even a single cell. Using DNA replication again as an example, there are roughly 25,000 "cloning machines" active at any given moment, each one zipping along at the 1000 additions/second rate mentioned in the prior paragraph. And when it comes to more simplified cellular processes, the number of simultaneously active chemical "machines" or components reaches millions or billions, depending on the type (keep in mind that in order to divide, a human cell must have on hand over a billion available nucleotides with which to construct a new billion-plus-basepair set of chromosomes).
So the amount and speed with which even a single cell can "shuffle" new configurations of molecules is boggling. A cell can do a *lot* of "molecular work" in one second.
Nonetheless, not every molecular operation is going to be such that it can produce novel content (much of the activity will be involved in just keeping the cell running), and selection events can't come into play every time there's a molecular rearrangement, so for now let's go with your "one per second" net rate of selection events. But keep in mind that chemical processes can produce staggering amounts of varying products under the right conditions (something to remember for "biotic soup" analyses).
The next step is to see how many evolutionary steps are necessary to explain the complexity we see.
The main problem is that the case you describe is not actually made up of "evolutionary steps". It's simply random trials from scratch each time until one "hits" a fully formed outcome. That's not evolution, that's a "brute force" random search.
I realize you're trying to calculate how "hard" it would be to develop a given structure if evolution had no choice *except* to "brute force" it by sheer luck all in one fell swoop, but that doesn't turn a "lucky shot" of the type you are analyzing here into a count of "evolutionary steps", because evolutionary steps are *not* the process which you are analyzing.
The next idea you probably will not like, and that is irreducible complexity.
As an "idea" I like it just fine, and so do evolutionary scientists. The problem is that Behe (and the creationists who follow him) have created a "straw man" version of "IC" which is quite simply incorrect -- but appears to give the conclusion they want.
The original notion of "IC" goes back to Darwin himself. He wrote:
"If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down."That's "Irreducible Complexity" in a nutshell. It's not as if Behe has pointed out anything that biologists (or Darwin) didn't already realize.
-- Charles Darwin, "On the Origin of Species", 1859
But let's examine Darwin's description of "IC" in a bit more detail (emphasis mine):
No doubt many organs exist of which we do not know the transitional grades, more especially if we look to much-isolated species, round which, according to my theory, there has been much extinction. Or again, if we look to an organ common to all the members of a large class, for in this latter case the organ must have been first formed at an extremely remote period, since which all the many members of the class have been developed; and in order to discover the early transitional grades through which the organ has passed, we should have to look to very ancient ancestral forms, long since become extinct.Darwin makes two critical points here:We should be extremely cautious in concluding that an organ could not have been formed by transitional gradations of some kind. Numerous cases could be given amongst the lower animals of the same organ performing at the same time wholly distinct functions; thus the alimentary canal respires, digests, and excretes in the larva of the dragon-fly and in the fish Cobites. In the Hydra, the animal may be turned inside out, and the exterior surface will then digest and the stomach respire. In such cases natural selection might easily specialise, if any advantage were thus gained, a part or organ, which had performed two functions, for one function alone, and thus wholly change its nature by insensible steps. Two distinct organs sometimes perform simultaneously the same function in the same individual; to give one instance, there are fish with gills or branchiae that breathe the air dissolved in the water, at the same time that they breathe free air in their swimbladders, this latter organ having a ductus pneumaticus for its supply, and being divided by highly vascular partitions. In these cases, one of the two organs might with ease be modified and perfected so as to perform all the work by itself, being aided during the process of modification by the other organ; and then this other organ might be modified for some other and quite distinct purpose, or be quite obliterated.
The illustration of the swimbladder in fishes is a good one, because it shows us clearly the highly important fact that an organ originally constructed for one purpose, namely flotation, may be converted into one for a wholly different purpose, namely respiration. The swimbladder has, also, been worked in as an accessory to the auditory organs of certain fish, or, for I do not know which view is now generally held, a part of the auditory apparatus has been worked in as a complement to the swimbladder. All physiologists admit that the swimbladder is homologous, or 'ideally similar,' in position and structure with the lungs of the higher vertebrate animals: hence there seems to me to be no great difficulty in believing that natural selection has actually converted a swimbladder into a lung, or organ used exclusively for respiration.
[Example snipped]
In considering transitions of organs, it is so important to bear in mind the probability of conversion from one function to another, that I will give one more instance. [Long detail of example snipped] If all pedunculated cirripedes had become extinct, and they have already suffered far more extinction than have sessile cirripedes, who would ever have imagined that the branchiae in this latter family had originally existed as organs for preventing the ova from being washed out of the sack?
-- Charles Darwin, "On the Origin of Species", 1859
1. A modern organ need not have evolved into its present form and function from a precursor which had always performed the same function. Evolution is quite capable of evolving a structure to perform one function, and then turning it to some other "purpose".
2. Organs/structures can reach their present form through a *loss* of function or parts, not just through *addition* of function or parts.
Despite the fact that these observations were laid out in 1859, Behe's version of "Irreducible Complexity" pretends they are not factors, and defines "IC" as something which could not have arisen through stepwise *ADDITIONS* (only) while performing the same function *THROUGHOUT ITS EXISTENCE*.
It's hard to tell whether Behe does this through ignorance or willful dishonesty, but the fact remains that *his* definition and analysis of "IC" is too restrictive. He places too many "rules" on how he will "allow" evolution to reach his examples of "Behe-style IC" structures, while evolution itself *IS NOT RESTRICTED TO THOSE RULES* when it operates. Thus Behe's conclusion that "Behe-style evolution" can not reach "Behe-style IC" hardly tells us anything about whether *real-world* evolution could or could not have produced them.
For specific examples, Behe's example of the "Behe-style IC" flagellum is flawed because flagella are composed of components that bacteria use FOR OTHER PURPOSES and were evolved for those purposes then co-opted (1, 2), and Behe's example of the "Behe-style IC" blood-clotting process is flawed because the biochemistry of blood-clotting is easily reached by adding several steps on top of a more primitive biochemical sequence, *and then REMOVING earlier portions which had become redundant* (1, 2).
Even Behe's trivial mousetrap example turns out to not actually be "IC".
The usual qualitative formulation is: "An irreducibly complex system cannot be produced...by slight, successive modifications of a precursor system, because any precursor to an irreducibly complex system, that is missing a part is by definition nonfunctional..."
Note the key error: By saying that it "breaks" if any part is "missing" (i.e. taken away), it is only saying that evolution could not have reached that endpoint by successively only ADDING parts. True enough, but Behe misses the fact that you can also reach the same state by, say, adding 5 parts one at a time, and then taking away 2 which have become redundant. Let's say that part "A" does the job, but not well. But starting with just "A" serves the need. Then add "B", which improves the function of "A". Add "C" which helps A+B do their job, and so on until you have ABCDE, which does the job very well. Now, however, it may turn out that CDE alone does just fine (conceivably, even better than ABCDE does with A+B getting in the way of CDE's operation). So A and B fade away, leaving CDE. Note that CDE was built in "one change at a time" fashion, with each new change improving the operation. HOWEVER, by Behe's definition CDE is "Irreducibly Complex" and "could not have evolved (been built by single steps)" because removing C or D or E from CDE will "break" it. Note that Behe's conclusion is wrong. His logic is faulty.
The other error in Behe's definition lies in this part: "...any precursor to an irreducibly complex system, that is missing a part is by definition nonfunctional". The problem here is that it may be "nonfunctional" for its *current* function, but perfectly functional for some *other* function helpful for survival (and therefore selected by evolution). Behe implicitly claims that if it's not useful for its *current* function, it's useless for *any* function. The flaw in this should be obvious.
"Since natural selection can only choose systems that are already working, then if a biological system cannot be produced gradually it would have to arise as an integrated unit, in one fell swoop, for natural selection to have anything to act on."
True as far as it goes, but but this is hardly the same as Behe's sleight-of-hand in the first part of his statement, which relies on the false premise that a precursor to a structure is 100% useless for *any* purpose if *taking away* (but not adding) one part from the current purpose makes it unsuitable for the current purpose. Two gaping holes in that one...
Behe (an anathematized name)
For reasons I've outlined above.
talks of the bacterial flagellum, which contains an acid-powered rotary engine, a stator, O-rings, bushings, and a drive shaft. The machinery of this motor requires approximately fifty proteins.
Except that it doesn't. As many biochemists have pointed out, other organisms have function flagella (even *as* flagella) with fewer proteins (and/or different proteins). That flagellum isn't even "IC" by Behe's own definition since you *can* remove proteins and have it still work as a flagellum.
Say each protein has a weight of 50,000 Daltons, made up of, say, 400 amino acids.
Actually, the median protein length for bacteria is considerably shorter than that:
And for all you know, the flagellar proteins might be unusually short, vastly changing the probability results. But let's stick with a more realistic size of about 200 instead of your 400. At first this may sound as if it would only make the odds of randomly producing one of the proteins only twice as large, but in fact it makes the probability 160,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000 times as likely to occur by chance... Your earlier assumptions were not critical to the outcome, but the size of the proteins *is*.
The protein is not a chain but a complex three-dimensional structure, but we will simplify by treating it as a linear chain.
This doesn't "simplify" it at all, since its "complex three-dimensional structure" is uniquely determined *by* its linear ordering. It's the same analysis.
The odds of the first protein being produced from amino acids (say 20 amino acids are needed for all of these simplified proteins) is (1/20)^400, about 1x10 to the -540.
Get your calculator checked, it's actually 3.87x10-521. You've understated the odds by a factor of about 100,000,000,000,000,000,000,000,000,000.
This immense inverse is larger than the odds of one protein being assembled through random chance since we have picked numbers almost surely so as to make our arithmetic conservative.
No you haven't, since you *overstated* the likely average length of the proteins.
Besides, we are talking about fifty separate proteins before the flagellum works.
Not true, see above.
The odds have this event arising through random chance
Remember this assumption upon which all your calculations are based, we'll come back to it later.
of 1x10 to the -540th power
-521st power, actually, if you get your calculator fixed, or more likely 6x10-261 using a more reasonable protein length.
raised to the 50th power, or about 1x10 to the -(540 raised to the 50th power) or about 1x10 raised to the power of 4x10 to the 136th power. (1x10)^(4^136) What is that, about the inverse of a decillion raised to the decillionth power?
No, that's in fact dead wrong, since (10^A)^B is sure as hell *NOT* equal to 10^(A^B) as you assert. It's actually 10^(AxB), which for large numbers gives a vastly smaller result.
1x10-521 to the 50th power is actually 1x10-26050. Still enormously slim odds to be sure, but practically a sure thing when compared against yours -- not to mention vastly more correct.
This single protein chain, this single tiny organelle, has to be produced in one completely in one timing cycle, and in one cell,
No it doesn't, and here lies the crux of the irrelevancy of your calculation.
so that it can make a bacterium move, and so be selected through natural selection. If one allows more timing cycles
I'd like to point out that you never did actually make use of your biomass/cellcount/cycle-total calculation...
then the odds of the event occurring in the 3x10 to the 31st timing cycles become even lower.
First, "huh"? If you allow more timing cycles, then the odds of the event improve. How do you figure they "become even lower"?
Second, "10 to the 31st"? What happened to your original "10 to the 41st" that you started out with (but never used in a calculation)?
Third, why have all of your errors been in the same direction (erroneously improving your point)? Coincidence?
So the odds against this organelle forming are are very much less than the inverse of 1x10 to the 100th power.
Correct even after your errors have been corrected. But again, one must realize that these are only "the odds against this organelle forming" *totally at random all at once" by sheer chance of jostling amino acids in a bucket. It is *not* the odds against the organelle forming by evolutionary processes.
Physicists call anything less likely than 1x10 to the 50th power "impossible".
Funny you should mention that... First, "physicists" don't say that, since events less likely than that easily occur. For example, shuffle a deck of cards, then spread the deck face-up on a tabletop. Congratulations, the odds of that particular arrangement of cards occurring as a result of a shuffle is less than 1.24x10-68, which is far less than 1x10-50 -- it's a miracle!
Second, even rare events chosen a priori occur easily enough when the number of trials is large enough. For example an atomic state which occurs in less than 1x10-50 atoms is a near certainty to occur in the Earth alone, which contains far more than 1x1050 atoms.
So the "law" you mention is incorrect as stated.
But what's really funny about you mentioning it is that it's an informal rule of thumb (for *human* watchable events, not meant to be applied universally), originated by Emil Borel in a couple of books he wrote in 1943 and 1950 to popularize science. It's sometimes affectionately known as "Borel's Law". And ironically, Borel himself wrote on the topic of biological probability calculations:
In conclusion, I feel it is necessary to say a few words regarding a question that does not really come within the scope of this book, but that certain readers might nevertheless reproach me for having entirely neglected. I mean the problem of the appearance of life on our planet (and eventually on other planets in the universe) and the probability that this appearance may have been due to chance. If this problem seems to me to lie outside our subject, this is because the probability in question is too complex for us to be able to calculate its order of magnitude. It is on this point that I wish to make several explanatory comments.So there.When we calculated the probability of reproducing by mere chance a work of literature, in one or more volumes, we certainly observed that, if this work was printed, it must have emanated from a human brain. Now the complexity of that brain must therefore have been even richer than the particular work to which it gave birth. Is it not possible to infer that the probability that this brain may have been produced by the blind forces of chance is even slighter than the probability of the typewriting miracle?
It is obviously the same as if we asked ourselves whether we could know if it was possible actually to create a human being by combining at random a certain number of simple bodies. But this is not the way that the problem of the origin of life presents itself: it is generally held that living beings are the result of a slow process of evolution, beginning with elementary organisms, and that this process of evolution involves certain properties of living matter that prevent us from asserting that the process was accomplished in accordance with the laws of chance.
Moreover, certain of these properties of living matter also belong to inanimate matter, when it takes certain forms, such as that of crystals. It does not seem possible to apply the laws of probability calculus to the phenomenon of the formation of a crystal in a more or less supersaturated solution. At least, it would not be possible to treat this as a problem of probability without taking account of certain properties of matter, properties that facilitate the formation of crystals and that we are certainly obliged to verify. We ought, it seems to me, to consider it likely that the formation of elementary living organisms, and the evolution of those organisms, are also governed by elementary properties of matter that we do not understand perfectly but whose existence we ought nevertheless admit.
Similar observations could be made regarding possible attempts to apply the probability calculus to cosmogonical problems. In this field, too, it does not seem that the conclusions we have could really be of great assistance.
-- Emil Borel, "Probability and Certainty", p. 124-126
Now you can say that there are other ways to get such things done, that this particular protein sequence, or this particular organelle, could be done differently. Well, if this event could have happened in 1 followed by fifty zeros different ways, then the odds against it happening are still grater than 1 followed by fifty zeros to one.
Ah, but what if it could demonstrably have happened in more than "1 followed by fifty zeros different ways"?
There's a thing called "protein functional redundancy". This means that because many amino acids in a protein can often be replaced with certain other (or in some cases, *any* other) amino acid without changing the function of the protein *at all* (because the change is in a portion of the protein which is merely a "placeholder" and does no "work"), there are actually many "alternate" forms for each protein used in organisms which would literally have worked just as well. How many? Far more than you'd possibly imagine.
For example, there are fully 3.8x1093 functionally equivalent variations of the cytochrome C protein (an essential component of the mitochondrial respiratory chain). For perspective, that's around a billion times larger than the number of atoms in the universe. See: [Yockey, H. P. (1992) Information Theory and Molecular Biology. New York, Cambridge University Press].
Presuming for the sake of estimation that your 50 "essential" flagellar proteins have an equal number of equivalents, then each protein is actually 1x1093 times more likely to arise than your original estimates would indicate, and the entire 50-protein set would be 1x104650 times as likely...
But there's more... Surely those "50 protein" aren't the *only* possible set of proteins which would have given rise to a working flagellum, right? How many other workable flagellum "designs" are there? Put another way, how many conceivable ways could God have given a bacteria any sort of functional "paddle"? Zillions, right? (And don't forget, there are *already* organisms with endless varieties of different flagella other than just Behe's one example case.) So the question becomes not just, "how likely is *this* one *precise* flagellar setup", but instead it's "how likely is *any* kind of remotely functional motor structure to arise out of the *countless* variations on a theme possible under every conceivable possibility which still would have worked in some fashion?"
Hmm, haven't calculated *those* odds yet, have you?
Finally, your entire exercise misses the boat entirely. Yes, as you've shown, the odds of a particular flagellar structure arising "from the ground up" in one "step" entirely by random chance of jostling amino acids is practically nil and probably never happened.
But so what? No one's suggesting that it *did* happen that way.
Instead, biologists believe that the flagella arose over a *long* period of accumulated improvements and increased complexity, through evolutionary processes (which weed out the failures while multiplying and accumulating successes), eventually producing that particular modern flagellum by a final combination of *pre-existing* proteins and components available in the cell which had been mostly developed for other purposes. For example, in the case of Behe's favorite flagella, the common bacterial type III export apparatus (used for transporting specific proteins out of the cytoplasm) is recognizably the core of the "motor" of the the flagellum. So the proteins which were developed (through evolution) to provide *that* functionality did *NOT* have to be randomly assembled "from scratch" in order to "luckily" produce the flagellum out of pure raw amino acids, as your calculation presumed. Instead, those proteins were *already present*. How much does that cut down the set of 50, eh? Likewise for other proteins and structures which were already present in the cell but later co-opted for flagellar use. Your entire "all from scratch all at once" calculation is quite simply irrelevant, since it totally fails to take into account things which we *know* were already present in the cell (for *other* purposes -- something Behe keeps forgetting or trying to hide), and therefore did not have to be "invented" randomly on the fly as your calculation requires.
For a far more realistic look at the evolutionary "invention" of the flagellum, see Evolution in (Brownian) space: a model for the origin of the bacterial flagellum , which I linked earlier in this post. From the abstract:
The model consists of six major stages: export apparatus, secretion system, adhesion system, pilus, undirected motility, and taxis-enabled motility. The selectability of each stage is documented using analogies with present-day systems. Conclusions include: (1) There is a strong possibility, previously unrecognized, of further homologies between the type III export apparatus and F1F0-ATP synthetase. (2) Much of the flagellum’s complexity evolved after crude motility was in place, via internal gene duplications and subfunctionalization. (3) Only one major system-level change of function, and four minor shifts of function, need be invoked to explain the origin of the flagellum; this involves five subsystem-level cooption events. (4) The transition between each stage is bridgeable by the evolution of a single new binding site, coupling two pre-existing subsystems, followed by coevolutionary optimization of components. Therefore, like the eye contemplated by Darwin, careful analysis shows that there are no major obstacles to gradual evolution of the flagellum.Now *that's* science. Behe's stuff is just hand-waving and ivory-tower blowhardedness.
Your refutation?
See above.