I still have not read any Intelligent Design articles, nor have I looked up how the evolution theory explains problems in evolution. There is a reason why I have not done so. One should think out of the box, look at the problem with fresh eyes. If so, then it would not be the best idea to start by brainwashing your fresh eyes with the existing approaches to the problem. What you should to is to first think about it yourself and in a simplest possible way. Only later you could (or even should) check what others have done. This is especially true with controversial topics because a controversy means that neither side has found convincing arguments.
I found this problem of how to get introns with random mutations. Let us continue with it, but in another setting. There are three major events when species appeared.
The Cenozoic era started after a major catastrophe 65 million years ago. This catastrophe destroyed dinosaurs and other saurs. In the beginning of the Cenozoic era fourteen new orders of mammals appeared in about 10-15 million years. A similar thing happened in the beginning of the Mesozoic era: five orders of dinosaurs appeared in a very short time, and there is the beginning of Paleozoic era when all major phylas appeared. Thus, there are three very strange times in the history of the life on the earth. Let us focus on the Cenozoic era.
The emergence of all these mammalian orders in a short time is a problem: they clearly derive from a smaller set of early mammals but each order is characterized by their own genes. Thus, in some way they got different genes.
How fast was it to create new exons my mutations? Let us recalculate it. The mutation rate is about 0.5*10-9 mutations per base pair (bp) per year. A gene is about 1000 bp and it contains about 8-10 exons, so an exon is about 100 bp. There are control parts between exons, but let us assume that the mammalian orders differ also by protein-coding parts, exons. If this is not the case with mammals, then we could take the two other times when new orders or phylas appeared in a short time, the beginnings of the Mesozoic and Paleozoic eras, but I think we can safely say that mammalian orders have good many different exons.
The mutation rate implies that it takes 2 billion years for all of the base pairs to change to new ones by mutations. That is, each base pair can be considered independent, so we only need to look at the mutation rate per bp per year.
How to calculate these probabilities? The mutations come with a Poisson process, that is, equal arrival probability, and the base pairs are independent. See how we can calculate how many mutations occur in a human: the human genome has 3 billion base pairs, the mutation rate is 0.5*10-9 mutations per bp per year, so one year there are 1.5 mutations and in a generation some 30-40. Actually there are some 20-30 mutations per generation in humans, the human mutation rate seems to be a bit smaller. But you can see that this is the way to calculate it.
Follow this example: there are about 1000 bp in a gene, thus there is one mutation in a gene in 2 million years and there are about 100 bp in an exon, thus there is one mutation in an exon in 200 million years.
How important is the population size? If we are interested in single mutations (SNPs, single nucleotide polymorphism), then it is very important. There is one mutation in any exon in 200 million years. If the population size is 200 million, then we get one mutated exon every year. In this case the population size clearly helps. It is quite easy to get new SNPs into the population. They are the alleles of a gene.
But this is not what we are interested in. Genes with SNPs (single mutation) are just alleles, also exons with a single mutation are just alleles. Alleles of a gene most probably still work. Alleles definitely are not different genes that do something completely different, but we assume that different orders of mammals have at least some completely different genes with different protein-coding sections.
In order to get a different gene one needs many mutations, quite many. After a few mutations in an exon a gene will not work and the organism dies as it does not get the correct protein. But it can happen that the gene (or exon) was duplicated: there still was the original working gene and the duplicate got many mutations. If this happens, the mutated duplicate will be so called pseudogene. They are not functional: they are so called Junk DNA. Mutations can happen in nonfunctional genes and selection does not act on them. The mechanism is that a gene gets duplicated, the duplicate gets mutated a bit too much and it becomes a pseudogene in the Junk DNA. There it can collect many mutations as it is nonfunctional. After a long time the duplicate may become functional again as a result of a lucky mutation and will again be a working gene.
This is the only way I invented how exons could get many mutations and turn from a gene of early mammals to genes of new mammalian orders. Maybe there are other methods, but I could not find them.
How many mutations are needed? I have no idea, but much more than one. I would imagine quite much since in order to be a working protein-coding gene it must produce proteins that are useful for some process. It seems quite unlikely that random mutations produce an exon that codes some useful protein in a new mammalian species.
But I have to make some guess. Let it be that we need a substantial fraction of base pairs changed in an exon, for instance 30% or so. That would be 30 mutations.
Let us say we want 30 mutations to a single exon. Does the size of the population make this any more probable. We saw that with single mutations the size of the population made things very much easier. But they do not help here. With the Poisson arrival process the probability of n arrivals in the time T is
Pn=((aT)n/n!)exp(-aT)
where a is the arrival rate, in our case the mutation rate. So, let us put T to 15 million years, n to 30 and a to 0.5*10-9. So aT=7.5*10-3≈10-2 and we get 10-60/30!. The number 30! is very large but we do not even need it. This probability is extremely small. It goes very fast to a very small number when n grows larger than one. With Poisson arrivals you get the average number of arrivals (mutations in our case) in a given time T and the number you get will vary a bit above or below this average, but it will not vary far from the average. The only way to get 30 mutations to one exon is to set the time T to such a value that the average number of mutations is from 29 to 31. In that case every exon in the population gets this about 30 mutations. The size of the population does not matter since to get about 30 mutations to each exon is the same as getting 30 mutations to an exon in the population size one.
So, what should be T in order to get 30 mutations?
It is just 30% of 2 billion years. Likewise, if we think we need 10 mutations in an exon, the time needed to make a single such exon in the population of any size through random mutations is 10% of 2 billion. As we only have 15 million years, that is 0.75% of 2 billion years, we can get only 0.75 base pairs changed in exons. That is, this time is sufficient only for single mutations. We can get lots of SNPs in this time, because if the population is larger, the number of mutations can vary a bit over the average of 0.75, but we cannot in any possible way get a substantial change of base pairs in even a single intron through mutations with this mutation rate.
How can these new mammalian orders then have been born? It is possible that they got infected by viruses or bacteria and DNA was transferred to them. These single cell organisms have been developing for 2 billion years. They may have in their Junk DNA any possible exon, but you better notice one thing. There are not that many single cell organisms on the Earth. Yes, I know, there are a huge number of them, only not so many compared to all possible exons.
Let us again estimate the number of single cell organisms. There are 10 million viruses in one drop of sea water. How big is a drop? Maybe ten drops is one milliliter. Does that sound correct? All of the earth surface is not seas, but let us say it is. The radius of the Earth is about 6000 km. The surface area is about 1014 m2. Seas can be quite deep, but most microbes do not live in great depths, but let’s say 1000 m. Thus, there is 1023 milliliters of sea water and in it lives 1030 viruses. Virus often has only a small DNA, but a bacterium has larger, even 10,000 genes and each gene has 10 exons. Thus, we might have 1035 or so exons somewhere on the earth. The number of all combinations of 100 base pairs is 4100=2200≈1065. If we try all 1035 exons, then we have only tried one in 1065/1035, that is, one in 1030. It is really lucky if life found useful exons by random mutations, unless the vast majority of all combinations are useful. How could they be useful for a specific purpose in a cell?
There is another solution. What if the mutation rate was much higher during these times when many new species were born? That would explain how new exons were made, but it creates a new problem: why would the mutation rate be so much higher. It would have to be about 100 times higher than now. Mutations are mainly caused by cosmic rays. It is indeed possible that the Earth was moving through a different part of the galaxy and there were much more mutations. Researchers have found a period on marine life extinctions that correlates rather well with the period of the Earth’s movement in the galaxy. See e.g.
https://arxiv.org/pdf/1309.4838.pdf
This is a very nice explanation attempt. Higher rate of cosmic rays not only explains how new exons can be made but also why the old species went extinct. The Milky Way has spiral arms and the sun circulates the Milky Way. At some times it passes through the arms and then there can be more meteors but conceivably also more cosmic rays. The article finds a good correlation with the Earth passing the spirals of the Milky Way and the five main mass extinction events. It also finds five smaller extinction events. This explanation seems to me very promising.
Mass extinctions are in this explanation not caused by the god of the sky getting angry with the dwellers on the Earth. They are a result of a natural process. This understandable process makes it possible to have many mutations in a short time.
This solves one central problem, but the other problem remains. How do we get useful exons? That is, life on the earth can try only an extremely small fraction of all possible combinations of exons. How can life hit to useful ones?
I think the best answer is to include the creator God to the solution. He must create life again after the mass extinctions, which are caused by the sun passing through the spirals of the Milky Way. So, God does not destroy life on the Earth. Life is destroyed by unavoidable cosmic radiation and meteor showers when the Earth passes the spirals.
In case you do not like the idea of a creator God, then by all means find a better solution to this second question. The solution is not natural selection since pseudogenes are not under natural selection and it hardly is possible to make many mutations to a protein-coding part and expect that the gene remains functional after each mutation. Mutations in pseudogenes are totally random, without any selection, and then finally the last mutation very luckily changes the pseudogene to a working gene. I think this requires a designer.
Let us look at the creation story in the Bible. It is the first creation story. The second one seems to me to reflect astrology: Adam and Eve are around the tree of life in the Paradise is the Milky Way around Deneb. The first creation story is more interesting.
The first story has six creation days. If we move day four to be day two, everything fits to the scientific view of how the life developed.
The first day there is light. That is the Big Bang, not light from the sun.
The second day (the 4th day in the Bible) the moon and the sun are created. The Earth gets its moon. It is interesting that from the earth both the moon and the sun have exactly the same visual sizes. This coincidence makes it possible to measure time precisely by eclipses.
The third day the atmosphere is created. First there is the early atmosphere with lots of water vapor in the air. Cyanobacteria create the oxygen in the atmosphere.
The fourth day is the Paleozoic era. Trees are created in this ear.
The fifth day is the Mesozoic era. Birds are created in this era.
The sixth is the Cenozoic era, the era of mammals and humans.
This is a very good fit to be a random fit. How could it be? Because the fourth day is in a wrong place, this is not dictated by God. If can be a vision by a prophet. But if it were a vision, then maybe the fourth day is in the correct place: in the vision the sun and the moon were seen only after the life rose from the sea to the land, which happened in about the mid point of the Paloezoic era.
Now I have even accepted visions by prophets. Is that reasonable? How could they know things that had passed long ago? The only way is to assume beings who remember, like angels. Angels let prophets to see visions from angel’s memory. Great!
Unfortunately I have to admit that the biblical explanation is quite sound. I will stop here and have to think about this. You in any case see that I start from the evolution theory, trying to get it working, and end up to the bliblical story.