1. Mutation Rates
Ultimately, the source of genetic variation observed among individuals in populations is gene mutation. Mutation generates new alleles, and these are the substance of all evolutionary change.
The mutation rate is defined as the probability that a copy of an allele changes to some other allelic form in one generation.
Mutation rates at the gene level depends on mutation rates at other levels:
Mutation rates for different kinds of mutations can be expressed as mutations per locus, per gene, per nucleotide, and per gamete. All of these indicate a specific type of mutation occurring per generation (higher eucaryotes) or per DNA replication (microorganisms), reflecting mutations arising anew in the unit time.
In addition, mutations rates may be expressed is relation to visible phenotypes or in relation to of DNA sequence changes
Therefore, it is useful distinguishing between mutation rates:
per base pair per generation (or replication)
per gene per generation (or replication)
per genome or gamete per generation (or replication)

2. Nucleotide mutation rate
Rates of spontaneous mutation seem to be determined by evolutionary balances between the deleterious consequences of too many mutations and the additional energy and time required to further reduce mutation rates.
In microorganisms, the rate of mutation for any nucleotide (point mutations) is generally included between 10-9 and per DNA replication.
Although this rate of mutation may seem exceedingly small, the total amount of new genetic variation introduced by spontaneous mutation at each DNA replication is significant. Consider the genome of E. coli, of the size of about 5 x 106 bp. With a mutation rate intermediate between those listed above (say 5 x 10-10), 25 x 10-4, or one every 400 cells carries a new point mutation.
This means that in a single large bacterial culture (1 litre), in which concentrations of 2 x 109 cells/ml are easily obtained (=2 x 1012 total cells), some 5 x 109 new mutations are present, corresponding to 1,000 mutations for each base pair.
In practice, all possible nucleotide substitutions and all possible single insertion/deletions, as well as many large rearrangements are represented in a moderately large bacterial population

3. Evolution in a glass
Experimental work with bacteria, eukaryotic micro-organisms and very small animals can tell us much about the occurrence and properties of mutations, including beneficial mutations. Over the last fifty years or so beneficial mutations have been observed to occur in a number of studies
Most of these experiments were done in a continuous culture system called a chemostat.
A chemostat consists of a bottle in which the organisms grow. Growth medium (i.e. food) is continuously pumped into the bottle and waste products, residual medium and organisms flow out. The contents of the bottle are well mixed so that each critter in the chemostat has an equal chance of getting at each bit of food. Factors that affect the growth of the organisms such as temperature are controlled, sometimes quite rigourously. Several variations of chemostats have been developed.

4. The chemostat
Continuous culture, in a device called a chemostat, can be used to maintain a bacterial population at a constant density, a situation that is, in many ways, more similar to bacterial growth in natural environments. In a chemostat, the growth chamber is connected to a reservoir of sterile medium. Once growth is initiated, fresh medium is continuously supplied from the reservoir. The volume of fluid in the growth chamber is maintained at a constant level by some sort of overflow drain. Fresh medium is allowed to enter into the growth chamber at a rate that limits the growth of the bacteria. The bacteria grow (cells are formed) at the same rate that bacterial cells (and spent medium) are removed by the overflow. The rate of addition of the fresh medium determines the rate of growth because the fresh medium always contains a limiting amount of an essential nutrient.
Schematic diagram of a chemostat, a device for the continuous culture of bacteria. The chemostat relieves the environmental conditions that restrict growth by continuously supplying nutrients to cells and removing waste substances and spent cells from the culture medium

5. Fluctuations of mutant strains
In an early study, resistance to a phage was used as a marker to follow the appearance of some mutations in a chemostat culture.
Novick and Szilard grew E. coli in a chemostat at a steady-state density of about 3 × 108 cells per ml. Periodically they assayed cells sampled from the chemostat for resistance to infection by bacteriophage T5 and calculated the density of T5 resistant cells in the culture.
At no time was phage T5 present in the chemostat nor had the cells in the chemostat been exposed to phage T5. They found that there was always a fraction of cells in the culture that was resistant to T5.
The density of resistant cells fluctuated betweeen 102 and 103 per ml.
The increases and decreases reflect the occurrence of mutations within strains in the chemostat. The initial increase in the frequency of resistant cells occurs because a mutation occurs within a T5 resistant strain that makes it (and its descendents) the fastest growing cells in the culture. As long as this strain remains the fastest growing one its representation in the population will increase. Eventually different favorable mutation occurs in a cell that is sensitive to T5 that makes it (and its descendents) the fastest growing cells in the culture. This causes the frequency of T5 resistance to decline. Later a different mutation occurs in a T5 resistant strain that makes it the fastest growing strain. Its frequency increases, and so on.

6. Neutral mutations
It is important to note that in this environment sensitivity and resistance to infection by T5 is a neutral trait here. Because there is no T5 in the environment, resistance does not provide an advantage.
But it doesn't seem to provide much disadvantage either. If it provided a disadvantage, the resistant cells would washout of the chemostat. In this environment, it is selectively neutral.
Mutations in other genes cause some cells to have a higher growth rate. It is just a matter of whether these mutations occur first in resistant or sensitive cells that determines whether the frequency of T5 resistant cells increases or decreases.
It's a hitchhiking effect - the T5 resistance gene just goes along for the ride with the genes causing the fluctuations.
Bacteria carrying neutral mutations constitute a fluctuating proportion of growing cultures. The fluctuations are attributed to periodic selection of fitter clones, with each successive sweep replacing less fit members of the population, including those with neutral mutations. The frequency of neutral mutations can also change in clonal populations as a consequence of hitchhiking with favorable mutations.

7. An example with yeast
Paquin & Adams (1983) studied haploid and diploid populations of yeast to estimate the relative rate that beneficial mutations would arise in an asexual population of each type.
Populations were kept in a chemostat (a fairly constant environment) at a population size of about 5 billion. Initially, the population was started from a single clone (one genotype).
A neutral marker, canavanine resistance then increased in frequency due to mutation pressure alone (amino acid mutation rate = 10-7), although the mutations always remained low in frequency (< 10-5) during the hundreds of generations of the experiment.
When a beneficial mutation occurred, it was most likely to arise in a canavanine sensitive cell.
The beneficial mutation would then sweep through the population. Canavanine sensitivity would "hitch-hike" along, driving back down the frequency of canavanine resistance.

8. Interpreting mutant fluctuations
This chart is an explanation of what happens in the chemostat

9. Genetic drift
Fluctuations of mutant-clone frequencies in the chemostat are examples of the process known as genetic drift.
If a population is finite in size (as all populations are) and if a given pair of parents of a diploid species has only a small number of offspring, then, even in the absence of all selective forces, the frequency of a gene will not be exactly reproduced in the next generation, because of sampling error.
If, in a population of 1000 individuals, the frequency of a is 0.5 in one generation, then it may by chance be or in the next generation because of the chance production of slightly more or slightly fewer progeny of each genotype. In the second generation, there is another sampling error based on the new gene frequency, so the frequency of a may go from to or back to This process of random fluctuation continues generation after generation, with no force pushing the frequency back to its initial state, because the population has no "genetic memory" of its state many generations ago. Each generation is an independent event.

10. Extinction of genetic variability by genetic drift
The final result of this random change in allelic frequency is that the population eventually drifts to p = 1 or p = 0. After this point, no further change is possible; the population has become homozygous. A different population, isolated from the first, also undergoes this random genetic drift, but it may become homozygous for allele A, whereas the first population has become homozygous for allele a. As time goes on, isolated populations diverge from each other, each losing heterozygosity. The variation originally present within populations now appears as variation among populations.
Computer simulation of genetic drift.  The frequency of an allele (e.g., A in a system with A and a) is shown for five replicate populations over the course of 100 generations, with a population size (N) of 20.  The effect of drift is inversely proportional to population size,  a fundamental driving force in many evolutionary divergences

11. Genetic drift over evolutionary time
The appearance, loss, and eventual incorporation of neutral mutations in the life of a population. If random genetic drift does not cause the loss of a new mutation, then it must eventually cause the entire population to become homozygous for the mutation.
At that point, the mutation has been fixed.
In the figure, 10 mutations have arisen, of which 9 (light red at bottom of graph) increased slightly in frequency and then died out. Only the fourth mutation eventually spread into the population.
Therefore, a steady substitution of one allele for another is expected to occur due to genetic drift alone

12. The probability of fixation of a neutral allele
It has been proven by matematical analysis (and it is quite intuitive) that the probability of fixation u of any neutral allele a is equal to its frequency in the population:
u = pa
In a finite population, pa takes discrete values only, starting from 1/(2N) in diploid species (when one copy only of allele a is present in the population), and incrementing at steps of 1/(2N).
In other words, the initial frequency of a mutant allele is, by definition, pa = 1/(2N)
Thus, the probability of ultimate fixation of any new neutral mutation is equal to the reciprocal of twice the population size.

13. The concept of gene substitution
It is important to distinguish between "Mutation" and "Substitution" with respect to individuals and populations:
The rate of gene substitution (K) is defined as the number of mutants reaching fixation per unit time.
If neutral mutations occur at a rate of μ per gene per generation, then the number of mutants arising in a diploid population of size N is 2N μ mutant alleles per generation. Since the probability of fixation for each of these mutations is 1/(2N), we obtain the result that
K = μ.
Thus theoretically, if the mutation rate μ is constant over time, neutral alleles accumulate at a fixed rate independently of population size, and their rate of accumulation can be used as an evolutionary clock to measure divergence times. This is one of the fundamental tenants of molecular evolution.
This result can be intuitively understood by noting that, in a large population, the number of mutations arising every generation is high but the fixation probability of each mutation is low. In comparison, in a small population, the number of mutations arising every generation is low, but the fixation probability of each mutation is high. As a consequence, the rate of substitution for neutral mutations is independent of population size.

14. Esimating nucleotide mutation rate in human
The result that the mutation rate for neutral mutations is equal to the rate of evolutionary substitution has been the basis of an approach to measuring the human mutation rate at the nucleotide level.
A direct comparison of stretches of DNA without function can provide an estimate of the mutation rate per generation between species whose divergence time and generation length are known.

15. Bacteria, Archae, and Eukaryotic microbes produce about one mutation per 300 chromosome replications. For E. coli this works out to be between 10-6 and 10-7 mutations per gene per generation, however it is important to note that there are certain "hot spots" or "cold spots" for spontaneous mutations. (A "hot spot" is a site that has a higher rate of mutations than predicted from a normal distribution, and a "cold spot" is a site with a lower rate of mutations than predicted from a normal distribution.) Higher eukaryotes have the same rate of spontaneous mutation, so that rates per sexual generation are about one mutation per gamete (close to the maximum compatible with life). RNA viruses have much higher mutation rates -- about one mutation per genome per chromosome replication -- and even small increases in their mutation rates are lethal.
Because a complex individual has a trillion or so nucleotides, each individual is likely to sustain one or more mutations.
Rates of expressed gene mutations average about 1 per 100,000 to 1 per million:
rates of expression of phenotypic effects are often higher because they are controlled by many genes

16. The mutation rate is a measure of the frequency of a given mutation per generation (or per gamete, which is equivalent). Ordinarily, rates are given for specific loci. Thus the mutation rate for achondroplasia is 6-13 mutants per million gametes. This means that each gamete has ca. 1 chance in 100,000 of carrying a new mutation for achondroplasia.
A. Mutation rates are based almost exclusively on rare autosomal dominant or X-linked recessive traits. It is virtually impossible to measure autosomal recessive traits accurately. B. The range of known mutation rates varies from 1 in 10,000 for Duchenne muscular dystrophy and neurofibromatosis type-1 (the largest genes known) to several genes in the range of 1 in 10,000,000.
C. Mutation rate studies never measure all the possible mutations at a locus. Many of the mutations cause no obvious phenotypic effect and could only be recognized by direct analysis of DNA sequences.
D. The rate of nucleotide substitutions is on the order of 1 per 100,000,000 nucleotides. Since there are 3 billion nucleotides per genome, that means that every gamete has about 30 new mutations involving nucleotide substitutions.

17. Mutation rates per generation
Per base pair ~
Per gene ~
Per genome ~
BUT -- These are highly variable from gene to gene, individual to individual, species to species

18. Mutation rates Mutation rate estimates:
From divergence among humans and chimpanzees: μ = 10-9 per site (bp) per year or 2*10-8 per site per generation
Phenotypic effect: μ = 10-6 to 10-5 per locus, gamete and generation
Viability: 0.5 per gamete per generation
At the molecular level, each human gamete genome may carry 200 new nucleotide substitutions
At the population level: A lot of new variation every generation!

21. The Rise and Fall of New Mutations
Even when a mutation confers a selective advantage the first few generations are dominated more by the whims of fate than by natural selection
The single individual carrying the mutation might die prematurely or its offspring might not find a mate
Seventy two years ago J.B.S. Haldane used an approach known as branching to calculate the probability that a new advantageous mutation will become fixed in a population
He found this to be approximately 2s, where s is the relative fitness advantage that those possessing the new mutation have relative to those who lack the mutation. Since selection is thought to be fairly weak on most amino acid variants, s~ , this probability could be quite low

22. Haldane’s Approach
Imagine that the new mutation sits at the root of a tree of descendants
Once the tree branches a few times, random pruning is unlikely to kill off all of the branches at once
The crucial phase is therefore the first few generations of existence when the number of branches is small

23. Central question is the probability that the mutation will persist through the initial branching process
Solving this problem yields the 2s result

24. Population size constant
time

25. Otto, S.P. and M.C. Whitlock The probability of fixation in populations of changing size. Genetics 146:
Generalized this approach to include cases in which population size rises and falls through time

26. Probability of Fixation
2(s + r), where r is the rate of population increase or decrease
Thus, a mutant that finds itself in a rapidly growing population is much more likely to be fixed than one in a shrinking population

27. Rate of gene substitution
1. Overall rate (K) of substitution of a gene by various successive
neutral mutations is the number of mutant alleles reaching fixation
per generation.