# How does mutation affect evolution?

##

# Model

## Start with chalcone synthetase locus

### Wildtype allele *A*_{1}
codes for blue flowers

### Suppose that *A*_{2} is a
mutant allele at this locus

#### Produces an ineffective enzyme that cannot produce pigment

#### A loss of function that codes for white flowers

## Initial allele frequencies

*p* = freq(*A*) allele

*q* = freq(*a*) allele

## Stipulate symbols for mutation rates

### Let *m* = rate/generation at which *A* mutates to *a*

### And *n* = rate/generation at which *a* mutates back to *A*

## Assume that mutations occur while gametes are in gamete pool (after
shedding, before fusion)

### So, gametes enter gamete pool in frequencies of *p* and *q*

### But then allele frequencies are altered by mutation

#### Frequency of forward mutations

##### Frequency of *A* gametes in
population times rate at which they can mutate to *a* = *p**m*

#### Frequency of backward mutations

##### Frequency of *a* gametes in
population times rate at which they can mutate to *A* = *q**n*

## Hence,

### Frequency of *A* gametes after
mutation events

*p**** = *p*
- *mp* + *nq*

### Frequency of *a* gametes after
mutation events

*q**** = *q*
+ *mp* - *nq*

### Note

*p**** + *q**
= *p* - *mp*
+ *nq* + *q* + *mp*
- *nq* = *p* + *q* = 1

## Calculate genotype frequencies for next generation (assume H-W equilibrium)

### Freq(*AA*)
= *(p**)^{2} = (*p* - *mp*
+ *nq* )^{2}

### Freq(*Aa*)
= 2*(p*q**) = 2(*p* - *mp* + *nq*
)(*q* + *mp*
- *nq*)

### Freq(*aa*)
= *(q**)^{2} = (*q* + *mp*
- *nq*)^{2}

## Calculate rate of change of frequencies

### D*p* = *p**
- *p* = (*p* - *mp* + *nq )* – *p* = *nq*
- *mp*

### D*q* = *q**
- *q* = (*q* + *mp* - *nq*)
– *q* = *mp*
- *nq*

# Is mutation an important evolutionary force?

## What would rate of evolution be if only evolutionary
forces was mutation?

### Suppose that new mutation arises

#### In a population fixed for *A* , new allele *a *arises
by new mutation

#### Mutation rate (*A*®*a*) is 11.2 x 10^{-6} = *m*

#### Back-mutation rate (*a*®*A*) is 2.5 x 10^{-6} = *n*

### Then,

#### D*p* = *p**
- *p* = (*p* - *mp* + *nq )* – *p* = *nq*
- *mp* = 2.5 x 10^{-6} (0) – 11.2 x 10^{-6}
(1) = -11.2 x 10^{-6}

#### D*q* = *q**
- *q* = (*q* + *mp* - *nq*)
– *q* = *mp*
- *nq* = 11.2 x 10^{-6} (1) – 2.5 x 10^{-6}
(0) = 11.2 x 10^{-6}

## Ignore back-mutation

#### D*p* = *p**
- *p* = -*mp*

#### D*q* = *q**
- *q* = *mp*

## Extra points on mutation as evolutionary force:

## Can derive more general equation if we assume that family size follows a
Poisson distribution with mean family size = *k*

_{}

##### I’m not prepared to derive this equation,
and don’t expect you to be able to do so

##### But, if you understand probability
distributions, you could do so

#### Where *x*_{1}
= probability mutation will persist to 2^{nd} generation

*k* = average family size when family size follows a
Poisson distribution

### When average family size (*k* = 2), then

_{}

##### Also, don’t expect you to be able to use
this equation

#### Approximately 36.8% of time, new mutant will
be lost in one generation

#### As family size increases, this probability decreases

##### So, more chance of mutant persisting in population

# Selection is major force that prevents detrimental alleles from increasing
in frequency

# Modeling mutation-selection balance

##

# First, assume that mutant is deleterious __recessive__

## Selection against deleterious recessive

_{}

## Increase in frequency due to mutation

_{}

## Because selection and mutation are opposing forces, they balance each other
to create an equilibrium

### So, at some point,

_{}

#### Or, _{}, _{}

### If we assume that the mutant is rare, then *q*^{2} is very small and denominator of right side of the
equation can be treated as unity

_{}

#### Then, cancel *p* from each side to get _{}

_{}

#### And, equilibrium allele frequency is

_{}

# Second, assume that mutant is deleterious __dominant__

## Selection against deleterious dominant

_{}

## Increase in frequency due to mutation

_{}

## Because selection and mutation are opposing forces, they balance each other
to create an equilibrium

_{}

#### Or, _{}, _{}

_{}

_{}

### If we assume that the mutant is rare, then *q*^{2} is very small and all term with *q*^{2} go to zero

_{}

_{}

_{}

### Also, if mutant is rare, then *q**m*
is vanishingly small

#### And equilibrium allele frequency is

_{}

## Think about the form of this equation

### Again, when selection is strong and mutation rate is low, equilibrium
frequency of mutant allele is low

# Apply these equations to the delphiniums studied by Nick Waser and Mary
Price

## Observed average frequency of albino plants = 7.4 x 10^{-4}

### i.e. less than one albino plant per thousand
individuals

### Assumed that this is the equilibrium phenotype frequency

## Also measured selection against albinos by pollinators

### Found that albinos produced from 20% (in artificial populations) to 70% (in
natural populations) the number of seed found on blue-flowered wildtypes

#### So, *s* = 0.3 to 0.8

## Next calculated expected equilibrium phenotype frequency at
mutation-selection equilibrium

### First, assumed that albinism is due to a single gene with a recessive white
mutant allele

#### So, assumed white-flowered plants are *aa* homozygous mutants

#### Calculated equilibrium phenotype frequency by assuming selection against
recessive homozygous mutant (equations above)

##### Designated frequency of mutant allele (*a*)
= *q*

##### Assumed H-W equilibrium

###### Frequency of mutant homozygote = *q*^{2}

##### Hence, equilibrium phenotype frequency at mutation selection balance = _{}

_{}

#### Plug in observed values

##### Low range of selection

_{}

_{}

_{}

##### High range of selection

_{}

_{}

_{}

### Second, assumed that albinism is due to a single gene with a partly or
completely dominant white mutant allele

#### So, assumed white-flowered plants are either homozygous (*aa*) or heterozygous (*Aa*)

#### Calculated equilibrium phenotype frequency by assuming selection against
dominant or co-dominant mutant

##### Designated frequency of mutant allele (*A*) = q

##### Important assumptions

###### Assumed H-W equilibrium

Frequency of the mutant
phenotype = 2*pq*

Mutant phenotype class
probably consists of only heterozygotes

Because the rarity of the
mutant ensures a low frequency of homozygous mutants

And because the dominant
homozygotes is probably lethal

###### Further, assumed (recognized) that
equilibrium frequency of wild type, recessive, allele is very nearly one _{}

##### Thus, the equilibrium frequency of the
mutant phenotype at mutation selection balance is

_{}

###### Because _{} and _{}

#### Plug in observed values

##### Low range of selection

_{}

_{}

_{}

##### High range of selection

_{}

_{}

_{}