Concepts and Rules

Question 1

Binomial Distribution for Population Genetics

Genotype Number

Answer 1

N^AA ~ Bi(N, (f^A)²)

E(N^AA) = N(f^A)²

E(f^AA) = (f^A)²

Var(f^AA) = (f^A)²(1-(f^A)²)/N

Question 2

Large Population Assumption

Answer 2

When N is large enough, the variability in f^AA, f^Aa, f^aa due to chance will be negligible

Question 3

Hardy-Weinburg Equilibrium

Answer 3

Under the assumptions: random mating, large population size, no mutation, no selection and no migration

• Allele frequencies never change
• After 1 generation, genotype frequencies don’t change and have the ratios AA:Aa:aa = p²:2pq:q²

f^A= p , f^a= q

Question 4

Including Sexual Reproduction

Answer 4

Allele frequency change over the first generation and become an average of the initial male and female pools. They are constant from generation 1 onwards. Genotype frequencies are constant from generation 2 onwards.

Question 5

Irreversible Mutation

Answer 5

Allele frequencies change over time (A decreases, a increases).

Genotypes are in HW each generation (but with changing p and q).

Question 6

First Order Difference Equation

Answer 6

Equation which gives the n + 1^th term of a sequence in terms of the n^th term.

Question 7

Qualitative behaviour of homogenous difference equations

Answer 7

a>1, x_n grows exponentially

a=1, x_n = C a constant

0 < a < 1, x_n decays exponentially

a = 0, x_n = 0, a constant solution independent of x₀

-1 < a < 0, x_n decays monotonically but x_n changes sign, converges with oscillation

a= -1, x_n = (-1)ⁿx_0, x_n oscillates with no decay: diverges with bound oscillation

a < -1, x_n diverges with unbounded oscillation monotonically

Question 8

What is a cobweb diagram?

Answer 8

A graphical method of determining stability

Graph y = x_n and y = g(x) where g(x) = x_{n + 1} on a graph of x_{n + 1} vs x_n

Draw line vertically to g(x) and cobweb!

Question 9

What are the equilibria of the hardy-weinburg equilibrium?

Answer 9

Every value for f^A_n+1 is an equilibrium as f^A_n+1 = f^A_n

Question 10

What is the basin of attraction?

Answer 10

The set of all initial conditions whose solutions approach the equilibrium in the limit.

Question 11

What is the linear stability criterion?

Answer 11

When a tangent line approximation is made (the curve y = g(x) is approximated by its tangent at that point) and teh instantaneous gradient is found:

If |g(x*)| < 1 then x* is stable

If |g(x*)| > 1 then x* is unstable

If |g(x*)| = 1 then the linear stability criterion is inconclusive

Question 12

How are X - linked genes modelled?

Answer 12

Both male and female gene pools are considered.

f^A_f(n+1) = 1/2[f^A_f(n) + f^A_f(n-1)]

using f^A_m(n) = f^A_f(n-1) (lagging females by one generation)

A second order difference equation

The general solution is:

f^A_f(n) = c₁ + c₂(-1/2)ⁿ

where c₁ and c₂ are constants that can be found using initial conditions

Question 13

How is selection modeled?

Answer 13

We use different quantities to distinguish which part of the life cycle we are at:

f^A_(n), f^AA_(n) etc for the zygotes

p^A_(n), p^AA_(n) etc for the adults about to mate

assume all genotypes have the same fertility, but each has a different, constant viability, giving a constant overall fitness:

w^AA, w^Aa, w^aa

where w^AA is the fraction of AA zygotes that survive to mate

population size can change from generation to generation

Question 14

What is the Fisher-Haldane-Wright equation?

Answer 14

pA(n + 1) = wAApA(n)2 + wAapA(n)[1 − pA(n)]

w_mean

which can be written solely in terms of p^A_(n)

Question 15

What is mean fitness?

Answer 15

Question 16

What is the model for lethal recessive genes?

Answer 16

w^AA = w^Aa

w^aa = 0

p^a_(n+1) = p^a_(n) / (1 + p^a_(n))

solution

p^a_(n) = p₀ / (1 + np₀)

Question 17

What is relative fitness?

Answer 17

The proportion of fitness compared to the fitness of AA

Question 18

What is the selection coeficient?

Answer 18

Determines how much fitter AA is compared to aa

Question 19

What is the heterozygous effect?

Answer 19

Compares fitness of Aa to that of aa

hs is selection pressure against Aa

Question 20

What are the attributes of dominance?

Answer 20

If A is completely dominant

w^AA = w^Aa > w^aa

h = 0

If a is completely dominant

w^Aa = w^aa < w^AA

h = 1

Question 21

What are the attributes of incomplete dominance?

Answer 21

w^AA > w^Aa >w^aa

0 < h < 1

called directed selection.

Question 22

What are the attributes of codominance?

Answer 22

Two possibilities:

1. w^Aa > w^AA ≥ w^aa

h < 0, heterozygote advantage

2. w^Aa < w^aa ≤ w^AA

h > 1, hetorozygote disadvantage

Question 23

What is the FHW in terms of h and s?

Answer 23

Question 24

What are the equilibrium solutions of the FHW equation?

Answer 24

p* = 0 (no A)

p* = 1 (A is fixed)

p* = (h - 1)/(2h-1) =P if h is not 1/2

Question 25

What happens in directed selection?

Answer 25

As 0<h></h>

<p>p* = 0 is unstable</p>

<p>p* = 1 is stable</p>

<p>So A increases, eventually becoming fixed while a becomes extinct</p>

</h>

Question 26

What happens in heterozygote advantage?

Answer 26

h < 0

All three equilibria are applicable

P = (h - 1)/(2h - 1) is a stable equilibrium

0, 1 are unstable - no fixation or extinction

The population reaches a stable equilibrium with both A and a present.

Question 27

What happens in heterozygote disadvantage?

Answer 27

h > 1

all three equilibria are applicable

P = (h-1)/(2h-1) is unstable

0, 1 are both stable

long term result depends on initial conditions

• if p₀ < P then p_n approaches 0
• if p₀ > P then p_n approaches 1

Question 28

What is the Wright-Fisher Model?

Answer 28

A model for population genetics in small population where N^A_(n) = i and generation n+1 is produced by 2N independent random samples from the allele pool, with probability of success i/2N

Markov chains are used to calculate the probabilities of different values of i

Question 29

What is a Markov Chain?

Answer 29

A markov chain is a process that is described by a sequence of random variables

Fixed Set S

probabilities for X_(n+1) depend only on the value of X_(n) and are independent of n

Question 30

What is a Monte Carlo Simulation?

Answer 30

A simulation where all possible paths to get to a particular destination are taken into account to get a frequency distribution and approximate the probability mass function.

Question 31

What is Genetic Drift?

Answer 31

Under the Wright-Fisher Model (Where N is small), the number of A alleles in a gene pool either goes to 0 (extinction) or goes to 2N (fixation) just due to random sample.

Fluctuations in N^A can cause extinction/fixation in finite time just by chance.

Question 32

What is the Moran Model?

Answer 32

The Moran model is a model for small populatio population genetics that considers a haploid population of size N

At each step, one allele is chosen to reproduce, one to die (with replacement). This creates a Markov Chain.

Question 33

What is Heterozygosity?

Answer 33

The probability that two randomly sampled alleles from generation n will be different.

Also tells us the expected frequency of heterozygotes