Basic Graduate Evolution

nucleotide ≅ base

Bases

• purines (A T), 3 H bonds, less stable (fewer bonds)
• pyrimidine (GC), 3 H bonds

Each strand has a 5' end and a 3' end. The convention is to write from the 5' and to the 3' end when transcribing DNA. The strands are anti-parallel.

RNA (as compared to DNA)

• single stranded
• much shorter (often 20-24 nucleotides long)
• Bases: (A C G U)

• traditionally, something that codes for a protein, i.e., gene → messenger-RNA → protein (this was previously the only functionality known for DNA segments)
• now, genes also include sequences which do other things, e.g., transcribe RNA, some RNA sequences are functional as themselves

Types of Genes

• protein coding
• RNA
• regularity

Eukaryotic protein coding genes comprise both transcribed () and non-transcribed () parts.

Gene's are transcribed by polymerase, which first anchors to a "promoter region" which flanks the gene, and then travels the gene transcribing its contents.

A "codon" is 3 nucleatides which are transcribed into 1 amino-acid.

Gene Structure (we're talking about protein coding gene)

TATA box

in the promoter region -19 from the gene start, in a GC-rich region of the DNA. Not required, but prevalent

A Gene with the promoter region.

  promoter region       Gene->
---->  --  <---  --     ----------   --------   ---------
GC    CAAT   GC  TATA    
box    box  box  box   

RNA polymerase

Composed of

• a central carbon
• an amino end
• a carboxyl end
• a hydrogen
• a side chain – this is the most variable and the most important

Simplest is glycine

• single H side chain
• fits in nooks and crannies of proteins

Five classes

• positively charged
• negatively charged
• hydrophobic
• neutral
• special

string of amino acid

• secondary structure – two most common 2-D structures for folded proteins
  • α helix
  • β pleated sheet
• tertiary structure – these combine into 3-D structures of the protein
• quartinary structure – two tertiary molecules

Losing the phase (correct codon alignment) can be caused by mutations and garbles the translation resulting in a garbled protein. The phase is determined by the start or initiation codon, and is called the "reading frame" or "open reading frame" (ORF).

Genetic code is degenerate but not ambiguous.

Multiple codons coding for an amino acid will often differ in the third position.

Terminology

fourfold-degenerate site

any nucleotide in this position will specify the same amino acid.

twofold-degenerate site

two of the four nucleotides at this position will specify the same amino acid.

non-degenerate site

every nucleotide here results in a different amino acid, called an "amino-acid substitution".

synonymous codons

different codons which code for the same amino acid.

In most codons the first two positions are non-degenerate sites.

numbers of codons

• 61 sense codons (remaining 3 are STOP codons)
• 549 possible codon nucleotide substitutions
  • 61 codons × 3 positions × (4-1) alternatives for each position
• assuming equally probable mutations (not true), then 70% of all 3^{rd} position changes are synonymous
• 2^{nd} site is the most sensitive to substitution, 0% of changes are synonymous
• in the 1^{st} site only 4% of changes are synonymous

Main evolutionary forces (on populations)

• mutation
• selection
• migration (new alleles from outside)
• drift

Mutations are hereditary changes in the genetic material due to errors in DNA replication or DNA repair.

Types of mutations

• substitutions
• recombination (crossing-over and gene conversion)
• deletions/insertions ("indels")
• duplication
• inversions

Evolutionarily mutations that occur in germ cells are the only hereditary ones.

Big "macro-level" changes (e.g., between species) are results of the same small "micro-level" changes between individuals.

Four major evolutionary forces

• mutation
• random genetic drift – ∃! animal, the Atlantic eel, which has an effectively infinite population size, because they all come together to one place annually to mate.
• natural selection
• gene flow

The study of gene changes in populations is population genetics.

• what influences mutant allele over time
• how is genetic variability maintained
• probability of going to fixation
• how fast will replacement take place
• influence of chance effects on molecular genetic change

(see Terms)

locus, allele, allele frequencies, genotype, phenotype, discrete trait, continuous or quantitative trait, homozygous vs. heterozygous, genotypic vs. phenotypic ratios, dominant vs. recessive allele, evolution, natural selection, fitness (w)

• BB is homozygous
• Bb is heterozygous

Mendel bred purple × purple plants and got both purple and white plants. Specifically 705 purple and 224 white.

Diploid parent will produce single-ploid gametes (else there would be a combinatorial explosion in the ploidy of the offspring).

A Punnett Square

• phenotypic ratio of above is
  • 3 purple
  • 1 white
• genotypic ratio of above is
  • 1 BB
  • 2 Bb
  • 1 bb

Problem

• 1000 peppered moths in Manchester
• dark melanic form of allele is dominant (M)
• ancestral is recessive (m)
• 825 melanic
• 175 peppered
• 512 of melanic are heterozygous

Some calculations

• Phenotypic ratio is 875/175 or 4.7 melanic to peppered
• Genotypic ratio is 313 MM, 512 Mm, 175 mm or 1.78 : 2.93 : 1
• Allele frequencies of M and m 616 + 512 / 2000 = 0.569 (2 * 175) + 512 / 2000 = 0.431

Allele frequencies are changed by

• selection
• drift
• migration

2 Mathematical approaches

deterministic

(analytic) can predict changes unambiguously. The first of these was "Harvey Weinberg".

stochastic

probabilistic, associates probability distributions with environmental conditions

Deterministic assumptions

• infinite population size
• constant environment

Needed for Darwinian selection (influenced by Menthusian principles)

• variation
• environmental limit to population size (carry capacity)
• differential reproduction (because of the above)

     +------------ mutation ------------------+
     |                  |                     |
     |                  |                     |
     |                  |                     |
deleterious          neutral            advantageous
     |                  |                     |
     |                  |                     |
     |                  |                     |
     |                  |                     |
purifying            chance             positive selection
selection            events                   or
                                         overdominant     
                                          selection

Normally selection reduces genetic variation, however "overdominant" selection can increase genetic variation. This is when the heterozygote has the highest fitness.

1 locus, 2 alleles (A_{1} A_{2})

• 3 possibly diploid genotypes
• allelic frequencies are
  • f(A_{1}) = p
  • f(A_{2}) = q
  • p + q = 1

genotypic frequencies

• f(A_{1}) = p^{2}
• f(A_{2}) = 2pq
• p + q = q^{2}

This is the null model.

Back to our problem.

• melanic is dominant and is 87% (could be heterozygotes)
• → 13% is non-melanic
• → q = 0.13
• → q = \sqrt{0.13} = 0.36
• → p = 1 - q = 0.64
• → f(Mm) = 2pq = 2 × 0.36 × 0.64

Graph (frequency of a, by frequency of genotype in population)

set xrange [0:1]
set xlabel 'frequency of A'
plot x * x title 'AA', (1-x) * (1-x) title 'aa', 2 * x * (1-x) title 'Aa'

change in frequency

Example

• heterozygous individuals have lighter eye spots (increased predation)
• relative fitness of genotypes

  SS	1
  Ss	0.9
  ss	0.6

• p(S) = 0.7

p = 0.7 q = 0.3

Frequencies in the original

Relative fitness

Next generation frequencies

• f(SS)' = p^{2}w_{11} = 0.49 × 1
• f(Ss)' = 2pqw_{12} = 0.42 × 0.9
• f(ss)' = q^{2}w_{22} = 0.09 × 0.6

Next generation's allelic frequencies

• f(SS) × 2 + f(Ss)
• f(ss) × 2 + f(Ss)

• Overdominance is also called heterozygote superiority.
• When the heterozygote has a higher fitness than either homozygote.
• this is one of the few times (along with frequency dependent selection) in which selection increases genetic diversity
• called "balancing" or "stabilizing" selection

The equilibrium frequency (\(\hat{p}\)) is given by

When w11=0.9, w12=1, and w22=0.8 then \(\hat{p} = 0.667\).

\(\bar{w}\) is the average fitness of the entire population.

"find them and grind them" ← experimental identification of overdominant selection. Cavalli-Sforza is prolific in this area.

The wild type allele is more dominance, but it is not a perfect dominance relation.

This is an unstable equilibrium, any deviation from equilibrium will fall away.

This is another instance of a valley which selection can not traverse.

• Changes in allele frequency due solely to chance effects.
• Moral is not to assume that every trait is adaptive.
• Especially important in our currently world of many species with severely reduced population sizes. Note: this will reduce the likelihood that these populations will be able to adapt to climate change.

Stochastic events from ecology have huge effects on small populations.

• alley effect
• catastrophe

;; the above generated with the following
(loop :for pop-size :in '(100000 1000000) :do
   (with-open-file (out (format nil "/tmp/pop~d.data" pop-size) :direction :output)
     (gen-drift out :pop-size pop-size)))

• depiction of the sampling process in populations of finite size
• the distribution of frequencies of gametes is expected to follow a binomial distribution

Process

1. N individuals in P_{0}produce ∞ gametes
2. 2N gametes are selected from the pool of ∞ gametes
3. N individuals in P_{1}

Consider a diploid population of N individuals w/2N genes

when 2N gametes are sampled from the ∞ gamete pool the probability P_{i} of i genes of type A is given by

Make some graphs of a population of a given size with some number of genes, should the frequency of each gene over a number of generations to show the increased effect of chance in smaller populations.

A pot of 100 seeds, 50 round and 50 wrinkled.

Enumerate all possible samples and the related probability of such a sample.

• Probability of four round seeds = \(\frac{4!}{4! 0!} 0.5^{4} 0.5^{0} = 2^{-4}\)
• Probability of three round seeds = \(\frac{4!}{3! 1!} 0.5^{3} 0.5^{1} = 4 \times 2^{-3} \times 2^{-1}\)
• Probability of two of each = \(\frac{4!}{2! 2!} 0.5^{2} 0.5^{2} = 4 \times 2^{-2} \times 2^{-2}\)

N individuals → ∞ gametes → N individuals → ∞ gametes

For an idealized population and makes the following assumptions

• all individuals contribute gametes equally to the next generation
• population size is constant
• non-overlapping generations

N_{e}: The size of an idealized population which would have the same effect of random sampling on gene frequency as that in the actual population.

N: The observed actual population size (census size).

Generally N_{e} << N, or roughly \(\frac{N}{3}\), because of

• Pre- and post-reproductive individuals should not be counted.
• difference in ratio of males to females, in which case \begin{eqnarray*} N_{e} &=& \frac{4N_{m}N_{f}}{N_{m}+N_{f}}, where\\ N_{m} &=& \text{num males}\\ N_{f} &=& \text{num females} \end{eqnarray*}

Also due to long-term variations in pop-size

• environmental catastrophes
• cyclical model of reproduction
• local extinction and re-colonization events

The long term size is the harmonic mean of population sizes (where n is the number of generations). Thus dips in population size affect the long term size more than temporary peaks.

Long bottlenecks have much more of an impact on maintained genetic diversity than short bottlenecks. It is impressive how much variation may be maintained even through very narrow short bottlenecks.

gene substitution

the process whereby a mutant allele completely replace the predominant or wild-type allele in the population

fixation time

time (usually measured in # generations) it takes for a mutant allele to become fixed in a population

fixation probability

the chance that a new mutant allele will reach fixation in a population

rate of gene substitution

number of substitutions or fixations per unit time

• fixation probability determined by
  1. initial frequency (often 1/2N)
  2. selective advantage
  3. the effective population size N_{e} ← very important

Fitness for an allele is equal to a combination of S and N_{e}

• Selection coefficient (S) or (w - S)

  S=0	neutral
  S>0	beneficial
  S<0	deleterious

• in small populations basically everything is neutral

P probability of fixating

neutral

\(p = \frac{1}{2N}\)

small selection coefficients

\(P = \frac{2s}{(1 - e^{-4Ns})}\)

positive values of s and large N

P = 2s

the time required for fixation or loss of a neutral allele depends upon

• initial frequency
• population size N

times shorten as frequency approaches 1 or 0

For a new mutation (Kimura and Ohta 1969) with initial frequency 1/2N, the mean time to fixation is ≈

neutral allele

\(\bar{t} = 4N generations\)

allele with a selective advantage of S

\(\bar{t} = \frac{2}{s}\ln{\left(\frac{2}{N}\right)}\)

Rate of gene substitutions (reaching fixations) per unit time.

neutral mutations

• neutral mutation rate = μ per gene per generation
• number of mutations arising at a locus in a diploid pop of size N = 2Nμ per generation
• but the probability of fixation of each mutation is P = 1/2N

Therefore the substitution rate is

K = (number of mutations)(probability of fixation)

or

substitution rate = mutation rate (thanks Kimura)

• advantageous mutation rate μ per gene per generation
• mutations per locus is 2Nμ per generation
• probability of fixation is P=2s

therefore

K = (num of mutations) (probability of fixation) (selection coefficient)

Rates and patterns of nucleotide substitutions, and molecular clock.

Darwin didn't know the mechanisms of heredity.

Mendel was treated as a crackpot because most people studied bio-metric continuous quantitative traits with normal distributions. So Mendel went back to gardening.

New-Darwinian theory Darwin and Mendel combined.

• mutation ultimate source of genetic variation
• natural selection given the dominant "creative" role in shaping genetic make-up of populations

Selectionism

• natural selection only evolutionary force
• polymorphism must be maintained by balancing selection
• gene substitution must be due to selection of advantageous mutations

• Gelatin pheresis?
• Finding large amounts of genetic variability in natural populations.
• how to explain this in terms of selection
• Fisher promoted eugenics.
• Haldane. 1960s ← awesome wild man

  The selective death that must occur for a gene to be substituted over another allele.

  Thus if all genetic variation was due to selection, this would require more death than the populations could tolerate

• calculated upper limit of 1 gene substitution per 300 generations

read the following papers

• The Cost of Selection Haldane 1954, Journal of Genetics
• Our load of mutations Muller HJ 1950, American Journal of Human Genetics
• Number of alleles that can be maintained in a finite population Kimura and Crow 1964, Genetics 49:725

Beginning of the neutral theory,

The Neutral Theory of Mutation Kimura 1968

most polymorphisms observed at the molecular level are selectively neutral, so that their frequency dynamics in a population are determined by random genetic drift.

Selection can not account for the following, drift can

1. Haldane's cost of selection – rate of change of amino-acids in haemoglobin over time, it was changing too fast to be selective, ~1000 fold over Haldane's limit
2. segregational load, when the most fit genotype is a heterozygote (overdominance). In combination with Mendelian Segregation ensures that in each generation inferior homozygous combinations will be produced. The segregational load is the decrease in population fitness due to these homozygotes. This load would be unbearable if all heterozygotes were selective.

Models the fate of selectively neutral (neutral or nearly neutral) mutations through drift.

1. The probability of fixation of a neutral allele in a population equals its frequency P_{0}.
2. The steady-state rate at which neutral mutations are fixed in a population equals the neutral mutation rate μ. Fixation is new mutations × probability of fixation. \begin{equation*} P_{fixation} = (2N\mu) \times (\frac{1}{2N}) = \mu \end{equation*}
3. Average time between neutral substitutions is \(\frac{1}{\mu}\).
4. New neutral mutations destined to be fixed require on average \(4N_{e}\) generations to come to fixation. Newly arising neutral mutations destined to be lost require on average \(\frac{2N_{e}}{N}\ln{2N}\)
5. If each neutral mutation creates a new universally unique allele, then at equilibrium, the average number of new alleles gained through mutation is exactly offset by the average number lost by genetic drift. At this point the amount of expected homozygosity equals \begin{equation*} \frac{1}{\theta+1}=\frac{1}{4N_{e}\mu + 1} \end{equation*}

   because

   \begin{equation*} \theta=4N_{e}\mu \end{equation*}
   • equilibrium average homozygosity = 1/(1+θ)
   • equilibrium average heterozygosity = θ/(1+θ)

   This is one of the first parameters you'll measure when finding a new population in the wild. Heterozygosity → genetic variation → evolvability. From this you can infer bottlenecks in the evolutionary history of the population.

   Excessive heterozygosity can be an indication that two previously isolated populations have recently mixed.

   More heterozygosity in larger populations.

mechanisms of substitution can be elucidated by comparing substitution rates among

1. genes
2. different DNA regions of genes
3. non-coding regions of the genome

Knowing the rate of nucleotide substitution may also enable us to date evolutionary events (e.g., divergence between species).

How variable is the rate of change between different evolutionary lineages? Is there a universal evolutionary clock? Definitely different for different genes and portions of DNA.

              /--- xxoxoxxxxx
 ancestral   /
xxxxxxxxxx --  total evolutionary time = 2t
 sequence    \
              \--- xxxxxoxxxx
            time t

L

total number of sites (10 here)

\(\hat{D}\)

proportion of sites that differ in the two sequences (3/10 here)

Formulas

• for relatively short intervals of evolutionary time, we can use \(\hat{D}\) (proportion of differences) in estimating the rate of molecular evolution
• over long periods of time the sequences becomes so saturated that multiple hits cannot be identified.
• \(\hat{K}\) takes these multiple hits into account \begin{equation*} \hat{K} = -\ln{(1-\hat{D})} \end{equation*}

  for nucleotides

  \begin{equation*} \hat{k} = -(\frac{3}{4}) ln (1-(\frac{4}{3}\hat{d})) \end{equation*}
• estimated rate of evolution \begin{equation*} \hat{\lambda} = \frac{\hat{K}}{s\hat{t}} \end{equation*}

Write out \hat{\lambda} as amino acid replacement per amino-acid site per year.

• amino acids called replacements
• nucleotides called substitutions

Rates of nucleotide substitution

• Rates of substitution complicated by cases in which different nucleotides mutation to being the same.
• several models of nucleotide substitution, differ primarily in rates of mutation between nucleotides
• simplest model is "Jukes Cantor" (all probabilities are equal)

Jukes-Cantor model

• L = length of sequence
• \hat{d} = proportion of sites that differ in the sequences
• \(\hat{k} = (\frac{3}{4})\ln{(1-4\frac{\hat{d}}{3})}\)
• \(\hat{\lambda} = \frac{\hat{k}}{2\hat{t}}\)

Kimura 2-parameter model

• 4 transitions possible and 8 transversions possible
• in the data despite numbers there were many more transitions than transversions
• updates the Jukes-Cantor model with two transition rates α and β depending on if the mutation is a transition or a transversion

The rate of mutation is surprisingly constant

The notion of the "Molecular Clock" was first attributed to Emile Zuckerkandi and Linus Pauling.

• across different times
• across different species
• across all genes

Used by Kimura as evidence of neutral molecular evolution.

• neutral mutation rate << overall mutation rate
• neutral mutation rate may vary widely across genes. Some genes are strongly conserved (e.g., histones) and some vary quickly (e.g., immunity with diversity at a premium – in fact humans pick mates with different genetic material at immune sites, leads to children with diverse immune system.).
• pattern of hereditary transmission leads to variable rates
  • mitochondrial DNA is very fast in animals
  • mitochondrial DNA is very slow in plants
  • X-linked genes have slow mutation rate than autosomal genes
• substitution rates can also vary across sites within genes

Genes have different neutral mutation rates depending on how well they are conserved.

Find graph of (corrected amino acid changes /100 residues) by millions of years since divergence.

Molecular evolution in the NS genes of influenza virus.

Rates vary widely! e.g., 10^{-9} for some organisms, but as high as 10^{-3} for some RNA viruses (RNA does not have a good proof-reading mechanism – but they don't mind as they evolve out from under our vaccines).

HIV started spreading in the 1920s in Africa caused by colonialism (see the book Tinderbox), had been in rural Africa for a long time, but was constrained by small village sizes.

• it would seem so
• a perfectly driven molecular clock driven by a random process should have a Poisson distribution (variance = mean)
• analysis of substitution rates for different genes for pairs of species violated Poisson expectations
  • perfect clock, variation in rate of ticking = average rate of ticking
  • Poisson, prob of a number of events in a fixed period of time (assumes events occur independently at some average rate)
  • in three different studies the Var >> Mean
• the excess variance in substitution rates has been called an "Episodic Clock", characterized by periods of stasis alternating with periods of rapid substitution

Drosophila adh gene (breaks down alcohol in rotting fruit)

• 3 adh alleles
• differ at 5 positions along 2379 base pairs
• nucleotide polymorphism ps=5/2379
• π calculation:
  • 3 alleles, 3 pairwise comparisons (\(\frac{n \times (n-1)}{2}\))
    • Ja-S(1)
    • Ja-s(2)
    • F1-F

Heterozygosity

• Ja-S(1) vs. Ja-s(2) = 0 ← (should be different: problem w/problem)
• Ja-S(1) vs. F1-F = 5/2379
• Ja-S(2) vs. F1-F = 5/2379
• average per site (or nucleotide diversity in the sample) = π = 10/(3×2379)

divergence

inter-specific, between species

polymorphism

intra-specific, within species

Both driven by the neutral mutation rate. So these should co-vary.

• Neutral Theory predicts both
  1. the rates of substitution over time
  2. the amount of polymorphism within a population
• Both of these are determined by the mutation rate of the gene
• Given this the HKA test detects natural selection
  • two different genetic regions in at least two different species
  • calculation of both within-species variation and between-species divergence
• expected values of above are a function of
  • time t since divergence
  • the relative effective population sizes
  • the θ=4N_{e}μ value for each gene

Two genetic regions relating to gene tb 1

• 5' regulatory non-transcribed NTR region
• coding region of tb 1

     |
 wild| -----              mutation rate along DNA region
     |      \--\          between maize and wild teosinte
     |          \
     |           \-
     |             \
   μ |              \
     |               \
     |                \-
maize| ------------------\
     |                   --------------------
     |
     +---------------------------------------
                      DNA
        non-coding            coding

95% reduction in variation in maize in this NTR region for maze as compared to teosinte (p=0.004).

This is limited to coding region because of the synonymous/non-synonymous bit.

• simpler statistical test to detect natural selection at the sequence level (compared to HKA test)
• Based on the same fundamental principles for non-synonymous (replacement) and synonymous (silent) changes in the coding region of a gene
• if observed substitutions are neutral, then non-synonymous to synonymous ration between species should equal the ratio within species

• 42 silent and 2 replacement changes within D. melanogaster
• 17 silent and 7 replacement changes between D. melanogaster and two other species

differences

• Fixed difference are between species
• polymorphic differences are intra-species

From the total ratio (0.867…) we can calculate the expected replacement/silent for fixed/polymorphism. We can then compare these expected values to the observed using a Χ^{2} test.

Note: The most powerful test for selection.

We have a df=1 with a p<0.01, we reject the null of neutral evolution, and know there is purifying selection within D. melanogaster.

Smallest known genome (non-viral): Bacterium Carsonella ruddii 160,000 base pairs; an endosymbiont, with 182 genes.

~500 minimum genes to be free living

Smallest genome for free-living organism: Bacterium Mycoplasma genitalium – 580,ooo base pairs (probably ~500 genes)

Largest known Genome: Amoeba Amoeba dubia – 670,000,000,000 base pairs

~115 million-fold difference!

find this graph

Smaller genomes fit close to the 100% coding line, then around 10^3 Megabases we start to see the % coding drops off and we have more junk DNA (small RNAs, regulatory stuff etc…).

Genome size is the amount of DNA in the Haploid gene (sperm or egg). This is commonly called the c-value.

• Larger genomes generally measured in pico-grams (pg) of DNA.
• Smaller genomes generally measured in base pairs or kilo-base-pairs

Among bacteria, 20-fold range in number of genes and 20-fold range in genome size, do not appear to contain large quantities of nongenetic DNA. Three fractions

chromosomal

protein coding 70-80%, spacers RNA 20-30%, RNA 1%

plasmid-derived

extra chromosomal and genes

transposable

common components, insertion sequences (more junky)

Explanation of distribution in genome size in bacteria

• major and minor peaks in the genome size of bacteria, these relate to gene duplication, and they all appear to have happened at roughly the same time (around the time when the atmosphere switched largely to oxygen)
• these genome duplication's occurred independently in many lineages
• small scale insertions and deletions
• duplicative transposition
• horizontal transfer of genes, derived mainly from plasmids
• loss of massive chunks of DNA in many parasitic lines

some (birds) have little variability, others (e.g., amoebas) have huge ranges (e.g., ~20,000 fold).

no real correlation between size and complexity

even some sister species have huge differences in genome sizes

• 20-fold range in number of genes
• 80,000-fold range in the size of the genome

non-genetic DNA is left of the sole culprit in the C-value paradox

most growth through duplication

• partial gene
• whole gene
• single chromosome
• whole genome

• Genes duplicate with great frequency
• it is possible to study genes using population models
  1. get a gene
  2. find all instances of it in a set of DNA
  3. use k_{S} to compare the relative ages and times since duplication
  4. use cohort analysis to find the typical ages of the genes

The duplication of genes is a primary driver of genetic evolution!

• unequal crossing-over
• retro-transposition, a transposon moves and takes a gene with it
• fragment capture by double strand break

                               2 regulatory    coding    
                              --====---===---============--
                           
                                        | duplication
                                        v
                           
                              --xxxx---===---============--
                              --====---===---============--
                           
                                new     | degeneration
                                 |      v
                                 v
                              --xxxx---===---============--
                              --====---===---============--

                                     complementation

--xxxx---===---============-- --====---===---============-- --====--knocked out========--
--====---xxx---============-- --====---YYY---============-- --====---===---============--
    sub-functionalization          neo-functionalization        non-functionalization
   (both needed to retain          (a new function after          (most common, one 
    ancestoral function)              mutation)                     is just worthless)

after duplication one element is free to adapt to a new purpose without losing the functionality of the original

Ohno 1970

1. complete gene duplication, regulatory and coding portions. one of these copies will stay under selection, the other will be free to drift
2. either neo-functionalization or non-functionalization, often that later occasionally the former
   • point mutations
   • short insertions and deletions

by Opsin gene duplication in Old World Primates

• gene duplication leading to red and blue ~500 Million years ago (using K_{s})
• splits (some monkeys still only have red and blue)
• in our line there is a red duplication, one of these duplicates evolves into a green sensor (15/348 sites have amino acid substitution)

• Arctic fish evolved their own antifreeze proteins, which bind to tiny ice crystals and prevent them from forming larger pieces of ice
• this novel anti-freeze gene was a partial duplication of a pancreatic (digestive) enzyme
• the small first exon is duplicated along with the first intron, and half of the second exon. 9 base pairs straddling the beginning of the second exon, are duplicated again and again and again forming a very large exon. This is "internal amplification".

Radically new functions often the product of moving around large chunks of a gene not through point mutations.

• TPA is derived from four genes
• TPA is a unique protein not found elsewhere, it protects red blood cells by breaking down blood clots and allows signaling between organs/tissues

Many exons code for functionality independent "domains" of a protein.

The TPA protein has 4 exons coming from 3 different genes.

This process is called "exon shuffling".

• evolution of hermaphrodism in 2 of many species of little worms
• parsimonious solution would be that it only evolved once, actual solution is convergent evolution

This depends upon both the duration of inbreeding and the closeness of the inbreeding pairs.

levels of closeness

1. self-fertilization
2. full-sib mating
3. double first-cousin mating

   1 | ---\-----
     |  \\-     \-----
     |  |  \          \-----     3
heter|   \  \               \----
ozygo|   |   \                   \-----
sity |    \   \-                       \-----
     |     \    \                            \--
     |     |     ----    2
     |      \        \-------
     |      |   1            \------
     |       ---                    \---
     |          \----
   0 +---------------\-------------------------
           5           10          15         20
                  Generations

full self'ing will lose heterozygosity by half every generation

• Inbreeding does not alter allele frequency, p/q stays the same.
• Inbreeding does push heterozygotes to homozygotes
  • relative to Hardy Weinberg frequencies
  • F = inbreeding coefficient
  • H_{i} = (2pq - 2pqF) = prob gene in an inbred individual I is heterozygous
  • H_{s} = proportion of heterozygous genotypes expected in random mating
  • F_{is} = proportion reduction in heterozygotes due to inbreeding relative to the subpopulation as a whole \begin{equation*} F_{is} = \frac{H_{s} - H_{i}}{H_{s}} = \frac{2pq - (2pq - 2pqF)}{2pq} = F \end{equation*}
• Inbreeding with natural selection can change allele frequencies through "Inbreeding Depression"
  • inbreeding depression has been shown for a long time in both laboratory and domesticated populations
  • most populations carry deleterious recessive alleles at low frequency at many loci, many of these are recessive and heterozygous
  • inbreeding makes many of these harmful recessive alleles homozygous, exposing their harmful effects to natural selection
• Inbreeding can increase "Linkage Disequilibrium"
  • lets take loci 1 and 2, with alleles Aa and Bb respectively.
  • linkage disequilibrium exists if certain combinations (say AB and ab) are represented in gametes more frequently than by chance and other (say Ab and aB) are less than by chance.

• organisms rarely exist as a single panmictic population in nature
  • naturally aggregate in flocks, herds, colonies, schools, prides
• genetic differentiation is inevitable in these cases
  • independent aggregate effects of mutation, natural selection and random drift
  • this can lead to speciation
• gene flow among populations via migration slows this divergence

In population genetics migration is often treated in the same way as mutation. Just a new allele in the population.

the reduction in heterozygosity in a population caused by subpopulation structure

• sometimes subdivision and isolation in nature are not obvious (e.g., D pseudoobscura and D. persimilis overlap in distribution)
  • they are subpopulations with limited gene-flow between them
  • moral: don't trust morphology to differentiate species, use genetics

nested subpopulations

stream < river < watershed < continent

F statistics

Results

• F_{SR} indicates there is a 10% reduction in heterozygosity due to subpopulations
• F_{RT} indicates there is more of a contribution to genetic differentiation by regional seperation than there is by subpopulation differentiation
• F_{ST} indicates there is a very great genetic variation due to sub-populations as compared to total pan-mixing

• quantify the inbreeding effect of population subdivision – the reduction in heterozygosity expected with random mating
• allows for an objective comparison of the effect of population structure among different orgamisms

Statistics

• F_{SR} fixation index of the subpopulations relative to the regional aggregates \begin{equation*} F_{SR} = \frac{H_{R} - H_{S}}{H_{R}} \end{equation*}
• F_{RT} fixation index of the regional level relative to the total heterozygosity \begin{equation*} F_{RT} = \frac{H_{T} - H_{R}}{H_{T}} \end{equation*}
• F_{ST} fixation index of the subpopulation relative to the total \begin{equation*} F_{ST} = \frac{H_{T} - H_{S}}{H_{T}} \end{equation*}

values and interpretation

F_{ST} = 0

no genetic divergence, panmictic

F_{ST} = 1

impossible in practice, maximum possible genetic divergence

F_{ST} = 0 to 0.05

little genetic differetiation

F_{ST} = 0.05-0.15

moderate genetic differentiation

F_{ST} = 0.15 to 0.25

strong genetic differentiation

F_{ST} > 0.25

very great genetic differentiation

• when two populations are genetically different gene flow will reduce that genetic differentiation
• goal is to quantify the this reduction in population differentiation
• requires both physical migration and breeding
• symbolized by "m" which is similar to μ (mutation), these are treated very similarly
• m = proportion of foreign alleles in a population which arrived in a given generation

Continent population size >> island population size

• consider a gene with two alleles, A and a, with respective frequencies p* and q* on the mainland and p and q on the island
• if p_{t} is the frequency of A in the island in generation t then \begin{equation*} p_{t} = p* + (1 - m)^{t}(p_{0} - p*) \end{equation*}
• thus with one-way migration, the allele frequency of A on the island subpopulation gradually approaches that of the mainland population
  • when t becomes large, p_{t} goes to p*

the continent island model of gene flow also applies to some wildlife management scenarios (e.g., continually moving fish from a hatchery to a river).

Example

• 0.8 allele frequency on mainland
• 0.2 allele frequency on island
• how many generations are required for the island population to achieve an allele frequency of 0.5
• migration rate of 0.01

• P* = 0.8
• P_{0} = 0.2
• m = 0.01

The Island Model describes a large number of islands each randomly exchanging individuals (genes) with all other islands.

• m is proportion of alleles on any island arriving from other islands
• the islands differ in allel frequency with an average allele frequency across all island of \bar{P}
• considering a single island, all of the other islands can be viewed as a single continent of frequency \bar{P}
• thus the frequency will converge to \bar{P} on all islands
• if P_{t} is the frequency on a given island at time t, then \begin{equation*} P_{t} = (m)(\bar{P}) + (1 - m) (P_{t-1}) \end{equation*}

  or

  \begin{equation*} P_{t} = \bar{P} + (1 - m)^{t}(P_{0}-\bar{P}) \end{equation*}

Example

• two populations with freqs of 0.8 and 0.2
• they exchange with each other at a rate of 10%
• what is the allele frequency of A in the two populations after 10 generations

answer is roughly 0.395 for pop with initial freq 0.2

(island-convergence '(0.8 0.2) .1 10)

The stepping stone model models isolation by distance. Any number of dimensions, e.g., at 2D you get a grid connected to your immediate neighbors along cardinal directions.

A variation of the two-dimensional model assumes that individuals are continuously distributed over a large geographic area and

• probability of two individuals mating decreases with distance
• individuals chosen from locations close together are likely to be more genetically similar

• opposing effects of genetic drift and gene flow/migration
• how much gene flow is required to prevent gene flow
  • first studied by Wright
  • infinite island model w/o selection or mutation
  • average allele frequency is \(\bar{P}\)
  • distribution of allele frequencies across all islands depends on the product of two factors
    1. effective population size N_{e}
    2. migration rate m
  • N_{e}m = Number of immigrant individuals in an island in each generation (this is just like θ)

Wright showed that

• if N_{e}m is large enough you get a humped distribution around \bar{P}, and gene flow dominates genetic drift
• if N_{e}m is small the distribution is U-shaped (bimodal), alleles approach fixation or extinction on most islands and drift dominates

Two magic numbers:

• 5 individuals per generation is enough to ensure subpopulations are identical
  • this is the case across all population sizes
  • this has less effect on larger populations, but they are also less vulnerable to drift
• 1 individual per generation is enough to prevent substantial drift

• equilibrium Heterozygosity \begin{equation*} \bar{H} \approx \frac{4N_{e}\mu}{4N_{e}\mu + 1} \end{equation*}
• migration is genetically equivalent to mutation, replacing μ with m we get the following \begin{equation*} \bar{H} \approx \frac{4N_{e}m}{4N_{e}m + 1} \end{equation*}
• conversely equilibrium homozygosity in the infinite-islands model of gene flow is given as \begin{equation*} \bar{F}_{st} \approx \frac{1}{4N_{e}m + 1} \end{equation*}

• if natural selection is acting differently on different sub-populations, how much migration is required to counteract natural selection
• just like mutation-selection balances

Haldane 1930, Wright 1931

Single locus, two allele

• a is the deleterious recessive allele
• fitness of various genotypes denoted as

  genotype	AA	Aa	aa
  fitness	1	1-hs	1-s

• s = selection coefficient against aa
• h = degree of dominance of a
  • h = 0 → a allele is completely recessive
  • h = 0.5 → additive effects of A and a
• m_{i} rate of in migration
• m_{o} rate of out migration

Example

• Nachman et al.
• Rock-pocket mice in southern Arizona
• dark and light rock; dark and light rock
• massive selection for melanic (dark) coat on volcanic (dark) rock
• Gene Mc1r encodes the melanocortin-recepter protein
  • DD and Dd have melanic coats
  • dd has a non-melanic (light) coat

Mice on a lava flow in New Mexico

• s = 0.29
• h = 0
• m_{i} = 0.4
• m_{0} = 0
• 40 DD, 5 Dd, 5 dd
  • p = 85%
  • q = 15%
• q^{*} = 1

(+ (/ (* -1 0.29 0.85 0.15 0.15)
      (- 1 (* 0.29 0.15 0.15)))
   (* 0.4 1)
   (* -1 0))

(+ change 0.15)

      |
      |                /-O
      |             /--
      |  X---------------X
trait |        /-
      |     /--
      |  O--        X is one genotype
      |             O is a different genotype
      |
      +----------------------
       env. 1       env. 2

ANOVA approach for G × E

significant genotype

overall additive genetic variation

significant environment

overall additive phenotypic plasticity

significant G × E

additive genetic variation in plasticity; plasticity can evolve

Graph things out when doing ANOVAs.

(Bradshaw 1965 – clarified thinking of plasticity)

plasticity can only be applied to particular combinations of traits and environments, not to whole genomes

There is a great deal of non-adaptive plasticity (e.g., shorter from worse diet)

One fun way to test, is to mismatch plastic versions of phenotypes in the wrong environment. This can determine if the particular phenotype is actually adaptive to the particular environment in which it is expressed.

Philosophically, it seems like the notion of plasticity requires a distinction between "being" and "action" of a phenotype.

some examples

• plants which react to caterpillar saliva with caterpillar defenses
• plants which notice the level of shade through red:far-red ratios, and then adjust their growth (e.g., up vs. out) appropriately

genes which control the "trait" of plasticity

reasons everything isn't plastic (see Weinig et al. 2006)

Only makes sense when a constant gene ↔ environment mapping is not possible.

Q_{ST} and G_{ST} are both analogs of F_{ST}, where

G_{ST}

differentiation due to drift in neutral traits

Q_{ST}

differentiation of quantitative traits

When Q_{ST} > G_{ST} then there is evidence of selection of sub-populations to their particular environments.

• selection differential (selective differential), diff in mean between selected and pop mean
• response to selection (r), movement of population mean in one generation
• generally we'll see a decrease in variance when using this "truncation selection"
• The breeders equation \begin{equation*} R = h^{2}S \end{equation*}
  • response in a trait is equal to the
  • heritability of that trait, times the
  • selection differential
  • h^{2}=v_{A}/v_{P}
• longest continually running artificial selection experiment U of Illinois Corn Selection Lines (started 1896)
• "cumulative selection differential" is the sum of the selection differentials (so you can have a metric of how far you've pushed the population)

The heritability is equal to the cumulative response / cumulative selection differential.

V_{x}

population variance of a trait

V_{p}

is the phenotypic variance for a trait

(variance (mapcar #'trait population))

this can be partitioned into

\begin{equation*} V_{P} = V_{G} + V_{E} \end{equation*}

V_{E}

Environmental variance

V_{G}

Genotypic variance, in non-clonal species we get \begin{equation*} V_{G} = V_{A} + V_{D} + V_{I} \end{equation*}

V_{A}

additive genetic variance

V_{D}

dominance variance

V_{I}

interaction or epistatic variance

The best way to predict breeding values is to use the "Best Linear Unbiased Prediction".

The (2×) is because the male only contributes half of the alleles to the children.

broad-sense heritability

is based on the genotypic variance, it includes the effects of dominance and epistatic variance and is most useful in clonal or highly self-fertilizing species \begin{equation*} h^{2}_{B} = \frac{V_{G}}{V_{P}} \end{equation*}

narrow-sense heritability

the proportion of total phenotypic variance determined by the additive genetic variance. This is more useful for outbreeding species \begin{equation*} h^{2}_{N} = \frac{V_{A}}{V_{P}} \end{equation*}

• should have 30 to 50 monogamously mated pairs

Linear regression on a plot of offspring values by parent midpoint values.

Steps

1. build a linkage map
   1. cross divergent parents
   2. find a number of neutral markers in each parent
   3. frequency of recombinant gametes (with markers from both parents) tells you how far apart these markers are on the chromosome – if very few, then these are close together, if many than they're far apart or on separate chromosomes
   4. map distance expressed in centiMorgans where 1cM - recomb-rate of 0.01
   5. linkage groups are groups of loci which are connected by recombinant rates under 0.5 meaning they are all connected
2. (slide 8)
   1. breed divergent parents → F_{1}
   2. follow on to F_{2} and see the outbreak of variation
   3. genotype all of these various individuals to find their neutral markers
   4. if we find significant variation in the trait across some of these neutral markers, then that marker's linkage group is "linked" with the genetic "cause" of the trait – this linkage from the neutral marker to the trait indicates the "strength" or "closeness" of the link

a QTL mapping program returns a map between markers and traits

• returns series of \(LOD=log_{10}{(\frac{L_{1}}{L_{2}})}\) scores
• can plot these LOD views along the length of the chromosome (see slide 14)

A hybrid sunflower is 14× as salt tolerant as its parent species.

Used QTL to identify areas related to survivorship and to mineral uptake. There was overlap indicating either pleiotropy or that these traits are related.

The "Beavis Effect" rule: one should have at least 500 individuals.

QTL is generally useful when one wants to get an idea of the high level architecture of an organism.

hw/sug.csv

in this file

• row 3 centiMorgans positions

library("qtl")
sug <- read.cross("csv", "http://www.rqtl.org", "sug.csv",
                  genotypes=c("CC", "CB", "BB"), alleles=c("C", "B"))

summary(sug)

These genotype percents are exactly what we want, about 1/2 BB, and quarters of the others

plot(sug)

Actually do some qtl analysis

sug <- calc.genoprob(sug, step=1)
out.em <- scanone(sug)
summary(out.em)

summary(out.em, threshold=3)

plot(out.em)

out.hk <- scanone(sug, method="hk")
plot(out.em, out.hk, col=c("blue", "red"))

Using Haley Knott regression to find what ld score we need for significance.

operm <- scanone(sug, method="hk", n.perm=1000)

We can view the summary of this to see what cutoffs we need for significance.

summary(operm)

If we want to use these thresholds against our prior qtl results, we can do the following.

summary(out.hk, perms=operm, alpha=0.2, pvalues=TRUE)

lodint(out.hk, chr=7)

QTL effects, what fraction of the variance in the blood pressure is explained by these loci.

max(out.hk)
mar <- find.marker(sug, chr=7, pos=47.7)
plotPXG(sug, marker=mar)

effectplot(sug, mname1=mar)

Looking at a different allele (this one is additive).

max(out.hk, chr=15)
mar2 <- find.marker(sug, chr=15, pos=12)
plotPXG(sug, marker=mar2)
effectplot(sug, mname1="15@12")

looking at epistasis

effectplot(sug, mname1="7@47.7", mname2="15@12")
effectplot(sug, mname2="7@47.7", mname1="15@12")

2D qtl analysis

sug <- calc.genoprob(sug, step=2)
out2 <- scantwo(sug, method="hk")
plot(out2)

A form of the breeders equation which can track multiple related traits with their correlations and pick out the effects of selection.

where

Δ Z

column vector of predicted change in traits

G

square matrix of genetic covariance between 2 traits, the diagonal is the additive genetic variance for a particular trait

β

column vector of selection gradient in traits

Where does G come from? You need to breed your organisms, then you get heritability (relation between offspring and parent), and you also get correlation.

The G matrix basically tells you how a species can evolve, what it is capable of doing.

(example from literature Campbell 1996 on slide 15 in lecture notes)

• a good deal of this evolve on the fossil time scales
• referred to as "genetic constraint"

one way to get a feel for what is possible

• look at the correlation between related traits (slide 17)

       |
       |      o
       |        o
  flower     o
       |       o    o
  size |         o    o   o
       |            o   o
       |                      o
       |                   o o
       |
       +-------------------------------
            flower number
  

  The "genetic trajectory" through these graphs shows where evolution could take these organisms.

• do it

maximum parsimony

you can often wind up with say 37 trees which are all equally parsimonious, you can either pick a representative tree, or construct a "consensus tree". this is a specific instance of maximum likelihood which uses unrealistic evolutionary assumptions

distance methods

first you build a distance matrix so you have all of the evo-distance number between pairs of taxa. builds trees which attempt to match all of these distances as closely as possible. Loses information about what characters change and is instead based solely on distance. Super Fast. The current best method is called "Neighbor Joining" (ninja in mesquite).

maximum likelihood

What set of model parameters m maximizes the likelihood of the observations x. \begin{equation*} L = P(x|m) \end{equation*}

Bayesian

Bayes Theorem \begin{equation*} P(m|x) = p(x|m) \frac{prior(m)}{prior(x)} \end{equation*}

MCMC comparing pairs of Bayesian models

\begin{equation*} \frac{P(M_{1}|x)}{P(M_{2}|x)} = \frac{P(x|M_{1}) prior(M_{1})}{P(x|M_{2}) prior(M_{2})} \end{equation*}

walks through fitness landscape, more time spent in any particular portion of the fitness landscape increases the likelihood of that portion of the landscape being the desirable.

People generally publish both a Bayesian-found tree and a MCMC-found tree. Why no GA?

neighbor joining

1. start phylogeny
2. recursively find the closest pair and join them
3. taking into account the averages distances between A and B and the rest of the tree, you decide where the join node u_{1} will live between A and B (alternately how far A and B are from u_{1})
4. replace A and B in your distance matrix with u_{1}

distance matrix construction

uses percent sequence distance to approximate evolutionary distance

All of these algorithms give you un-rooted trees.

midpoint rooting

for when you just don't know, pick a point roughly equidistant from all leafes

outgroup rooting

this is what people do – we know before-hand that some node is sister to the rest of the taxa, we then include this group so that it can be the root of our final tree

Can't do this when you're doing the whole tree of life.

bootstrap

resample alignment columns many times w/replacement, build a tree w/each re-sampled data matrix. Examine the resulting collection of trees.

jackknife

is bootstrapping w/o replacement

similarly with Bayesian and MCMC you end up with piles of trees.

This pile of trees can be used to

• break out clade confidence (% of trees)
• consider a clade well supported when it appears in
  • >70% of bootstrap or jackknife trees
  • >95% of Bayesian trees

Consensus tree which only reflects those arrangements which appear in 100% of your conflicting trees.