Varriation Within and Between Species

Introduction to

Bioinformatics

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Introduction to Bioinformatics.
LECTURE 5: Variation within and between
species

* Chapter 5: Are Neanderthals among us?

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Neandertal, Germany, 1856
Initial interpretations:

* bear skull
* pathological idiot
* Old Dutchman ...

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Introduction to Bioinformatics
LECTURE 5: INTER- AND INTRASPECIES VARIATION

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5.1 Variation in DNA sequences

* Even closely related individuals differ in genetic sequences

* (point) mutations : copy error at certain location

* Sexual reproduction – diploid genome

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5.1 VARIATION IN DNA SEQUENCES

Diploid chromosomes

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Mitosis: diploid reproduction

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Meiosis: diploid (=double) → haploid (=single)

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* typing error rate very good typist: 1 error / 1K typed letters

* all our diploid cells constantly reproduce 7 billion letters

* typical cell copying error rate is ~ 1 error /1 Gbp

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GERM LINE
Reverse time and follow your cells:

• Now you count ~ 1013 cells
• One generation ago you had 2 cells ‘somewhere’ in your parents body
• Small T generations ago you had (2T – multiple ancestors) cells
• Large T generations ago you counted #(fertile ancestors) cells
• Congratulations: you are 3.4 billion years old !!!

Fast-forward time and follow your cells:
• Only a few cells in your reproductive organs have a chance to live on
in the next generations

• The rest (including you) will die … 12


GERM LINE MUTATIONS
This potentially immortal lineage of (germ) cells is
called the GERM LINE

All mutations that we have accumulated are en route on
the germ line

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* Polymorphism : multiple possibilities for a nucleotide: allelle

* Single Nucleotide Polymorphism – SNP (“snip”) point mutation
example: AAATAAA vs AAACAAA

* Humans: SNP = 1/1500 bases = 0.067%

* STR = Short Tandem Repeats (microsatelites)
example: CACACACACACACACACA …

* Transition - transversion

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Purines – Pyrimidines

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Transitions – Transversions

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5.2 Mitochondrial DNA

* mitochondriae are inherited only via the maternal line!!!

* Very suitable for comparing evolution, not reshuffled

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5.2 MITOCHONDRIAL DNA

H.sapiens mitochondrion 18


EM photograph of H. Sapiens mtDNA
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5.3 Variation between species

* genetic variation accounts for morphological-
physiological-behavioral variation

* Genetic variation (c.q. distance) relates to phylogenetic
relation (=relationship)

* Necessity to measure distances between sequences: a
metric

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5.3 VARIATION BETWEEN SPECIES

Substitution rate
* Mutations originate in single individuals

* Mutations can become fixed in a population

* Mutation rate: rate at which new mutations arise

* Substitution rate: rate at which a species fixes new mutations

* For neutral mutations

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5.3 VARIATION BETWEEN SPECIES

Substitution rate and mutation rate
* For neutral mutations

* ρ = 2Nμ*1/(2N) = μ

* ρ = K/(2T)

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5.4 Estimating genetic distance

* Substitutions are independent (?)
* Substitutions are random
* Multiple substitutions may occur
* Back-mutations mutate a nucleotide back to an earlier value

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5.4 ESTIMATING GENETIC DISTANCE

Multiple substitutions and Back-mutations
conceal the real genetic distance

GACTGATCCACCTCTGATCCTTTGGAACTGATCGT
TTCTGATCCACCTCTGATCCTTTGGAACTGATCGT
TTCTGATCCACCTCTGATCCATCGGAACTGATCGT
GTCTGATCCACCTCTGATCCATTGGAACTGATCGT
observed : 2 (= d)
actual : 4 (= K)

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* Saturation: on average one substitution per site

* Two random sequences of equal length will match
for approximately ¼ of their sites

* In saturation therefore the proportional genetic
distance is ¼

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* True genetic distance (proportion): K

* Observed proportion of differences: d

* Due to back-mutations K ≥ d

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SEQUENCE EVOLUTION is a Markov process: a
sequence at generation (= time) t depends only the
sequence at generation t-1

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The Jukes-Cantor model
Correction for multiple substitutions

Substitution probability per site per second is α

Substitution means there are 3 possible replacements
(e.g. C → {A,G,T})

Non-substitution means there is 1 possibility
(e.g. C → C)

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5.4 THE JUKES-CANTOR MODEL

Therefore, the one-step Markov process has the following
transition matrix:

A C G T
A 1-α α/3 α/3 α/3
C α/3 1-α α/3 α/3
MJC =
G α/3 α/3 1-α α/3
T α/3 α/3 α/3 1-α

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After t generations the substitution probability is:

M(t) = MJCt

Eigen-values and eigen-vectors of M(t):

λ1 = 1, (multiplicity 1): v1 = 1/4 (1 1 1 1)T

λ2..4 = 1-4α/3, (multiplicity 3): v2 = 1/4 (-1 -1 1 1)T
v3 = 1/4 (-1 -1 -1 1)T
v4 = 1/4 (1 -1 1 -1)T

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Spectral decomposition of M(t):

MJCt = ∑i λitviviT

Define M(t) as:
r(t) s(t) s(t) s(t)
s(t) r(t) s(t) s(t)
MJCt = s(t) s(t) r(t) s(t)
s(t) s(t) s(t) r(t)

Therefore, substitution probability s(t) per site after t
generations is:

s(t) = ¼ - ¼ (1 - 4α/3)t 32


substitution probability s(t) per site after t generations:

s(t) = ¼ - ¼ (1 - 4α/3)t
observed genetic distance d after t generations ≈ s(t) :

d = ¼ - ¼ (1 - 4α/3)t
For small α : 3
t≈− ln (1 − 4 d )
4α 3

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For small α the observed genetic distance is:
3
t≈− ln (1 − 4 d )
4α 3

The actual genetic distance is (of course):
K = αt

So:
K ≈ − 3 ln (1 − 4 d )
4 3

This is the Jukes-Cantor formula : independent of α and t.

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The Jukes-Cantor formula : K ≈ − 3 ln (1 − 4 d )
4 3

For small d using ln(1+x) ≈ x : K≈d
So: actual distance ≈ observed distance

For saturation: d ↑ ¾ : K →∞
So: if observed distance corresponds to random sequence-
distance then the actual distance becomes indeterminate

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Variance in K
2
 ∂K   ∂K 
If: K = f(d) then: 2δK =  δd ⇒ δK 2 =   δd
2

 ∂K   ∂d   ∂d 
So: Var ( K ) =  ∂d  Var(d )
 

Generation of a sequence of length n with substitution rate
n k
d is a binomial process: Prob(k ) =  d (1 − d ) n − k
k 
 
and therefore with variance: Var(d) = d(1-d)/n
∂K 1
Because of the Jukes-Cantor formula: =
∂d 1 − 4 d
3

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Variance in K

Variance: Var(d) = d(1-d)/n

∂K 1
Jukes-Cantor: =
∂d 1 − 4 d
3

So: d (1 − d )
Var ( K ) ≈
n(1 − 4 d ) 2
3

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EXAMPLE 5.4 on page 90

* Create artificial data with n = 1000: generate K* mutations
* Count d
* With Jukes-Cantor relation reconstruct estimate K(d)
* Plot K(d) – K*

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5.4 EXAMPLE 5.4 on page 90

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5.4 EXAMPLE 5.4 on page 90 (= FIG 5.3)

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The Kimura 2-parameter model
Include substitution bias in correction factor

Transition probability (G↔A and T↔C) per site per second
is α

Transversion probability (G↔T, G↔C, A↔T, and A↔C)
per site per second is β

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5.4 THE KIMURA 2-PARAM MODEL

The one-step Markov process substitution matrix
now becomes:

A C G T
A 1-α-β β α β
MK2P = C β 1-α-β β α
G α β 1-α-β β
T β α β 1-α-β

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After t generations the substitution probability is:

M(t) = MK2Pt

Determine of M(t):

eigen-values {λi}

and eigen-vectors {vi}

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Spectral decomposition of M(t):

MK2Pt = ∑i λitviviT

Determine fraction of transitions per site after t
generations : P(t)

Determine fraction of transitions per site after t
generations : Q(t)

Genetic distance: K ≈ - ½ ln(1-2P-Q) – ¼ ln(1 – 2Q)

Fraction of substitutions d = P + Q → Jukes-Cantor
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Other models for nucleotide evolution
* Different types of transitions/transversions

* Pairwise substitutions GTR (= General Time Reversible) model

* Amino-acid substitutions matrices

*…

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Other models for nucleotide evolution
DEFICIT:

all above models assume symmetric substitution probs;

prob(A→T) = prob(T→A)

Now strong evidence that this assumption is not true

Challenge: incorporate this in a self-consistent model

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5.5 CASE STUDY: Neanderthals

* mtDNA of 206 H. sapiens from different regions

* Fragments of mtDNA of 2 H. neanderthaliensis, including
the original 1856 specimen.

* all 208 samples from GenBank

* A homologous sequence of 800 bp of the HVR could be
found in all 208 specimen.

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* Pairwise genetic difference – corrected with Jukes-Cantor
formula

* d(i,j) is JC-corrected genetic difference between pair (i,j);

* dT = d

* MDS (Multi Dimensional Scaling): translate distance table
d to a nD-map X, here 2D-map

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distance map d(i,j)

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MDS

ted
se para
well- H. neanderthaliensis

H. sapiens

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phylogentic tree

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END of LECTURE 5

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