7. Introduction to Bioinformatics
LECTURE 5: INTER- AND INTRASPECIES VARIATION
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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11. Introduction to Bioinformatics
5.1 VARIATION IN DNA SEQUENCES
* 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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5.1 VARIATION IN DNA SEQUENCES
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
13. Introduction to Bioinformatics
5.1 VARIATION IN DNA SEQUENCES
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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5.1 VARIATION IN DNA SEQUENCES
* 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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LECTURE 5: INTER- AND INTRASPECIES VARIATION
5.2 Mitochondrial DNA
* mitochondriae are inherited only via the maternal line!!!
* Very suitable for comparing evolution, not reshuffled
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LECTURE 5: INTER- AND INTRASPECIES VARIATION
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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22. Introduction to Bioinformatics
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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LECTURE 5: INTER- AND INTRASPECIES VARIATION
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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25. Introduction to Bioinformatics
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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5.4 ESTIMATING GENETIC DISTANCE
* 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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5.4 ESTIMATING GENETIC DISTANCE
* True genetic distance (proportion): K
* Observed proportion of differences: d
* Due to back-mutations K ≥ d
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5.4 ESTIMATING GENETIC DISTANCE
SEQUENCE EVOLUTION is a Markov process: a
sequence at generation (= time) t depends only the
sequence at generation t-1
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29. Introduction to Bioinformatics
5.4 ESTIMATING GENETIC DISTANCE
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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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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
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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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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5.4 THE JUKES-CANTOR MODEL
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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45. Introduction to Bioinformatics
5.4 ESTIMATING GENETIC DISTANCE
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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46. Introduction to Bioinformatics
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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5.4 THE KIMURA 2-PARAM MODEL
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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5.4 THE KIMURA 2-PARAM MODEL
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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49. Introduction to Bioinformatics
5.4 ESTIMATING GENETIC DISTANCE
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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50. Introduction to Bioinformatics
5.4 ESTIMATING GENETIC DISTANCE
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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51. Introduction to Bioinformatics
LECTURE 5: INTER- AND INTRASPECIES VARIATION
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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5.5 CASE STUDY: Neanderthals
* 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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