Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process Federico Squartini and Peter. F . Arndt Max Planck Institute for Molecular Genetics Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.1
We will talk about disequilibrium and irreversibility. . . Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.2
Markovian Sequence Evolution Nucleotide substitution models: i.i.d Markov models of evolution, i.e. a master equation: ∂ � ∂tρ β ( t ) = Q βα ρ α ( t ) α, β ∈ { A , G , C , T } α A T G C Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.3
Markovian Sequence Evolution Nucleotide substitution models: i.i.d Markov models of evolution, i.e. a master equation: ∂ � ∂tρ β ( t ) = Q βα ρ α ( t ) α, β ∈ { A , G , C , T } α A C G T · Q AC Q AG Q AT A · Q CA Q CG Q CT C Q = . · Q GA Q GC Q GT G · Q TA Q TC Q TG T Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.3
Markovian Sequence Evolution Nucleotide substitution models: i.i.d Markov models of evolution, i.e. a master equation: ∂ � ∂tρ β ( t ) = Q βα ρ α ( t ) α, β ∈ { A , G , C , T } α The solution to this equation, with initial condition ρ 0 , is: � � e Qt ρ 0 ρ β ( t ) = β P ( t ) = e Qt Such a model is not complete. . . Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.3
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Widely used models: T C A G · µ µ µ T · µ µ µ C Q = . · µ µ µ A · µ µ µ G Jukes-Cantor Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Widely used models: T C A G · α β β T · α β β C Q = . · β β α A · β β α G Kimura Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Widely used models: T C A G · π T π T π T T · π C π C π C C Q = . · π A π A π A A · π G π G π G G Felsenstein Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Widely used models: T C A G · kπ T π T π T T · kπ C π C π C C Q = . · π A π A kπ A A · π G π G kπ G G Hasegawa-Kishino-Yano Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Choosing Parameters Specifying an evolutionary mode ⇒ postulating a form for the rate matrix: A C G T 0 1 Q AC Q AG Q AT A · Q CA Q CG Q CT C B C · Q = A . B C B C Q GA Q GC Q GT G · @ Q TA Q TC Q TG T · Widely used models: T C A G · k 1 π T π T π T T · k 1 π C π C π C C Q = . · π A π A k 2 π A A · π G π G k 2 π G G Tamura-Nei Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.4
Two Evolutionary Models All preceding models are nested into the following: A G T C · aπ A bπ A cπ A A · aπ G dπ G eπ G G Q GTR = . · bπ T dπ T fπ T T cπ C eπ C fπ C · C A possible alternative: A C G T · r AC r AG r AT A · r GT r CG r CT C Q RCS = . · r CT r CG r GT G · r AT r AG r AC T Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.5
Two Evolutionary Models - 2 0 No 4 Assumption 2 3 1 sister1 outgroup sister2 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.6
Two Evolutionary Models - 2 0 Equilibrium + 4 Time Rev. 2 3 1 sister1 outgroup sister2 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.6
Two Evolutionary Models - 2 0 4 No selection 2 3 1 sister1 outgroup sister2 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.6
Estimating Parameters α i of nucleotide sequences from 3 species, For a given triple alignment � the likelihood of the alignment is: N � � ρ 0 α 0 [ P 30 ] α 3 k α 0 [ P 40 ] α 4 α 0 [ P 24 ] α 2 k α 4 [ P 14 ] α 1 L = k α 4 k =1 α 0 ,α 4 ∈{ A , C , G , T } The vector ρ 0 represents the ancestral nucleotide distribution at the root node. 0 one Q for 4 each branch? 2 3 1 sister1 outgroup sister2 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.7
Equilibrium Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.8
The stationarity index The equilibrium distribution of a Markov process is defined by: Qπ = 0 Just taking the difference between present and stationary distribution: ∆ α = ρ α − π α And rearrange the terms: = ∆ C + ∆ G = ρ GC − π GC STI 1 = ∆ A − ∆ T STI 2 = ∆ C − ∆ G , STI 3 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.9
The STI - Reverse complement symmetry Substituting the equilibrium distribution: (1 − π CG , π CG , π CG , 1 − π CG ) Where: r GT + r CT π CG = r AC + r AG + r GT + r CT For the reverse complement symmetric model the STI has a simple form: = ρ GC − π GC STI 1 = ( ρ A − ρ T ) STI 2 = ( ρ C − ρ G ) . STI 3 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.10
Analysis of the Fly Genome Results about the time reversal properties for the evolution of the fly genome: Alignment of 3 Drosophilas: sechellia, simulans and melanogaster Removed annotated coding regions Rates have been estimated using a maximum likelihood algorithm Sliding window analysis, 50kbp length For each window we have calculated the stationarity index in the simulans lineage D.Melanogaster D.Sechellia D.Simulans Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.11
Analysis of the Fly Genome - Stationarity 0.4 STI1 STI2 STI3 0.3 Freq 0.2 0.1 0.0 −0.1 0.0 0.1 STI Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.12
Reversibility Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.13
Time Reversibility: the Detailed Balance Time reversibility is usually defined in terms of the detailed balance conditions : Q ji π i = Q ij π j From which one can derive the General Time Reversible (GTR) Parameterization: A G T C · aπ A bπ A cπ A A · aπ G dπ G eπ G G Q GTR = · bπ T dπ T fπ T T · cπ C eπ C fπ C C Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.14
Time reversibility: Kolmogorov Cycle Conditions A lesser known formulation of time reversibility: Definition. A Markov process is said to satisfy the Kolmogorov cycle conditions if the following equality on generators holds: Q i 1 i n Q i n i n − 1 . . . Q i 2 i 1 = Q i 1 i 2 . . . Q i n − 1 i n Q i n i 1 ∀ i 1 , . . . , i n ∈ C ( -2) i_1 Q Q i_6 i_2 Q Q i_5 i_3 Q Q i_4 Quantifying the Equilibrium and Irreversibility Properties of the Nucleotide Substitution Process – p.15
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