A Lie algebraic classification ofcontinuous-time Markov models

Jeremy SumnerSchool of Maths and Physics, University of Tasmania

Joint work with Jesus Fernandez-Sanchez and Peter Jarvis

Phylogenetics: New data, new Phylogenetic challengesFollow-up Meeting, June 2011

A Lie algebraic classification ofcontinuous-time Markov models

Jeremy SumnerSchool of Maths and Physics, University of Tasmania

Joint work with Jesus Fernandez-Sanchez and Peter Jarvis

Phylogenetics: New data, new Phylogenetic challengesFollow-up Meeting, June 2011

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.

Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2).

What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average?

Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

Closure of Markov models

Sequence evolution, homogeneous cont-time Markov chain:

Rate-matrix Q with “free parameters” and M = eQt .

Examples: JC, K2ST, K3ST, F81, GTR and GMM.

Model M=eL where L={Q ∈ GMM + extra constraints}.Closure: What if we consider an inhomogeneous process?

M = M1M2 = eQ1t1eQ2t2 = eQ(t1+t2). What is Q???

This is relevant because:

i. Models are being used with different Q’s on each edge.

ii. In truth rates are not constant anyway.

Is Q an average? Does Q ∈ L?

BCH formula: eAeB = eA+B+ 12

[A,B]+ 112

[A,[A,B]]..., with[A,B] := AB − BA.

BCH implies model is closed if Q + Q ′ and [Q,Q ′] ∈ L.

ie. L forms a Lie algebra.

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.

1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.

3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.

4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.

[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)X

What about GTR?

So what models are closed?

Consider Q as a vector Q =∑

i αiLi .

Checking generators is enough: [Li , Lj ] ∈ LX

LJC ={L := Lag + Lac + . . .+ Lct}.1D and commutator is trivial. X

LK3ST ={αLα + βLβ + γLγ |α, β, γ ≥ 0}.3D and “abelian”: [Lx , Ly ] = 0 X

LF81 ={πaRa + πgRg + πcRc + πtRt |πi ≥ 0}.4D and non-abelian: [Ri ,Rj ] = Ri − RjX

LGMM ={∑

i 6=j αijLij |αij ≥ 0}.[Lij , Lkl ] = (Lil − Ljl) (δjk − δjl)− (Lkj − Llj) (δil − δjl)XWhat about GTR?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.

Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.

Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

The “GTR must die” slide.

Under any sensible definition, GTR is not closed.

“Grand-daddy” version:

GTR ={Q ∈ GMM : ∃π and QD(π) = D(π)QT}.Fixed stationary distribution version:

GTRπ={Q ∈ GMM : QD(π) = D(π)QT}.Neither version forms a Lie algebra or gives a closed model. 7

Thus GTR has serious problems of interpretation in moregeneral settings.

What are the alternatives? Can we generate a full list?

Just a matter of finding (stochastic) subalgebras of GMM?

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.

But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition:

A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

What do we mean by a symmetry of a model?

A model has symmetry if when we permute nucleotidessomething doesn’t change.

Consider graphical representation of K3ST.

This graph is invariant underZ2 × Z2

∼= {e, (ac)(gt), (ag)(ct), (at)(gc)}.But statistically, the parameter labels are irrelevant.

Our condition: A model has a symmetry if a permutationleaves its graph invariant up to a corresponding permutationof the parameters.

That is, there is a group homomorphism from nucleotidepermutations to parameter permutations:

ρ : S4 → Sd .

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Using symmetry to find Lie Markov Models.

We can use the symmetry requirement to dramaticallysimplify the “brute force” approach.

Choose a symmetry G < S4.

Want L = 〈L1, L2, . . . , Ld〉C s.t. σ · L = 〈Lσ(1), . . . , Lσ(d)〉C.

⇒ L = a union of G-orbits (as a finite set)

= sum of G-irreps (as a vector space)

Orbit/stablizer thm: Any orbit is isomorphic to G/Gx , Gx < G.

We can thus write out a list of G-orbits.

Decompose GMM into a sum of irreps of G.

Working up in dimension, look for models as union of orbits.

Check for Lie algebra (the stochastic condition is tricky!)

Example: Models with S4 symmetry.

Under S4, GMM= id ⊕ 2{31} ⊕ {22} ⊕ {212}.

Model Dim Orbit Decomp Lie algebraGMM 12 S4/Z2 – You saw it already.

K3ST + F81 6 S4/Z3 id ⊕ {31} ⊕ {22}[L(ij)(kl),Rj

]= Rj − Ri

F81 4 S4/S3 id ⊕ {31} [Ri − Rj ] = Ri − Rj

K3ST 3 S4/ (Z2wrZ2) id ⊕ {22} [Lx , Ly ] = 0

JC 1 S4/S4 id trivial

We are still organizing the list of Lie Markov models for thecase of Z2wrZ2 symmetry...

Example: Models with S4 symmetry.

Under S4, GMM= id ⊕ 2{31} ⊕ {22} ⊕ {212}.

Model Dim Orbit Decomp Lie algebraGMM 12 S4/Z2 – You saw it already.

K3ST + F81 6 S4/Z3 id ⊕ {31} ⊕ {22}[L(ij)(kl),Rj

]= Rj − Ri

F81 4 S4/S3 id ⊕ {31} [Ri − Rj ] = Ri − Rj

K3ST 3 S4/ (Z2wrZ2) id ⊕ {22} [Lx , Ly ] = 0

JC 1 S4/S4 id trivial

We are still organizing the list of Lie Markov models for thecase of Z2wrZ2 symmetry...

Example: Models with S4 symmetry.

Under S4, GMM= id ⊕ 2{31} ⊕ {22} ⊕ {212}.

Model Dim Orbit Decomp Lie algebraGMM 12 S4/Z2 – You saw it already.

K3ST + F81 6 S4/Z3 id ⊕ {31} ⊕ {22}[L(ij)(kl),Rj

]= Rj − Ri

F81 4 S4/S3 id ⊕ {31} [Ri − Rj ] = Ri − Rj

K3ST 3 S4/ (Z2wrZ2) id ⊕ {22} [Lx , Ly ] = 0

JC 1 S4/S4 id trivial

We are still organizing the list of Lie Markov models for thecase of Z2wrZ2 symmetry...

Example: Models with S4 symmetry.

Under S4, GMM= id ⊕ 2{31} ⊕ {22} ⊕ {212}.

Model Dim Orbit Decomp Lie algebraGMM 12 S4/Z2 – You saw it already.

K3ST + F81 6 S4/Z3 id ⊕ {31} ⊕ {22}[L(ij)(kl),Rj

]= Rj − Ri

F81 4 S4/S3 id ⊕ {31} [Ri − Rj ] = Ri − Rj

K3ST 3 S4/ (Z2wrZ2) id ⊕ {22} [Lx , Ly ] = 0

JC 1 S4/S4 id trivial

We are still organizing the list of Lie Markov models for thecase of Z2wrZ2 symmetry...

“Lie Markov models”, Jeremy Sumner, JesusFernandez-Sanchez and Peter Jarvis.arxiv: Just google it.

Thanks and please consider coming to Phylomania2011.

“Lie Markov models”, Jeremy Sumner, JesusFernandez-Sanchez and Peter Jarvis.arxiv: Just google it.

Thanks and please consider coming to Phylomania2011.