Table 2 Conditions required for a 4-state Markovian process to be lumpable (in terms of s and π), and transformations to obtain $\tilde{\pi }$ and $\tilde{s}$ such that the lumpability holds.

{{A, G}, C, T}

s₁₂ = s₂₃

s₁₄ = s₃₄

${\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{23}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{14}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$

{A, G, {C, T}}

s₁₂ = s₁₄

s₂₃ = s₃₄

${\tilde{s}}_{\mathsf{\text{12}}}={\tilde{s}}_{14}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{23}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$

{A, {C, G}, T}

s₁₂ = s₁₃

s₂₄ = s₃₄

${\tilde{s}}_{\mathsf{\text{12}}}={\tilde{s}}_{\mathsf{\text{13}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{13}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{24}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{24}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$

{C, G, {A, T}}

s₁₂ = s₂₄

s₁₃ = s₃₄

${\tilde{s}}_{\mathsf{\text{12}}}={\tilde{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$

{{A, C}, G, T}

s₁₃ = s₂₃

s₁₄ = s₂₄

${\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{23}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{14}}}={\tilde{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$

{A, C, {G, T}}

s₁₃ = s₁₄

s₁₂ = s₁₃

${\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{14}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{2}}$

${\tilde{s}}_{\mathsf{\text{23}}}={\tilde{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$

{A, {C, G, T}}

s₁₂ = s₁₃

s₁₃ = s₁₄

${\tilde{s}}_{\mathsf{\text{12}}}={\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{14}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{3}}$

{C, {A, G, T}}

s₁₂ = s₂₃

s₂₃ = s₂₄

${\tilde{s}}_{\mathsf{\text{12}}}={\tilde{s}}_{\mathsf{\text{23}}}={\tilde{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{3}}$

{G, {A, C, T}}

s₁₃ = s₂₃

s₂₃ = s₃₄

${\tilde{s}}_{\mathsf{\text{13}}}={\tilde{s}}_{\mathsf{\text{23}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{3}}$

{T, {A, C, G}}

s₁₄ = s₂₄

s₂₄ = s₃₄

${\tilde{s}}_{\mathsf{\text{14}}}={\tilde{s}}_{\mathsf{\text{24}}}={\tilde{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{24}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{3}}$

{{A, G}, {C, T}}

s₁₂π₂ + s₁₄π₄ = s₂₃π₂+ s₃₄π₄

s₁₂π₁ + s₂₃π₃ = s₁₄π₁+ s₃₄π₃

${\tilde{s}}_{23}=\frac{{\tilde{s}}_{12}\left({\widehat{\pi }}_2{\widehat{\pi }}_3-{\widehat{\pi }}_1{\widehat{\pi }}_4\right)+{\tilde{s}}_{14}{\widehat{\pi }}_4\left({\widehat{\pi }}_1+{\widehat{\pi }}_3\right)}{{\widehat{\pi }}_3\left({\widehat{\pi }}_2+{\widehat{\pi }}_4\right)}$

${\tilde{s}}_{34}=\frac{{\tilde{s}}_{12}{\widehat{\pi }}_1+{\tilde{s}}_{23}{\widehat{\pi }}_3-{\tilde{s}}_{14}{\widehat{\pi }}_1}{{\widehat{\pi }}_3}$

{{A, T}, {C, G}}

s₁₂π₂ + s₁₃π₃ = s₂₄π₂+ s₃₄π₃

s₁₂π₁ + s₂₄π₄ = s₁₃π₁+ s₃₄π₄

${\tilde{s}}_{13}=\frac{{\tilde{s}}_{12}\left({\widehat{\pi }}_1{\widehat{\pi }}_3-{\widehat{\pi }}_2{\widehat{\pi }}_4\right)+{\tilde{s}}_{24}{\widehat{\pi }}_4\left({\widehat{\pi }}_2+{\widehat{\pi }}_3\right)}{{\widehat{\pi }}_3\left({\widehat{\pi }}_1+{\widehat{\pi }}_4\right)}$

${\tilde{s}}_{34}=\frac{{\tilde{s}}_{12}{\widehat{\pi }}_2+{\tilde{s}}_{13}{\widehat{\pi }}_3-{\tilde{s}}_{24}{\widehat{\pi }}_2}{{\widehat{\pi }}_3}$

{{A, C}, {G, T}}

s₁₃π₃ + s₁₄π₄ = s₂₃π₃ + s₂₄π₄

s₁₃π₁ + s₂₃π₂ = s₁₄π₁ + s₂₄π₄

${\tilde{s}}_{23}=\frac{{\tilde{s}}_{13}\left({\widehat{\pi }}_2{\widehat{\pi }}_3-{\widehat{\pi }}_1{\widehat{\pi }}_4\right)+{\tilde{s}}_{14}{\widehat{\pi }}_4\left({\widehat{\pi }}_1+{\widehat{\pi }}_2\right)}{{\widehat{\pi }}_2\left({\widehat{\pi }}_3+{\widehat{\pi }}_4\right)}$

${\tilde{s}}_{24}=\frac{{\tilde{s}}_{13}{\widehat{\pi }}_1+{\tilde{s}}_{23}{\widehat{\pi }}_2-{\tilde{s}}_{14}{\widehat{\pi }}_1}{{\widehat{\pi }}_2}$