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❉◆❆ ❡✈♦❧✉t✐♦♥✱ ❆✉t♦♠❛t❛ ❛♥❞ ❈❧✉♠♣s P✐❡rr❡ ◆✐❝♦❞è♠❡ ▲■P◆ ❚❡❛♠ ❈❆▲■◆✱ ❯♥✐✈❡rs✐t② P❛r✐s ✶✸✱ ❱✐❧❧❡t❛♥❡✉s❡

Pr♦❜❧❡♠ s❡tt✐♥❣ ◮ ❆❧♣❤❛❜❡t A = { A , C , G , T } ✭❉◆❆✮ t✐♠❡ = 0 S n (0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = T S n ( T ) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨ ◮ n = ❧❡♥❣t❤ ♦❢ r❛♥❞♦♠ s❡q✉❡♥❝❡s S n (0) , . . . , S n ( T ) ( n ≈ 2000) ◮ b = ❋❋✳✳❋❋ ∈ A k ❚r❛♥s❝r✐♣t✐♦♥ ❋❛❝t♦r (5 ≤ k = | b | ≤ 10) ◮ b ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ S n (0) ◮ b ♦❝❝✉rs ❢♦r t❤❡ ✜rst t✐♠❡ ❜② ❡✈♦❧✉t✐♦♥ ❛t t✐♠❡ T ✐♥ ❛ s❡q✉❡♥❝❡ ❡✈♦❧✈✐♥❣ ❢r♦♠ S n (0) ❆✐♠✿ ❈♦♠♣✉t❡ T ▼❛② ✸✵✱ ✷✵✶✸

■♥✐t✐❛❧ ν ( α ) ❛♥❞ ❙✉❜st✐t✉t✐♦♥ Pr♦❜❛❜✐❧✐t✐❡s π α → β ❆ ❆ 0 . 9999999763 � 4 . 54999994943 × 10 − 9 α ν ( α ) ❆ ❈ � 1 . 57499995613 × 10 − 8 ❆ ● ✵✳✷✸✽✽✾ ❆ � 3 . 40000001733 × 10 − 9 ✵✳✷✻✷✹✷ ❆ ❚ ❈ � 6 . 14999993408 × 10 − 9 ✵✳✷✺✽✻✺ ❈ ❆ ● � ✵✳✷✹✵✵✹ ❈ ❈ 0 . 99999996495 ❚ � 7 . 14999984731 × 10 − 9 ❈ ● � 2 . 17499993935 × 10 − 8 ❈ ❚ � 2 . 17499993935 × 10 − 8 ● ❆ � 7 . 14999984731 × 10 − 9 ● ❈ � � ● ● 0 . 99999996495 � 6 . 14999993408 × 10 − 9 ● ❚ s✉❜st✐t✉t✐♦♥ ♣r♦❜✳ � 3 . 40000001733 × 10 − 9 ❚ ❆ P (1) = π α → β � 1 . 57499995613 × 10 − 8 ❢♦r ♦♥❡ ❣❡♥❡r❛t✐♦♥ ❚ ❈ � 4 . 54999994943 × 10 − 9 ❚ ● ✭ 20 ②❡❛rs✮ � ❚ ❚ 0 . 9999999763 � ▼❛② ✸✵✱ ✷✵✶✸

P♦✇❡rs ♦❢ P (1) r❡♠❛✐♥s ❝❧♦s❡ t♦ t❤❡ ■❞❡♥t✐t② ▼❛tr✐①   1 − 3 m m m m 1 − 3 m m m m   ✇✐t❤ m ≈ 10 − 8 P (1) ≈   m m m 1 − 3 m   1 − 3 m m m m  1 − 3 mN  mN mN mN mN 1 − 3 mN mN mN  + O ( m 2 N ) P N (1) ≈     1 − 3 mN mN mN mN  mN mN mN 1 − 3 mN ❚❤❡r❡❢♦r❡ ❢♦r N ≈ 10 6 ❛♥❞ N < 10 6 P N (1) × ν ≈ ν P ∞ (1) × ν = (0 . 25 , 0 . 25 , 0 . 25 , 0 . 25) t ▼❛② ✸✵✱ ✷✵✶✸

♦❝❝✉rs ✐♥ ♥♦ ✐♥ ♦❝❝✉rs ✐♥ ♥♦ ✐♥ ❙❡tt✐♥❣ ♦❝❝✉rs ✐♥ ♥♦ ✐♥ ✱ ●❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡ ❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν ✱ ❛ss✉♠✐♥❣ T ∈ N � � ♥♦ b ✐♥ S n ( j + 1) | ♥♦ b ✐♥ S n ( j ) P � � = P ♥♦ b ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � = 1 − P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) ▼❛② ✸✵✱ ✷✵✶✸

❙❡tt✐♥❣ ♦❝❝✉rs ✐♥ ♥♦ ✐♥ ✱ ●❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡ ❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν ✱ ❛ss✉♠✐♥❣ T ∈ N � � ♥♦ b ✐♥ S n ( j + 1) | ♥♦ b ✐♥ S n ( j ) P � � = P ♥♦ b ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � = 1 − P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � b ♦❝❝✉rs ✐♥ S n ( T ) | ♥♦ b ✐♥ S n ( T − 1) P � � = P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) ▼❛② ✸✵✱ ✷✵✶✸

●❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡ ❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν ✱ ❛ss✉♠✐♥❣ T ∈ N � � ♥♦ b ✐♥ S n ( j + 1) | ♥♦ b ✐♥ S n ( j ) P � � = P ♥♦ b ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � = 1 − P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � b ♦❝❝✉rs ✐♥ S n ( T ) | ♥♦ b ✐♥ S n ( T − 1) P � � = P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � ❙❡tt✐♥❣ p n = P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) ✱ (1 − p n ) i × p n = 1 � E ( T ) ≈ i p n i ≥ 0 ▼❛② ✸✵✱ ✷✵✶✸

❘❡♥❡✇✐♥❣ t❤❡ ❆✐♠ ❲❡ ♥❡❡❞ ♥♦✇ ❝♦♠♣✉t✐♥❣ � � p n = P b ♦❝❝✉rs ✐♥ S n (1) | ♥♦ b ✐♥ S n (0) � � b ♦❝❝✉rs ✐♥ S n (1) ❆◆❉ ♥♦ b ✐♥ S n (0) = P P ( no b ✐♥ S n (0)) ▼❛② ✸✵✱ ✷✵✶✸

❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ p n ✶✳ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ✭✷✵✶✵✮ ◮ ❆♣♣r♦❛❝❤ ♥❡❣❧❡❝t✐♥❣ ✇♦r❞s ❝♦rr❡❧❛t✐♦♥ ✳ ◮ ❊✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ p n ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❛ss✉♠♣t✐♦♥✳ ✷✳ ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✭✷✵✶✷✮ ◮ ❘✐❣♦r♦✉s ❛♥❞ ❡✣❝✐❡♥t ❛♣♣r♦❛❝❤ ❜② ❛✉t♦♠❛t❛ ✳ ✸✳ ◆ ✭◆❈▼❆✷✵✶✷✮ ◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ ❜② ❝❧✉♠♣ ❛♥❛❧②s✐s ✱ ❡✐t❤❡r ❜② ❝♦♠❜✐♥❛t♦r✐❝s ♦❢ ✇♦r❞s ♦r ❜② ❛✉t♦♠❛t❛ ❛♥❞ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳ ✹✳ ◆ ✭✷✵✶✸✮ ◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤✱ ❛❞❛♣t❛t✐♦♥ ♦❢ t❤❡ ❘é❣♥✐❡r✲❙③♣❛♥❦♦✇s❦✐ ❡q✉❛t✐♦♥s ❛♥❞ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛♣♣r♦①✐♠❛t✐♥❣ p n ▼❛② ✸✵✱ ✷✵✶✸

❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✉s❡ ❛♥ ❛✉t♦♠❛t♦♥ ♦♥ t❤❡ ❛❧♣❤❛❜❡t t❤❛t s❝❛♥s s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✳ ❚❤✐s ❛✉t♦♠❛t♦♥ ✐s ❛ ❦✐♥❞ ♦❢ ♣r♦❞✉❝t ♦❢ t✇♦ ❑♥✉t❤✲▼♦rr✐s✲Pr❛tt ❛✉t♦♠❛t❛ ✳ ◆ ✭✷✵✶✷✮ ❛ss✉♠❡s t❤❛t ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥ ♦❝❝✉rr❡❞ ❛♥❞ ❝♦♥s✐❞❡rs t❤❡ ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ♦❢ ❛t ❞✐st❛♥❝❡ ✐♥ ✳ ❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ p n t✐♠❡ = 0 S n (0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = 1 S n (1) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨ ◮ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t b ♦❝❝✉rs ✐♥ S n (1) ✭ ✇✐t❤♦✉t ❛❧❧♦✇✐♥❣ ♦✈❡r❧❛♣s ♦❢ ♦❝❝✉rr❡♥❝❡s ✮✱ ❛♥❞ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t S n (0) ❡✈♦❧✈❡s t♦ S n (1) ▼❛② ✸✵✱ ✷✵✶✸

◆ ✭✷✵✶✷✮ ❛ss✉♠❡s t❤❛t ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥ ♦❝❝✉rr❡❞ ❛♥❞ ❝♦♥s✐❞❡rs t❤❡ ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ♦❢ ❛t ❞✐st❛♥❝❡ ✐♥ ✳ ❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ p n t✐♠❡ = 0 S n (0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = 1 S n (1) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨ ◮ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t b ♦❝❝✉rs ✐♥ S n (1) ✭ ✇✐t❤♦✉t ❛❧❧♦✇✐♥❣ ♦✈❡r❧❛♣s ♦❢ ♦❝❝✉rr❡♥❝❡s ✮✱ ❛♥❞ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t S n (0) ❡✈♦❧✈❡s t♦ S n (1) ◮ ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✉s❡ ❛♥ ❛✉t♦♠❛t♦♥ ♦♥ t❤❡ ❛❧♣❤❛❜❡t A × A t❤❛t s❝❛♥s s✐♠✉❧t❛♥❡♦✉s❧② S n (0) ❛♥❞ S n (1) ✳ ❚❤✐s ❛✉t♦♠❛t♦♥ ✐s ❛ ❦✐♥❞ ♦❢ ♣r♦❞✉❝t ♦❢ t✇♦ ❑♥✉t❤✲▼♦rr✐s✲Pr❛tt ❛✉t♦♠❛t❛ ✳ ▼❛② ✸✵✱ ✷✵✶✸