❊✣❝✐❡♥t ❯♥❢♦❧❞✐♥❣ ♦❢ ❈♦❧♦✉r❡❞ P❡tr✐ ◆❡ts ✉s✐♥❣ ■♥t❡r✈❛❧ ❉❡❝✐s✐♦♥ ❉✐❛❣r❛♠s ▼❛rt✐♥ ❙❝❤✇❛r✐❝❦✱ ❈❤r✐st✐❛♥ ❘♦❤r✱ ❋❡✐ ▲✐✉✱ ●❡♦r❣❡ ❆ss❛❢✱ ❏❛❝❡❦ ❈❤♦❞❛❦ ❛♥❞ ▼♦♥✐❦❛ ❍❡✐♥❡r ❇r❛♥❞❡♥❜✉r❣ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② P❡tr✐ ◆❡ts ✷✵✷✵ ✲ P❛r✐s ✷✺ ❏✉♥❡ ✷✵✷✵ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶ ✴ ✸✼
❖✉t❧✐♥❡ ❲❤② ❝♦❧♦✉r❡❞ P❡tr✐ ◆❡ts❄ ✶ ❙t❛t❡ ♦❢ t❤❡ ❆rt ✷ Unfolding ❚❤❡ Pr♦❜❧❡♠ ✸ ❲❤❛t ❛r❡ ■❉❉❄ ✹ ■❉❉ ❜❛s✐❝ ♣r✐♥❝✐♣❧❡s ✺ ■❉❉✲❜❛s❡❞ ✉♥❢♦❧❞✐♥❣ ❛❧❣♦r✐t❤♠ ✻ ❊①♣❡r✐♠❡♥ts ✼ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷ ✴ ✸✼
P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ✲ ❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸ ✴ ✸✼
❈♦❧♦✉r❡❞ P◆s ✲ P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿ zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦✉r❡❞ P◆s ✲ P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿ zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦✉r❡❞ P◆s ✲ P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿ zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦✉r❡❞ P◆s ✲ P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿ zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦✉r❡❞ P◆s ✲ P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿ zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ colour-dependent rate function ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ a=A & b=B : MassAction(k1) ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
P♦✇❡r❢✉❧ ♠♦❞❡❧❧✐♥❣ ✲ ❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts Guards Guard ❝♦❧♦✉r❡❞ ❝♦♥t✐♥✉♦✉s P❡tr✐ ♥❡t ❝♦❧♦✉r❡❞ st♦❝❤❛st✐❝ P❡tr✐ ♥❡t ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✺ ✴ ✸✼
❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳ ❙t❛t❡ ♦❢ t❤❡ ❆rt ❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳ ❙t❛t❡ ♦❢ t❤❡ ❆rt ❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t Ñ ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
❙t❛t❡ ♦❢ t❤❡ ❆rt ❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t Ñ ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳ ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
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