Reaction rules
We are interested in the reaction probability: the probability that two entities interact during a time step given that they can interact with reaction rate k, their diffusion rates are D1 and D2 and they start a distance d apart at the beginning of the time interval Δt. This probability is illustrated in Figure 1C. The reaction between two diffusing particles can be considered to occur in two steps: firstly the encounter of the two entities through diffusion, followed by the actual chemical reaction. Let us consider two freely diffusing chemical entities, A and B, starting a distance d0 from each other at time t0. At any time t later the rate κAB (t|d0, t0) of the reaction between entities A and B can be expressed as:
κ A B   ( t | d 0  , t 0  ) = p C A B   ( t | d 0  , t 0  ) ⋅ k R A B         ( 2 )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF6oWAdaahaaWcbeqaaiabdgeabjabdkeacbaakiabcIcaOiabdsha0jabcYha8jabdsgaKnaaBaaaleaacqaIWaamaeqaaOGaeiilaWIaemiDaq3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkcqGH9aqpcqWGWbaCdaqhaaWcbaGaem4qameabaGaemyqaeKaemOqaieaaOGaeiikaGIaemiDaqNaeiiFaWNaemizaq2aaSbaaSqaaiabicdaWaqabaGccqGGSaalcqWG0baDdaWgaaWcbaGaeGimaadabeaakiabcMcaPiabgwSixlabdUgaRnaaDaaaleaacqWGsbGuaeaacqWGbbqqcqWGcbGqaaGccaWLjaGaaCzcamaabmaabaGaeGOmaidacaGLOaGaayzkaaaaaa@567F@
where pCAB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaem4qameabaGaemyqaeKaemOqaieaaaaa@3169@ (t|d0, t0) is the probability of the two entities coming into contact at time t and kRAB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemOuaifabaGaemyqaeKaemOqaieaaaaa@317D@ is the rate of the reaction once in contact, averaged over all possible orientations of the two entities relative to each other. Both parts of this equation: pCAB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaem4qameabaGaemyqaeKaemOqaieaaaaa@3169@ (t|d0, t0) and kRAB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemOuaifabaGaemyqaeKaemOqaieaaaaa@317D@ can be estimated as described below.
The reaction rate κAB (t|d0, t0) can be integrated over a simulation timestep Δt to provide the probability of at least one reaction taking place in that timestep. We are interested in the probability of at least one event taking place during the time interval Δt, i.e. 1 - P(no event during Δt). The process under consideration is a Poisson process with a time dependent rate of the event taking place. Given the rate κ(t) of an event taking place at a time t, the probability, PAB, of at least one reaction taking place during that time interval takes the general form [24,25]:
PAB (Δt) = 1 - e-I(Δt)     (3)
where
I ( Δ t ) = ∫ 0 Δ t   κ ( t ) d t          ( 4 )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGjbqscqGGOaakcqqHuoarcqWG0baDcqGGPaqkcqGH9aqpdaWdXaqaaGGaciab=P7aRjabcIcaOiabdsha0jabcMcaPiabbsgaKjabdsha0bWcbaGaeGimaadabaGaeuiLdqKaemiDaqhaniabgUIiYdGccaWLjaGaaCzcamaabmaabaGaeGinaqdacaGLOaGaayzkaaaaaa@44AD@
Such that the probability of a reaction taking place during timestep δt can be expressed as:
P A B   ( Δ t ) = 1 − e − ∫ 0 Δ t   κ A B   ( t ) d t            ( 5 )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaahaaWcbeqaaiabdgeabjabdkeacbaakiabcIcaOiabfs5aejabdsha0jabcMcaPiabg2da9iabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0Yaa8qmaeaaiiGacqWF6oWAdaahaaadbeqaaiabdgeabjabdkeacbaaliabcIcaOiabdsha0jabcMcaPiabbsgaKjabdsha0badbaGaeGimaadabaGaeuiLdqKaemiDaqhaoiabgUIiYdaaaOGaaCzcaiaaxMaadaqadaqaaiabiwda1aGaayjkaiaawMcaaaaa@4DA9@
where κAB (t) is given in equation (2).