Numerical approximation methods for stochastically modeled biochemical reaction networks DavidF.Anderson∗ ∗[email protected] DepartmentofMathematics UniversityofWisconsin-Madison VirginiaTechColloquium February11th,2011 Outline 1. DiscussPoissonprocessandtimechanges. 2. Describe/developmodel–stochasticallymodeledreactionnetworks. 3. Numericalmethodsanderrorapproximation. 4. Mathematicalquestionfortoday: howdoweeffectivelyquantifyhowwell differentmethodsapproximatethecontinuoustimeMarkovchainmodel. 5. Discuss/usemulti-scalenatureofbiochemicalreactionnetworks. The Poisson process (cid:73) ThemodelsIamgoingtodiscussareasubsetoftheclassofmodels termedcontinuoustimeMarkovchains. (cid:73) ThesimplestcontinuoustimeMarkovchainisthePoissonprocess. (b) Now,putpointsdownonalinewithspacingequaltotheξ. i x x x x x x x x ↔ ↔ ←→ ξ ξ ξ ··· t 1 2 3 (cid:73) LetY(t)denotethenumberofpointshitbytimet. (cid:73) Inthefigureabove,Y(t)=6. Intuition: TheunitratePoissonprocessissimplythenumberofpointshit whenwerunalongthetimeframeatrateone. 25 20 λ = 1 15 10 5 00 5 10 15 20 The Poisson process APoissonprocess,Y(·),isamodelforaseriesofrandomobservations occurringintime. (a) Let{ξ}bei.i.d. exponentialrandomvariableswithparameterone. i 25 20 λ = 1 15 10 5 00 5 10 15 20 The Poisson process APoissonprocess,Y(·),isamodelforaseriesofrandomobservations occurringintime. (a) Let{ξ}bei.i.d. exponentialrandomvariableswithparameterone. i (b) Now,putpointsdownonalinewithspacingequaltotheξ. i x x x x x x x x ↔ ↔ ←→ ξ ξ ξ ··· t 1 2 3 (cid:73) LetY(t)denotethenumberofpointshitbytimet. (cid:73) Inthefigureabove,Y(t)=6. Intuition: TheunitratePoissonprocessissimplythenumberofpointshit whenwerunalongthetimeframeatrateone. The Poisson process APoissonprocess,Y(·),isamodelforaseriesofrandomobservations occurringintime. (a) Let{ξ}bei.i.d. exponentialrandomvariableswithparameterone. i (b) Now,putpointsdownonalinewithspacingequaltotheξ. i x x x x x x x x ↔ ↔ ←→ ξ ξ ξ ··· t 1 2 3 (cid:73) LetY(t)denotethenumberofpointshitbytimet. (cid:73) Inthefigureabove,Y(t)=6. Intuition: TheunitratePoissonprocessissimplythenumberofpointshit whenwerunalongthetimeframeatrateone. 25 20 λ = 1 15 10 5 00 5 10 15 20 60 50 λ = 3 40 30 20 10 00 5 10 15 20 The Poisson process Let (cid:73) Y beaunitratePoissonprocess. (cid:73) DefineY (t)≡Y(λt), λ ThenY isaPoissonprocesswithparameterλ. λ Intuition: ThePoissonprocesswithrateλissimplythenumberofpointshit (oftheunit-ratepointprocess)whenwerunalongthetimeframeatrateλ. Thus,wehave“changedtime”toconvertaunit-ratePoissonprocesstoone whichhasrateλ. The Poisson process Let (cid:73) Y beaunitratePoissonprocess. (cid:73) DefineY (t)≡Y(λt), λ ThenY isaPoissonprocesswithparameterλ. λ Intuition: ThePoissonprocesswithrateλissimplythenumberofpointshit (oftheunit-ratepointprocess)whenwerunalongthetimeframeatrateλ. Thus,wehave“changedtime”toconvertaunit-ratePoissonprocesstoone whichhasrateλ. 60 50 λ = 3 40 30 20 10 00 5 10 15 20 Keyproperty: Y(T +∆)−Y(T)=Poisson(∆). Thus  Z t+∆t ff P{Y (t+∆t)−Y (t)>0|F}=1−exp − λ(t)dt ≈λ(t)∆t. λ λ t t Points: 1. Wehave“changedtime”toconvertaunit-ratePoissonprocesstoone whichhasrateorintensityλ(t). 2. Willusemorecomplicatedtimechangesofunit-rateprocessestobuild modelsofinterest. The Poisson process Thereisnoreasonλneedstobeconstantintime,inwhichcase „Z t « Y (t)≡Y λ(s)ds λ 0 isaninhomogeneousPoissonprocess. Thus  Z t+∆t ff P{Y (t+∆t)−Y (t)>0|F}=1−exp − λ(t)dt ≈λ(t)∆t. λ λ t t Points: 1. Wehave“changedtime”toconvertaunit-ratePoissonprocesstoone whichhasrateorintensityλ(t). 2. Willusemorecomplicatedtimechangesofunit-rateprocessestobuild modelsofinterest. The Poisson process Thereisnoreasonλneedstobeconstantintime,inwhichcase „Z t « Y (t)≡Y λ(s)ds λ 0 isaninhomogeneousPoissonprocess. Keyproperty: Y(T +∆)−Y(T)=Poisson(∆).