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Integration schemes for biochemical systems unconditional positivity and mass conservation Jorn Bruggeman Hans Burchard, Bob Kooi, Ben Sommeijer Theoretical Biology Vrije Universiteit, Amsterdam Background Master Theoretical biology (2003) Start PhD study (2004) “Understanding the ‘organic carbon pump’ in mesoscale ocean flows” Focus: details in 1D water column turbulence and biota, simulation in time Tool: General Ocean Turbulence Model (GOTM) modeling framework that hosts biota Life is complex: aggregate! individual Kooijman (2000) population functional group ecosystem Aim: single model for population of ‘universal species’ One parameter per biological activity, e.g. – – Bruggeman (2009) nutrient affinity detritus consumption Parameter probability distributions = ecosystem biodiversity Example Functional group ‘phytoplankton’: light light harvesting + structural biomass nutrient + nutrient uptake maintenance Start in end of winter: – – deep mixed layer little primary productivity uniform trait distribution, low biomass for all ‘species’ No predation Results structural biomass light harvesting biomass nutrient harvesting biomass Integration schemes Biochemical criteria: – – State variables remain positive Elements and energy are conserved Even if model meets criteria, integration results may not GOTM: different schemes for different problems: – – – Advection (TVD schemes) Diffusion (modified Crank-Nicholson scheme) Production/destruction Mass conservation Model building block: reaction r (...) 6 CO2 6 H 2 O 6 O2 1C6 H12O6 Conservation – Property of conservation – – for any element, sums on left and right must be equal is independent of r(…) does depend on stoichiometric coefficients Conservation = preservation of stoichiometric ratios Systems of reactions Integration scheme operates on ODEs Reaction fluxes distributed over multiple ODEs: dcCO2 r (...) 6 CO2 6 H 2O 6 O 2 C6 H12 O6 dt dcH 2O dt dcO2 dt dcC6 H12O6 dt 6r (...) 6r (...) 6r (...) r (...) Forward Euler, Runge-Kutta cn 1 cn t f t n , c n Conservative – all fluxes multiplied with same factor Δt Non-positive Order: 1, 2, 4 etc. Backward Euler, Gear cn 1 cn t f t n 1 , c n 1 Conservative – all fluxes multiplied with same factor Δt Positive for order 1 (Hundsdorfer & Verwer) Generalization to higher order eliminates positivity Slow! – – requires numerical approximation of partial derivatives requires solving linear system of equations Modified Patankar: concepts Burchard, Deleersnijder, Meister (2003) – “A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations” Approach – – – – Compound fluxes in production, destruction matrices (P, D) Pij = rate of conversion from j to i Dij = rate of conversion from i to j Source fluxes in D, sink fluxes in P Modified Patankar: structure n 1 i c Flux-specific multiplication factors cn+1/cn Represent ratio: (source after) : (source before) Multiple sources in reaction: – I c nj 1 I cin 1 c t Pij n Dij n j 1 c j j 1 ci n i multiple, different cn+1/cn factors Then: stoichiometric ratios not preserved! Modified Patankar: example/conclusion r (...) 6 CO2 6 H 2O 6 O 2 C6 H12 O6 n 1 c CO n 1 n 2 cCO2 cCO2 t 6r (...) n cCO2 n 1 c H n 1 n 2O cH 2O cH 2O t 6r (...) n cH 2O 2. n cCO 2 cHn 21O cHn 2O Conservative only if 1. n 1 cCO 2 every reaction contains ≤ 1 source compound source change ratios are identical (and remain so during simulation) Positive Order 1, 2 Requires solving linear system of equations Typical MP conservation error Total nitrogen over 20 years: MP 1st order 600 % increase! MP-RK 2nd order New 1st order scheme: structure cn1 cn t f t n , cn p with p jJ n c nj 1 c nj J n i : fi (t n , cn ) 0, i {1,..., I } Non-linear system of equations Positivity requirement fixes domain of product term p: p0 p 1 c nj p minn n n jJ t f j t , c New 1st order scheme: solution Non-linear system can be simplified to polynomial: g ( p) 1 a j p p 0 with a j t f j t n , c n jJ n Polynomial in p: – positive at left bound p=0, negative at right bound Derivative dg/dp < 0 within p domain: – c nj only one valid p Bisection technique is guaranteed to find p Test case: linear system Test case: non-linear system New schemes: conclusion Conservative – all fluxes multiplied with same factor pΔt Positive Extension to order 2 available Relatively cheap – – – ±20 bisection iterations = evaluations of polynomial Always cheaper than Backward Euler Cost scales with number of state variables, favorably compared to Modified Patankar Not for stiff systems (unlike Modified Patankar) – unless stiffness and positivity problems coincide Plans Publish new schemes – Short term – – – Bruggeman, Burchard, Kooi, Sommeijer (submitted 2005) Explore trait-based models (different traits) Trait distributions single adapting species Modeling coagulation (marine snow) Extension to 3D global circulation models The end Test cases Linear system: Non-linear system: