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Integration schemes for biochemical systems unconditional positivity and mass conservation

Integration schemes for biochemical systems unconditional positivity and mass conservation. Jorn Bruggeman Hans Burchard, Bob Kooi, Ben Sommeijer Theoretical Biology Vrije Universiteit, Amsterdam. Start PhD study (2004) “Understanding the ‘organic carbon pump’ in mesoscale ocean flows”.

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Integration schemes for biochemical systems unconditional positivity and mass conservation

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  1. Integration schemes for biochemical systemsunconditional positivity and mass conservation Jorn Bruggeman Hans Burchard, Bob Kooi, Ben Sommeijer Theoretical Biology Vrije Universiteit, Amsterdam

  2. 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 Background Master Theoretical biology (2003)

  3. Kooijman (2000) Bruggeman (2009) Life is complex: aggregate! • Aim: single model for population of ‘universal species’ • One parameter per biological activity, e.g. • nutrient affinity • detritus consumption • Parameter probability distributions = ecosystem biodiversity individual population functional group ecosystem

  4. light harvesting light + structural biomass nutrient nutrient uptake + maintenance Example • Functional group ‘phytoplankton’: • Start in end of winter: • deep mixed layer  little primary productivity • uniform trait distribution, low biomass for all ‘species’ • No predation

  5. Results structural biomass light harvesting biomass nutrient harvesting biomass

  6. 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

  7. Mass conservation • Model building block: reaction • Conservation • for any element, sums on left and right must be equal • Property of conservation • is independent of r(…) • does depend on stoichiometric coefficients • Conservation = preservation of stoichiometric ratios

  8. Systems of reactions • Integration scheme operates on ODEs • Reaction fluxes distributed over multiple ODEs:

  9. Forward Euler, Runge-Kutta • Conservative • all fluxes multiplied with same factor Δt • Non-positive • Order: 1, 2, 4 etc.

  10. Backward Euler, Gear • 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

  11. Modified Patankar: concepts • 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 • Burchard, Deleersnijder, Meister (2003) • “A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations”

  12. Modified Patankar: structure • Flux-specific multiplication factors cn+1/cn • Represent ratio: (source after) : (source before) • Multiple sources in reaction: • multiple, differentcn+1/cn factors • Then: stoichiometric ratios not preserved!

  13. Modified Patankar: example/conclusion • Conservative only if • 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

  14. Typical MP conservation error Total nitrogen over 20 years: MP 1st order 600 % increase! MP-RK 2nd order

  15. New 1st order scheme: structure • Non-linear system of equations • Positivity requirement fixes domain of product term p:

  16. New 1st order scheme: solution • Non-linear system can be simplified to polynomial: • Polynomial in p: • positive at left bound p=0, negative at right bound • Derivative dg/dp < 0 within p domain: • only one valid p • Bisection technique is guaranteed to find p

  17. Test case: linear system

  18. Test case: non-linear system

  19. 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

  20. Plans • Publish new schemes • Bruggeman, Burchard, Kooi, Sommeijer (submitted 2005) • Short term • Explore trait-based models (different traits) • Trait distributions  single adapting species • Modeling coagulation (marine snow) • Extension to 3D global circulation models

  21. The end

  22. Test cases Linear system: Non-linear system:

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