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Integration schemes for biochemical systems unconditional positivity and mass conservation

Explore numerical integration schemes for biochemical systems focusing on positivity and mass conservation with the help of General Ocean Turbulence Model (GOTM). Understand the 'organic carbon pump' in ocean flows. Implement conservative integration schemes for accurate simulations.

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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. Focus: 1D discretized water column turbulence and biota, simulation in time Tool: General Ocean Turbulence Model (GOTM) Modeling framework, split integration of advection, diffusion, production/destruction Background Master Theoretical biology (2003) Start PhD study (2004) “Understanding the ‘organic carbon pump’ in mesoscale ocean flows”

  3. Outline • Biochemical systems • reaction-based framework • conservation (of elements) • positivity • Traditional integration schemes • Euler, Runge-Kutta • Modified Patankar • New 1st and 2nd order schemes

  4. Corresponding system of ODEs: Generalized for I state variables: Biochemical systems: the reaction • chemical compounds = state variables c • sources (left) are destroyed to produce sinks (right) • constantstoichiometric coefficients (unit: compound/reaction) • variablereaction rate (unit: reactions/time)

  5. Corresponding system of ODEs: Generalized for I state variables, R reactions : Systems of reactions

  6. C O H for 1 conservative reaction: The conservative reaction Compounds consist of chemical elements: Conservation: in reaction, no elements are created or destroyed!

  7. With biochemical framework: microscopic conservation: in any reaction, no elements are created or destroyed Without biochemical framework: macroscopic conservation: in (closed) system, no elements are created or destroyed Conservative systems

  8. Macroscopic conservation: within system, quantities of element species are constant: Microscopic conservation? View on reaction-level is gone… ‘Biochemical integrity’: state variables change through known reactions only: for some vector If satisfied, implies microscopic/macroscopic conservation Conservative integration schemes

  9. Integration scheme must satisfy: biochemical integrity/conservation: if given positivity: Criteria for integration schemes Given a positive definite, conservative biochemical system:

  10. Forward Euler, Runge-Kutta • Conservative: • Non-positive • Order: 1, 2, 4 etc.

  11. Backward Euler, Gear • Conservative: • 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

  12. 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”

  13. 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!

  14. 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 (higher possible?) • Requires solving linear system of equations

  15. Typical MP conservation error Total nitrogen over 20 years: MP 1st order MP-RK 2nd order

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

  17. New 1st order scheme: solution Component-wise, dividing by cn: Left and right, product over set Jn: • Polynomial for p: • positive at left bound p=0, negative at right bound • Derivative of polynomial < 0 within p domain: • only one valid p • Bisection technique is guaranteed to find p

  18. New 1st order scheme: conclusion • Positive • Conservative: • ±20 bisection iterations (evaluations of polynomial) • Always cheaper than Backward Euler • >4 state variables? Then cheaper than Modified Patankar • Note: not suitable for stiff systems (unlike Modified Patankar)

  19. Extension to 2nd order

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

  21. Test case: linear system

  22. Test case: non-linear system

  23. Order tests Linear system: Non-linear system:

  24. Plans • Publish new schemes • Bruggeman, Burchard, Kooi, Sommeijer (submitted 2005) • Short term • Modeling ecosystems • Aggregation into functional groups • Modeling coagulation (marine snow) • Extension to 3D global circulation models

  25. The end

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