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

2. 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

3. 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 Kooijman (2000) Bruggeman (2009)

4. Example Functional group ‘phytoplankton’: nutrient uptake structural biomass nutrient light + + maintenance light harvesting 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 biomassnutrient 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 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 ) – P ij = rate of conversion from j to i – D ij = rate of conversion from i to j – Source fluxes in D, sink fluxes in P

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

13. Modified Patankar: example/conclusion Conservative only if 1. every reaction contains ≤ 1 source compound 2. 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-RK 2 nd order MP 1 st order 600 % increase!

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

16. New 1 st order scheme: solution 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 Non-linear system can be simplified to polynomial:

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: