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: 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

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

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

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

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

Conservative systems 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 integration schemes If satisfied, implies microscopic/macroscopic conservation 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

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

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

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

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 –

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!

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 (higher possible?) Requires solving linear system of equations

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

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

New 1 st order scheme: solution 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 Component-wise, dividing by c n : Left and right, product over set J n :

New 1 st 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)

Extension to 2 nd order

Test cases Linear system: Non-linear system:

Test case: linear system

Test case: non-linear system

Order tests Linear system:Non-linear system:

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

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