Integration schemes for biochemical systemsunconditional positivity and mass conservation

Jorn BruggemanHans Burchard, Bob Kooi, Ben Sommeijer

Theoretical BiologyVrije 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 columnturbulence and biota, simulation in time

Tool: General Ocean Turbulence Model (GOTM)modeling framework that hosts biota

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)

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

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

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

(...)2 2 2 6 12 66 6 6CO H O O C H O1r

Systems of reactions

Integration scheme operates on ODEs Reaction fluxes distributed over multiple ODEs:

2

2

2

6 12 6

6 (...)

6 (...)

6 (...)

(...)

CO

H O

O

C H O

dcr

dtdc

rdtdc

rdt

dcr

dt

(...)2 2 2 6 12 66 6 6CO H O O C H Or

Forward Euler, Runge-Kutta

1 ,n n n nt t c c f c

Conservative– all fluxes multiplied with same factor Δt

Non-positive Order: 1, 2, 4 etc.

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 1,nn nn t t c c f c

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

1 1

1

1 1

I In ni

n n

i ij ijj j

j in nj i

c cc c t P D

c c

Flux-specific multiplication factors cn+1/cn

Represent ratio: (source after) : (source before) Multiple sources in reaction:

– multiple, different cn+1/cn factors

Then: stoichiometric ratios not preserved!

Modified Patankar: example/conclusion

2

2 2

2

2

2 2

2

11

11

6 (...)

6 (...)

nCOn n

CO CO nCO

nH On n

H O H O nH O

cc c t r

c

cc c t r

c

Conservative only if1. every reaction contains ≤ 1 source compound2. source change ratios are identical (and remain so during simulation)

Positive Order 1, 2 Requires solving linear system of equations

2 2

2 2

1 1n nCO H O

n nCO H O

c c

c c

(...)2 2 2 6 12 66 6 6CO H O O C H Or

Typical MP conservation error

Total nitrogen over 20 years:

MP-RK 2nd order

MP 1st order

600 % increase!

11 , with

: ( , ) 0, {1,..., }

n

njn n n nn

j J j

n n ni

ct t p

c

J i f t i I

p

c c f c

c

New 1st order scheme: structure

Non-linear system of equations Positivity requirement fixes domain of product term p:

0

1

min,n

nj

n nj Jj

p

p

cp

t f t

c

New 1st order scheme: solution

,( ) 1 0 with

n

n nj

j j nj J j

t f tg p a p p a

c

c

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:

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– 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

The end

Test cases

Linear system:

Non-linear system: