integration schemes for biochemical systems unconditional positivity and mass conservation
DESCRIPTION
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”. - PowerPoint PPT PresentationTRANSCRIPT
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: