Integration schemes for biochemical systems unconditional positivity and mass conservation
description
Transcript of Integration schemes for biochemical systems unconditional positivity and mass conservation
![Page 1: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/1.jpg)
Integration schemes for biochemical systemsunconditional positivity and mass conservation
Jorn BruggemanHans Burchard, Bob Kooi, Ben Sommeijer
Theoretical BiologyVrije Universiteit, Amsterdam
![Page 2: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/2.jpg)
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
![Page 3: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/3.jpg)
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)
![Page 4: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/4.jpg)
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
![Page 5: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/5.jpg)
Results
structural biomass
light harvesting biomass nutrient harvesting biomass
![Page 6: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/6.jpg)
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
![Page 7: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/7.jpg)
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
![Page 8: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/8.jpg)
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
![Page 9: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/9.jpg)
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.
![Page 10: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/10.jpg)
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
![Page 11: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/11.jpg)
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
![Page 12: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/12.jpg)
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!
![Page 13: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/13.jpg)
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
![Page 14: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/14.jpg)
Typical MP conservation error
Total nitrogen over 20 years:
MP-RK 2nd order
MP 1st order
600 % increase!
![Page 15: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/15.jpg)
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
![Page 16: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/16.jpg)
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:
![Page 17: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/17.jpg)
Test case: linear system
![Page 18: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/18.jpg)
Test case: non-linear system
![Page 19: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/19.jpg)
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
![Page 20: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/20.jpg)
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
![Page 21: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/21.jpg)
The end
![Page 22: Integration schemes for biochemical systems unconditional positivity and mass conservation](https://reader035.fdocuments.in/reader035/viewer/2022062517/56813a8e550346895da288db/html5/thumbnails/22.jpg)
Test cases
Linear system:
Non-linear system: