integration schemes for biochemical systems unconditional positivity and mass conservation jorn...
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![Page 1: Integration schemes for biochemical systems unconditional positivity and mass conservation Jorn Bruggeman Hans Burchard, Bob Kooi, Ben Sommeijer Theoretical](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d415503460f94a1c5a7/html5/thumbnails/1.jpg)
Integration schemes for biochemical systemsunconditional positivity and mass conservation
Jorn BruggemanHans Burchard, Bob Kooi, Ben Sommeijer
Theoretical BiologyVrije Universiteit, Amsterdam
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Background
Master Theoretical biology (2003)
Start PhD study (2004)“Understanding the ‘organic carbon pump’
in mesoscale ocean flows”
Focus: 1D discretized water columnturbulence and biota, simulation in time
Tool: General Ocean Turbulence Model (GOTM)Modeling framework, split integration of advection,
diffusion, production/destruction
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Outline
Biochemical systems– reaction-based framework– conservation (of elements)– positivity
Traditional integration schemes– Euler, Runge-Kutta– Modified Patankar
New 1st and 2nd order schemes
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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)
(...)carbon-dioxide water oxygen glu6 ose6 6 c1r
Corresponding system of ODEs:
carbon-dioxide
water
oxygen
glucose
( )
( )d( )d
( )
6
6
6
1
...
c t
c t
t
t
rct
c
d( )
d, ( )t
tr tt s cc
Generalized for I state variables:
I
I
r
c
s
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Systems of reactions
carbon-dioxide
water
oxygen
glucose
ethanol
1
2
( )
( )d
( )d
( )
(
...
.
6 2
6 0
6 0
1 1
0)
..
1
c t
c t
c tt
c t
c t
r
r
Corresponding system of ODEs:
1
2
(...)
(...)
carbon-dioxide water oxygen glucose
glucose ethanol carbon-dioxide
6 6 6 1
1 2 2
r
r
d( ) ,
d( )t
tt t S r c
c
Generalized for I state variables, R reactions :
I
I R
R
S
c
r
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The conservative reaction
(...)2 2 2 6 12 66 6 6CO H O O C H O1r
(...)carbon-dioxide water oxygen glu6 ose6 6 c1r
61 0
0
0
0 0 6
2 1 2 6
01
0 12
6
26
sE
Conservation: in reaction, no elements are created or destroyed!
Compounds consist of chemical elements:
E s 0for 1 conservative reaction:
O
C
H
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Conservative systems
E S 0
With biochemical framework:microscopic conservation: in any reaction, no elements are created or destroyed
( ),t t f cE 0
Without biochemical framework:macroscopic conservation: in (closed) system, no elements are created or destroyed
) , (, )(t t t t S r cE Ef c 0
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Conservative integration schemes
1 1 1( , , , )n n n n n nt t t c c c c
If satisfied, implies microscopic/macroscopic conservation
E S 0
1n n E c c 0
Macroscopic conservation:within system, quantities of element species are constant:
Microscopic conservation? View on reaction-level is gone…
1n n nt c S rc
‘Biochemical integrity’: state variables change through known reactions only:
for some vector nr
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Criteria for integration schemes
Given a positive definite, conservative biochemical system:
d( ) , ( )
d, ( )t tt t t
t rS c
cf c
1 1( , , , )n n n n nt t c c S r
1 0n c if given 0n c
1 1 1( , , , )n n n n n nt t t c c c c
biochemical integrity/conservation:
positivity:
Integration scheme must satisfy:
( ) 0 0t t c
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Forward Euler, Runge-Kutta
1 ,n n n nt t c c f c
Conservative: Non-positive Order: 1, 2, 4 etc.
( , )( , )
n
n n n ntt r
cc S r
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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
1 1 1,n n n nt t c c f c
1 11 1 ( ), ,( )
n
n n n ntt r
c cS r
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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– , ( )t t S rP cD
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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!( , )n n nt c S r
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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 (higher possible?) 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
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Typical MP conservation error
Total nitrogen over 20 years:
MP-RK 2nd order
MP 1st order
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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
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New 1st order scheme: solution
11 ,1 with
,1
n
n
n n nni ji
n n nj Ji i j
n nj
nj J j
t f t ccp p
c c c
t f tp p
c
c
c
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 cn:
Left and right, product over set Jn:
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New 1st 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)
,( (, ) )
n
n n n np tt r
cc S r
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(1)(1)
11 1 (1)
(1)
1 (1)
,
, ,2
: ( , ) 0, {1,..., }
: ( , ) ( , ) 0, {1,..., }
n
n
jn n nn
j J j
nn n n n n k
k K k
n n ni
n n n ni i
ct t
c
ctt t
c
J i f t i I
K i f t f t i I
c c f c
c c f c f c
c
c c
Extension to 2nd order
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Test cases
Linear system:
Non-linear system:
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Test case: linear system
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Test case: non-linear system
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Order tests
Linear system: Non-linear system:
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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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The end