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: 1D discretized water columnturbulence 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 1st and 2nd 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)

(...)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

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

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

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

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

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

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

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

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

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

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

Typical MP conservation error

Total nitrogen over 20 years:

MP-RK 2nd order

MP 1st order

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

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:

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

(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

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

The end