Guy SOULAS

UMR Œnologie-AmpélologieUniversité Victor Segalen Bordeaux 2351 cours de la Libération, TALENCE CedexFrance

KINETIC MODELS

Where does first-order come from ?

Reason 1: the experience

Experience shows that many biotic and abiotic processes in environmental compartments such as soil effectively follow single first order kinetics (exponential decay)

Reason 2: pragmatism

    The equation is simple and has only two parameters

    It is easy to fit the equation to experimental data

     DT50 and DT90 values are easy to calculate

     Parameters are theoretically independent of concentration and time … and appropriate for use as input for pesticide leaching models.

Reason 3: scientific justifications

    abiotic hydrolytic processes often follow first-order reaction kinetics

    biotic degradation processes may be approximated by first-order reaction : ex. when responsible microbial agents (or enzymes) are in excess compared to the chemical (pseudo first order reaction kinetics).

S0<<Ks S0Ks S0>>Ks

First-order Monod without growth Zero -order

S0<<X0

s

0m

K

Xk

kSdtdS

0m

s

Xk

SKkS

dtdS

0mXk

kdtdS

Logistic growth Monod with growth Logarithmic

S0>>X0

s

m

00

Kk

)SXS(kSdtdS

)SXS(SK

SdtdS

00s

m

)SXS(

dtdS

00m

S0: initial substrate concentration; X0: initial biomass concentration

A phylogeny for the disappearance modelsThe « Metabolism » case

Reason 1: heterogeneity

The Gustafsson and Holden assumption:

The soil can be divided into a large number of independent compartments whith distributed first order rate constants.

If pdf = Gamma distribution …

Equation (integrated form) Underlying differential equationtk

0 eMM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical present at time t=0

k = Rate constant

Mkdt

dM

Parameters to be determined

M0, k

Endpoints

k

10lnDT

k

2lnDT

kx100

100ln

DT

90

50

x

Equation (integrated form) Underlying differential equationtk

0 eMM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical present at time t=0

k = Rate constant

Mkdt

dM

Parameters to be determined

M0, k

Endpoints

k

10lnDT

k

2lnDT

kx100

100ln

DT

90

50

x

Parameters to be determined

M0, k

Endpoints

k

10lnDT

k

2lnDT

kx100

100ln

DT

90

50

x

0

20

40

60

80

100

0 20 40 60 80 100Time (days)

Co

nce

ntr

atio

n (

% o

f in

itia

l)k = 0.005

k = 0.020

k = 0.050

Single first order kinetics (SFO)

But

First-order reaction kinetics may not be obeyed

Equation (integrated form) Differential equation(to be used only for parameter estimation)

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

= Shape parameter determined by coefficient of variation of k values

= Location parameter

Parameters to be determined

M0, ,

Endpoints

1t

MM 0 1

1t

Mdt

dM

110DT

12DT

1x100

100DT

1

90

1

50

1

x

Note: The proposed equation differs slightly from that in the original Gustafson and Holden (1990) reference. The parameter corresponds to 1 / in the original equation.

Equation (integrated form) Differential equation(to be used only for parameter estimation)

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

= Shape parameter determined by coefficient of variation of k values

= Location parameter

Parameters to be determined

M0, ,

Endpoints

1t

MM 0 1

1t

Mdt

dM

110DT

12DT

1x100

100DT

1

90

1

50

1

x

Note: The proposed equation differs slightly from that in the original Gustafson and Holden (1990) reference. The parameter corresponds to 1 / in the original equation.

The bi-phasic Gustafson & Holden model (FOMC)

alpha = 0.2 , beta = 5.00alpha = 0.2 , beta = 1.00alpha = 0.2 , beta = 0.05alpha = 1.0 , beta = 5.00alpha = 2.0 , beta = 5.00

Time

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60

Con

cent

ratio

n

Reason 2: limited availability.

In the soil, pesticides are distributed between a solid phase and a liquid phase where they are available for degradation. This partition induces a bi-phasic pattern of degradation

Equation (integrated form) Underlying differential equation

tk0

1eMM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

k1 = Rate constant until t=tbk2 = Rate constant from t=tbtb = Breakpoint (time at which rate constant changes)

Endpoints

for ttb

Parameters to be determined

M0, k1, k2, tb

2

b1

bx

1x

k

tkx100

100ln

tDT

kx100

100ln

DT

if DTxtb

if DTx>tb

for t>tb b2b1 ttktk

0 eeMM

Mkdt

dM1 for ttb

Mkdt

dM2 for t>tb

Equation (integrated form) Underlying differential equation

tk0

1eMM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

k1 = Rate constant until t=tbk2 = Rate constant from t=tbtb = Breakpoint (time at which rate constant changes)

Endpoints

for ttb

Parameters to be determined

M0, k1, k2, tb

2

b1

bx

1x

k

tkx100

100ln

tDT

kx100

100ln

DT

if DTxtb

if DTx>tb

for t>tb b2b1 ttktk

0 eeMM

Mkdt

dM1 for ttb

Mkdt

dM2 for t>tb

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60

Time

Con

cen

tra

tion

k1 = 0.05 , k2 = 0.01 , tb = 10k1 = 0.07 , k2 = 0.01 , tb = 10k1 = 0.09 , k2 = 0.01 , tb = 10k1 = 0.09 , k2 = 0.01 , tb = 15k1 = 0.09 , k2 = 0.02 , tb = 15

The bi-phasic Hockey Stick model (HS)

Equation (integrated form) Differential equation(to be used only for parameter estimation)

tk2

tk1

21 eMeMM

where

M = Total amount of chemical present at time t

M1 = Amount of chemical applied to compartment 1 at time t=0

M2 = Amount of chemical applied to compartment 2 at time t=0

M0 = M1 + M2 = Total amount of chemical applied at time t=0

g = fraction of M0 applied to compartment 1

k1 = Rate constant in compartment 1

k2 = Rate constant in compartment 2

Parameters to be determined

M1, M2, k1, k2 or M0, g, k1, k2

Endpoints

An analytical solution does not exist.

DTx values can only be found by an iterative procedure

tktk0

21 eg1egMM

or

M

eg1eg

eg1kegk

dt

dMtktk

tk2

tk1

21

21

Equation (integrated form) Differential equation(to be used only for parameter estimation)

tk2

tk1

21 eMeMM

where

M = Total amount of chemical present at time t

M1 = Amount of chemical applied to compartment 1 at time t=0

M2 = Amount of chemical applied to compartment 2 at time t=0

M0 = M1 + M2 = Total amount of chemical applied at time t=0

g = fraction of M0 applied to compartment 1

k1 = Rate constant in compartment 1

k2 = Rate constant in compartment 2

Parameters to be determined

M1, M2, k1, k2 or M0, g, k1, k2

Endpoints

An analytical solution does not exist.

DTx values can only be found by an iterative procedure

tktk0

21 eg1egMM

or

M

eg1eg

eg1kegk

dt

dMtktk

tk2

tk1

21

21

k1 = 0.03 , k2 = 0.001 , M1 = 75k1 = 0.06 , k2 = 0.001 , M1 = 75k1 = 0.09 , k2 = 0.001 , M1 = 75k1 = 0.09 , k2 = 0.010 , M1 = 75k1 = 0.09 , k2 = 0.010 , M1 = 90

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60

Time

Con

cen

tra

tion

The bi-phasic bi-exponential model (DFOP)

Reason 3: microbial behaviour

Different environmental factors affect the activity of the microbial degraders.

Respective substrate concentration and cell density may induce very different degradation patterns

S0<<Ks S0Ks S0>>Ks

First-order Monod without growth Zero -order

S0<<X0

s

0m

K

Xk

kSdtdS

0m

s

Xk

SKkS

dtdS

0mXk

kdtdS

Logistic growth Monod with growth Logarithmic

S0>>X0

s

m

00

Kk

)SXS(kSdtdS

)SXS(SK

SdtdS

00s

m

)SXS(

dtdS

00m

S0: initial substrate concentration; X0: initial biomass concentration

A phylogeny for the disappearance models(1) Metabolism

True lag phase:

The logistic model

Equation (integrated form) Differential equation(to be used only for parameter estimation)

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t = 0

amax = Maximum value of degradation constant (reflecting microbial activity)

a0 = Initial value of degradation constant

r = Microbial growth rate

Note:

For a0 =amax (i.e. activity of degrading microorganisms is already at its maximum at the start of the experiment) the model reduces to SFO kinetics with rate constant amax

Parameters to be determined:

M0, amax, a0, r

Endpoints

r

a

)tr(00max

max0

max

]eaaa

a[MM

)tr(0max0

max0

e)aa(a

aaa

)]21(a

a1[ln

r

1DT maxa/r

0

max50

)]101(a

a1[ln

r

1DT maxa/r

0

max90

)]x100

1001(

a

a1[ln

r

1DT

maxa/r

0

maxx

Equation (integrated form) Differential equation(to be used only for parameter estimation)

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t = 0

amax = Maximum value of degradation constant (reflecting microbial activity)

a0 = Initial value of degradation constant

r = Microbial growth rate

Note:

For a0 =amax (i.e. activity of degrading microorganisms is already at its maximum at the start of the experiment) the model reduces to SFO kinetics with rate constant amax

Parameters to be determined:

M0, amax, a0, r

Endpoints

r

a

)tr(00max

max0

max

]eaaa

a[MM

)tr(0max0

max0

e)aa(a

aaa

)]21(a

a1[ln

r

1DT maxa/r

0

max50

)]101(a

a1[ln

r

1DT maxa/r

0

max90

)]x100

1001(

a

a1[ln

r

1DT

maxa/r

0

maxx

a0 = 0.0001 , r = 0.2a0 = 0.0001 , r = 0.4a0 = 0.0001 , r = 0.8a0 = 0.001 , r = 0.2a0 = 0.08 , r = 0.2

1000

20

40

60

80

100

0 20 40 60 80Time

Co

nce

ntr

atio

n

aMdt

dM

kx100

100ln

DTx

Equation (integrated form) Underlying differential equation

0MM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

k = Rate constant from t=tbtb = Breakpoint (time at which decline starts)

Endpoints

or

for ttb

Parameters to be determined

M0, k, tb

for t>tb bttk

0 eMM

0dt

dM for ttb

Mkdt

dM for t>tb

bx tk

x100100

lnDT

kx100

100ln

DTx

Equation (integrated form) Underlying differential equation

0MM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

k = Rate constant from t=tbtb = Breakpoint (time at which decline starts)

Endpoints

or

for ttb

Parameters to be determined

M0, k, tb

for t>tb bttk

0 eMM

0dt

dM for ttb

Mkdt

dM for t>tb

Equation (integrated form) Underlying differential equation

0MM

where

M = Total amount of chemical present at time t

M0 = Total amount of chemical applied at time t=0

k = Rate constant from t=tbtb = Breakpoint (time at which decline starts)

Endpoints

or

for ttb

Parameters to be determined

M0, k, tb

for t>tb bttk

0 eMM

0dt

dM for ttb

Mkdt

dM for t>tb

bx tk

x100100

lnDT

0

20

40

60

80

100

0 10 20 30 40 50 60Time

Co

nce

ntr

atio

n

Lag phase:

The Hockey stick model (HS)

A phylogeny for the disappearance models(1) Metabolism

S0<<Ks S0Ks S0>>Ks

First-order Monod without growth Zero -order

S0<<X0

s

0m

K

Xk

kSdtdS

0m

s

Xk

SKkS

dtdS

0mXk

kdtdS

Logistic growth Monod with growth Logarithmic

S0>>X0

s

m

00

Kk

)SXS(kSdtdS

)SXS(SK

SdtdS

00s

m

)SXS(

dtdS

00m

S0: initial substrate concentration; X0: initial biomass concentration