KINETIC MODELS Guy SOULAS UMR Œnologie-Ampélologie Université Victor Segalen Bordeaux 2 351 cours de la Libération, TALENCE Cedex France

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).

: initial substrate concentration; X initial biomass concentration A phylogeny for the disappearance models The « Metabolism » case S <<K s » K >>K First - order Monod without growth Zero <<X m X k kS dt dS = + Logistic growth Monod with growth Logarithmic >>X ) ( : initial substrate concentration; X : initial biomass concentration

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 …

Single first order kinetics (SFO) 20 40 60 80 100 Time (days) Concentration (% of initial) k = 0.005 k = 0.020 k = 0.050

But First-order reaction kinetics may not be obeyed

The bi-phasic Gustafson & Holden model (FOMC) alpha = 0.2 , beta = 5.00 alpha = 0.2 , beta = 1.00 alpha = 0.2 , beta = 0.05 alpha = 1.0 , beta = 5.00 alpha = 2.0 , beta = 5.00 Time 10 20 30 40 50 60 70 80 90 100 Concentration

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

The bi-phasic Hockey Stick model (HS) 10 20 30 40 50 60 70 80 90 100 Time Concentration k1 = 0.05 , k2 = 0.01 , tb = 10 k1 = 0.07 , k2 = 0.01 , tb = 10 k1 = 0.09 , k2 = 0.01 , tb = 10 k1 = 0.09 , k2 = 0.01 , tb = 15 k1 = 0.09 , k2 = 0.02 , tb = 15

The bi-phasic bi-exponential model (DFOP) k1 = 0.03 , k2 = 0.001 , M1 = 75 k1 = 0.06 , k2 = 0.001 , M1 = 75 k1 = 0.09 , k2 = 0.001 , M1 = 75 k1 = 0.09 , k2 = 0.010 , M1 = 75 k1 = 0.09 , k2 = 0.010 , M1 = 90 10 20 30 40 50 60 70 80 90 100 Time Concentration

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

: initial substrate concentration; X initial biomass concentration A phylogeny for the disappearance models (1) Metabolism S <<K s » K >>K First - order Monod without growth Zero <<X m X k kS dt dS = + Logistic growth Monod with growth Logarithmic >>X ) ( : initial substrate concentration; X : initial biomass concentration

True lag phase: The logistic model a0 = 0.0001 , r = 0.2 a0 = 0.0001 , r = 0.4 a0 = 0.0001 , r = 0.8 a0 = 0.001 , r = 0.2 a0 = 0.08 , r = 0.2 100 20 40 60 80 Time Concentration

The Hockey stick model (HS) Lag phase: The Hockey stick model (HS) 20 40 60 80 100 10 30 50 Time Concentration

: initial substrate concentration; X initial biomass concentration A phylogeny for the disappearance models (1) Metabolism S <<K s » K >>K First - order Monod without growth Zero <<X m X k kS dt dS = + Logistic growth Monod with growth Logarithmic >>X ) ( : initial substrate concentration; X : initial biomass concentration