FRCGC, JAMSTEC ECEM07,

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1 Comparing Optimal Uptake Kinetics and Michaelis-Menten by Assimilating Data from the SERIES Fe-Enrichment Experiment into a Marine Ecosystem Model S. Lan Smith, Naoki Yoshie 2 and Yasuhiro Yamanaka Ecosystem Change Research Program, JAMSTEC, Yokohama, Japan 2 Tohoku National Fisheries Research Inst., Fisheries Res. Agency, Shiogama, Japan, & Japan Society for the Promotion of Science Outline Brief Review of Uptake Kinetics Application to SERIES Iron fertilization expt. Introduction of the New Ecosystem Model QeNEMURO: Quota-based, variable composition Results & Conclusions S. Lan Smith, FRCGC, JAMSTEC ECEM7, Trieste

2 The Question: What Happens Between the two Extremes? At the Extremes, Consumption is determined either by Supply or by the Consumers' Preference (Ideal Ratio). Beer Before Before Before Tea "Researchers" After: All Empty!? After:? After: More Tea Left (4X as much beer as tea consumed) "Supply Limited" Intermediate "Uptake Limited" or "Kinetically Limited" Supply determines The rate expression Preference determines consumption ratio. for uptake determines consumption ratio. They "are what consumption ratio, They "drink what they they drink". depends on both. need [or like]".

3 The Michaelis-Menten (MM) Equation Uptake Rate, U(S) = Rate Expressions for Nutrient Uptake V max S [ K s + S ] U S Affinity-based Equation (Aksnes & Egge, MEPS, 99) V(S) = [ (A s S) + (V max ) ] Different Parameter, A s = Affinity More general Reduces to MM as a special case Optimal Uptake (OU) Equation (Pahlow, MEPS, 25) Extended the above equation, assuming Optimal Acclimation of N Uptake Sites more sites => Greater Affinity, A (lower K s ) Internal Enzymes more enzymes => Greater V max Ion Channels = Uptake Sites Nutrient Ions Cell Internal Enzymes Low Nutrient Conc. f A = fractional allocation of internal N: A = A f A V max = V ( f A ) Acclimation Both are mostly protein & contain lots of N. High Nutrient Conc.

4 Affinity-based Uptake & Michaelis-Menten Uptake of Nutrient, S Affinity-based V S = [ (AS S) + (V max ) ] V V S = max S [ Vmax /A S + S ] Michaelis-Menten V max S U S = [ Ks + S ] K s = V max /A S For the Opitmization Approach (Pahlow, 25) A S = f A A & V max = ( f A ) V ( f V S = A ) V S [ ( fa ) V /(f A A ) + S ] Equivalent to Michaelis-Menten with K s = ( f A ) V f A A Optimizing V S in terms of f A yields: f A = So f A depends on S + ( A S ).5 V from Pahlow (25) V max = ( f A ) V for any fixed value of f A K s & V max increase with S

5 Observed Increases in V max & K s with Substrate Concentration Pahlow's model is equivalent to MM with K s = ( f A ) V = ( Α S ).5 f A A V Total Nitrate Conc. after enrichment (mg m -3 ) from Kudela and Dugdale (DSR II, 47, 2) Data from enrichment expts. using samples from MontereyBay, California. Hyperbolic increase in V max with S, as predicted by OU kinetics. data of McCarthy et al (DSR II, 46, 999) from an observation line in the Arabian Sea.

6 Simple Phytoplankton Optimal Nutrient Gathering Equations (SPONGE) Optimize only for Limiting nutrient, L Pahlow's single-nutrient Optimal Uptake Equations: V Lim = with conc. S L f A = [( f A )V, L ] + [f A A, L S L ] ( A, L S L + V )/2, L for any Non-Limiting Nutrient, n with conc. S n Affiinity-based equation, with the same value of f A => Sub-optimal uptake of Non-limiting nutrients V non = [( f A )V, n ] + [f A A, n S n ] V non = f (S n, S L ) f A = same as above NOTE: Limiting => Growth Limiting: For a quota model, the Limiting Nutrient is determined NOT by the uptake parameters, but by the cell quotas, q, of nutrients. Growth (in terms of C or # of cells) is limited by the internal nutrient concentration: µ = µ inf ( q /q), where q is the minimum quota (Droop's quota model).

7 Reducing the SPONGE to Michaelis-Menten (MM) kinetics Optimize only for Limiting nutrient, L Pahlow's single-nutrient Optimal Uptake Equations: V Lim = with conc. S L To get MM kinetics, f A = [( f A )V, L ] + [f A A, L S L ] simply ( A set, L S L + f A = constant )/2 V, L for any Non-Limiting Nutrient, n with conc. S n Affiinity-based equation, with the same value of f A => Sub-optimal uptake of Non-limiting nutrients V non = [( f A )V, n ] + [f A A, n S n ] V non = f (S n, S L ) Affinity-based kinetics with constant coefficients f A = same as above is equivalent to MM (Aksnes & Egge, MEPS 99). NOTE: Limiting => Growth Limiting: For a quota model, the Limiting Nutrient is determined NOT by the uptake parameters, but by the cell quotas, q, of nutrients. Growth (in terms of C or # of cells) is limited by the internal nutrient concentration: µ = µ inf ( q /q), where q is the minimum quota (Droop's quota model).

8 Essence of the SPONGE: Dynamic Physiology for Efficient Nutrient Uptake Assume a fixed total amount of internal N for Uptake Hardware Uptake Hardware for each nutrient, respectively Surface Uptake Sites more sites => Greater Affinity, A (lower Ks) Internal Enzymes more enzymes => Greater Vmax Phytoplankton try to maximize uptake of the growth-limiting nutrient, without reference to concentrations of non-limiting nutrients. They allocate N for uptake hardware in the same proportion for all nutrients based only on the concentration of the growth-limiting nutrient. Low Nutrient Concentration Uptake Sites Nutrient Ions Cell Enzymes Many uptake sites, few enzymes for two nutrients, &, each with its own set of uptake sites & enzymes Both are mostly protein, contain lots of Nitrogen. High Nutrient Concentration Few uptake sites, many enzymes Cell

9 Key Points about the SPONGE SPONGE = Simple Phytoplankton Optimal Nutrient Gathering Equations Smith and Yamanaka. Optimality-based model for multinutrient uptake kinetics. Limnology & Oceanography 52: , 27 At extreme nutrient ratios, it describes uptake well: much better than Michaelis-Menten kinetics under limitation by N, P, and vitamin B2 Also agrees with data at typical nutrient ratios Has 2 parameters per nutrient (same number as Michaelis-Menten) In fact, over narrow ranges of concentrations (or ratios), it gives results very similar to Michaelis-Menten. The key is the optimization: Different behaviour for Limiting versus Non-Limiting nutrients.

10 SERIES Expt. Iron-fertilization Expt. in the NE subarctic Pacific Under Fe stress Si:N drawdown ratio increased Takeda et al (26, DSR II 53) modeled it using a modified version of the NEMURO model NEMURO assumes fixed ratios e.g., N:Si, C:N They applied two Si:N ratios for diatoms Fe-replete: Si:N = Fe-stress: Si:N = 3 & specified a value of Nutrient limitation factor for switching => Added 2 parameters to NEMURO Fe Stress & Changes in Si:N drawdown Net Growth Rate (d ) Nutrient Drawdown (µmol L d ) Molar Drawdown Ratio (Si : N) (A) (B) (C) [Si(OH) 4 ] [NO 3 - ] Time (days) Fe Stress

11 QeNEMURO model: Cell Quotas for Variable Composition Separate compartments for C, N, Fe and Si of phytoplankton 2 versions (SPONGE or MM uptake kinetics) applied to SERIES

12 Fitting Method Markov Chain Monte Carlo Method (MCMCM) J. C. Hargreaves and J. D. Annan (FRSGC), Climate Dynamics, 9, p. 37, 22. Starts with a guess for each parameter, runs the model, calculates error, Perturbs each parameter, then re-runs the model & repeats... ALWAYS accepts a parameter set with lower error & SOMETIMES accepts one with higher error (to avoid local minima) Parameters Varied (to be determined by fitting) chosen iteratively, based on Assimilations & Sensitivity Analyses Nutrient Uptake Rate Parameters 8 Grazing Rate (Large Zoo grazing diatoms) total no. 9. Fit each version (SPONGE & MM) of QeNEMURO to all data (IN & OUT of Fe-patch) 2. Compare fits to data (concentrations) & also modeled values of material flows (e.g., uptake rates), phytoplankton composition S. Lan Smith, FRSGC ECRP

13 Comparing Best-fits to all data The two versions of the model are the same, except for the uptake kinetics. MM version SPONGE version The model is -D (Mixed-layer only). Both versions fit the data well Vertical bars are Std. Deviations as assumed for weights in the fitting. These data are assumed to be averages over the mixed-layer. Diatoms Growth Rate IN Patch Specific Growth Rate (h - ) Time (days) µm µm 5 5 µg/l QeNEMURO model IN Patch OUT of Patch NH4 Chl Nitrate SiOH Time (Days) NH4 Chl Nitrate SiOH 4

14 Best-fits to ALL data (IN & OUT) Cell Quotas of Nutrients in QeNEMURO Comparing Simulations IN & OUT of fertilized patch. Cell Quotas are different. MM version SPONGE version Despite very similar fits to nutrients & Chlorophyll No Data Fe:C N:C (µmol : mol) (mol : mol) IN Patch a b OUTSIDE d e Which model is correct? Or are they both wrong? It would be very nice to have Observations of Cell Quotas! or at least bulk ratios in POM Si:C (mol : mol) c 2 2 Time (Days).5..5 f S. Lan Smith, FRCGC October, 27

15 Best-fits to ALL data (IN & OUT) Comparing Simulations IN & OUT of fertilized patch. Uptake Rates are different. MM version SPONGE version Despite very similar fits to nutrients & Chlorophyll In General, compared to MM kinetics, the SPONGE gives: faster uptake for limiting nutrient except for Fe OUTSIDE patch slower uptake for non-limiting except for Si OUTSIDE patch Nutrient Uptake Rates in QeNEMURO IN Patch OUTSIDE N limitation Time (Days) Fe Fe limitation S. Lan Smith, FRCGC October, 27 Fe N Si Si uptake (μmol L - day - ) N uptake (μmol L - day - ) Fe uptake (μmol L - day - )

16 Best-fits to ALL data (IN & OUT) Comparing Simulations IN & OUT of fertilized patch. Uptake Rates are different. MM version SPONGE version Despite very similar fits to nutrients & Chlorophyll In General, compared to MM kinetics, the SPONGE gives: faster uptake for limiting nutrient except for Fe OUTSIDE patch slower uptake for non-limiting except for Si OUTSIDE patch Nutrient Uptake Rates in QeNEMURO IN Patch OUTSIDE N limitation Time (Days) Fe Fe limitation S. Lan Smith, FRCGC October, 27 Fe N Si Si uptake (μmol L - day - ) N uptake (μmol L - day - ) Fe uptake (μmol L - day - )

17 Conclusions The QeNEMURO model fits the data from SERIES well, using either Michaelis-Menten kinetics or the SPONGE at least in this -D (box) model there were no constraints on fluxes or vertical profiles We cannot say which is better based only on the data from SERIES. Very different Nutrient Uptake Rates and Compositions for Phytoplankton! We need more DATA to test the SPONGE vs. MM kinetics Vertical profiles of nutrients & organic matter with simultaneous measurements for multiple elements Large-scale distributions, with wide ranges of concentrations Longer time series Although Michaelis-Menten kinetics has been the standard for years, Optimal Uptake (incl. SPONGE) has the advantage theoretically and when compared to laboratory (chemostat) experiments.