Comparing Optimal Uptake Kinetics and Michaelis-Menten by …doga.ogs.trieste.it/doga/echo/ecem07/Main/30Nov/Smith_SL.pdf · 1 Ecosystem Change Research Program, JAMSTEC, Yokohama,

S. Lan Smith, FRCGC, JAMSTEC ECEM07, Trieste

Comparing Optimal Uptake Kinetics and Michaelis-Menten by Assimilating Data from the SERIES Fe-Enrichment Experiment into a Marine Ecosystem Model

S. Lan Smith1, Naoki Yoshie2 and Yasuhiro Yamanaka1

1 Ecosystem Change Research Program, JAMSTEC, Yokohama, Japan2 Tohoku National Fisheries Research Inst., Fisheries Res. Agency, Shiogama, Japan, & Japan Society for the Promotion of Science

OutlineBrief Review of Uptake KineticsApplication to SERIES Iron fertilization expt.Introduction of the New Ecosystem Model QeNEMURO: Quota-based, variable compositionResults & Conclusions

S. Lan Smith, FRCGC ECRP

"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]".

At the Extremes, Consumption is determined either by Supply or by the Consumers' Preference (Ideal Ratio).

Beer

Tea

Before

After:

All Empty!

"Researchers"

Before

After:More Tea Left(4X as much beer as tea consumed)

Before

After: ??

The Question: What Happens Between the two Extremes?


1V(S) = [ (As S)−1 + (Vmax)−1 ]

Different Parameter, As = AffinityMore generalReduces to MM as a special case

Affinity-based Equation (Aksnes & Egge, MEPS, 1991)

Optimal Uptake (OU) Equation (Pahlow, MEPS, 2005) Extended the above equation, assuming Optimal Acclimation of N

Uptake Sites more sites => Greater Affinity, A (lower Ks)Internal Enzymes more enzymes => Greater Vmax

Both are mostly protein& containlots of N.

US

The Michaelis-Menten (MM) Equation

Uptake Rate, U(S) = [ Ks + S ]Vmax S

Rate Expressions for Nutrient Uptake

fA = fractional allocation of internal N: A = A0 fA Vmax= V0 (1 − fA)

Acclimation

NutrientIons

Ion Channels= Uptake Sites

Cell

Internal Enzymes

Low Nutrient Conc. High Nutrient Conc.


Ks = (1 − fA) V0

For the Opitmization Approach (Pahlow, 2005)

AS = fA A0 & Vmax = (1 − fA) V0

VS = [ (1 − fA) V0 /(fAA0) + S ]Vmax = (1 − fA) V0

Ks = Vmax/AS

fA A0

Vmax SVS = [ (ASS)−1 + (Vmax)−1 ]

VS = [ Vmax /AS + S ]

US = [ Ks + S ]1

Affinity-based Michaelis-Menten

Vmax S

(1 − fA) V0 S

Equivalent to Michaelis-Menten with

for any fixed value of fA

Optimizing VS in terms of fA yields:

fA = 1 + (A0S)0.5 1

V0

So fA depends on S

Ks & Vmax increase with Sfrom Pahlow (2005)

Affinity-based Uptake & Michaelis-Menten Uptake of Nutrient, S


Pahlow's model is equivalent to MM with

Ks = (1 − fA) V0 = (Α0S)0.5

fA A0 V0

data of McCarthy et al (DSR II, 46, 1999)from an observation line in the Arabian Sea.

0 10 20 30 40 50 60

Total Nitrate Conc. after enrichment (mg m-3)

from Kudela and Dugdale (DSR II, 47, 2000) Data from enrichment expts. using samples from MontereyBay, California.

Hyperbolic increase in Vmax with S, as predicted by OU kinetics.

Observed Increases in Vmax & Ks with Substrate Concentration


Simple Phytoplankton Optimal Nutrient Gathering Equations (SPONGE)

Optimize only for Limiting nutrient, L with conc. SL Pahlow's single-nutrient Optimal Uptake Equations:

VLim = 1 fA = 1

[(1−fA)V0, L ]−1 + [fAA0, LSL]−1

(A0, LSL)1/2 + 1 V0, L

for any Non-Limiting Nutrient, n with conc. Sn Affiinity-based equation, with the same value of fA => Sub-optimal uptake of Non-limiting nutrients

Vnon= 1 fA = same as above

[(1−fA)V0, n ]−1 + [fAA0, nSn]−1

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 = µinf (1 − q0/q), where q0 is the minimum quota (Droop's quota model).

Vnon = f (Sn, SL)


Reducing the SPONGE to Michaelis-Menten (MM) kinetics

Optimize only for Limiting nutrient, L with conc. SL Pahlow's single-nutrient Optimal Uptake Equations:

VLim = 1 fA = 1

[(1−fA)V0, L ]−1 + [fAA0, LSL]−1

(A0, LSL)1/2 + 1 V0, L

for any Non-Limiting Nutrient, n with conc. Sn Affiinity-based equation, with the same value of fA => Sub-optimal uptake of Non-limiting nutrients

Vnon= 1 fA = same as above

[(1−fA)V0, n ]−1 + [fAA0, nSn]−1

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 = µinf (1 − q0/q), where q0 is the minimum quota (Droop's quota model).

Vnon = f (Sn, SL )

To get MM kinetics, simply set fA = constant

Affinity-based kineticswith constant coefficientsis equivalent to MM (Aksnes & Egge, MEPS 1991).


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 High Nutrient Concentration

Many uptake sites, few enzymes Few uptake sites, many enzymes

for twonutrients, & ,each withits own set ofuptake sites& enzymes

Both are mostlyprotein, contain lots of Nitrogen.

Cell

EnzymesNutrientIons

Uptake SitesCell

Essence of the SPONGE: Dynamic Physiology for Efficient Nutrient Uptake


SPONGE = Simple Phytoplankton Optimal Nutrient Gathering Equations

Smith and Yamanaka. Optimality-based model for multinutrient uptake kinetics. Limnology & Oceanography 52: 1545-1558, 2007

At extreme nutrient ratios, it describes uptake well: much better than Michaelis-Menten kinetics under limitation by N, P, and vitamin B12

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.

Key Points about the SPONGE


0.01.0

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-]

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Iron-fertilization Expt. in the NE subarctic PacificUnder Fe stress Si:N drawdown ratio increased

Takeda et al (2006, DSR II 53) modeled it using a modifiedversion 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 = 1 Fe-stress: Si:N = 3& specified a value ofNutrient limitation factor for switching

=> Added 2 parameters to NEMURO

Net Growth

Rate (d−1)

NutrientDrawdown

(µmol L−1 d−1)

MolarDrawdown

Ratio(Si : N)

Fe Stress SERIES Expt. Fe Stress & Changes in Si:N drawdown


QeNEMURO model: Cell Quotas for Variable Composition

2 versions (SPONGE or MM uptake kinetics) applied to SERIES

Separate compartments for C, N, Fe and Si of phytoplankton

S. Lan Smith, FRSGC ECRP

1. 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

Nutrient Uptake Rate Parameters 8Grazing Rate (Large Zoo grazing diatoms) 1 total no. 9

Fitting Method

Parameters Varied (to be determined by fitting) chosen iteratively, based on Assimilations & Sensitivity Analyses

Markov Chain Monte Carlo Method (MCMCM)J. C. Hargreaves and J. D. Annan (FRSGC), Climate Dynamics, 19, p. 371, 2002.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)


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SiOH4

Nitrate Nitrate

NH4 NH4

Chl

SiOH4

Chl

µM

µM

µg/L

IN Patch OUT of Patch

MM version SPONGE version

The two versions of the model are the same, except for the uptake kinetics.

The model is 0-D (Mixed-layer only).

Both versions fit the data well

Vertical bars are Std. Deviationsas assumed for weights in the fitting.

These data areassumed to be averages over the mixed-layer.

Comparing Best-fits to all dataQeNEMURO model

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Spec

ific

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Diatoms Growth Rate IN Patch

S. Lan Smith, FRCGC October, 2007

Comparing Simulations

IN & OUT of fertilized patch.

Cell Quotas are different.


Despite very similar fits to nutrients & Chlorophyll

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

IN Patch OUTSIDENo Data

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Fe:C

N:C

(µm

ol :

mol

)(m

ol :

mol

)(m

ol :

mol

)

Best-fits to ALL data (IN & OUT)Cell Quotas of Nutrients in QeNEMURO


Comparing Simulations IN & OUT of fertilized patch.

Uptake Rates are different.



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

IN Patch OUTSIDE

N limitation

Fe limitationFe

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ptak

e (μ

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ay-1

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upta

ke (μ

mol

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upta

ke (μ

mol

L-1 d

ay-1

)0 10 20 0 10 20

Time (Days)

Si

Fe

N

Best-fits to ALL data (IN & OUT) Nutrient Uptake Rates in QeNEMURO


Comparing Simulations IN & OUT of fertilized patch.

Uptake Rates are different.



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

IN Patch OUTSIDE

N limitation

Fe limitationFe

0

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ptak

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mol

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upta

ke (μ

mol

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)

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upta

ke (μ

mol

L-1 d

ay-1

)0 10 20 0 10 20

Time (Days)

Si

Fe

N

Best-fits to ALL data (IN & OUT) Nutrient Uptake Rates in QeNEMURO


The QeNEMURO model fits the data from SERIES well, using either Michaelis-Menten kinetics or the SPONGE at least in this 0-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.

Conclusions

Documents

Comparing Optimal Uptake Kinetics and Michaelis-Menten by …doga.ogs.trieste.it/doga/echo/ecem07/Main/30Nov/Smith_SL.pdf · 1 Ecosystem Change Research Program, JAMSTEC, Yokohama,