affinity: the meaningful trait-based alternative to the half-saturation constant
DESCRIPTION
Presentation at the Japan Oceanographic Society Meeting, September, 2012TRANSCRIPT
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith
Affinity: the meaningful alternative to the ‘half-saturation constant’
S. Lan Smith, James D. Annan, and Julia C. HargreavesResearch Institute for Global Change
Japan Agency for Marine-Earth Science & TechnologyYokohama, Japan
JOS Autumn Meeting 2012 Shimizu, Japan
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 2
Michaelis-Menten / Monod (Michaelis & Menten 1913, Monod 1942, Dugdale 1967)
VMM = VmaxS Ks + S
Affinity-based (Button & Robertson 1989, Aksnes & Egge 1991)
VAff = VmaxaS
Vmax + aS
Two Equations for the Same Curve
a is just the initial slope,
which is determines competitive ability at low nutrient concentrations (Healey. Micrbial Ecology 1980).
Ks defines the concentration at which rate is half-saturated.
Vmax is the maximum uptake rate.
Concentration, S
Vmax
α
V A
Concentration, S
Vmax
Ks
V MM
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 3
Affinity-based
VAff = VmaxaS
Vmax + aS
Affinity and Ks are related:
a = Vmax Ks
The initial slope, a, of the MM eq.measures competitive ability at lownutrient concentrations, but neither Vmax nor Ks alone does so. (Button Deep-Sea Res. 25, 1978; Healey Microb. Ecol. 5, 1980).
They’re really the same shape.
Michaelis-Menten/ Monod
VMM = VmaxS Ks + S
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 4
What difference does this make?
0.0
0.5
1.0
1.5
0 5 10 15
−0.4
−0.2
0.0
0.2
0.4
nutrient concentration (mol m-3)
fract
iona
l diff
eren
ce
R
ate
(d-1
)
MM / Monod equation
0.0
0.5
1.0
1.5
−0.4
−0.2
0.0
0.2
0.4
0 5 10 15nutrient concentration (mol m-3)
fract
iona
l diff
eren
ce
R
ate
(d-1
)
Affinity-based equation
Effect of varying only Vmax
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 5
What difference does this make?
0.0
0.5
1.0
1.5
0 5 10 15
−0.4
−0.2
0.0
0.2
0.4
nutrient concentration (mol m-3)
fract
iona
l diff
eren
ce
R
ate
(d-1
)
MM / Monod equation
0.0
0.5
1.0
1.5
−0.4
−0.2
0.0
0.2
0.4
0 5 10 15nutrient concentration (mol m-3)
fract
iona
l diff
eren
ce
R
ate
(d-1
)
Affinity-based equation
Effect of varying only Vmax
Changing Vmax has the same effect at low & high nutrient concentrations.
Model response is more sensitive to Vmax.
=> after tuning Vmax must tune Ks too.
This may also cause poor perfor-mance for some data assimilation alogirthms.
Changing Vmax has no effect at low nutrient concentrations.
Model response is comparatively less sensitive to Vmax.
=> Vmax & a can be tuned separately.
Easier to tune models.
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 6
But does a positive Vmax vs. Ks relationship reveal a trade-off? Affinity, not Ks, quantifies competitive ability at low nutrients. So, let’s transform the data: a = Vmax Ks
Trade-off or Not Trade-off?from Litchman et al. (Ecology Letters 10, 2007) per cell basis vs. per mol C basis
Fig. 1a,b of Litchman et al. (Ecol. Lett. 10:1170-1181, 2007)
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 7
There is no Trade-off!
Positive relationship between Vmax and a per cell basis per mol C basis
α (L (μmol C)-1 d-1)1e−04 0.01 1
0.001
0.01
0.1
1
10
V max
(μm
ol (μ
mol
C)-1
d-1
)
α (L cell-1 d-1)
V max
(μm
ol c
ell-1
d-1
)
1e−08 1e−06 1e−04
1e−09
1e−07
1e−05
0.001
r2 = 0.92, p < 0.001
r2 = 0.80, p < 0.001
Data from Litchman et al. (EL 2007, Fig. 1ab), transformed to affinity.
This constrasts with the following from Litchman et al. (2007): “Significant positive correlations between ... Vmax and K found in our data analysis imply inherent physiological trade-offs between these physiological traits.”
But Ks is NOT a physiological trait!
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 8
The Mathematical relationship alone implies correlations
V max
(μm
ol (μ
mol
C)-1
d-1
)
α (L (μmol C)-1 d-1)1e−04 0.01 1
0.001
0.01
0.1
1
10
V max
(μm
ol (μ
mol
C)-1
d-1
)
0.001
0.01
0.1
1
10
0.1 1.0 10
Kn (μmol L-1)
α (L cell-1 d-1)
V max
(mm
ol c
ell-1
d-1
)
1e−08 1e−06 1e−04
1e−09
1e−07
1e−05
0.001
V max
(mm
ol c
ell-1
d-1
)
Kn (μmol L-1)
1e−09
1e−07
1e−05
0.001
0.1 1.0 10
red dots transformed
red dots generated as independent Gaussian variables, same mean & s.d. as data
red dots generated as independent Gaussian variables, same mean & s.d. as data
log-log slope = 0.66
less steep than in the data, slope = 2.3
log-log slope = 0.76
the same as for data, slope = 0.71 +/- 0.09
a = Vmax Kn
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 9
No overall relationship between Vmax & Ks
Only 2 significant intra-species rels.
Strong overall positive relationship between Vmax & a
4 significant intra-species rels., all positive
No Trade-off.
An independent data setDauta (Ann. Limnol. 18:263–292,1982) measured nitrate uptake parameters for 8 species, each at various temperatures
0.2 0.5 2.0 5.0 20.0 50.0
1
2
5
10
20
50
100
200
V max
(μg
atom
s N (1
09 cel
ls h
)-1)
Kn (mmol m-3)
V max
(μg
atom
s N (1
09 cel
ls h
)-1)
α (m3 μg atoms N (mmol 109 cells h)-1)
0.5 1.0 2.0 5.0
1
2
5
10
20
50
100
200
Anabaena cylindricaCoelastrum microsporumDictyosphaerium pulchellumFragillaria bidensPediastrum boryanumMonoraphidium minutumScenedesmus crassusScenedesmus quadricauda
a = Vmax Kn
Transforming as before to affinity.
r2 = 0.89, p < 0.001
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 10
‘Half-saturation’... but half of what? Ks alone does not tell us.
Using the power-law relationship from Litchman et al. (Ecology Letters 2007)
Vmax = 6 x 10-7 Kn2.8
nutrient conc. (μM)
Upt
ake
Rat
e (d
-1)
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0Small PhyKs = 1
Large PhyKs = 2
Vmax = 1for both
Small Phywins at low nutrientconc.
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
nutrient conc. (μM)U
ptak
e R
ate
(d-1
)
Small PhyKs = 1Vmax = 1
Large PhyKs = 2Vmax = 22.8 = 7
Species with lower Ks will grow faster at low nutrient concentrations,
If both species have the same Vmax
“Half of what?” really matters! Now Large Phy wins at low nutrient concentrations, despite its greater Ks, be-cause of much greater Vmax.
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 11
In this data set, species that com-pete better at low nutrient concen-trations also tend to compete better at higher concentrations.
Strong overall relationship between Vmax & aHere the log-log slope = 0.57
No Trade-off.
What does this mean in terms of the response?
0.2 0.5 2.0 5.0 20.0 50.0
1
2
5
10
20
50
100
200
V max
(μg
atom
s N (1
09 cel
ls h
)-1)
α (m3 μg atoms N (mmol 109 cells h)-1)
r2 = 0.89, p < 0.001
0
20
40
60
80
100Rate vs. Concentration Response
nutrient concentration
Rat
e
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 12
log NO3 (in seawater)lo
g K N
O3
-2.5 -1.0 0.0
-3
-2
-1
0
n = 61 data pts.
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 0 200 400 600 800 1000
Upt
ake
Rat
e
NO3 in incubation expts.
Adaptive Response
Smith et al. (MEPS 2009)
Trade-off
V maxα
But, Optimal Uptake kinetics IS based on a trade-off : Vmax vs. a
OU kinetics predicts a shape-changing response in short-term expts., i.e., MM param-eters that depend on nutrient concentration.
This does NOT imply a universal negative relation-ship between Vmax & a.
Low Nutrient Conc. High Nutrient Conc.
This physiological trade-off was postulated specifically for accli-mation (or adaptation) to ambient nutrient concentrations.
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 13
PO4 (mmol m-3)
Gro
wth
Rat
e (d
-1)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.0
0.5
1.0
1.5
2.0
Small Phy
Large Phy
Relevance for Parameterizing Trade-offs in Models
KPO4 (mmol m-3)
Max
. Gro
wth
Rat
e (d
-1)
Small Phy
Large Phy
log-log regression liner2 = 0.65 (n = 16) p < 0.001
Max
. Gro
wth
Rat
e (d
-1)
αPO4 (d-1 (mmol m-3)-1)
log-log regression liner2 = 0.17 (n = 16)p < 0.07
0.01 0.02 0.05
1.0
1.5
2.0
2.5
20 40 60 80 100 120
1.0
1.5
2.0
2.5
For example, Follows et al. (Science 2007) simulated many different phytoplankton species, and allowed the model environment to select the winners. They chose the parameters for diffwerent species based on trade-offs in terms of Monod kinetics, i.e., using half-saturation constants.
Re-drawn from their supplementary material:
In terms of Kp this looks like a strong trade-off.
But in terms of affinity, it is clear that there is a great deal of overlap and only a weak nega-tive relationship.
It would be easier to pa-rameterize trade-offs clear-ly and correctly in terms of affinity, rather than in terms of Ks.
Japan Oceanographic Society Meeting, September 16, 2012S. Lan Smith p. 14
Affinity-based kinetics clearly separates the traits relevant at high vs. low nutrient concentrations. This makes it easier to tune models & interpret results, compared to MM/Monod kinetics using Ks.
A postive relationship between Vmax & Ks does NOT constitute a trade-off. Analyses in terms of Ks have ‘found’ trade-offs where none exist.
Affinity, a, as a trait-based quantity, more clearly and simply reveals relationships between kinetic parameters.
Affinity is a better choice for modeling trade-offs and their impact on large-scale biodiversity & biogeochemistry, as in e.g., Follows & Dutkiewicz (Ann. Rev. Mar. Sci. 2011) & Smith et al. (L&O 2011).
Conclusions
Ks