Aquatic Sciences Meeting, July 9, 2012S. Lan Smith
Affinity: a clearly superior alternative to the obsolete obfuscation known as the
‘half-saturation constant’
S. Lan SmithEnvironmental Biogeochemical Cycles Research Program
Research Institute for Global ChangeJapan Agency for Marine-Earth Science & Technology
Yokohama, Japan
ASLO Aquatic Sciences Meeting 2012 Lake Biwa, Japan
Aquatic Sciences Meeting, July 9, 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
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 3
Affinity-based
VAff = VmaxaS
Vmax + aS
Affinity and Ks are related:
a = Vmax Ks
the initial slope of the MM eq.is a better measure of competitive ability than either Vmax or Ks alone (Button Deep-Sea Res. 25, 1978; Healey Microb. Ecol. 5, 1980).
They’re really the same shape.
Michaelis-Menten/ Monod
VMM = VmaxS Ks + S
Aquatic Sciences Meeting, July 9, 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
Aquatic Sciences Meeting, July 9, 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.
Aquatic Sciences Meeting, July 9, 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)
Aquatic Sciences Meeting, July 9, 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.”
Ks is NOT a physiological trait.
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 8
Mathematical relationship 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
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 9
No overall relationship between Vmax & Ks
Only 2 significant intra-species rels.
Strong overall 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
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 10
Species that compete better at low nutrient concentrations also tend to compete better at higher concentra-tions.
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
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 11
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.
Aquatic Sciences Meeting, July 9, 2012S. Lan Smith p. 12
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 necessarily identify 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