1) The document presents an argument that affinity, represented by the variable "a", is a superior metric to the half-saturation constant "Ks" for modeling nutrient uptake kinetics in aquatic systems. Affinity separates the traits relevant for uptake at high versus low nutrient concentrations in a clearer way. 2) Analysis of multiple data sets shows relationships between maximum uptake rate (Vmax) and affinity, but these relationships do not necessarily indicate a physiological trade-off as relationships between Vmax and Ks had been interpreted. In some cases there was a strong positive correlation between Vmax and affinity. 3) Adopting affinity over Ks allows models to be more easily tuned and better reveals relationships between kinetic parameters,

1. ASLO Aquatic Sciences Meeting 2012 Lake Biwa, Japan
Affinity: a clearly superior alternative
to the obsolete obfuscation known as the
‘half-saturation constant’
S. Lan Smith
Environmental Biogeochemical Cycles Research Program
Research Institute for Global Change
Japan Agency for Marine-Earth Science & Technology
Yokohama, Japan
S. Lan Smith Aquatic Sciences Meeting, July 9, 2012

2. Two Equations for the Same Curve
Affinity-based Michaelis-Menten / Monod
(Button & Robertson 1989, (Michaelis & Menten 1913,
Aksnes & Egge 1991) Monod 1942, Dugdale 1967)
VmaxaS VmaxS
VAff = VMM =
Vmax + aS Ks + S
Vmax Vmax
VMM
VA
Ks
α Concentration, S Concentration, S
a is just the initial slope, Ks defines the concentration at
which rate is half-saturated.
which is determines competitive
ability at low nutrient concentrations Vmax is the maximum uptake rate.
(Healey. Micrbial Ecology 1980).
S. Lan Smith p. 2 Aquatic Sciences Meeting, July 9, 2012

3. They’re really the same shape.
Affinity-based Michaelis-Menten/ Monod
VmaxaS VmaxS
VAff = VMM =
Vmax + aS Ks + S
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).
S. Lan Smith p. 3 Aquatic Sciences Meeting, July 9, 2012

4. What difference does this make?
Effect of varying only Vmax
Affinity-based equation MM / Monod equation
1.5 1.5
Rate (d-1)
Rate (d-1)
1.0 1.0
0.5 0.5
0.0 0.0
fractional difference
fractional difference
0.4
0.4
0.2 0.2
0.0 0.0
−0.2 −0.2
−0.4 −0.4
0 5 10 15 0 5 10 15
nutrient concentration (mol m-3) nutrient concentration (mol m-3)
S. Lan Smith p. 4 Aquatic Sciences Meeting, July 9, 2012

5. What difference does this make?
Effect of varying only Vmax
Affinity-based equation MM / Monod equation
1.5 1.5
Rate (d-1)
Rate (d-1)
1.0 1.0
0.5 0.5
0.0 0.0
fractional difference
fractional difference
0.4
0.4
Changing Vmax has no effect at low Changing Vmax has the same effect at
0.2
nutrient concentrations. low & 0.2
high nutrient concentrations.
0.0
Model response is comparatively less Model0.0
response is more sensitive
sensitive to Vmax.
−0.2 to Vmax.
−0.2
=> Vmax & a can be tuned separately.
−0.4 => after tuning Vmax must tune Ks too.
−0.4
Easier to tune models. 10
0 5 15 This may 0 also cause poor perfor-
5 10 15
nutrient concentration (mol m-3) -3
mance nutrient concentration (mol m )
for some data assimilation
alogirthms.
S. Lan Smith p. 5 Aquatic Sciences Meeting, July 9, 2012

6. 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)
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
S. Lan Smith p. 6 Aquatic Sciences Meeting, July 9, 2012

7. There is no Trade-off!
Positive relationship between Vmax and a
per cell basis per mol C basis
0.001 10
Vmax (μmol (μmol C)-1 d-1)
r2 = 0.92, r2 = 0.80,
Vmax (μmol cell-1 d-1) p < 0.001 1
p < 0.001
1e−05
0.1
1e−07
0.01
1e−09 0.001
1e−08 1e−06 1e−04 1e−04 0.01 1
α (L cell-1 d-1) α (L (μmol C)-1 d-1)
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.
S. Lan Smith p. 7 Aquatic Sciences Meeting, July 9, 2012

8. Mathematical relationship implies correlations
Vmax (μmol (μmol C)-1 d-1)
0.001 10
red dots
Vmax (mmol cell-1 d-1)
log-log 1
generated as
slope = 0.66 1e−05 independent
less steep than 0.1 Gaussian
in the data, 1e−07
variables,
slope = 2.3 0.01 same mean &
s.d. as data
1e−09 0.001
0.1 1.0 10 0.1 1.0 10
Kn (μmol L-1) Kn (μmol L-1)
red dots Vmax
transformed
a=
Kn
Vmax (μmol (μmol C)-1 d-1)
0.001 10
Vmax (mmol cell-1 d-1)
red dots log-log
generated as 1e−05
1
slope = 0.76
independent
the same as for data,
Gaussian
0.1
slope = 0.71 +/- 0.09
variables, 1e−07
0.01
same mean &
s.d. as data 1e−09 0.001
1e−08 1e−06 1e−04 1e−04 0.01 1
α (L cell-1 d-1) α (L (μmol C)-1 d-1)
S. Lan Smith p. 8 Aquatic Sciences Meeting, July 9, 2012

9. An independent data set
Dauta (Ann. Limnol. 18:263–292,1982)
measured nitrate uptake parameters for 8 species, each at various temperatures
Vmax (μg atoms N (109 cells h)-1)
200 No overall relationship between
100 Vmax & Ks
50
20
10 Anabaena cylindrica
Only 2 significant intra-species rels.
Coelastrum microsporum
5 Dictyosphaerium pulchellum
Fragillaria bidens
Transforming
Pediastrum boryanum
2 Monoraphidium minutum
Scenedesmus crassus
1 Scenedesmus quadricauda as before to
0.5 1.0
Kn (mmol
2.0
m-3)
5.0
affinity.
Vmax (μg atoms N (109 cells h)-1)
200
r2 = 0.89, Strong overall relationship between
100
p < 0.001 Vmax & a
50
Vmax
20
a=
10 Kn 4 significant intra-species rels.,
5
all positive
2
1
0.2 0.5 2.0
α
5.0 20.0 50.0
No Trade-off.
(m3 μg atoms N (mmol 109 cells h)-1)
S. Lan Smith p. 9 Aquatic Sciences Meeting, July 9, 2012

10. What does this mean in terms of the response?
Rate vs. Concentration Response
100
80
Species that compete better at low
60
nutrient concentrations also tend to
Rate
40
20
compete better at higher concentra-
0
tions.
nutrient concentration
Vmax (μg atoms N (10 cells h) )
-1
200
Strong overall relationship between
100
Vmax & a
50
9
Here the log-log slope = 0.57
20
10
5
r2 = 0.89,
2 p < 0.001
1
0.2 0.5 2.0 5.0 20.0 50.0
α No Trade-off.
(m3 μg atoms N (mmol 109 cells h)-1)
S. Lan Smith p. 10 Aquatic Sciences Meeting, July 9, 2012

11. But, Optimal Uptake kinetics is based on a trade-off : Vmax vs. a
This does NOT imply a OU kinetics predicts a shape-changing
universal negative relation- response in short-term expts., i.e., MM param-
ship between Vmax & a. eters that depend on nutrient concentration.
Trade-off Adaptive Response
This physiological 1.0
trade-off was postulated 0.8
Uptake
Rate
0.6
specifically for accli-
Vmax
0.4
0.2
mation (or adaptation) 0.0
0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000
to ambient nutrient NO3 in incubation expts.
concentrations. α
0
-1
log KNO3
-2
Low Nutrient Conc. High Nutrient Conc. n = 61 data pts.
-3
-2.5 -1.0 0.0
Smith et al. (MEPS 2009) log NO3 (in seawater)
S. Lan Smith p. 11 Aquatic Sciences Meeting, July 9, 2012

12. Conclusions
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).
Ks
S. Lan Smith p. 12 Aquatic Sciences Meeting, July 9, 2012