Affinity: the meaningful trait-based alternative to the half-saturation constant

JOS Autumn Meeting 2012 Shimizu, Japan

Affinity: the meaningful alternative
to the ‘half-saturation constant’

S. Lan Smith, James D. Annan, and Julia C. Hargreaves
Research Institute for Global Change
Japan Agency for Marine-Earth Science & Technology
Yokohama, Japan

S. Lan Smith Japan Oceanographic Society Meeting, September 16, 2012

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 Japan Oceanographic Society Meeting, September 16, 2012

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, a, of the MM eq.
measures competitive ability at low
nutrient concentrations, but
neither Vmax nor Ks alone does so.
(Button Deep-Sea Res. 25, 1978;
Healey Microb. Ecol. 5, 1980).


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

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)


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

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.


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


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.”
But Ks is NOT a physiological trait!

The Mathematical relationship alone implies correlations

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

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)

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 positive relationship
100
p < 0.001 between 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)


‘Half-saturation’... but half of what? Ks alone does not tell us.

Species with lower Ks will grow faster Using the power-law relationship from
at low nutrient concentrations, Litchman et al. (Ecology Letters 2007)
If both species have the same Vmax Vmax = 6 x 10-7 Kn2.8

1.0 7
Small Phy Vmax = 1 Large Phy

Uptake Rate (d-1)
Uptake Rate (d-1)

6
0.8 Ks = 1 for both Ks = 2
5

0.6 Vmax = 22.8 = 7
Small Phy 4

0.4
Large Phy wins at low 3
Ks = 2 nutrient 2 Small Phy
conc.
0.2
1 Ks = 1
0.0 0 Vmax = 1
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

nutrient conc. (μM) nutrient conc. (μM)

“Half of what?” really matters! Now Large Phy wins at low
nutrient concentrations,
despite its greater Ks, be-
cause of much greater Vmax.

What does this mean in terms of the response?
Rate vs. Concentration Response
100

80
In this data set, species that com-
60
pete better at low nutrient concen-
Rate

40

20
trations also tend to compete better
0
at higher concentrations.
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)


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)

Relevance for Parameterizing Trade-offs in Models
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:

Max. Growth Rate (d-1)
2.5
Large Phy
In terms of Kp this
Large Phy
Growth Rate (d-1)

2.0
2.0
looks like a strong
1.5
1.5 trade-off.
1.0
Small Phy Small Phy
1.0
log-log regression line
0.5
r2 = 0.65 (n = 16)
0.0
p < 0.001
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.01 0.02 0.05

PO4 (mmol m-3) KPO4 (mmol m-3)

But in terms of affinity,
Max. Growth Rate (d-1)

2.5

It would be easier to pa- 2.0
it is clear that there is
rameterize trade-offs clear-
a great deal of overlap
ly and correctly in terms
1.5

and only a weak nega-
of affinity, rather than in log-log regression line
1.0
r2 = 0.17 (n = 16) tive relationship.
terms of Ks. p < 0.07
20 40 60 80 100 120

αPO4 (d-1 (mmol m-3)-1)


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 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).

Ks

Affinity: the meaningful trait-based alternative to the half-saturation constant

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Affinity: the meaningful trait-based alternative to the half-saturation constant