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