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