Dynamic Energy Budget Theory - I

Dynamic Energy Budget Theory - I

Tânia Sousa with contributions from : Bas Kooijman

Physical length

where is the volumetric length and the shape coefficient

What are the shape coefficients of a sphere with a diameter of and a cube with length ?

Physical volume

Weight

Measurements vs. DEB variables

𝐿=𝑉 1 /3=δ𝑀 𝐿𝑤

E E 1V V Rw E

E E

wd

E E W V Rw V E

E

d w

K 293K; 6400

}exp{)(

1

11

TTTT

TTkTk

A

AA ln ra

te

104 T-1, K-1

Daphnia magna

Metabolic rates: the effect of temperature

The Arrhenius relationship has good empirical support The Arrhenius temperature is given by minus the slope:

the higher the Arrhenius temperature the more sensitive organisms are to changes in temperature

reproductionyoung/d

ingestion106 cells/h

growth, d-1

aging, d-1

Arrhenius relationship:

The Arrhenius relationship is valid in the

temperature tolerance range At temperatures too high the organism usually dies At temperatures too low the rates are usually

lower than predicted by the Arrhenius relationship, e.g., the black bears spend the winter months in a state of hibernation. Their body temperatures drop, their metabolic rate is reduced, and they sleep for long periods.

Many extinctions are tought to be related with to changes in temperature late Pleistocene, 40,000 to 10,000 years

ago, North America lost over 50 percent of its large mammal species. These species include mammoths, mastodons, giant ground sloths, among many others.

Metabolic rates: temperature range

All parameters that have units time-1 depend

on temperature


How do feeding, assimilation, somatic maintenance and maturity maintenance powers depend on temperature?

Xm Xm 11exp A Ap p T T T T

Am Am 11exp A Ap p T T T T

11expT T A Ap p T T T T

11expM M A Ap p T T T T

1 1expJ J A Ak k T T T T

1 1exp A Av v T T T T


on temperature


Why do all these metabolic rates depend on temperature on the same way?







�̇�𝑋= �̇� 𝑋 𝐴𝜇𝑋= 𝑓 (𝑋 ) {�̇�𝑋𝑚 }𝑉 2/3

�̇�𝐴= �̇� 𝐸𝐴𝜇𝐸= 𝑓 (𝑋 ) {�̇�𝐴𝑚 }𝑉 2 /3

�̇�𝑆= �̇� 𝐸𝑆𝜇𝐸=[�̇�𝑀 ]𝑉 + {�̇�𝑇 }𝑉 2 /3

=


on temperature


Why do all these metabolic rates depend on temperature on the same way? Otherwise it would be difficult for organisms to cope with

changes in temperature (evolutionary principle)

�̇�







What is the effect of temperature on dL/dt?


𝑑𝐿𝑑𝑡 =1

3𝑘𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑘𝑚𝐸 [𝑀𝑉 ]+[𝑀𝑉 ] 𝑦𝐸𝑉


How do the von Bertallanfy growth rate and ultimate length depend on temperature?


𝑑𝐿𝑑𝑡 =1

3𝑘𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }




How do growth and catabolic powers depend on temperature?


𝑑𝐿𝑑𝑡 =1

3𝑘𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }




How do growth and catabolic powers depend on temperature?


𝑑𝐿𝑑𝑡 =1

3𝑘𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }


�̇�𝐶= �̇�𝐸𝐶𝜇𝐸=𝐸( �̇�𝐿 − �̇�)�̇�𝐺= �̇� 𝐸𝐺𝜇𝐸=[𝐸𝐺 ] 𝑑𝑉𝑑𝑡

Does the level of food that sets f(X)=1 change

with temperature?


�̇�𝑋= 𝑓 ( 𝑋 ) {�̇�𝑋𝑚 }𝑉23 ={�̇�𝑋𝑚 }𝑉

23

Does the level of food that sets f(X)=1 change

with temperature?

Yes. At a higher temperature the organism has

a higher maximum ingestion rate which means that the same absolute amount of food in the environment corresponds a lower f(x)


�̇�𝑋= 𝑓 ( 𝑋 ) {�̇�𝑋𝑚 }𝑉23 ={�̇�𝑋𝑚 }𝑉

23

At a higher temperature the organism has

a higher maximum ingestion rate which means that the same absolute amount of food in the environment corresponds a lower f(x)

What is the relationship between the reserve densities of 2 organisms (same species) living with the same absolute amount of food in the environment at different temperatures?


𝑚𝐸𝑚{ �̇� 𝐸𝐴𝑚}�̇� [𝑀𝑉 ]

( )E Emm f X m

What is the relationship between the

ultimate length of 2 organisms (same species) living with the same absolute amount of food in the environment at different temperatures?


ETE V

EM EM

Jm v ML

J J

What is the relationship between the ultimate

length of 2 organisms (same species) living with the same absolute amount of food in the environment at different temperatures?

Summarizing: Two organisms that live with the same f(X) at different

temperatures have the same ultimate length Two organisms that live with the same absolute

amount of food at different temperatures have different ultimate lengths


ETE V

EM EM

Jm v ML

J J

DEB prediction: ultimate

size does not depend on temperature

Lei de Bergmann: numa espécie que tenha uma distribuição que se extenda ao longo de diferentes latitudes as espécies com maior tamanho e mais pesadas estão junto dos polos

Lei de Bergmann (1847)

How can we explain this rule using DEB theory? At a higher temperature the same absolute amount of food in the

environment corresponds a lower f(x) Ultimate size is proportional to mE implying that for the same

absolute amounts of food the organism reaches a smaller ultimate length in higher temperatures

Ornitorrinco na Austrália

Two aspects of shape are relevant for

energetics: surface areas (acquisition processes) and volume (maintenance processes)

Shape defines how these measures relate to each other

An individual that does not change in shape during growth is an isomorph, i.e.,

In an isomorph surface area is proportional to volume2/3

Energetics: the importance of shape

𝐴1

𝐴2=𝐿1

2

𝐿22

𝑉 1

𝑉 2=𝐿1

3

𝐿23

Isomorph: surface area proportional to

volume2/3

V0-morph: surface area proportional to volume0

the dinoflagelate Ceratium with a rigid cell wall V1-morph: surface area proportional to

volume1

The cyanobacterial colony Merismopedia

Change in body shape

Chorthippus biguttulus Psammechinus miliaris

To judge weather or not an organism is isomorphic,

we need to compare shapes at different sizes. All shapes can grow isomorphically

Are these organisms isomorphic? Sphere with an increasing diameter:

Rectangle with constant width and high and an increasing length:

Energetics: the importance of shape

For non-isomorphs the surface V2/3 (the

isomorphic surface area) should be replaced by the real surface area = Where is the shape correction function volume

Prove that for: Isomorph: V0-morph: where vd is the volume at which the

surface area is equal to the surface area of an isomorph

V1-morph:

Shape correction function

𝑀 (𝑉 )=surface area / isomorphic surface area

Scales of life: the importance of size

Life span10log a

Volume10log m3

earth

whale

bacterium

water molecule

life on earth

whale

bacteriumATP molecule

30

20

10

0

-10

-20

-30

Scales of life: the importance of

size Specific oxygen consumption decreases with

body weigth in mammals

Life-span increases with weigth in mammals

Differences between species are reduced to differences

between parameters values Scaling relationships: the parameter values tend to co-

vary across species Constant Primary Parameters: There are parameters

that are similar across (related) species because they characterize biochemical processes that are similar across species: Cells of equal size have similar growth, maintenance and

maturation costs, i.e., are similar Energy partioning of energy mobilized from reserves is

done at the level of somatic and reproductive cells, i.e., is similar

Two individuals of different but related species with the same size and reserve density have similar metabolic needs, i.e., is similar

Scaling Relations I

Empirical support: Cells are very similar independent of size of the organism

Differences between species are reduced to differences

between parameters values Scaling relationships: the parameter values tend to co-

vary across species Design Primary Parameters: There are parameters that

are similar across (related) species because they characterize biochemical processes that are similar across species: Cells of equal size have similar specific maturation

thresholds, i.e., and are proportional to Lm3.

How do the following parameters vary across related species?

mEm

Scaling Relations II

- maximum length- maximum reserve density

Interspecies comparisons are done for:

Fully grown organism Abundant food f(X)=1 Null heating length LT=0

The relationship between maximum sizes is the zoom factor:

Differences between intra and terspecies comparisons:

Inter vs. Intra species comparisons

Primary parameters standard DEB model

Kooijman 1986J. Theor. Biol. 121: 269-282

Dynamic Energy Budget Theory - I

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