Dynamic Energy Budget Theory - I

Tânia Sousa with contributions from : Bas Kooijman

Metabolism in a DEB

individual. Rectangles are state

variables Arrows are flows of food

JXA, reserve JEA, JEC, JEM, JET , JEG, JER, JEJ or structure JVG.

Circles are processes The full square is a fixed

allocation rule (the kappa rule)

The full circles are the priority maintenance rule.

A DEB organism: growth

MV - Structure

Feeding

MH - Maturity

XAJ EAJ

Assimilation

ME - ReserveMobilisation

ECJ

Offspring MER

Somatic Maintenance

Growth

Maturity Maintenance

Reproduction

Maturation

ESJ

EGJEJJ

ERJ

Von Bertallanffy growth in DEB theory

DEB theory predicts: decreases with specific maintenance needs and

increases with the reserve density (food level) decreases with

𝑑𝐿𝑑𝑡 =�̇� 𝐵 (𝐿−𝐿 )

Von Bertalanffy: growth at constant food

1�̇�𝐵

=3𝐿�̇� +

(3 �̇� [𝑀𝑉 ] 𝑦𝐸𝑉+3𝐿𝑇 [ �̇� 𝐸𝑀 ] )�̇� [ �̇�𝐸𝑀 ]

Von Bertalanffy: growth at constant food

time, dultimate length, mm

leng

th, m

m

Von

Ber

t gro

wth

rate

-1, d

A lower the food level implies a smaller ultimate size and a shorter time to reach it. Empirical fact: organisms of the same species at different food levels

exhibit von Bertallanfy growth rates that are inversely proportional to ultimate length

Extremes in relative growth rate in insects

Buprestis splendens (jewel beetle)Juveniles live in wood for 20-40 a

Antheraea polyphemus (polyphemus moth)Juveniles increase weight 80000 × in 48 h

Represent dL/dt as a function of L. Represent dL/dt as a funtion of L for two

organisms of the same species living at different food densities.

Exercise

𝑑𝐿𝑑𝑡 =�̇� 𝐵 (𝐿−𝐿 )

Growth in DEB:

What happens to the reserve density in an egg?

Egg and foetal development: differences

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+ [𝑀𝑉 ] 𝑦𝐸𝑉

Growth in DEB:

What happens to the reserve density in an egg? It decreases in time

What happens to the reserve density in a foetus?

Egg and foetal development: differences

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+ [𝑀𝑉 ] 𝑦𝐸𝑉

Growth in DEB:

What happens to the reserve density in an egg? It decreases in time

What happens to the reserve density in a foetus? It tends to infinity

Obtain V(t) for a foetus

Egg and foetal development: differences

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+ [𝑀𝑉 ] 𝑦𝐸𝑉

Growth in DEB:

What happens to the reserve density in an egg? It decreases in time

What happens to the reserve density in a foetus? It tends to infinity

Obtain V(t) for a foetus

Empirical pattern: Volume is proportional to cubed time in a foetus

Egg and foetal development: differences

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+ [𝑀𝑉 ] 𝑦𝐸𝑉

V (𝑡 )=( �̇� 𝑡3 )3

Egg & Foetal development

If growth is supply driven when does growth

stops?

Competition between growth and somatic maintenance

MV - Structure

Feeding

MH - Maturity

XAJ EAJ

Assimilation

ME - ReserveMobilisation

ECJ

Offspring MER

Somatic Maintenance

Growth

Maturity Maintenance

Reproduction

Maturation

ESJ

EGJEJJ

ERJ

If growth is supply driven when does growth stops?

When all the energy that goes for somatic maintenance plus growth is used for maintenance

As the organism gets bigger it gets more food ( to V2/3) but it grows slower because somatic maintenance ( to V) is competing with growth

The higher the specific somatic maintenance needs the lower the ultimate size

Competition between growth and somatic maintenance

MV - Structure

Feeding

MH - Maturity

XAJ EAJ

Assimilation

ME - ReserveMobilisation

ECJ

Offspring MER

Somatic Maintenance

Growth

Maturity Maintenance

Reproduction

Maturation

ESJ

EGJEJJ

ERJ

�̇�𝐸𝐴=𝑦𝐸𝑋 �̇� 𝑋𝐴= 𝑓 (𝑋 ) { �̇� 𝐸𝐴𝑚}𝑉 2/3

Metabolism in a DEB

individual. Rectangles are state

variables Arrows are flows of food

JXA, reserve JEA, JEC, JEM, JET , JEG, JER, JEJ or structure JVG.

Circles are processes The full square is a fixed

allocation rule (the kappa rule)

The full circles are the priority maintenance rule.

A DEB organism: maturity maintenance

MV - Structure

Feeding

MH - Maturity

XAJ EAJ

Assimilation

ME - ReserveMobilisation

ECJ

Offspring MER

Somatic Maintenance

Growth

Maturity Maintenance

Reproduction

Maturation

ESJ

EGJEJJ

ERJ

Collection of processes that maintain the level of

maturity Defense and regulating systems

Maturity maintenance is paid from flux (1-)JE,C:

maturity level It does not increase after the onset of reproduction

Maturity maintenance

Specific maturity maintenance costs are constant because of the strong homeostasis

The complexity would decrease in the absence of energy spent in its maintenance (2nd Law of thermodynamics)

Empirical pattern: no reproduction occurs at very low food densities

�̇�𝐸 𝐽=𝑘 𝐽 𝑀𝐻

- maturity maintenance rate coefficient

Metabolism in a DEB

individual. Rectangles are state

variables Arrows are flows of food

JXA, reserve JEA, JEC, JEM, JET , JEG, JER, JEJ or structure JVG.

Circles are processes The full square is a fixed

allocation rule (the kappa rule)

The full circles are the priority maintenance rule.

A DEB organism: maturation/reproduction

MV - Structure

Feeding

MH - Maturity

XAJ EAJ

Assimilation

ME - ReserveMobilisation

ECJ

Offspring MER

Somatic Maintenance

Growth

Maturity Maintenance

Reproduction

Maturation

ESJ

EGJEJJ

ERJ

The use of reserve to increase the state of

maturity (embryo and juvenile) or to reproduce (adult)

Allocation to maturation in a juvenile (MH <MH

p) or to reproduction in na adult (MH >=MH

p) (supply driven):

Maturation/Reproduction

Empirical pattern: organisms kept at low food density never reach puberty implying that they will not reproduce

Stage transitions should not be linked with size

�̇�𝐸𝑅=(1−) �̇� 𝐸𝐶− �̇�𝐸 𝐽

MHb- threshold of maturity at birth

MHp- threshold of maturity at puberty

Extremes in relative maturity at birth in

mammals

Ommatophoca rossii (Ross Seal) ♂ 1.7-2.1 m, 129-216 kg♀ 1.3-2.2 m, 159-204 kgAt birth: 1 m, 16.5 kg; ab = 270 d

Didelphus marsupiales (Am opossum) ♂, ♀ 0.5 + 0.5 m, 6.5 kgAt birth: <2 g; ab = 8-13 d10-12 (upto 25) young/litter, 2 litters/a

Extremes in relative maturity at birth in fish

Latimeria chalumnae (coelacanth) ♂, ♀ 1.9 m, 90 kgEgg: 325 gAt birth: 30 cm; ab = 395 dFeeds on fish

Mola mola (ocean sunfish) ♂,♀ 4 m, 1500 (till 2300) kgEgg: 3 1010 eggs in bufferAt birth: 1.84 mm g; ab = ? dFeeds on jellyfish & combjellies

The amount of energy continuously invested

in reproduction is accumulated in a buffer and then it is converted into eggs providing the initial endowment of the reserve to the embryo

Initial amount of reserve follows from Initial structural vol. and maturity are negligibly

small and maturity at birth is given Empirical fact: reserve density at birth equals that of

mother at egg formation (egg size covaries with the nutritional state of the mother)

Reproduction

𝑅 �̇� 𝐸𝑅=𝑅 ((1−) �̇� 𝐸𝐶− �̇� 𝐸𝐽 )=𝑑𝑀𝐸𝑅

𝑑𝑡

- initial amount of reserve of the egg - reproduction efficiency

Rules for handling the reproduction buffer are

species-specific (different evolutionary strategies) Some species reproduce when

enough energy for a single egg has been accumulated

Some species reproduce a large clutch (some fishes have thousands of eggs)

Some species use environmental triggers for spawning (e.g., moluscs)

Reproduction: buffer handling rules

Energy flows vs. Mass flows

Fluxes Parameters�̇�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

=�̇�𝑅= �̇� 𝐸𝑅𝜇𝐸

{�̇�𝑋𝑚 }=𝜇𝑋 { �̇� 𝑋𝑚 }{�̇�𝐴𝑚 }=𝜇𝐸 { �̇� 𝐴𝑚}

State Variables

[�̇�𝑀 ]=𝜇𝐸 [ �̇�𝐸𝑀 ]�̇�𝐶= �̇�𝐸𝐶𝜇𝐸=𝐸( �̇�𝐿 − �̇�) {�̇�𝑇 }=𝜇𝐸 { �̇� 𝐸𝑇 }

�̇�𝐺= �̇� 𝐸𝐺𝜇𝐸=[𝐸𝐺 ] 𝑑𝑉𝑑𝑡

[𝐸𝐺 ]=𝑦𝐸𝑉 𝜇𝐸 [𝑀𝑉 ]

EHb- threshold of maturity at birth

EHp- threshold of maturity at puberty