Dynamic Energy Budget Theory - I

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

Growth is the increase of the amount of

structure (conversion of reserve into structure) Allocation to growth (supply driven):

Growth

Strong homeostasis imposes a fixed conversion efficiency

Strong homeostasis imposes a constant density

- number of C-moles per unit of structure body volume -yield of reserve on structure

Obtain expressions that depend only on state

variables and parameters for 1) growth and 2) growth at constant food (weak homeostasis)

Suggestion use the: following equations for 1)

Use the following definition for 2)

Exercises

- reserve density

The expressions that depend only on state

variables and parameters for 1) growth is

Exercises

𝑑𝑉𝑑𝑡 =

𝑀𝐸�̇�𝐿 − [ �̇�𝐸𝑀 ]𝑉 − { �̇� 𝐸𝑇 }𝑉 2/3

𝑀𝐸

𝑉 +[𝑀𝑉 ]𝑦𝐸𝑉

𝑑𝑉𝑑𝑡 =

𝑚𝐸𝑉 2/3 �̇� [𝑀𝑉 ]− [ �̇�𝐸𝑀 ]𝑉 − { �̇�𝐸𝑇 }𝑉 2/3

𝑚𝐸 [𝑀𝑉 ]+[𝑀𝑉 ]𝑦𝐸𝑉

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+[𝑀𝑉 ]𝑦𝐸𝑉

Is this Von Bertallanffy growth?

Yes, with

Exercises

𝑑𝐿𝑑𝑡 =�̇� 𝐵 (𝐿−𝐿 )

- heating length

𝑑𝐿𝑑𝑡 =

[ �̇� 𝐸𝑀 ]

3(𝑚𝐸 [𝑀𝑉 ]+ [𝑀𝑉 ]𝑦𝐸𝑉

) (𝑚𝐸 �̇� [𝑀𝑉 ]− { �̇�𝐸𝑇 }

[ �̇�𝐸𝑀 ] −𝐿  )

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 ]+𝐿𝑇 𝑦𝐸𝑉 [ �̇� 𝐸𝑀 ] )�̇� 𝑦𝐸𝑉 [ �̇� 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.

Growth in DEB:

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

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

Egg and foetal development: differences

Empirical fact: Foetal weigth is proportional to cubed time

𝑑𝐿𝑑𝑡 =1

3𝑚𝐸 �̇� [𝑀𝑉 ]− [ �̇� 𝐸𝑀 ]𝐿− { �̇�𝐸𝑇 }

𝑚𝐸 [𝑀𝑉 ]+[𝑀𝑉 ]𝑦𝐸𝑉

V (𝑡 )=( �̇� 𝑡3 )3

Egg & Foetal development

As the organism gets bigger it gets more food

(proportional to V2/3) but it grows slower because somatic maintenance (proportional to V) is competing with growth

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

Competition between growth and somatic maintenance

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

Obtain an expression for the dynamics of the

reserve density mE Suggestion use the equations for the dynamics of ME and

MV and following equations:

Obtain na expression for the maximum reserve density mEm

Set dmE/dt=0 (weak homeostasis). What is the value of mE? What is the maximum value of mE?

Rewrite using mEm. What is the meaning of ?

Exercises

- maximum length- maximum reserve density

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

�̇�𝐸𝑅=(1−) �̇� 𝐸𝐶− �̇�𝐸 𝐽=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

�̇�𝑋= �̇� 𝑋 𝐴𝑋= 𝑓 (𝑋 ) {�̇�𝑋𝑚 }𝑉 2 /3

�̇�𝐴= �̇� 𝐸𝐴𝐸=𝑦 𝐸𝑋 �̇� 𝑋𝐴𝐸=𝐸 𝑓 (𝑋) {�̇�𝐴𝑚 }𝑉 2 /3

�̇�𝐶= �̇�𝐸𝐶𝐸=𝐸( �̇�𝐿 − �̇� )�̇�𝑆= �̇� 𝐸𝑆𝐸= [�̇�𝑀 ]𝑉 + {�̇�𝑇 }𝑉 2/3

�̇�𝐺= �̇� 𝐸𝐺𝐸=[𝐸𝐺 ] 𝑑𝑉𝑑𝑡=

�̇�𝑅= �̇� 𝐸𝑅𝐸