1- maturitymaintenance

maturityoffspring

maturationreproduction

Standard DEB model

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

Feeding

Feeding has two aspects• disappearance of food (for food dynamics): JX,F

• appearance of substrate for metabolic processing: JX,A= JX,F

Faeces • cannot come out of an animal, because it was never in it• is treated as a product that is linked to assimilation: JP,F= yPX JX,F

Feeding

time

time

bind

ing

prob

.bi

ndin

g pr

ob.

fast SU

slow SU

arrival events of food items

raten associatio:rate;on dissociati:density; food:

:response functionalwith ;/

:ratefeeding

:fraction mequilibriu;)1(:SUoffractionunbounded

,*

,

*

bkXXK

XfJf

Xbk

kXbXθJ

bXk

kθbXθθkθ

dt

d

FmXFX

0

0

Busy periods not only include handling but also digestion and other metabolic processing

AssimilationDefinition:Conversion of substrate(s) (food, nutrients, light) into reserve(s)Energy to fuel conversion is extracted from substratesImplies: products associated with assimilation (e.g. faeces, CO2)

Depends on:• substrate availability• structural (fixed part of) surface area (e.g. surface area of gut)

Consequence of strong homeostasis:Fixed conversion efficiency for fixed composition of substrate

However, biomass composition is not fixed many species feed on biomass

EXAXEXAE yJyJ fixedfor,,

Assimilation

KX

Xf

JyJJfJ

VJJ

AmXEXAmEAmEAE

AEAE

responsefunctionalscaledand

}{}{fixedfor}{}{with

}{onassimilatitolinkedfluxreserve

,,,,

3/2,,

EXyEVKX food density

saturation constantstructural volumereserveyield of E on X

Reserve dynamics & allocation

Increase: assimilation structural surface areaDecrease: mobilisation reserve-structure interface Change in reserve density structural length-1

Reserve dynamics follows from weak homeostasis of biomass = structure + reserve

-rule for allocation to soma: constant fraction of mobilisation rate

Reserve dynamics

time, h

PH

B d

ensi

ty,

mol

/mol

in starving active sludge

Data fromBeun, 2001

Yield of biomass on substrate

1/spec growth rate, h-1

cusStreptococ mg

glucose mg

Data fromRussel & Cook, 1995

maintenance

reserve

-rule for allocation

Age, d Age, d

Length, mm Length, mm

Cum

# of young

Length, m

mIngestion rate, 105

cells/h

O2 consum

ption,

g/h

• large part of adult budget to reproduction in daphnids• puberty at 2.5 mm• No change in ingest., resp., or growth • Where do resources for reprod. come from? Or:• What is fate of resources in juveniles?

Respiration Ingestion

Reproduction

Growth:

32 LkvL M2fL

332 )/1( pMM LkfgLkvL

)( LLrLdt

dB

Von Bertalanffy

Somatic maintenanceDefinition of maintenance (somatic and maturity):Collection of processes not associated with net productionOverall effect: reserve excreted products (e.g. CO2, NH3)

Somatic maintenance comprises:• protein turnover (synthesis, but no net synthesis)• maintaining conc gradients across membranes (proton leak)• maintaining defence systems (immune system)• (some) product formation (leaves, hairs, skin flakes, moults)• movement (usually less than 10% of maintenance costs)

Somatic maintenance costs paid from flux JE,C: • structural volume (mosts costs), pM

• surface area (specific costs: heating, osmo-regulation), pT

Maturity maintenanceDefinition of maturity maintenance:Collection of processes required to maintain current state of maturity

Maturity maintenance costs paid from flux (1-)JE,C: • maturity• constant in adults (even if they grow)

Else: size at transition depends on history of food intake

p

MEpJE

Vκ

κjVVj

sizefixedatoccurstransitionstage

1)/,1min(If ,,

0

num

ber

of d

aphn

ids

Maintenance first

106 cells.day-1

300

200

100

01206030126

max

num

ber

of d

aphn

ids

30 35

400

300

200

100

8 11 15 18 21 24 28 32 37time, d

30106 cells.day-1

Chlorella-fed batch cultures of Daphnia magna, 20°Cneonates at 0 d: 10winter eggs at 37 d: 0, 0, 1, 3, 1, 38

Kooijman, 1985 Toxicity at population level. In: Cairns, J. (ed) Multispecies toxicity testing. Pergamon Press, New York, pp 143 - 164

Maitenance requirements:6 cells.sec-1.daphnid-1

GrowthDefinition:Conversion of reserve(s) into structure(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with growth (e.g. CO2, NH3)

Allocation to growth:

Consequence of strong homeostasis:Fixed conversion efficiency

][fixedfor][

fixedfor,,,

VVV

EVGEEVVVGVVGV

MVMM

yJyMrMjMdt

dJ

constantandwith ,,,,,, MEVMEMEMECEGE jMjJJJκJ

Mixtures of V0 & V1 morphs

volu

me,

m

3vo

lum

e,

m3

volu

me,

m

3

hyph

al le

ngth

, mm

time, h time, min

time, mintime, min

Fusarium = 0Trinci 1990

Bacillus = 0.2Collins & Richmond 1962

Escherichia = 0.28Kubitschek 1990

Streptococcus = 0.6Mitchison 1961

Growth

Growth at constant food

time, dultimate length, mm

leng

th, m

m

Von

Ber

t gro

wth

rat

e -1, d

Von Bertalanffy growth curve:

Mouse goes preying 2.1c

On the island Gough, the house mouse Mus musculus

preys on chicks of seabirds, Tristan albatross Diomedea dabbenena

Atlantic petrel Pterodroma incerta

The bird weights are 250 the mouse weight of 40 g,Mice typically weigh 15 g

99% of these bird speciesbreed on Gough and are

now threatened with extinction

Metamorphosis

The larval malphigian tubes are clearly visible in this emerging cicadaThey resemble a fractally-branching space-filling tubing system, according to Jim Brown, but judge yourself …. Java, Nov 2007

ReproductionDefinition:Conversion of adult reserve(s) into embryonic reserve(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with reproduction (e.g. CO2, NH3)

Allocation to reproduction in adults:

Allocation per time increment is infinitesimally smallWe therefore need a buffer with buffer-handling rules for egg prod (no buffer required in case of placental mode)

Strong homeostasis: Fixed conversion efficiencyWeak homeostasis: Reserve density at birth equals that of motherReproduction rate: follows from maintenance + growth costs, given amounts of structure and reserve at birth

constantwith)1( ,,,, JEJECERE JJJκJ

eggpercostswith/ 00, EEJκR RER0E

Reproduction at constant food

length, mm length, mm

103

eggs

103

eggs

Gobius paganellusData Miller, 1961

Rana esculentaData Günther, 1990

Maturity & its maintenanceDEB implementation is motivated by 4 observations 1 Contrary to age, volume at birth or puberty hardly depends on food density. So stage transitions cannot be linked to age. 2 Some species continue growing after puberty. Other species, such as birds, only reproduce well after the growth period. So stage transitions cannot be linked to size.

3 Total cumulative energy investment in development at any given size of the individual depends on food density;

this can be removed by allowing for maturity maintenance. 4 Ultimate reproduction rate is a continuous function of food density This demonstrates the existence of maturity maintenance.

Maintenance ratio 2.5.3b

Extremes in relative maturity at birth in mammals 2.5.2a

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 birds 2.5.2b

Apteryx australis (kiwi) ♂ 2.2 kg; ♀ 2.8 kgEgg: 12×8 cm, 550 g; ab = 63-92 d

Cuculus canorus (cuckoo) ♂,♀ 115 gEgg: 3.3 g; ab = 12 d

Extremes in relative maturity at birth in fish 2.5.2c

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 jellfish & combjellies

Short juvenile period 2.5.2d

Hemicentetes semispinosus (streaked tenrec )

ap - ab = 35 d

Lemmus lemmus (Norway lemming ) ap - ab = 12 d

Embryonic development

time, d time, d

wei

ght,

g

O2 c

onsu

mpt

ion,

ml/h

l

ege

dτ

d

ge

legl

dτ

d

3

3,

3, l

dτ

dJlJJ GOMOO

; : scaled timel : scaled lengthe: scaled reserve densityg: energy investment ratio

Crocodylus johnstoni,Data from Whitehead 1987

yolk

embryo

Diapauze 2.6.2c

seeds of heather Calluna vulgaris can germinate after 100 year

Foetal developmentw

eigh

t, g

time, d

Mus musculus

Foetes develop like eggs, but rate not restricted by reserve (because supply during development)Reserve of embryo “added” at birth Initiation of development can be delayed by implantation egg cellNutritional condition of mother only affects foetus in extreme situations

33/20 )3/()(;0)0(;:For vttVVvVV

dt

dE

Data: MacDowell et al 1927

High age at birth 2.6.2f

Sphenodon punctatus (tuatara)Adult: 45-60 cm, Wm = 0.5 – 1 kg, ♂ larger than ♀10 eggs/litter, life span 60 - >100 aBody temp 20-25 °C, ap = 20 a, Wb = 4 g, ab = 450 d.

ReproductionDefinition:Conversion of adult reserve(s) into embryonic reserve(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with reproduction (e.g. CO2, NH3)

Allocation to reproduction in adults:

Allocation per time increment is infinitesimally smallWe therefore need a buffer with buffer-handling rules for egg prod (no buffer required in case of placental mode)

Strong homeostasis: Fixed conversion efficiencyWeak homeostasis: Reserve density at birth equals that of motherReproduction rate: follows from maintenance + growth costs, given amounts of structure and reserve at birth

constantwith)1( ,,,, JEJECERE JJJκJ

eggpercostswith/ 00, EEJκR RER0E

Reproduction at constant food

length, mm length, mm

103

eggs

103

eggs

Gobius paganellusData Miller, 1961

Rana esculentaData Günther, 1990

General assumptions• State variables: structural body mass & reserve & maturity structure reserve do not change in composition; maturity is information• Food is converted into faeces Assimilates derived from food are added to reserves, which fuel all other metabolic processes Three categories of processes: Assimilation: synthesis of (embryonic) reserves Dissipation: no synthesis of biomass Growth: synthesis of structural body mass Product formation: included in these processes (overheads)• Basic life stage patterns dividers (correspond with juvenile stage) reproducers embryo (no feeding initial structural body mass is negligibly small initial amount of reserves is substantial) juvenile (feeding, but no reproduction) adult (feeding & male/female reproduction)

Specific assumptions

• Reserve density hatchling = mother at egg formation foetuses: embryos unrestricted by energy reserves• Stage transitions: cumulated investment in maturation > threshold embryo juvenile initiates feeding juvenile adult initiates reproduction & ceases maturation

• Somatic maintenance structure volume & maturity maintenance maturity (but some somatic maintenance costs surface area) maturity maintenance does not increase after a given cumulated investment in maturation• Feeding rate surface area; fixed food handling time• Body mass does not change at steady state• Fixed fraction of mobilised reserve is spent on somatic maintenance + growth (-rule)• Starving individuals: priority to somatic maintenance do not change reserve dynamics; continue maturation, reprod. or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure

Primary DEB parameters 2.8a

time-length-energy time-length-mass