Basic DEB scheme
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1- maturitymaintenance
maturityoffspring
maturationreproduction
Basic DEB scheme
food faecesassimilation
reserve
feeding defecation
structurestructure
somaticmaintenance
growth
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Feeding 3.1
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
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Feeding 3.1
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
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Assimilation 3.3
Definition: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,,
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Assimilation 3.3
KX
Xf
JyJJfJ
VJJ
AmXEXAmEAmEAE
AEAE
responsefunctionalscaledand
}{}{fixedfor}{}{with
}{onassimilatitolinkedfluxreserve
,,,,
3/2,,
EXyEVKX food density
saturation constantstructural volumereserveyield of E on X
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Reserve dynamics 3.4
Increase: assimilation surface areaDecrease: catabolism reserve density (= reserve/structure)
First order process on the basis of densities follows from• weak homeostasis of biomass = structure + reserve• partitionability of reserve dynamics (essential for symbioses)
Mechanism: structural & local homeostasis
-rule for allocation to growth + somatic maintenance: constant fraction of catabolic rate
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AA
AA
pκpκ )1(
Reserve partitioning 3.4
EκA )1(reserve
EκA
reserve stru
ctur
e, V
GA pκ )1(
GA pκ
MA pκ )1(
MA pκIf reserves are partitioned e.g. into lipids and non-lipids maintenance and growth are partitioned as wellPartitioning requirement for catabolic power ( use of reserves, [pM] = pM/V and [EG] constant)
for some function [pC]= pC/V of state variables [E],V
)],[],[|],[]([)],[],[|],]([[
θEκpκVEκpθEpVEpκ
GAMAAC
GMCA
vectorparameter factorarbitrary
costgrowth spec][powergrowth power maint.power assim. volumestructuraldensity reserve][
θκEpppVE
A
G
G
M
A
]1,0[somefor Aκ
Cp
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Reserve dynamics 3.4
• Relationship assimilation, growth and maintenance
• Weak homeostasis
• Partionability
• Conclusions Function H is first degree homogeneous:
Function is zero-th degree homogeneous in [E]: : So may depend on V, but not on [E]• Result
]/[]][[])/[][1]([][][
][)],([
/][with,ln][][][][
GMGCA
GMC
AACA
EpEEEκppEdt
d
Vdt
dEppVEκ
VppVdt
dEppE
dt
d
3/1][with0][at0][ VpEdt
dE
dV
dA
]/[])[|],([1
]/[)](][[)|]([][
3/1
G
GMC EEθVEκ
EVpEθEHVp
)],[],[|],[]([)],[],[|],]([[ θEκpκVEκpθEpVEpκ GAMAACGMCA
][
}{,
][
][for)(or
][
][}{][
3/13/1m
Am
mm
Am
E
pv
E
Eeef
V
ve
dt
d
E
Ef
V
pE
dt
d
vEvEHθEκHθEHκ EE constantfor][])([so),|][()|]([
)],([)],[( VEκVEκκ E
θfvκpppp
VEEE
C
M
A
A
G
m
}{
][][
][ reserve densitymax reserve densityspec growth coststructural volumespec assim powerassim powermaint. powercatabolic powerfraction catabol.energy conductancescaled funct. resp.parameter vector
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Reserve dynamics 3.4
XK
XffVppppE
dt
dAmACA
;}{; 3/2
V
EE
E
pv
E
Eeef
V
ve
dt
d
E
Ef
V
pE
dt
d
m
Am
mm
Am
][;
][
}{,
][
][for)(or
][
][}{][
3/13/1
Isomorphs
V1-morphs
)]([ 3/2 Vdt
dvVEpC
3/1)/()(with{multiply dAm VVVp M}
][
][for)(;][
m
AmEEAmA E
pkefke
dt
dfVpp
C
A
ppVEX
][][},{
m
AmAm
Epp
Kf
Ekvfood density
reserve energystructural volumeassimilation powercatabolic power
scaled functional responsesaturation constantmax spec assimilation powermax reserve capacity
energy conductancereserve turnover rate
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Reserve dynamics
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Reserve dynamics
• reserve & structure: spatially segregated
• reserve mobilized at rate surface area of reserve-structure interface
• rejected reserve flux returns to reserve
• SU-reserve complex dissociates to demand-driven maintenance supply-driven growth (synthesis of structure)
• abundance of SUs such that local homeostasis is achieved
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Reserve dynamics
SU abundance, relative to DEB value
sd s
peci
fic
use
of r
eser
ve
for assimilation being an alternating Poisson process
10 h-1
50 h-1
2 h-1
assim = 0 assim = 1
0
1
time
assimilation
10 h-1
10 h-1
10 h-1
hazard rates
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Reserve dynamics
time, h
PH
B d
ensi
ty, m
ol/m
ol
in starving active sludge
Data fromBeun, 2001
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Yield of biomass on substrate
1/spec growth rate, h-1
cusStreptococ mg
glucose mg
Data fromRussel & Cook, 1995
maintenance
reserve
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-rule for allocation 3.5
Age, d Age, d
Length, mm Length, mm
Cum
# of young
Length,
mm
Ingestion rate, 105
cells/h
O2 consum
ption,
g/h
• 80% 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
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Somatic maintenance 3.6
Definition 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
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Maturity maintenance 3.6
Definition of maturity maintenance:Collection of processes required to maintain current state of maturity
Main reason for consideration:making total investment into maturation independent of food intake
Maturity maintenance costs paid from flux (1-)JE,C: • structural volume in embryos and juveniles, pJ
• 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 ,,
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0
num
ber
of d
aphn
ids
Maintenance first 3.6
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
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Growth 3.7
Definition: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
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Mixtures of V0 & V1 morphs 3.7.2
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
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structural volume
reserve density max res densityspec assim powerspec heating costsspec som maint costsspec growth costsfraction catabolic p
Growth 3.7
ge
VVVVVeVvVeV
dt
d mmh
3/13/13/23/2 /)/(
),(
κEppp
E
Ee
V
G
M
T
A
m
][][}{}{
][
][
][
}{][
][][
][][
}{][
}{
3/1
3/1
m
Am
m
G
G
MM
MM
Amm
M
Th
E
pv
Eκ
Eg
E
pk
gk
v
p
pκV
p
pV
heating length
max length
maint rate coeff
en investment ratio
energy conductance
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Growth at constant food 3.7
time, d ultimate length, mm
leng
th, m
m
M
M
δfVfLLvLδkr
trLLLtL
mm
MB
Bb
//33
)exp()()(
3/1
11
LLLt
b
Mδkvr
M
BtimeLengthL. at birthultimate L.
von Bert growth rateenergy conductancemaint. rate coefficientshape coefficient
vδ /3 M
Von
Ber
t gro
wth
rat
e -1, d
13 Mk
Von Bertalanffy growth curve:
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Embryonic development 3.7.1
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
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Foetal development 3.7.1
wei
ght,
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
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Maturation 3.8
Definition:Use of reserve(s) to increase the state of maturityThis, however, does not increase structural massImplies: products associated with maturation (e.g. CO2, NH3)
Allocation to maturation in embryos and juveniles:
This flux is allocated to reproduction in adults
Dissipating power: with R = 0 in embryos and juvenilesNotice that power also dissipates in association with
constantandwith)1( ,,,,,, JEVJEJEJECERE jMjJJJκJ
RRJTMD pκpppp )1(
MD
GA
pppp
e,maintenanc n,dissipatio :morphs-V1For ,growth,,onassimilati
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Reproduction 3.9.1
Definition: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
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Reproduction at constant food 3.9.1
length, mm length, mm
103 e
ggs
103 e
ggs
Gobius paganellusData Miller, 1961
Rana esculentaData Günther, 1990
)(foodconstantat
)()1(),(
332
3/2
0
feLδv
k
f
fgLδ
v
kL
VgkVkvVeg
geκ
Ve
κVeR
pMM
pMMm
R
ΜΜ
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General assumptions 3.10
• State variables: structural body mass & reserves they do not change in composition• 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)
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Specific assumptions 3.10
• 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 & maturity maintenance structure volume (but some maintenance costs surface area) maturity maintenance does not increase after a given cumulated investment in maturation• Feeding rate surface area; fixed food handling time• Partitioning of reserves should not affect dynamics comp. body mass does not change at steady state• Fixed fraction of catabolic energy 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