Transcript Mass balance 4.3

Mass balance
0  nMJ M  nO J O
1
 J M  n M
nO J O
compounds
JX
JC
1 0
0 2
JV
JH
JM 
; JO 
; nM  
J

J
JO
E
ER
2 1
0 0
JN
JP

M
C
H
O
N
4.3
minerals
carbon dioxide
water
dioxygen
nitrogen-waste
O
X
V
E
P
0
0
2
0
nCN 
 nCX
n
nHN 
; n O   HX
nON 
 nOX
n
nNN 
 NX
organics
food
structure
reserve
product
DEB model specifies organic fluxes
Mineral fluxes follow from mass balance
Extendable to more elements/compounds
nCV
nHV
nOV
nNV
nCE
nHE
nOE
nNE
nCP 
nHP 
nOP 
nNP 
flux of compound i
chemical index for
element i in compound j
nCj  1 for all compounds j
Ji
nij
Mass-energy coupling
4.3
1
1
J O  ηOp; ηXA  μ AX
; μ AX  y XE μE ; ηVG  μVG
; μVG  yVE μE
O
X
V
E
P
organics
food
structure
reserve
product
powers
compounds
JX
  ηXA
pA
 0
JV
JO 
; p  pD ; ηO   1
J E  J ER
 μE
pG
 η
JP
 PA
A assimilation
D dissipation
G growth
0
0
 μE1
ηPD
0 
ηVG 
 μE1 
ηPG 
Organic fluxes
are linear combinations
of 3 energy fluxes
μ E chemical potential of E
yij yield of compound i on j
ηij coupler of compound i to power j
for faeces: ηPD  ηPG  0
Decomposition of mineral fluxes into contributions from 3 basic energy fluxes:
1
1
1
JM  nM
nOJO  nM
nOηOp  ηMp for ηM  nM
nOηO
Energy balance
4.9.1
For dissipating heat pT  :
T
T
1
0  pT   μM
J M  μOT J O  pT   (μOT  μM
nM
nO ) ηOp
Dissipating heat can be decomposed
into contributions from 3 basic energy fluxes
μ
η
J
p
n
chemical potentials (energy-mass couplers)
mass-energy couplers
M minerals
fluxes of compounds
O organics
3 basic energy fluxes (powers)
chemical indices
Method of indirect calorimetry
4.9.2
Empirical origin (multiple regression): Lavoisier 1780
Heat production =
wC CO2-production + wO O2-consumption + wN N-waste production
DEB-explanation:
Mass and heat fluxes =
wA assimilation + wD dissipation + wG growth
Applies to CO2, O2, N-waste, heat, food, faeces, …
For V1-morphs:
dissipation  maintenance
Mass fluxes
 JX
food
40J V
st ruct ure
4.1
10( J E  J ER )
reserve
JP
faeces
 flux
allocation to
reproduction
0 lb
2J H
wat er
 2JO
dioxygen
1
10J N
ammonia
 flux
2JC
carbon dioxide
lp
 scaled lengthl
0 lb
lp
At abundant food: growth ceases at l = 1
1
use of reserve
not balanced by
feeding in embryo
notice small dent
due to transition
maturation 
reproduction
Methanotrophy
4.3.1
CH4  YCX CO2  YOX O2  YNX NH3  YWX CHn HW OnOW Nn NW
symbol
process
X: methane
C: carbon dioxide
H: water
O: dioxygen
N: ammonia
E: reserve
V: structure
Yield coefficientsT
AC
Assim (catabolic)
-1
1
2
-2
0
0
0
GA
Growth (anabolic)
0
0
Chemical indices
Yield coefficients Y and chemical indices n depend on (variable) specific growth rate r
For reserve density mE = ME/MV (ratio of amounts of reserve and structure),
the macroscopic transformation can be decomposed into 5 microscopic ones
j X
jEA  EAm
with fixed coefficients
C
Carbon
1
1
0
0
0
1
1
H
Hydrogen
4
0
2
0
3
nHE
nHV
O
Oxygen
0
2
1
2
0
N
Nitrogen
0
0
0
0
1
nOE
n NE
nOV
n NV
jEM
AA
Assim (anabolic)
-1
0
YHXA
YOXA
YNXA
1
0
M
OE
M
OE
Y
n NE
-1
0
Y
n NE
-1
0
G
OE
G
NE
-1
1
M
Maintenance
0
1
GC
Growth (catabolic)
0
1
M
YHE
M
YHE
G
HE
Y
Y
Y
KX
 y EV k M
rate
jEG  y EV r
( y XE  1) jEA
j EA
jEM
(1  yVE ) jEG
yVE jEG
r  M V1
d
m k  jEM
MV  E E
dt
mE  y EV
From 0  n Y
A
YNX
 nNE
A
A
YHX  2  YNX
3 / 2  nHE / 2
YOXA  YHXA / 2  nOE / 2
M
YHE
 nNE 3 / 2  nHE / 2
M
M
YOE  1  nOE / 2  YHE
/2
G
YNE
 nNE  nNV
G
G
YHE  nHE / 2  nHV / 2  YNE
3/ 2
G
G
YOE  nOE / 2  nOV / 2  YHE / 2
E
N
4.3.1
flux ratio, mol.mol-1
spec flux, mol.mol-1.h-1
Methanotrophy
C
X/O
N/O
C/O
X
O
spec growth rate, h-1
X: methane
C: carbon dioxide
O: dioxygen
N: ammonia
E: reserve
jEAm = 1.2 mol.mol-1.h-1
yEX = 0.8
yVE = 0.8
kM = 0.01 h-1
kE = 2 h-1
spec growth rate, h-1
chemical indices
nHE = 1.8 nHV = 1.8
nOE = 0.3 nOV = 0.3
nNE = 0.3 nNV = 0.3
Kooijman, Andersen &
Kooi 2004. Ecology,
to appear
Biomass composition
4.3.4
nOW
nNW
Spec growth rate, h-1
h-1
kE 2.11
kM 0.021
yEV 1.135 yXE 1.490
rm 1.05 h-1 g = 1
h-1
Sousa et al 2004
Interface, subm
Weight yield, mol.mol-1
Reserve 74.9
Structure 52.0
Spec prod, mol.mol-1.h-1
Relative abundance
Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C
•μE-1
nHW
Entropy J/C-mol.K
Glycerol 69.7
JC
pA
pM
pG
0.14
1.00
-0.49
JH
1.15
0.36
-0.42
JO
-0.35
-0.97
0.63
JN
-0.31
0.31
0.02
O2
CO2
Spec growth rate
nHE 1.66 nOE 0.422 nNE 0.312
nHV 1.64 nOV 0.379 nNV 0.189
Spec growth rate, h-1
Product Formation
4.7
pyruvate, mg/l
According to
Dynamic Energy Budget theory:
Product formation rate =
wA . Assimilation rate +
wM . Maintenance rate +
wG . Growth rate
For pyruvate: wG<0
throughput rate, h-1
Glucose-limited growth of Saccharomyces
Data from Schatzmann, 1975
1 Reserve – 1 Structure
2 Reserves – 1 Structure
Reserve Capacity & Growth
low turnover rate: large reserve capacity
high turnover rate: small reserve capacity
Multivariate extensions 5
animal
symbiosis
heterotroph
phototroph
plant
Interactions of substrates 5.1
Photosynthesis 5.1.3
2 H2O + 4 h  O2 + 4 H+ + 4 eCO2 + 4 H+ + 4 e-  CH2O + H2O
CO2 + H2O + light  CH2O + O2
yCH E CH 2O  yNO E NO3 
CH nHE OnOE N nNE  yCE CO 2  yHE H 2O  yOE O2  yN H E NH3
Simultaneous nutrient limitation 5.2.3
Specific growth rate of Pavlova lutheri as function
of intracellular phosphorus and vitamine B12 at 20 ºC
Data from Droop 1974
Note the absence of high contents for both compounds
due to damming up of reserves, and
low contents in structure (at zero growth)
Vitamin B12
kE
1.19
1.22 d-1
yXV
0.39 10-15 2.35 mol.cell-1
jEAm
4.91 10-21 76.6 10-15 mol.cell-1. d-1
κE
0.69
0.96
kM
0.0079
0.135 d-1
K
0.017
0.12 pM, μM
5.2.4
1.44
68
14.4
6.8
1.44
20.4
1.44
6.8
B12-conc, pM
P
P-conc, μM
P-content, fmol.cell-1
Data from Droop 1974 on Pavlova lutheri
B12(pM)
B12-cont., 10-21.mol.cell-1
Reserve interactions
P(μM)
Spec growth rate, d-1
Spec growth rate, d-1
Spec growth rate, d-1
Steps in food
7.1.2
0d
7d
14 d
21 d
Steps up
length, mm
Only curves at
0 d are fitted
Notice
• slow response
• gut content in
down steps
length, mm
Growth of Daphnia magna at 2 constant food levels
Steps down
time, d
time, d
time, d
time, d
7.1.3
Conc. potassium, mM
Optical Density at 540 nm
Growth on reserve
time, h
Potassium limited growth of E. coli at 30 °C
Data Mulder 1988; DEB model fitted
OD increases by factor 4 during nutrient starvation
internal reserve fuels 9 hours of growth
Growth on reserve
7.1.3
Growth in starved Mytilus edulis at 21.8 °C
growth rate, mm.d-1
Data Strömgren & Cary 1984; DEB model fitted
internal reserve fuels 5 days of growth
time, d
RNA/dry weight, μg.μg-1
scaled elongation rate
Protein synthesis 7.5
Data from Koch 1970
Data from
Bremer & Dennis 1987
spec growth rate, h-1
scaled spec growth rate
RNA
= wRV MV + wRE ME
dry weight = wdV MV + wdE ME
Scales of life
Life span
10log a
8.0
earth
Volume
3
10log m
life on earth
whale
whale
bacterium
ATP
bacterium
water molecule
Invariance property
8.1
The parameters of two individuals can differ in a very special way
such that both individuals behave identically at constant food density
if they start with the same values for the state variables
(reserve, structure, damage)
At varying food density, two individuals only behave identically
if all their parameters are equal
Inter-species body size scaling
8.2
• parameter values tend to co-vary across species
• parameters are either intensive or extensive
• ratios of extensive parameters are intensive
• maximum body length is Lm  { pA} /[ pM ]
 allocation fraction to growth + maint. (intensive)
[ pM ] volume-specific maintenance power (intensive)
{ p A} surface area-specific assimilation power (extensive)
• conclusion : { pA}  Lm (so are all extensive parameters)
• write physiological property as function of parameters
(including maximum body weight)
• evaluate this property as function of max body weight
Kooijman 1986
Energy budgets can explain body size scaling relations
J. Theor. Biol. 121: 269-282
Primary scaling relationships
invariance
property
(at food density X)
primary
scaling
parameters
8.1
K2=K1z+X(z-1)
{JXm}2={JXm}1z
[pM]2=[pM]1
{pT}2={pT}1
Lb2 = Lb1
{pAm}2={pAm}1z
[EG]2=[EG]1
ha2 = ha1
Lp2 = Lp1
[Em]2=[Em]1z
2=  1
R2=  R1
K2=K1z
{JXm}2={JXm}1z
[pM]2=[pM]1
{pT}2={pT}1
Lb2 = Lb1z
{pAm}2={pAm}1z
[EG]2=[EG]1
ha2 = ha1
Lp2 = Lp1z
[Em]2=[Em]1z
2=  1
R2=  R1
K saturation
constant
{JXm} max spec
feeding rate
[pM] spec maint.
costs
{pT} spec heating
costs
Lb length at birth
{pAm} max spec
assim rate
[EG] spec growth
costs
ha aging
acceleration
Lp length at
puberty
[Em] max reserve
capacity
 partitionning
R reprod.
efficiency
fraction
z: arbitrary zoom factor for species 2 relative to species 1: z = Lm2/Lm1
8.2.1
Lp, cm
Length at puberty
Clupoid fishes
 Clupea
• Brevoortia
° Sprattus
 Sardinops
Sardina
 Sardinella
+ Engraulis
* Centengraulis
 Stolephorus
Data from Blaxter & Hunter 1982
L, cm
Length at first reproduction Lp  ultimate length L
Body weight
8.2.2
Body weight has contribution from structure and reserve
If reserves allocated to reproduction hardly contribute:
f 1

W  dV V  E wE μ  V dV  [ Em ] wE μE1
W  V dV  [ Em 0 ] (V/V0 )1/ 3 wE μE1
 V  V 4 / 3 LW1

W
W
V
V
E
LW
1
E
intra-spec body weight
inter-spec body weight
intra-spec structural volume
Inter-spec structural volume
reserve energy
compound length-parameter
dV
wE
μE
[ Em ]


specific density for structure
molecular weight for reserve
chemical potential of reserve
maximum reserve energy density
Feeding rate
8.2.2
Filtration rate, l/h
slope = 1
Mytilus edulis
poikilothermic tetrapods
Data: Winter 1973
Data: Farlow 1976
Length, cm
Intra-species: JXm  V2/3
Inter-species: JXm  V
Scaling of metabolic rate
8.2.2
Respiration: contributions from growth and maintenance
Weight: contributions from structure and reserve
3
Structure  l ; l = length; endotherms lh  0
intra-species
inter-species
maintenance
 lhl   l 3
 lhl   l 3
growth
 l 2  vl 3
0
 l0
l
ls l 2  l 3

dl3
lh l 2  l 3

dV l 3  d E l 4
reserve
structure
respiration
weight
Metabolic rate
8.2.2
slope = 1
0.0226 L2 + 0.0185 L3
0.0516 L2.44
Log metabolic rate, w
O2 consumption, l/h
2 curves fitted:
endotherms
ectotherms
slope = 2/3
unicellulars
Length, cm
Intra-species
(Daphnia pulex)
Log weight, g
Inter-species
Von Bertalanffy growth rate
von Bert growth rate, a-1
8.2.2
10log
25 °C
TA = 7 kK
10log

rB  3 / kM  3 V
1/ 3

ultimate length, mm
/v

1

10log
 3 / kM  3V
1/ 3
/v
ultimate length, mm

1
V1/ 3
At 25 °C :
maint rate coeff kM = 400 a-1
energy conductance v = 0.3 m a-1
V 1/ 3
↑
V 1/ 3 (a)  V1/ 3  (V1/ 3 Vb1/ 3 ) exp(rB a)
Vb1/ 3
rB1
↑
0
a