Supplementary Material for the Whale HCB Model
The Supplemental Data includes detailed descriptions of the NPZD and DEB models, and
details of the PBPK model not presented in the manuscript. It also includes the output
produced by the NPZD model.
NPZD Model Equations and Parameters
Ecosystem component
The state variables of the ecosystem component of the NPZD model are nutrient (N),
phytoplankton (P), zooplankton (Z) and detritus (D):
dN
N
= uD - m
P,
dt
N +k
(1)
dP
N
=m
P - j Z PZ - s P P ,
dt
N +k
(2)
dZ
= j Z (1- y Z )PZ - s Z Z ,
dt
(3)
dD
= s P P + s Z Z + j Zy Z PZ - u D .
dt
(4)
We impose conservation of mass of nutrient on the ecosystem model, so:
N + P + Z + D = NT
Û
dN dP dZ dD
+
+
+
= 0.
dt dt dt dt
(5)
Parameter definitions and values used in the ecosystem dynamics simulations are
provided in Table S1.
Table S1. Parameters for Ecosystem model obtained by tuning measured parameter
values to reproduce satellite chlorophyll data. Measured values were nondimensionalised for model simulations.
Parameter
Unit
Description
Value
𝜐
h-1
Detritus remineralisation rate
4.17 x 10-3
𝜇
h-1
Maximum phytoplankton growth rate
1.92 x 10-2
𝜅
mgN m-3
Phytoplankton nutrient half-saturation
10.62
𝜑𝑍
m3 mgN-1 h-1
𝜎𝑃
h-1
Zooplankton grazing rate on phytoplankton
2.26 x 10-3
Phytoplankton mortality rate
6.23 x 10-6
1
𝜎𝑍
h-1
yZ
-
NT
mgN m-3
Zooplankton mortality rate
1,75 x 10-3
Zooplankton assimilation efficiency
7.96 x 10-1
Maximum nutrient
11.68
HCB Fugacity Component
The state variables of this component of the NPZD model are the fugacities of HCB in
each compartment. f A is the fugacity of HCB in air, fW is the fugacity of HCB in water,
f S is the fugacity of HCB in sediment, f P is the fugacity of HCB in phytoplankton, f Z is
the fugacity of HCB in zooplankton and f D is the fugacity of HCB in detritus:
VA Z A
(
)
dfA
dZ
= DAW ( fW - fA ) - fA FAW UqQ + U p Z x - VA fA A ,
dt
dt
dfW
= DAW ( fA - fW ) + DSW ( fS - fW ) + DPW ( fP - fW ) + DZW ( fZ - fW )
dt
,
+DDW (1 - w ) ( fD - fW ) + fA FAW U qQ + U p Z x
VW ZW
(
+ fSU r AS Z S - fW U d AS ZW - VW fW
VS Z S
)
VZ Z Z
VD Z D
dfP
N
æ
= DPW ( fW - fP ) - fP rP Z P ç m
è N +k
dt
dZ
ö
P ÷ - VP fP P ,
ø
dt
dfZ
dZ
= DZW ( fW - fZ ) + ( fP rP Z P - fZ rZ Z Z )j PZ - VZ fZ Z ,
dt
dt
df D
= DDSw ( fS - fD ) + DDW (1 - w ) ( fW - fD )
dt
+ ( fP rP Z P - fD rD Z D ) s P P + ( fZ rZ Z Z - fD rD Z D ) (jy PZ + s Z Z ) ,
+ fD rD Z Du D - VD f D
(7)
dZW
dt
dfS
dZ
= DSW ( fW - fS ) + DDSw ( fD - fS ) + fW Ud AS ZW - fSUr AS Z S - VS fS S ,
dt
dt
VP Z P
(6)
(8)
(9)
(10)
(11)
dZ D
dt
dN
N
= uD - m
P,
dt
N +k
(12)
dP
N
=m
P - j PZ - s P P ,
dt
N +k
(13)
dZ
= j (1 - y )PZ - s z Z ,
dt
(14)
2
dD
= s p P + s z Z + jy PZ - u D ,
dt
(15)
Here, the following flux notations are defined:
DAW = kV AW ZW ,
(16)
DSW = kT AS ZW ,
(17)
DPW = kPD rP PZP ,
(18)
DZW = kZD rZ ZZ Z ,
(19)
DDW = kDW rD DZ D ,
(20)
DDS = kDS AS ZW ,
(21)
We impose conservation of mass of HCB on the ecosystem HCB model, so:
VA Z A fA + VW ZW fW + VS ZS fS + VP ZP fP + VZ ZZ fZ +VD Z D fD = POPT
.
(22)
In a constant environment this implies:
dfA
df
df
df
df
df
+ VW ZW W + VS Z S A + VP Z P P + VZ Z Z Z + VD Z D D
dt
dt
dt
dt
dt
dt
,
dVP
dVZ
dVD
= - fP Z P
- fZ Z Z
- fD Z D
dt
dt
dt
VA Z A
(23)
and in an environment where the air and water temperatures change:
dfA
df
df
df
df
df
+ VW ZW W + VS Z S A + VP Z P P + VZ Z Z Z + VD Z D D
dt
dt
dt
dt
dt
dt
dV
dV
dV
.
= - f P Z P P - fZ Z Z Z - f D Z D D
dt
dt
dt
dZ
dZ
dZ
dZ
dZ
dZ
- f AVA A - fW VW W - fSVS A - fPVP P - fZVZ Z - fDVD D
dt
dt
dt
dt
dt
dt
VA Z A
(24)
Parameter definitions and values used in the ecosystem HCB dynamics simulations are
provided in Table S2.
Table S2: Description of parameters, units, values, and derivation where
appropriate for the fugacity equations.
Par
Parameter Description
Derivation
Value
Units
T
Temperature
273.15
K
H
HCB Henry’s Law Constant
172 [1]
Pa m3
mol-1
H1
HCB Henry’s Law Constant
27.71
3
Pa m3
(corrected)
Kow
mol-1
HCB Octanol-water partition
537032 [1]
-
1.31x106
-
220,000 [2]
-
coefficient
Kow1
HCB Octanol-water partition
coefficient (corrected)
Koc
Organic carbon partition
0.41Kow
coefficient
P ls
HCB Sub-cooled liquid vapour
0.267 [1]
Pa
0.0246
Pa
pressure
Pls1
HCB Sub-cooled liquid vapour
pressure (corrected)
foc
Fraction organic carbon in
0.02 [3]
sediment
rS
Density of sediment
2.3 [4]
AW
Air-water interface area
1
m2
AS
Sediment-water interface
1
m2
kg L-1
area
VA
Volume of air
1000
m3
VW
Volume of water
100
m3
VS
Volume of sediment
0.05
m3
Q
Scavenging ratio
200 000 [5]
-
f
Volume fraction occupied by
5x10-12 [5]
-
0.00044 [6]
mol m-3
aerosols
ZA
Fugacity capacity for air
1
RT
ZW
Fugacity capacity for water
1
Pa-1
0.0361 [6]
H
ZS
Fugacity capacity for
sediment
ZP
Fugacity capacity for
K oc f oc r s
H
K ow Z w
Pa-1
892 [6]
Fugacity capacity for
mol m-3
Pa-1
47275 [7]
phytoplankton
ZZ
mol m-3
mol m-3
Pa-1
K ow Z w
4
47275 [7]
mol m-3
zooplankton
ZD
Fugacity capacity for detritus
Pa-1
K oc Z w
19383 [7]
mol m-3
Pa-1
Zx
kv
Fugacity capacity for aerosols
6 ´ 10 6 Z a
Pl s
Air-water
107317 [6]
mol m-3
Pa-1
0.000117 [8]
m h-1
0.0001 [9]
m h-1
4.57 [10]
h-1
5.67 [11]
h-1
0.2 [12]
h-1
volatilization/absorption
mass transfer coefficient
kt
Sediment-water diffusion
mass transfer coefficient
kpu
Phytoplankton-water uptake
rate constant
kzu
Zooplankton-water uptake
rate constant
kdd
Detritus-water depuration
rate constant
Up
Dry deposition velocity
10 [3]
m h-1
Uq
Rain rate
8.06x10-6
m h-1
Ur
Sediment resuspension rate
1.1x10-8 [5]
m h-1
Ud
Sediment deposition rate
1.1x10-8 [5]
m h-1
gP
Phytoplankton mass-volume
5.33x10-8 [13]
m3 mgN-1
5.33x10-8 [14]
m3 mgN-1
5.33x10-8
m3 mgN-1
conversion
gZ
Zooplankton mass-volume
conversion
gD
Detritus mass-volume
conversion
xP
Phytoplankton lipid
0.1 [13]
-
proportion
xZ
Zooplankton lipid proportion
0.045 [15]
kL2
Diffuse attenuation coefficient
0.05
m-1
for seawater
TMAX2 Maximum seasonal sea
2
surface temperature from
5
0C
AVHRR
ISAT2
Saturating photosynthetically
35
E m-2 d-1
available irradiance
1
Values corrected for temperature
2
Values obtained from satellite data
DEB Model Equations and Parameters
This model is taken from Klanjscek et al. [16]. The model has four state variables: the
energy content of the blood (EB), the energy content of the lipid reserve (blubber, EL) ,
the energy content of structural lipids (ES) and the volume of the whale (V):
dV 1
= [ bG EB - mV ]+ ,
dt g
(25)
dEB
= I MAX fV 2 3 + b L kL EL - b L EB - mV - [ bG EB - mV ]+ ,
dt
(26)
dEL
dV
,
= b L EB - b L kL EL - eS0
dt
dt
(27)
dES
dV
.
= eS0
dt
dt
(28)
The feeding function ( f ) is a smoothed square wave function generated by piecewise
combining sections of the error function:
ì
1
ï
t
æ
ö
ï
2
2
10.5
1+
e- x dx ÷
ï
ò
ç
p 0
è
ø
ï
ï
f =í
0
ï
t
ï 0.5 æ 1+ 2 e- x 2 dx ö
ç
÷
ï
p ò0
è
ø
ï
1
ïî
if 1 £ t < 58
if 58 £ t < 62
if 62 £ t < 298
.
(29)
if 298 £ t < 302
if 302 £ t < 365
Parameter definitions and values used in the whale DEB dynamics simulations are
provided in Table S3.
Table S3. Parameters for the whale DEB model
6
Parameter
Value
Units
m
2.07 x 107
kJ m-3 yr-1
kL
0.012
-
g
1.587 x 107
kJ m-3
bL
730
yr-1
Energy conductivity
bG
104
yr-1
Rate of utilization of lipids in blood
I MAX
2.07 x 108
kJ m-2 yr-1
eS
1.08 x 106
kJ m-3
Energy density of structural blubber
eD
2.0712 x 107
kJ m-3
Energy density of blubber
f
0–1
-
0
Description
Cost of maintenance of a unit volume of
structure
Equilibrium ratio of energy in blood and lipid
Energetic cost of growing structure
Energy acquisition rate per biometric area
Square wave feeding / migration cycle
Ecosystem-Whale Interaction
The interaction between the ecosystem and the whale is mediated by the whale dynamic
energy budget (DEB) model, which prescribes what volume of lipid, and hence HCB, is
consumed (FF), taken up (FIB) and excreted (FEF) by the whale during feeding:
FF =
FIB
yW
,
(30)
FEF = (1- y W ) FIB ,
(31)
where the energy intake via food from the environment ( FIB ) is given by:
FIB =
I max f V
eD
2
3
(32)
PBPK Model Equations and Parameters
The PBPK model solves for the fugacity of the HCB in different tissues in the whale. Each
tissue compartment has its blood component differentiated from the blood distribution
network, which is divided into arterial and venous blood. The whale compartments
include the lungs ( f P , f PB ), rapidly perfused tissue ( f R , f RB ), muscle ( f M , f MB ), blubber (
f L , f LB ) and gut ( fG , fGB ). The model includes the fugacity of compartments that
7
influence the partitioning of HCBs between the whale and its environment, including air
(fA) and food (fF).
The PBPK model equations are then:
VP Z P
VPB Z PB
dfP
dV
= DAP ( fA - fP ) + DPPB ( fPB - fP ) + DPBP
¢ fPB - fP Z P P ,
dt
dt
dfPB
dV
= DB ( fVB - fPB ) + DAPB ( fP - fPB ) - DPBP
¢ fPB - fPB Z PB PB ,
dt
dt
VR Z R
dfR
dV
= DRRB ( fRB - fR ) + DRBR
¢ fRB - fR Z R R ,
dt
dt
dfRB
dV
= a RB DB ( fAB - fRB ) + DRRB ( fR - fRB ) - DRBR
¢ fRB - fRB Z RB RB
dt
dt ,
VRB Z RB
VM Z M
VMB Z MB
dfM
dV
= DMMB ( fMB - fM ) + DMBM
¢ fMB - fM Z M M ,
dt
dt
dfMB
dV
= a MB DB ( fAB - fMB ) + DMMB ( fM - fMB ) - DMBM
¢ fMB - fMB Z MB MB ,
dt
dt
(34)
(35)
(36)
(37)
(38)
dfL
dV
= DLLB ( fLB - fL ) + DLBL
¢ fLB - DLLB
¢ fL - fL Z L L ,
dt
dt
(39)
dfLB
dV
= a LB DB ( fAB - fLB ) + DLLB ( fL - fLB ) + DLLB
¢ fL - DLBL
¢ fLB - fLB Z LB LB ,
dt
dt
(40)
dfG
dV
= DGGB ( fGB - fG ) + y DGGL ( fF - fG ) + DF¢ fGB - fG ZG G ,
dt
dt
(41)
VL Z L
VLB Z LB
(33)
VG ZG
dfGB
dV
= a GB DB ( fAB - fGB ) + DGGB ( fG - fGB ) + DIB
¢¢ fF - fGB ZGB GB ,
dt
dt
(42)
VAB Z AB
dfAB
dV
= DB fPB - fAB (a RB DB + a MB DB + a LB DB + a GB DB ) - fAB Z AB AB
dt
dt ,
(43)
VVB ZVB
dfVB
dV
= DB (a RB fRB + a MB fMB + a LB fLB + a GB fGB ) - DB fVB - fVB ZVB VB
dt
dt .
(44)
VGB ZGB
Lipid transfer ( Di¢ =
a i Fi
eD
Z L ) where eD is the energy density of the lipid represents the
uptake of lipid into tissue to fuel metabolism and the mobilization of lipid from blubber
reserves to be transported to tissues requiring energy. Specifically, we include lipid
uptake into the gastric blood from food:
DGGB
¢ =
FIB
Z ,
eD F
8
(45)
uptake into blubber (lipid reserve) from the blubber blood supply:
DLBL
¢ = DGGB
¢ =
FBL
Z ,
eD F
(46)
mobilization from the lipid reserve to fuel growth and maintenance:
DLLB
¢ =
FLB
Z ,
eD L
(47)
and uptake by structure to fuel growth and maintenance:
g R ( FBG + FM )
DRBR
¢ =
D¢MBM =
DPBP
¢ =
DGBG
¢ =
ZL,
(48)
g M ( FBG + FM )
ZL ,
eD
(49)
eD
g P ( FBG + FM )
eD
g G ( FBG + FM )
eD
ZL,
(50)
ZL ,
(51)
The subscript in the Di and Di¢ terms shall denote the source and sink for these
movements, i.e. DLBL
¢ denotes movement from the lipid blood compartment (LB) to the
lipid compartment (L). The coefficient g i denotes the fraction of the whale’s total energy
demand consumed by compartment i and is derived from empirical data [17].
We treat the intake of POP by the whale due to consumption of food by considering the
concentration of POP in the food:
DF¢¢ f F =
FF Z F
F Z
F Z C
F
f F = IB F f F = IB F F = IB CF
eD
y W eD
y W eD Z F y W eD ,
(52)
where
CF = Z Z f Z
(53)
,
which is obtained from the NPZD model. We represent the processes associated with
HCB in the fraction of the food ingested that passes through the gut ( y ) by
y DGGL ( fF - fG ) which assumes that food enters the gut with the fugacity of the krill (in
this case we assume the whale feeds solely on krill so that fF = fZ ) and is excreted at the
fugacity of the gut ( fG ).
9
Volume corrections
The whale HCBs model uses four volume corrections to implement conservation of
mass:

corrections to volumes of lipid in structural components (muscle, RPT, gut, lung)
using appropriate proportions of
dV
dV
(i.e. g X
where g X converts from energy
dt
dt
to lipid volume and includes the fraction of total volume that compartment X
constitutes),
correction to the volume of blubber lipid using g L

dEL
(where g L converts from
dt
energy to volume),
correction to the volume of blood lipid using g B

dEB
(where g B converts from
dt
energy to volume),

corrections to the volume of lipid in structural components (muscle, RPT, gut,
lung) using appropriate proportions of g X mV to correct for consumption of
lipids to provide energy for maintenance (where again g X converts from energy
to lipid volume and includes the fraction of total volume that compartment X
constitutes).
Parameter definitions and values used in the ecosystem HCB dynamics simulations are
provided in Table S4.
Table S4. Pharmacokinetic model parameter values
Parameter
Value
Units
ZL
1062
fA
0
Pa
FA
31,536
m3 yr-1
Air flow through lungs
FB
262.8
m3 yr-1
Lipid flow in blood flow.
y
0.20
-
bH
0.10
-
mol m-3
Pa-1
Description
Fugacity capacity for lipid at 309K
Fugacity of HCB in air (assumed)
Fraction of food intake that is not taken up by
whale
Fraction of tissue occupied by blood for highly
10
perfused tissue (pulmonary, RPT, gut)
Fraction of tissue occupied by blood for
bN
0.01
-
lP
0.05
-
lR
0.05
-
lM
0.04
-
lG
0.05
-
VP
0.2000
-
VR
0.1077
-
VM
0.3462
-
VG
0.0385
-
g AB
0.375
Fraction of blood volume that is arterial
g VB
0.625
Fraction of blood volume that is venous
gP
V P (1- b H ) l P
m3 lipid
gR
V R (1- b H ) l R
m3 lipid
gM
V M (1- b N ) l M
m3 lipid
gG
V G (1- b H ) lG
m3 lipid
g PB
V Pb H lP
m3 lipid
g RB
V Rb H lR
m3 lipid
g MB
V M bN lM
m3 lipid
normally perfused tissue (muscle, blubber)
Fraction of pulmonary tissue that is lipid (by
volume)
Fraction of RPT tissue that is lipid (by volume)
Fraction of muscle tissue that is lipid (by
volume)
Fraction of gut tissue that is lipid (by volume)
Fraction of structural volume that is pulmonary
tissue
Fraction of structural volume that is RPT tissue
Fraction of structural volume that is muscle
tissue
Fraction of structural volume that is gut tissue
Proportion of structural volume occupied by
pulmonary tissue
Proportion of structural volume occupied by
RPT tissue
Proportion of structural volume occupied by
muscle tissue
Proportion of structural volume occupied by gut
tissue
Proportion of structural volume occupied by
pulmonary blood
Proportion of structural volume occupied by
RPT blood
Proportion of structural volume occupied by
muscle blood
11
g GB
V G b H lG
m3 lipid
g LB
bN
-
Proportion of structural volume occupied by gut
blood
Proportion of blubber volume occupied by
blubber blood
* Note: diffusion values are for the low diffusion case. See text for other values used in
sensitivity analysis.
NPZD Model Outputs
Chlorophyll concentration predicted by the NPZD model was calibrated to NASA satellite
data, and produced the predictions of krill and detritus biomass shown in SM Figure 1.
The concentrations of HCB predicted in the physical and biological compartments of the
NPZD model are shown in SM Figure 2.
SM Figure 1. Seasonal dynamics of the plankton compartments of the NPZD model.
The dots are satellite measurements of chlorophyll-a concentration converted to a
nitrogen equivalent concentration (mgN m3) using Redfield ratios [18]. The solid
line is phytoplankton concentration predicted by the model. The dotted line is
predicted zooplankton (krill) concentration and the dashed line predicted
detritus concentration. Detritus is predicted by the NPZD model in mgN m3 and is
converted to an organic carbon equivalent using Redfield ratios when simulating
12
the partitioning of HCB in the ecosystem. The whale is feeding during days 1 – 60
and 300 – 365 (i.e. the Antarctic summer) and is migrating, and hence fasting,
during days 61 – 299.
SM Figure 2. Seasonal variations of the concentration of HCB in air (a), water (b),
sediment (c), phytoplankton (d), zooplankton (krill) (e) and detritus (f) predicted
by the NPZD model. The whale is feeding during days 1 – 60 and 300 – 365 (i.e. the
Antarctic summer) when the concentration of HCB in the krill population is
relatively low, in part due to its relatively large biomass. The whale is migrating,
and hence fasting, during days 61 – 299.
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