Supplementary Material
1
Reduced-order equations
This section includes material from Gawthrop and Crampin [4, §3(c)] about using reduced-order equations for simulation.
Given the reaction flows V of Equation (17), the rate of change of the internal states is given by:
(63)
Ẋi = Ni V
As discussed by number of authors the presence of conserved moieties leads to potential numerical
difficulties with the solution of Equation (63) [65, 66]. Using the notation of Gawthrop and Crampin
[4, §3(c)], the reduced-order state x and the internal state Xi are given by:
ẋ = LxX Ni V
Xi = LXx x + GX Xi (0)
(64)
Equation (64) was used to generate all of the simulation figures in this paper.
2
Conversion of kinetic data
Reaction
ADK
CK
ALD
TPI
ENO
Fout
PGM
GAPDH
LDH
PGK
PK
GPa
GPb
PGI
PGLM
PFK
ATPase
K eq
2.210e+00
2.330e+02
9.500e-05
1.923e+01
4.900e-01
1.000e+00
1.800e-01
8.900e-02
1.620e+04
5.711e+04
1.030e+04
4.200e-01
4.200e-01
4.500e-01
1.662e+01
2.420e+02
2.497e+05
Vmaxf
8.800e-01
5.000e-01
1.040e-01
1.200e+01
1.920e-01
2.000e+02
1.120e+00
1.265e+00
1.920e+00
1.120e+00
1.440e+00
2.000e-02
3.000e-02
–
4.800e-01
5.600e-02
7.500e+02
Vmaxr
–
–
–
–
–
–
–
–
–
–
–
–
–
8.800e-01
–
–
–
KaM
8.640e-02
1.330e+01
5.000e-02
3.200e-01
1.000e-01
1.000e+06
2.000e-01
6.525e-05
6.700e-04
1.600e-05
2.400e-02
4.000e+00
2.000e-01
4.800e-01
6.300e-02
1.440e-02
1.000e+06
KbM
1.225e-01
1.499e-01
2.000e+00
6.100e-01
3.700e-01
1.000e+06
1.400e-02
2.640e-06
1.443e+01
4.200e-01
7.966e+00
2.700e+00
1.500e+00
1.190e-01
3.000e-02
1.085e+01
1.000e+12
Table 1: Parameters from Lambeth and Kushmerick [23, Tables 1&2].
Using the methods of § 3)3.1, the equilibrium constants quoted by Lambeth and Kushmerick
[23, Tables 1&2] (Table 1) were converted into the free-energy constants required by the bond graph
formulation and are listed in Table 2.
Using the methods of § 3)3.2, the reaction constants quoted by Lambeth and Kushmerick [23]
(Table 1) were converted into the reaction constants required by the bond graph formulation of § 3)3.2
and are listed in Table 3.
1
Species
ADP
AMP
ATP
Cr
P
PCr
DHAP
FBP
GAP
DPG
LAC
LACo
NAD
NADH
P2G
P3G
PEP
PYR
F6P
G1P
G6P
GLY
K
7.677e-01
2.776e-03
4.692e+02
6.174e-01
2.447e-03
1.620e+00
1.038e+00
1.968e-03
1.996e+01
1.512e+06
1.748e-18
1.748e-18
1.659e+03
6.026e-04
2.406e-01
4.330e-02
4.910e-01
7.795e-08
7.792e-04
5.827e-03
3.506e-04
1.000e+00
µ0
RT
-2.643e-01
-5.887e+00
6.151e+00
-4.822e-01
-6.013e+00
4.822e-01
3.718e-02
-6.231e+00
2.994e+00
1.423e+01
-4.089e+01
-4.089e+01
7.414e+00
-7.414e+00
-1.425e+00
-3.140e+00
-7.114e-01
-1.637e+01
-7.157e+00
-5.145e+00
-7.956e+00
0.000e+00
Table 2: Bond graph species parameters used in simulation (See § 3) 3.1
Reaction
ADK
CK
ALD
TPI
ENO
Fout
PGM
GAPDH
LDH
PGK
PK
GPa
GPb
PGI
PGLM
PFK
ATPase
vmax
3.439e+02
2.418e-02
1.040e+02
1.082e+03
1.695e+02
1.000e+05
3.136e+02
3.953e+02
1.096e+03
3.527e+02
4.494e+01
1.233e+01
2.841e+01
5.674e+02
1.337e+01
4.239e+01
6.001e+05
kv
4.398e-02
1.863e-01
9.840e-05
5.760e-01
2.124e-02
8.738e-13
2.425e-03
1.653e-03
1.796e-14
5.847e+00
2.823e-04
6.035e-03
4.635e-04
5.978e-05
1.023e-05
3.985e-03
3.755e+08
ρv
6.092e-01
1.000e+00
2.375e-06
9.098e-01
1.169e-01
5.000e-01
7.200e-01
6.875e-01
4.292e-01
6.851e-01
9.688e-01
3.836e-01
5.303e-02
6.448e-01
9.721e-01
2.430e-01
1.998e-01
Table 3: Bond graph reaction parameters used in simulation (See § 3) 3.2
2
Section 2.1 looks at mass-action reactions as used for AT P ase, § 2.2 looks at the relationship
of the approach in § 33.2 to the direct-binding modular rate law of Liebermeister et al. [11], § 2.3 to
the common modular rate law of Liebermeister et al. [11] and § 2.4 to the computational model of
Lambeth and Kushmerick [23].
2.1
Mass-action reactions
The mass-action formulation of chemical equations reveals key issues encountered in converting kinetic data from enzymatic models into the form required by a bond graph model. Enzyme catalysed
reactions are discussed in § 33.2.
The mass-action formulation presented by Gawthrop and Crampin [4, Equation 2.6] uses the
Marcelin formulation rewritten here as:
�
�
f
r
where V0+ = eA /RT
and V0− = eA /RT
(65)
v = κ V0+ − V0−
�
In terms of the reaction A −
−B
V0+ = Ka xa
V0− = Kb xb
(66)
V0− = (Kc xc )2
(67)
−
In terms of the reaction A + B �
− 2C
V0+ = Ka xa Kb xb
�
One standard way of writing the Mass-action rate of A −
−B
v = κeq
�
1
xa −
xb
Keq
�
Ka
Kb
(68)
Ka Kb
Kc2
(69)
where Keq =
−
similarly, A + B �
− 2C can be rewritten as
v = κeq
�
1 2
xa xb −
x
Keq c
�
where Keq =
Define K f as the constant on the substrate side and K r as the constant on the product side. In the
−
case of A �
−B
K f = Ka
and K r = Kb
(70)
−
and in the case of A + B �
− 2C
K f = Ka K b
and K r = Kc2
(71)
Kf
Kr
(72)
It follows that:
κeq = K f κ
and Keq =
As Kf can be computed from K, which in turn can be deduced as discussed in § 33.1, it follows
that κ can be deduced from κeq .
3
2.2
Relation to the Direct Binding Modular Rate Law of Liebermeister et al. [11]
It is now shown using an example that Equation (53) is of the same form as the direct binding modular
�
rate law of Liebermeister et al. [11]. Consider the enzyme catalysed reaction A + B −
− 2C In this
case:
(73)
V0+ = Ka Kb xa xb , V0− = Kc2 x2c
Equation (53) is of the form:
2 2
v=
+ Ka Kb xa xb − v − Kc xc
vmax
max K −
K+
v
1+
(74)
v
Ka Kb xa xb
Kv+
+
Kc2 x2c
Kv−
The direct binding modular rate law is given by Equation (4) of Liebermeister et al. [11] and in the
notation of this paper is:
�
�2
xb
xc
−
a
k + KxM
−
k
M
KcM
a Kb
v=u
(75)
�
�2
xa xb
xc
1 + KM KM + KM
a
c
b
Equations (74) and (75) are identical if we set:
+
vmax
= uk +
2.3
−
vmax
= uk −
KaM KbM
Ka Kb
Kv+ =
Kv− =
�
KcM
Kc
�2
(76)
Relation to the Common Modular Rate Law of Liebermeister et al. [11]
However, Equation (53) is not the same as the common modular rate law of Liebermeister et al. [11].
�
In the context of the reaction A + B −
− 2C, the common modular rate law of Liebermeister et al. [11]
is of the form:
�
�2
xb
xc
−
a
k + KxM
−
k
M
M
Kc
a Kb
v=u
(77)
�2
�
xb
xc
xc
a
b
a
1 + KxM
+ KxM
+ KxM
+
2
+
M
M
M
K
K
K
a
a
b
c
b
c
As discussed by Liebermeister et al. [11], the additional denominator terms imply that Equation (77)
is not the same as Equation (75) but can be considered an approximation to it. However, in the case
�
of reaction A −
− B, the common modular and direct binding modular reaction rates are identical.
2.4
Relation to the computational model of Lambeth and Kushmerick [23]
�
The general enzyme catalysed reaction between two species is given by A −
− B and the corresponding
rate is written by Lambeth and Kushmerick [23] as:
v=
a
b
− Vmaxr KxM
Vmaxf KxM
a
1+
xa
KaM
+
xb
KbM
b
where Vmaxr = Vmaxf
M
KbM Keq
KaM
(78)
In this case V0+ = Ka Xa and V0− = Kb Xb hence Equation (53) becomes:
v=
vmax
Ka Xa − Kb Xb
�
kv 1 + (1 − ρ ) Ka X + ρ
v
4
kv
a
Kb
v kv X b
�
(79)
Comparing Equations (78) and (79) and using Equation (54) it follows that they are identical if:
KaM =
or:
1 kv
1 kv
and KbM =
1 − ρv K a
ρv Kb
(80)
kv = (1 − ρv )Ka KaM and kv = ρv Kb KbM
(81)
Having deduced ρv from the given data using Equation (54), kv can then be deduced from KaM using
Ka , as shown in § 33.1.
Alternatively comparing Equation (52) with Equation (78) gives:
kv+ =
KM
KaM
and kv− = b
Ka
Kb
(82)
Using the expressions for kv+ and kv− (52) gives the same result.
2.5
TPI
The equilibrium constant is given as Keq = 0.052. However this gives the wrong value of Keqcombo .
Because the reaction is specified in the “wrong” direction, it is assumed that Keq should be the reciprocal of the given value, ie Keq = 19.23. Beard [41] quotes Keq = 19.87; so this alteration seems to
be correct.
3
Hierarchical modelling
The bond graphs of the subsystems GAP2LAC and CrAMP appear in Figures 1 and 2.
C:NAD
0
SS:[P]
1
0
C:DPG
C:P3G
C:P2G
SS:[ADP]
C:PYR
C:PEP
C:LAC
C:LACo
[ain]
[ain]
[aout]
0
[aout]
[aout]
0
Re:Fout
0
Rea:PK
Re:ENO
0
Rea:LDH
[ain]
0
Re:PGM
0
Rea:PGK
Rea:GAPDH
SS:[GAP]
0
[ain]
[aout]
C:NADH
0
0
SS:[ATP]
Figure 1: Submodel: GAP2LAC
The ODEs, and corresponding flows, automatically generated from the Bond Graph are given by
the following equations
5
C:AMP SS:[ATP]
0
1
C:P
Re:ADK
0
TF:tf2
C:ATP
SS:[ADP]
0
1
0
Re:CK
1
SS:[P]
C:Cr
1
0
C:ADP
C:PCr
0
Figure 2: Submodel: CrAMP
Ẋadp = 2Vadk − Vpgk − Vpk + Vpf k + Vatpase + Vck
Ẋamp = −Vadk
Ẋatp = −Vadk + Vpgk + Vpk − Vpf k − Vatpase − Vck
Ẋcr = −Vck
Ẋp = −Vgpa − Vgpb + Vatpase − Vgapdh
Ẋpcr = Vck
Ẋdhap = Vald + Vtpi
Ẋf bp = Vpf k − Vald
Ẋgap = Vald − Vtpi − Vgapdh
Ẋdpg = −Vpgk + Vgapdh
Ẋlac = −Vf out + Vldh
(83)
Ẋlaco = Vf out
Ẋnad = −Vgapdh + Vldh
Ẋnadh = Vgapdh − Vldh
Ẋp2g = −Veno + Vpgm
Ẋp3g = Vpgk − Vpgm
Ẋpep = −Vpk + Veno
Ẋpyr = Vpk − Vldh
Ẋf 6p = Vpgi − Vpf k
Ẋg1p = Vgpa + Vgpb − Vpglm
Ẋg6p = −Vpgi + Vpglm
Ẋgly = 0
6
Vadk = �
Vck =
Vald =
Vtpi =
Veno =
Vf out =
Vpgm =
Vgapdh =
Vldh =
Vpgk =
Vpk =
Vgpa =
Vgpb =
Vpgi =
Vpglm =
Vpf k =
Vatpase =
3.1
kadk +
�
�
��
2 X2 + k
vadk −kadp
k
X
X
amp
atp
amp
atp
adp
2 X2 ρ
kadp
adp adk
− kamp katp Xamp Xatp ρadk + kamp katp Xamp Xatp
�
(vck (−kadp kpcr Xadp Xpcr + katp kcr Xatp Xcr ))
(kadp kpcr Xadp Xpcr ρck − katp kcr Xatp Xcr ρck + katp kcr Xatp Xcr + kck )
(vald (−kdhap kgap Xdhap Xgap + kf bp Xf bp ))
(kald + kdhap kgap Xdhap Xgap ρald − kf bp Xf bp ρald + kf bp Xf bp )
(vtπ (−kdhap Xdhap + kgap Xgap ))
(kdhap Xdhap ρtπ − kgap Xgap ρtπ + kgap Xgap + ktπ )
(veno (kp2g Xp2g − kpep Xpep ))
(keno − kp2g Xp2g ρeno + kp2g Xp2g + kpep Xpep ρeno )
(vf out (klac Xlac − klaco Xlaco ))
(kf out − klac Xlac ρf out + klac Xlac + klaco Xlaco ρf out )
(vpgm (−kp2g Xp2g + kp3g Xp3g ))
(kp2g Xp2g ρpgm − kp3g Xp3g ρpgm + kp3g Xp3g + kpgm )
(vgapdh (−kdpg knadh Xdpg Xnadh + kgap knad kp Xnad Xp Xgap ))
(kdpg knadh Xdpg Xnadh ρgapdh − kgap knad kp Xnad Xp Xgap ρgapdh + kgap knad kp Xnad Xp Xgap + kgapdh )
(vldh (−klac knad Xlac Xnad + knadh kpyr Xnadh Xpyr ))
(klac knad Xlac Xnad ρldh + kldh − knadh kpyr Xnadh Xpyr ρldh + knadh kpyr Xnadh Xpyr )
(vpgk (−kadp kdpg Xadp Xdpg + katp kp3g Xp3g Xatp ))
(kadp kdpg Xadp Xdpg ρpgk − kadp kdpg Xadp Xdpg − katp kp3g Xp3g Xatp ρpgk − kpgk )
(vpk (−kadp kpep Xadp Xpep + katp kpyr Xpyr Xatp ))
(kadp kpep Xadp Xpep ρpk − kadp kpep Xadp Xpep − katp kpyr Xpyr Xatp ρpk − kpk )
(kgly Xgly vgpa (−kg1p Xg1p + kp Xp ))
(kg1p kgly Xg1p Xgly ρgpa − kgly kp Xgly Xp ρgpa + kgly kp Xgly Xp + kgpa )
(kgly Xgly vgpb (−kg1p Xg1p + kp Xp ))
(kg1p kgly Xg1p Xgly ρgpb − kgly kp Xgly Xp ρgpb + kgly kp Xgly Xp + kgpb )
(vpgi (−kf 6p Xf 6p + kg6p Xg6p ))
(kf 6p Xf 6p ρpgi − kg6p Xg6p ρpgi + kg6p Xg6p + kpgi )
(vpglm (−kg1p Xg1p + kg6p Xg6p ))
(kg1p Xg1p ρpglm − kg1p Xg1p − kg6p Xg6p ρpglm − kpglm )
(vpf k (−kadp kf bp Xadp Xf bp + katp kf 6p Xf 6p Xatp ))
(kadp kf bp Xadp Xf bp ρpf k − katp kf 6p Xf 6p Xatp ρpf k + katp kf 6p Xf 6p Xatp + kpf k )
(vatpase (−kadp kp Xadp Xp + katp Xatp ))
(kadp kp Xadp Xp ρatpase − katp Xatp ρatpase + katp Xatp + katpase )
(84)
Simulation
Further figures corresponding to Section 4.3 appear in Figures 3, 4 and 5.
Closed system. Figure 4(a) provides another validity check, showing that the conserved moieties
are constant; because the reduced order equations (§ 1) were used, this constraint is automatically
7
4
10
3
4
10
2
10
1
101
0
100
10
3
10
2
10
10
X (mM)
X (mM)
10
-1
10
-2
10
-3
10
10
10-5
-6
10
10-1
100
101
-2
10
10-3
ATP
ADP
P
GLY
LAC
-4
10-1
ATP
ADP
P
GLY
LAC
10-4
10-5
-6
10
102
10-1
100
t (min)
101
102
t (min)
(a) Closed system
(b) Open system
Figure 3: Simulated concentrations. Evolution of the concentrations for AT P , ADP , P , GLY and
LAC corresponds to the situation in Lambeth and Kushmerick [23, Figure 2] except that they use unit
initial states. Following an initial transient, the species concentrations reach steady-state values for
both the closed and open systems (cf Figure 2).
3
3
10
Conserved Moieties (mM)
Conserved Moieties (mM)
10
2
10
1
10
100
10-1
10-1
100
101
2
10
1
10
100
10-1
10-1
102
t (min)
100
101
t (min)
(a) Closed system
(b) Open system
Figure 4: Conserved Moieties. The sum of each conserved moiety remains constant.
8
102
1
8
ATPase
in
out
6
V (mM/min)
V (mM/min)
0.5
0
-0.5
-1
-1
10
ATPase
in
out
4
2
0
0
1
10
10
-2
-1
10
2
10
t (min)
0
1
10
10
2
10
t (min)
(a) Closed system
(b) Open system
Figure 5: Simulated mass flows. Both plots show the input mass flow (though SS:GLYo), the output
mass flow (though Re:Fout) and the ATP flux (though Re:ATPase) (a) In the closed system, the flows
are zero. (b) In the open system, the three flows reach a constant steady state. The final value of ATP
flow is 5.9mM/min; this is close to to value of 6.1mM/min quoted in the “Moderate exercise” column
of Lambeth and Kushmerick [23, Table 4]
enforced and thus numerical drift is avoided. Figure 5(a) shows mass flows within the system.
Open system. Figure 3(b) shows the evolution of simulated concentrations for AT P , ADP , P ,
GLY and LAC which reach new steady state values due to flows induced by the AT P ase reaction.
As with Figure 4(a), Figure 4(b) shows the conserved moieties are constant for the open system. In
contrast to Figure 5(a), Figure 5(b) shows mass flow which, in this open-system context settle to
non-zero values.
4
Virtual Reference Environment
The software required to generate all of the simulation figures shown in this paper from the bond
graph representation is packaged in the form of a Virtual Reference Environment [43]. It is available
at https://sourceforge.net/projects/hbgm/.
9