Rate expressions for kinetic mechanisms and
metabolite elasticities
Kinetic mechanism: A1
This mechanism follows uni-uni reversible Michaelis-Menten kinetics,
v f  vmax, f
vb  vmax, b
s
1 s  p ,
p
1 s  p .
where s is the scaled concentrations of the substrate, and
elasticities have the forms
 f ,s 
1 p
1 s  p ,
 f ,p 
p
1 s  p ,
 b,s 
s
1 s  p ,
 b, p 
1 s
1 s  p .
p
is the scaled concentrations of the product. The
Kinetic mechanism: A1CSP
The elasticities for the substrate, and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the compensated species are modeled in the following way:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
In general, the suffix CSnPm in the name of the kinetic mechanism indicates that n species are
compensated substrate side and m species are compensated product side.
Kinetic mechanism: A1CSPh13
The elasticities for the substrate, and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the compensated species are modeled in the following way:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 3
Kinetic mechanism: A1CS2P2
The elasticities for the substrate, and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the two compensated substrates c1 and c2 and for the two compensated products c3 and c4 are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 1
e b, c2 = 0
e f , c3 = 0
e b, c3 = 1
e f , c4 = 0
e b, c4 = 1
Kinetic mechanism: A1CSP2h112
The elasticities for the substrate, and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the compensated substrate c1 and for the two compensated products c2 and c3 are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
e f , c3 = 0
e b, c3 = 2
Kinetic mechanism: A1CSP3
The elasticities for the substrate, and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the compensated substrate c1 and for the three compensated products c2, c3 and c4 are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
e f , c3 = 0
e b, c3 = 1
e f , c4 = 0
e b, c4 = 1
Kinetic mechanism: A1CP2i
The elasticities for the substrate and for the product have the same expressions as in the kinetic mechanism A1. The
elasticities for the two compensated products c1 and c2 are:
e f , c1 = 0
e b, c1 = 1
e f , c2 = 0
e b, c2 = 1
The small letter “i” in the name of the kinetic mechanism denotes that this reaction is kinetically favorable in the
forward direction. To model this effect we have imposed that the probability of the occupancy of the active site by the
products is very low.
Kinetic mechanism: A2
Generalized reversible Hill with two substrates and two products and with the coefficient
v f  vmax, f
s1 s2
1  s1  p1 1  b2  p2  ,
vb  vmax, b
p1 p2
1  s1  p1 1  s2  p2  .
s
s
where 1 and 2 are the scaled concentrations of substrates, while
products. The elasticities can be expressed as
 f ,s 
1  p1
1  s1  p1 ,
 f ,s 
1  p2
1  s2  p2
,
 p1
1  s1  p1
,
1
2
 f ,p 
1
 f ,p 
2
 b,s 
1
 b ,s 
2
 b, p 
1
 p2
1  s2  p2
 s1
1  s1  p1
 s2
1  s2  p2
,
,
,
1  s1
1  s1  p1 ,
p1
and
p2
h = 1:
are the scaled concentrations of
 b, p 
2
1  s2
1  s2  p2
.
Kinetic mechanism: A2CSP
The elasticities for the substrates and for the products have the same expressions as in the kinetic mechanism A2.
The elasticities for the compensated species are modeled in the following way:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
Kinetic mechanism: A2CSPh22
The elasticities for the substrates, and for the products have the same expressions as in the kinetic mechanism A2.
The elasticities for the compensated species are modeled in the following way:
e f , c1 = 2
e b, c1 = 0
e f , c2 = 0
e b, c2 = 2
Kinetic mechanism: A2CSPh43
The elasticities for the substrates, and for the products have the same expressions as in the kinetic mechanism A2.
The elasticities for the compensated species are modeled in the following way:
e f , c1 = 4
e b, c1 = 0
e f , c2 = 0
e b, c2 = 3
Kinetic mechanism: A2CS2P2
The elasticities for the substrates, and for the products have the same expressions as in the kinetic mechanism A2.
The elasticities for the two compensated substrates c1 and c2 and for the two compensated products c3 and c4 are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 1
e b, c2 = 0
e f , c3 = 0
e b, c3 = 1
e f , c4 = 0
e b, c4 = 1
Kinetic mechanism: A6CS
It is the ordered bi-bi kinetics, with cofactor binding first, and with one compensated substrate
v f  vmax, f
vb  vmax, b
where
and
s1 and s1
s2
s1 s2
1  s1  s2  p1  p2  s1 s2  s1 p1  s2 p2  p1 p2  s1s2 p 1  s2 p 1 p2
p1 p2
1  s1  s2  p1  p2  s1 s2  s1 p1  s2 p2  p1 p2  s1s2 p 1  s2 p 1 p2
,
,
are the scaled concentration of substrate 1 with respect to two different enzymatic constants,
are the scaled concentration of substrate 2 with respect to two different enzymatic constants,
p
s2
p1
p1 and
p2 are the
2 and
are the scaled concentration of product 1 with respect to two different enzymatic constants,
scaled concentration of product 2 with respect to two different enzymatic constants. The elasticities can be expressed
as
e f ,S1 =
1+ s2¢ + p¢1 + p2 + s2¢ p2 + p1 p2 + s2 p1 p2¢
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e f ,S 2 =
1+ s1 + p¢1 + p2 + s1p¢1 + p1 p2
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e f ,P1 =
- ( p¢1 + s1p¢1 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ )
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e f ,P2 =
- ( p2 + s2¢ p2 + p1 p2 + s2 p1 p¢ )2
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e b,S1 =
- ( s1 + s1s2 + s1p¢1 + s1¢s2 p1 )
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e b,S 2 =
- ( s2¢ + s1s2 + s2¢ p2 + s1¢s2 p1 + s2 p1 p¢ )
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e b,P1 =
1+ s1 + s2¢ + p2 + s1s2 + s2¢ p2
1+ s1 + s2¢ + p¢1 + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ ,
e b,P 2 =
1+ s1 + s2¢ + p¢1 + s1s2 + s1p¢1 + s1¢s2 p1
1+ s1 + s2¢ + p1¢ + p2 + s1s2 + s1p¢1 + s2¢ p2 + p1 p2 + s1¢s2 p1 + s2 p1 p2¢ .
We generate random independent samples of scaled metabolite concentrations,
p1 , p2 , p2 . These samples are used to calculate the elasticities.
The elasticities for the compensated substrate c1 are modeled in the following way:
e f , c1 = 1
s1 , s1 , s2 , s2 , p1 ,
e b, c1 = 0
Kinetic mechanism: A99
Generalized reversible Hill with two substrates and two products, and with the allosteric regulation:
¾¾
¾
® F1,6bP + ADP
F6P + ATP ¬
¾
PEP acts as external inhibitor and ADP as activator.
We rename the concentrations of F6P, ATP, F1,6bP, ADP and PEP as
S1 , S2 , P1, P2 and M , respectively.
vi,PFK
vd ,PFK
The reversible Hill rate law for the PFK in the case of independent and dependent modifiers (
respectively) can thus be written as:
V f ,max S1 S2W1W2
vi,PFK =
Q1T1 + Q2T2T3 + T4
vd ,PFK =
V f ,max S1 S2W1W2
(1- G )
(1- G )
Z1 + Z2T3 + T4
,
where
W1 = (S1 + P1 )3 ,
W2 = (S2 + P2 )3,
1+ M 4
1+ m2 M 4 ,
Q1 =
T1 =
1+ P2
Q2 =
T2 =
4
1+ p 2 P2
4
,
1+ mM 4
1+ m2 M 4 ,
1+ pP2
4
1+ p 2 P2
4
,
T3 = (S1 + P1 )4 + (S2 + P2 )4 ,
T4 = (S1 + P1 )4 (S2 + P2 )4 ,
Z1 =
Z2 =
1+ M 4 + P2
4
1+ m2 M 4 + p 2 P2
1+ mM 4 + pP2
4
,
4
1+ m2 M 4 + p 2 P2
,
4
.
and
,
The effects of inhibition by PEP and activation by ADP are given by the two parameters
0 < m <1 and p > 1.
D = QT +Q T T +T
i
1 1
2 2 3
4 , the expressions for the irreversible forward metabolite elasticities of PFK
Given that
in the case of independent modifiers are the following:
e Si = 1+
1
3S1
S1 + P1
e Si = 1+
2
e Pi =
1
-
3S2
-
S2 + P2
3P1
S1 + P1
-
4S1W1 (W2 S2 +W2 P2 + Q2T2 )
Di
,
4S2W2 (W1 S1 +W1 P1 + Q2T2 )
Di
4P1W1 (W2 S2 +W2 P2 + Q2T2 )
Di
,
,
3
4P2 éê P2 (Q1 (1- p 2T1 ) + pQ2T3 (1- pT2 ))
T4 ùú
e =
+ Q2T2W2 +
4
2
S2 + P2 Di ê
S2 + P2 ú
1+
p
P
ë
2
û,
i
P2
3P2
4 M 4 é T1 (1- m Q1 ) + T2T3m(1- mQ2 ) ù
e =ê
ú
Di ë
1+ m2 M 4
û
2
i
M
.
D = Z + Z T +T
d
1
2 3
4 , the expressions for the irreversible forward metabolite elasticities of PFK in the
Given that
case of dependent modifiers are the following:
e Sd = 1+
1
e Sd = 1+
2
e Pd =
1
3S1
S1 + P1
3S2
S2 + P2
3P1
S1 + P1
-
-
4S1W1 (W2 S2 +W2 P2 + Z2 )
Dd
4S2W2 (W1 S1 +W1 P1 + Z2 )
Dd
-
4P1W1 (W2 S2 +W2 P2 + Z2 )
Dd
,
,
,
3
4P2 éê P2 (1- p 2 Z1 + pT3 - p 2T3Z2 )
T4 ùú
e =
+
Z
W
+
2 2
4
2
4
2
S2 + P2 Dd ê
S2 + P2 ú
1+
m
M
+
p
P
ë
2
û,
d
P2
3P2
é
ù
2
2
4 M 4 ê 1- m Z1 + mT3 - m T3Z2 ú
e =Dd ê 1+ m2 M 4 + p 2 P 4 ú
ë
2
û.
d
M
Kinetic mechanism: B4
It is the reversible Michaelis-Menten type kinetics as
v f = v max,f
mt
1+ 2d + m + t ,
v b = v max,b
where
d , m , and t
d2
1+ 2d + m + t ,
represent the scaled concentrations of
P1, S1 , and S2 , respectively. Their elasticities are
expressed in the following equations:
e f ,S1 =
1+ 2d + t
,
1+ 2d + m + t
e f ,S 2 =
1+ 2d + m
,
1+ 2d + m + t
e f ,P1 =
-2d
,
1+ 2d + m + t
e b,S1 =
-m
,
1+ 2d + m + t
e b,S 2 =
-t
,
1+ 2d + m + t
e b,P1 =
2 (1+ d + m + t )
.
1+ 2d + m + t
Kinetic mechanism: BTCS
Bi-Ter kinetics mechanism modeled using the convenience kinetics
v=
v max,f s1 s2 - v max,b p1 p2 p3
(1+ s1 )(1+ s2 ) + (1+ p1 )(1+ p2 )(1+ p3 ) -1
The elasticities are calculated as
e f , s1 = 1-
s1 (1+ s2 )
D ,
s2 (1+ s1 )
D
- p (1+ p2 )(1+ p3 )
= 1
D
,
e f , s 2 = 1ef, p
1
ef,p =
- p2 (1+ p1 ) (1+ p3 )
D
2
ef,p =
,
- p3 (1+ p1 ) (1+ p2 )
D
s (1+ s2 )
e b , s1 = - 1
D ,
3
eb , s2 = -
s2 (1+ s1 )
D ,
p1 (1+ p2 )(1+ p3 )
D
p2 (1+ p1 )(1+ p3 )
= 1D
.
e b , p = 11
eb , p
2
=
v max,f s1 s2 - v max,b p1 p2 p3
D
.
eb , p 3 = -
p3 (1+ p1 )(1+ p2 )
D
.
The elasticities for the compensated substrate c1 are modeled in the following way:
e f , c1 = 1
e b, c1 = 0
Kinetic mechanism: BTCSP
This kinetic mechanism is like BTCS, but it has one compensated substrate c1 and one compensated product c2. For
the compensated species, the elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
Kinetic mechanism: CHEM2
The elasticities for this reaction are as follows:
e f ,s = 1
1
ef,p = 0
1
ef,p = 0
2
eb, s = 0
1
e b, p = 2
1
e b, p = 2
2
Kinetic mechanism: CONVKIN
The general convenience kinetics form is:
The elasticities are provided for general stoichiometry and for any possible coefficient of substrates and products.
Kinetic mechanism: RH3CS2P2
Ter-Ter kinetic mechanism modeled by the generalized reversible Hill kinetics.
3
vmax, f Õ
v=
i=1
si
(1- G)
ks
i
s
p
(1+ i + i )
Õ
ks k p
i=1
3
i
The forward and backward metabolite elasticities are given by:
i
.
e f ,s =
1+ pi
,
1+ si + pi
e f ,p =
- pi
1+ si + pi
i
i
,
e b,s =
-si
,
1+ si + pi
e b, p =
1+ si
.
1+ si + pi
i
i
In addition, this Kinetic mechanism has two compensated substrates c1 and c2 and two compensated products c3
and c4. For the compensated species, the elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 1
e b, c2 = 0
e f , c3 = 0
e b, c3 = 1
e f , c4 = 0
e b, c4 = 1
Kinetic mechanism: RHUB
We model this uni-bi kinetic mechanism with the reversible Hill kinetic approximation:
v = v max,f
æ
ö
æ
ö
s1 ç1- G K ÷
s1 ç1- G K ÷
eq ø
eq ø
è
è
= v max,f
1+ s1 + p1 + p2 + p1 p2
D
,
where s1 is the scaled concentration of the substrate,
product. The elasticities are calculated as
e f , s = 1ef, p =
1
ef,p =
s1
D,
- p1 (1+ p2 )
D
,
- p2 (1+ p1 )
2
eb, s =
,
-s1
D,
e b , p = 11
D
p1 (1+ p2 )
D
,
p1
and
p2
are the scaled concentrations of first and second
p2 (1+ p1 )
D
.
e b , p = 12
Kinetic mechanism: RHUBCSP
This kinetic mechanism is like RHUB, but it has one compensated substrate c1 and one compensated product c2,
whose elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
Kinetic mechanism: RHUBCS2P
This kinetic mechanism is like RHUB, but it has two compensated substrates c1 and c2 and one compensated
product c3, whose elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 1
e b, c2 = 0
e f , c3 = 0
e b, c3 = 1
Kinetic mechanism: RHBU
We model this bi-uni kinetic mechanism with the reversible Hill kinetic approximation:
v = v max,f
where
æ
ö
æ
ö
s1 s2 ç1- G K ÷
s1 s2 ç1- G K ÷
eq ø
eq ø
è
è
= v max,f
,
1+ s1 + p1 + s2 + s1 s2
D
s1 and s2 are the scaled concentration of first and second substrate, p1
product. The elasticities are calculated as
e f , s1 = 1-
s1 (1+ s2 )
,
D
e f , s 2 = 1-
s2 (1+ s1 )
,
D
ef,p =
1
- p1
,
D
e b , s1 = -
s1 (1+ s2 )
,
D
is the scaled concentrations of the
eb , s2 = -
s2 (1+ s1 )
,
D
p1
.
D
e b , p = 11
Kinetic mechanism: TBCSP
Ter-bi kinetics mechanism modeled by the convenience kinetics
v=
v max,f s1 s2 s3 - v max,b p1 p2
(1+ s1 )(1+ s2 )(1+ s3 ) + (1+ p1 )(1+ p2 ) -1
=
v max, f s1 s2 s3 - v max,b p1 p2
D
,
The elasticities are calculated as
e f , s1 = 1-
s1 (1+ s2 )(1+ s3 )
D
,
s2 (1+ s1 )(1+ s3 )
D
s (1+ s1 )(1+ s2 )
e f , s 3 = 1- 3
D
- p (1+ p2 )
ef, p = 1
D
,
e f , s 2 = 1-
1
ef,p =
- p2 (1+ p1 )
D
,
s1 (1+ s2 )(1+ s3 )
e b , s1 = D
,
2
eb, s2 = -
s2 (1+ s1 )(1+ s3 )
D
,
eb, s3 = -
s3 (1+ s1 )(1+ s2 )
D
,
p1 (1+ p2 )
D
p (1+ p1 )
= 1- 2
D
.
e b , p = 11
eb, p
2
The model has one compensated substrate c1 and one compensated product c2, whose elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 0
e b, c2 = 1
Kinetic mechanism: TBCS2P2
This mechanism is like TBCSP, but it has two compensated substrates c1 and c2 and two compensated products c3
and c4. For the compensated species, the elasticities are:
e f , c1 = 1
e b, c1 = 0
e f , c2 = 1
e b, c2 = 0
e f , c3 = 0
e b, c3 = 1
e f , c4 = 0
e b, c4 = 1
Kinetic mechanism: IRRX
It is assumed that the kinetic mechanism from each of substrates (precursors for the biomass) to the biomass follows
the irreversible Michelis-Menten kinetics:
v = v max
where
s1
1+ s1
s1 represents the scaled concentration of substrates mentioned above. The elasticities can then be
expressed as
e f ,s =
1
1
1+ s1
e b ,s = 0
1