Doc S1. Further mathematical models of existing enzyme models
The forms of the product concentrations given in section 2 are not in the conventional forms
given in the literature. That is the enzyme kinetic models are typically shown as a reaction
velocity rather than the equilibrium product concentration. In this supporting section, we
express the results we derived in terms of the reaction velocity as well as the total enzyme
concentration as a comparison to the classical textbook derivations. The reaction velocity
and the equilibrium product concentration can be linked using the following expression.
Here v as the reaction velocity of the enzymatic reaction and the left hand side is set to
zero to indicate equilibrium.
[ P]  v  k2 [ P]
(S1)
The free enzyme product concentration can then be converted to the total enzyme
concentration using the conservation law given in Eqn. 6.
Reversible enzyme kinetic model
The complex concentrations can be found by setting Eqn. 2 and 3 to zero and solving
simultaneously. This can be done using mathematica with the following command:
FullSimplify[ Solve[{0 == kfa*S*EE + krc*EP - (kfc + kfd)*ES,
0 == kra*P*EE + kfc*ES - (krc + krd)*EP}, {EP, ES}]]
This produces the following expressions for the enzyme substrate and enzyme product
complexes.
[ ES ] 
[ EP] 
[S ][ E ](krc  krd )k af  kra krc [ E ][ P]
(krc  krd )k df  krd k cf
[ P][ E ](k cf  k df )kra  k af k cf [ E ][ S ]
(k cf  k df )krd  k df krc
(S2)
(S3)
By substituting the Michaelis constant given in Eqn. 9 into Eqn. S2 and S3 and rearranging,
leads to the following expression.
kra krc [ E ][ P]
[ E ][ S ]k 
 krc  krd 
a
f
[ ES ] 
k 
d
f
k
[ E ][ P ]kra 
[ EP ] 
krd 
k
krd k cf
c
r
(S4)
 krd 
k af k cf [ E ][ S ]
k
c
f
d
f
c
r
k k
c
f
 k df
 k df


Dividing throughout by k cf  k df in Eqn. S4 and krc  krd in Eqn. S5 leads to:
(S5)
Enzyme kinetics model of complex biochemical systems
[ E ][ S ]k af
k
[ ES ] 
c
f
k
c
f
[ EP ] 
 k df
k
d
f
2
kra krc [ E ][ P ]
 c
k f  k df krc  krd
 


k k
c
f

k af k cf [ E ][ S ]
[ E ][ P ]kra

krc  krd
krc  krd k cf  k df



c
f
 k

 krd
 k df
 k
d
r
 k df
 
c
r

d
f
c
r
k k
krd
 c
c
d
kr  kr
kr  krd k cf  k df
 

(S6)
(S7)

Subtituting in the Michaelis Constant and α and β from Eqn. 8, leads to:
[ ES ] 
[ EP] 
krc
[ E ][ S ]
[ E ][ P]
 c
m
d
KS
 k f  k f  K Pm
 f   f r
k cf
[ E ][ P]
[ E ][S]
 c
m
d
KP
K Sm
kr  kr


r   r  f
(S8)
(S9)
Using these results, the standard form of the reaction velocity becomes:
k cf [ E ]T [ S ]
v  k cf [ ES ]  krc [ EP ] 
where,
H ij 
and
J ij 

j

j
krc [ E ]T [ P]

K Sm H fr
K Pm H rf
[S ]
[ P]
1 m
 m
K S J fr K P J rf
 i j  i   j i 
 j 1   i j 
(S10)
(S11)
 i j  i   j i 
(S12)
 kic 
kic
 j  1  c   j i   i j  j c
 kj 
kj


On a side note, it can be seen that the parameters can only be grouped into a minimum of
six parameters. So this formulation doesn’t simplify the kinetic compared to the basic mass
action implementation, which means the dQSSA approach is unnecessary here.
Coupled and fully Irreversible Enzyme Kinetic model
The complex concentrations can be found by setting Eqn. 2 and 3 to zero and solving
simultaneously. This can be done using mathematica with the following command:
Enzyme kinetics model of complex biochemical systems
3
FullSimplify[ Solve[{0 == kfa*S*EE - (kfc + kfd)*ES, 0 ==
kra*P*EE - (krc + krd)*EP}, {EP, ES}]]
This produces the following expressions for the enzyme substrate and enzyme product
complexes.
[ EP ] 
[ P][ E ]
K Pm
(S13)
[ EP ] 
[ P][ E ]
K Pm
(S14)
This is the standard form for the complex concentration based on the standard MichaelisMenten derivation, which hence results in the following for the reaction velocity.
k cf [ E ]0 [ S ]
v
krc [ E ]0 [ P]
K Sm
K Pm
[ S ] [ P]
1 m  m
KS KP

(S15)
Enzyme kinetics model of complex biochemical systems
4
Doc S2. Rigorous Derivation of the Quasi-Steady State Assumption
The conventional derivation of quasi-steady state enzyme kinetics is achieved by setting
the rate of change of the complex concentration to zero and solving for the complex
concentration. For the irreversible half reaction, this is:


[ ES ]  k a [ E ][ S ]  k c  k d [ ES ]  0
[ ES ] 
[ E ][ S ]
Km
(S16)
(S17)
Setting [ ES ] to zero then implies the complex concentration is constant at all times. This
has been shown to be false independently by Chen and Tzafriri: the complex concentration
does change whilst the system is in quasi-steady state [1,2]. This is shown in figure S5. In
Fig. S 1a, we show the difference in the time course of the complex, generated with the
tQSSA and mass action models. There is clearly a difference at the early transient phase,
but after a short time the two models converge. The initial transient phase is termed region
A. Fig. S 1b shows the time course of the complex only. What is interesting is region A can
be clearly identified, with the transition out of region A marked by a large change in the
complex concentration. Once out of region A the complex concentration is constant, which
we call region B, where from Fig. S 1a it can be seen that the two kinetics are consistent.
However, for large values of k a it can be seen that the there is a later region of change in
the complex concentration between region B and C. Here, there are only small differences
between the two models. Thus the tQSSA has modelled a change in the complex
concentration without deviating from the mass action model, which contradicts the quasisteady state assumption of Eqn. S16.
Fig. S 1: Temporal differences between the tQSSA and dQSSA. Heat map of (a) the fractional
difference and hence accuracy of the complex concentrations predicted by the tQSSA model vs mass
action model for different ka over the time course: (b) the mass action calculated complex concentration
for different ka over the time course. The regions marked are A: initial mass action transient phase
(difference between tQSSA and mass action), B: intermediate equilibrium, and C: the late quasi-steady
transient phase (no difference between tQSSA and mass action).
Enzyme kinetics model of complex biochemical systems
5
This is a peculiar mathematical contradiction. In fact, this contradiction leads to some
unrealistic implications. For example, in the Michaelis-Menten model, using the [ ES ]  0
assertion, the following logical steps can be made:
[ S ]0  [ S ]  [ P]  [ ES ]
(S18)
[ S ]0  [ S ]  [ P]  [ ES ]
(S19)
0  [ S ]  [ P]
(S20)
[S ]  [ P]
(S21)
As such the rate of change of the substrate concentration is exactly equal to the rate of
change of the product concentration. This is the way systems modellers typically implement
Michaelis-Menten kinetics in their models [1]. However, this leads to the conclusion that the
complex concentration has never changed from its initial condition. That is the final state of
an irreversible enzyme reaction, based on the mathematics is:
[S ]  0
(S22)
[ S ][ E ]T
K m  [S ]
(S23)
[ P]  [S ]0  [ ES ]
(S24)
[ ES ] 
However, we know intuitively that the final state for an idealised irreversible enzyme
reaction must be:
[S ]  0
(S25)
[ ES ]  0
(S26)
[ P ]  S0
(S27)
This is because all substrates (both bound and free) must be converted to product at
equilibrium. However, this cannot be arrived at mathematically unless [ E ]T
[S ]  K m .
Even then it will only be approximately true. Under this condition, the two final states are
equal because the condition forces [ ES ]  0 . However, this is not a rigorous solution to the
contradiction.
For a resolution, the definition of the quasi-steady state must be carefully considered.
Strictly speaking, the quasi-steady state means the kinetics of one state is far quicker than
the others, which means when non quasi-steady states moves, the quasi-steady state
reacts so quickly that it changes directly in response to slower transient states. Quasisteady states have only a dependence on other states and no time dependence of its own.
To mathematically demonstrate this, we will deal entirely using the state variables [ S ]T (total
substrate), [ E ]T (total enzyme), [ ES ] and [ P ] . So the quasi-steady state implies that all other
states besides the complex are fixed in time. In other words:
[ ES ]QSSA  [ ES ]
(S28)
ST , ET
Enzyme kinetics model of complex biochemical systems
6
In fact, this is the basic definition of a partial derivative. So:

[ ES ]
[ S ]T ,[E]T
[ ES ]
0
t
(S29)
This can be related to the complex rate equation (Eqn.10) by noting that [ES] is not just a
function of time (implicitly due to it being a function of itself), but a function of [ S ]T and [ E ]T .
As such, the total time derivative is.
[ ES ] 
[ ES ] [ ES ] d [ S ]T [ ES ] d [ E ]T


t
[ S ]T dt
[ E ]T dt
(S30)
Substituting this into Eqn. 10 gives the full equation:
[ ES ] 
[ ES ] [ ES ] d [ S ]T [ ES ] d [ E ]T


 k f [ E ][ S ]  k c  k r [ ES ]
t
[ S ]T dt
[ E ]T dt


(S31)
To determine the quasi-steady state, the relevant condition is given by Eqn. S29.
Essentially what is done here is determining the complex concentration if all other states
are fixed. That is:
d [ S ]T d [ E ]T

0
dt
dt
(S32)
Now note that this is not universally true. It is simply a condition we have placed on the
quasi-steady state condition. In this condition:
[ ES ]  0 
[ES ]
[ ES ]
0
0
[ S ]T
[ E ]T
(S33)
In practice, this becomes the same method as the conventional derivation of the quasisteady state. The crucial difference though is in the interpretation. We have noted that [ S ]T
and [ E ]T are zero when solving the quasi-steady state. However, when the system is
evolved past that steady state (i.e. when the slow states are allowed to change again), [ S ]T
and [ E ]T can no longer be universally assumed to be zero. As such:
[ ES ] 
[ ES ] d [ S ]T [ ES ] d [ E ]T

[ S ]T dt
[ E ]T dt
(34)
This no longer assumes that universally [ ES ]  0 , which no longer contradicts the results
seen in Fig. S 1 and the results of Chen and Tzafriri [1,2]. The ultimate interpretation of this
derivation is that the quasi-steady state assumption is accurate, even temporally, when the
system can be split into two phases. A fast phase when the catalytic step can be assumed
to not be proceeding while the complex formation occurs, followed by a slow phase where
the catalytic step proceeds and the complex concentration changes dynamically based on
the reagents remaining. Because the complex concentration equilibrates extremely quickly
though, in latter phase it can be assumed to instantaneously equilibrate to what it should be
at quasi-steady state. The quasi-steady state model simply ignores the first phase and
Enzyme kinetics model of complex biochemical systems
7
begins modelling the system from the second phase. This is essentially the interpretation
given by Chen in their analysis of Michaelis-Menten kinetics [1].
As was seen in the body of this paper, there are circumstances where the temporal profile
between the quasi-steady state model and the mass action model are different. These are
the cases when the true system cannot be cleanly divided into the two temporal regimes.
That is, the complex formation rate is slow enough that a significant amount of catalysis has
occurred before the complex pool has reach quasi-steady state. During these cases, the
true time course has a grey changeover point between the initial transient phase and the
latter quasi-steady state. It is during this changeover point that the quasi-steady state
model is inaccurate, because the true model is not truly in quasi-steady state, which implies
the full transient model, i.e. the mass action model, must be used.
Implicit modelling of the enzyme-substrate complex in the dQSSA
In the body of the paper, the complex is explicitly modelled. This increases the number of
state variables in the system as well as the dimensionality of the tensors compared to a
system that does not model the complex explicitly. It is possible to express Eqn. 30 and 31
as functions entirely of [S] and [E] by substituting in [ES] from Eqn. 17. Following this
derivation, the resulting differential equations are:
[ P] 
k c [ E ][ S ]
Km
k c [ E ][ S ] [ S ]
[E]
 m [ E]  m [S ]
m
K
K
K
[E]
[S ]
[ E ]   m [S ]  m [ E ]
K
K
[S ]  
(S35)
(S36)
(S37)
Eqn. 33 can then be excluded as [C] is no needed for the calculation for any other state
variable. Again using tensor notation, the differential equations are transformed into:
x2 
x1
k c x1 x2

1

x

x


2

m  1
Km
Km
 K 
x2
x

x  1  1m
m 1
K
 K
x3  

 x2  0

k c x1 x2
Km
(S38)
(S39)
(S40)
Which can then be vectorised into a single equation:

ij
 G ijk x k  xk  Vijk x j xk
(S41)
Leading to the rules:
1/ K m
Gijk  
 0
if [i,j,k] = [1,1,2], [1,2,1], [2,1,2], [2,2,1]
Else
(S42)
Enzyme kinetics model of complex biochemical systems
k c / K m

Vijk   k c / K m

0

8
if [i,j,k] = [1,1,2]
if [i,j,k] = [3,1,2]
(S43)
Else
Readers should note that preliminary testing with the two different implementations suggest
that the complex implicit implementation is less stable for ODE solvers as the explicit
reliance of [E] and [S] on the velocity of [P] means for small K
stiffness.
m
the system is vulnerable to
Enzyme kinetics model of complex biochemical systems
9
Doc S3. Additional figures for in silico validation
Comparison of progress curves between the mass action, Michaelis Menten and dQSSA
models using different randomly generated parameters.
Mass Action rate parameters
Fig. S 2: Example 1 of time course generated from
random assignment of parameters in in silico
model.
Km1f
kc1f
Km1r
kc1r
k2
k3
k4
[A]0
1.21
ka1f
84.0
kc7f
0.776
kd1f
2750
ka7r
235
kc1f
27.5
kd7r
77.6
ka1r
10800
kc7r
0.4
kd1r
2750
k8
0.821
kc1r
0.229
ka9f
23600
k2
3.32
kd9f
884
k3
0.416
kc9f
8.84
k4
98.4
ka9r
658
ka5f
4320
kd9r
884
kd5f
1340
kc9r
3.36
kc5f
13.4
ka10f
0.343
ka5r
45.9
kd10f
2550
kd5r
1340
kc10f
25.5
kc5r
3.66
ka10r
181
k6
1.39
kd10r
2550
ka7f
8480
kc10r
3.25
kd7f
77.6
k11
3.14
Km10f
kc10f
Km10r
kc10r
k11
7510
25.5
14.1
3.25
3.14
Michaelis-Menten and dQSSA rate parameters
Km5f
Km7r
33.1
0.31
0.332
c
c
k 5f
k 7r
27.5
13.4
0.4
m
K 5r
k8
0.26
29.3
0.821
c
m
k 5r
K 9f
0.229
3.66
0.0378
k6
kc9f
3.315
1.39
8.84
m
m
K 7f
K 9r
0.416
0.00925
1.35
c
c
k 7f
k 9r
98.37
0.776
3.36
Concentrations
[B]0
[C]0
[D]0
0.0283
73.0
87.3
[I]0
100
Enzyme kinetics model of complex biochemical systems
10
Mass Action rate parameters
ka1f
2400
kc7f
4.50
kd1f
75.0
ka7r
31.3
kc1f
0.750
kd7r
450
ka1r
324
kc7r
0.00888
kd1r
75.0
k8
0.356
kc1r
0.374
ka9f
2.81
k2
0.295
kd9f
798
k3
0.128
kc9f
7.98
k4
0.224
ka9r
137
ka5f
553
kd9r
798
kd5f
243
kc9r
1.39
kc5f
2.43
ka10f
4030
ka5r
151
kd10f
252
kd5r
243
kc10f
2.52
kc5r
0.122
ka10r
147
k6
2.12
kd10r
252
ka7f
1980
kc10r
0.356
kd7f
450
k11
5.48
Km10f
kc10f
Km10r
kc10r
k11
0.0631
2.52
1.71
0.356
5.48
Fig. S 3: Example 2 of time course generated from
random assignment of parameters in in silico
model.
Michaelis-Menten and dQSSA rate parameters
Km5f
Km7r
0.0316
0.443
14.4
c
c
k 5f
k 7r
0.750
0.43
0.00888
m
K 5r
k8
0.232
1.60
0.356
c
m
k
K
0.374
0.122
287
5r
9f
k6
kc9f
k2
0.295
2.12
7.98
Km7f
Km9r
k3
0.128
0.23
5.81
c
c
k4
k 7f
k 9r
0.224
4.5
1.39
Concentrations
[A]0
[B]0
[C]0
[D]0
7.94
0.0294
3.20
3.94
Km1f
kc1f
Km1r
kc1r
[I]0
100
Enzyme kinetics model of complex biochemical systems
11
Mass Action rate parameters
Fig. S 4: Example 3 of time course generated from
random assignment of parameters in in silico
model.
ka1f
160
kc7f
3950
kd1f
174
ka7r
58.4
kc1f
1.74
kd7r
395000
ka1r
0.844
kc7r
43.2
kd1r
174
k8
9.56
kc1r
0.457
ka9f
3390
k2
0.414
kd9f
3.02
k3
0.0186
kc9f
0.0302
k4
2.30
ka9r
1200
ka5f
34100
kd9r
3.02
kd5f
39.0
kc9r
0.00109
kc5f
0.39
ka10f
4.10
ka5r
0.00395
kd10f
1050000
kd5r
39.0
kc10f
10500
kc5r
0.00163
ka10r
0.336
k6
0.169
kd10r
1050000
ka7f
1130
kc10r
0.024
kd7f
395000
k11
1.12
Michaelis-Menten and dQSSA rate parameters
Km5f
Km7r
1.10
0.00115
6770
c
c
k 5f
k 7r
1.74
0.390
43.2
m
K 5r
k8
207
9860
9.56
c
m
k 5r
K 9f
0.457
0.00163
0.000898
k2
k6
kc9f
0.414
0.169
0.0302
m
m
k3
K 7f
K 9r
0.0186
355
0.00251
c
c
k4
k 7f
k 9r
2.30
3950
0.00109
Concentrations
[A]0
[B]0
[C]0
[D]0
2.03
0.115
4.23
1.33
Km1f
kc1f
Km1r
kc1r
Km10f
kc10f
Km10r
kc10r
k11
259000
10500
3130000
0.0240
1.12
[I]0
3.39
Enzyme kinetics model of complex biochemical systems
12
Parameter fitting of noisy time course using the dQSSA
To test the performance of the dQSSA when fitted to noisy time course data, synthetic data
was generated using the mass action model of the in silico hypothetical network of section
3.2. The default parameters used in this simulation is given in table S. 1. Noise was
introduced into the generated time course using the following equation:
y
randi
(44)
5
Where y’ is the noisy version of the simulated time course y. This formula result in a signal
to noise ratio of 5. The ratio of 5 was chosen such that the noise is not large enough that
the signal of the time course is completely lost.
The hypothetical model was only operated in the Ioff regime to reduce the dimensionality of
the parameter fitting because only reactions 1 (forward and reverse), 2, 3 and 4 are active.
Once the mass action time course was generated, complexes were dissociated into their
constituent components to emulate real data as complex concentrations cannot easily be
measured directly. Noise was then applied, and then the dQSSA was fitted by minimising
the following objective function.
y' y 
    ydQSSA  y 'MA 
2
(45)
The curve fitting was performed using Levenberg-Marquardt algorithm using the “correct”
quasi-steady state parameters as the seed point, which generates the fitted parameter set.
The time course generated from this parameter set is shown in Fig. S 5. From the fitted
parameter set, each parameter was varied and the change in the goodness of fit was
calculated. A probability distribution function across each parameter, with the other
parameters fixed, was calculated using the following equation:
exp[ ( xi )]
PDF ( xi )  
(46)
 exp[ ( x )]dx
i
i
0
The generated probability distribution function loosely gives the uncertainty of the fitted
parameter due to the noise in the data. The result of this for each parameter is shown in Fig.
S 6.
Table S. 1 Parameters used to test estimation fitting accuracy. The first two rows gives parameters
used to generate the mass action simulation time course. The fourth row gives the equivalent dQSSA
parameters. The last row gives the fitted parameter values for the seven parameters found by optimising
to the mass action time course with noise.
Mass Action
ka1f
kd1f
kc1f
ka1r
kd1r
kc1r
k2
k3
k4
192.1
773.4
7.344
55.28
773.4
1.252
8.018
0.425
0.653
Km1f
kc1f
Km1r
kc1r
k2
k3
k4
dQSSA Correct
4.066
7.344
14.013
1.252
8.018
0.425
0.653
dQSSA Fitted
3.214
7.586
9.921
1.448
9.084
0.564
0.964
Enzyme kinetics model of complex biochemical systems
13
Fig. S 5: Time course of the three parameter sets. Species used for curve fitting were total
concentrations for each species (that is complexes have been dissociated to their constituent
components). The mass action (MA) time course with added noise is shown with crosses. The green line
shows the time course generated from the fitted parameters while the red curve shows the time course
generated from the parameters used to the generate the mass action curve.
Enzyme kinetics model of complex biochemical systems
14
Fig. S 6: Local sensitivity analysis of fitted parameters. Exponentiated residuals with variation across
each parameter showing uncertainty of each parameter arising from the noise of the fitted time course.
Location of the fitted parameter and correct underlying parameter (dot dashed and dotted lines
respectively) are shown as vertical lines on the distributions.
Enzyme kinetics model of complex biochemical systems
15
Doc S4. Curve fitting data for in vitro validation
Fig. S 7. Model of drop in NADH fluorescence with NAD+. The drop in fluorescence is modelled to an
exponential decay with a decay constant of 0.0065mM-1. n = 8-24 for the data points obtained.
Enzyme kinetics model of complex biochemical systems
16
Fig. S 8. Curve fitting of the Michaelis-Menten and dQSSA model to the time course data for determining
the NADH-LDH binding dissociation constant. Title of each subfigure is the initial condition of that
experiment. Replicate 1.
Enzyme kinetics model of complex biochemical systems
17
Fig. S 9. Curve fitting of the Michaelis-Menten and dQSSA model to the time course data for determining
the NADH-LDH binding dissociation constant. Title of each subfigure is the initial condition of that
experiment. Replicate 2. Legend as Fig S.8.
Enzyme kinetics model of complex biochemical systems
18
Fig. S 10. Curve fitting of the Michaelis-Menten and dQSSA model to the time course data for
determining the NADH-LDH binding dissociation constant. Title of each subfigure is the initial condition of
that experiment. Replicate 3 (Lactate to pyruvate experiment). Legend as in Fig S.8.
Enzyme kinetics model of complex biochemical systems
19
Fig. S 11. Curve fitting of the Michaelis-Menten and dQSSA model to the time course data for
determining the NADH-LDH binding dissociation constant. Title of each subfigure is the initial condition of
that experiment. Replicate 3 (Pyruvate to lactate reaction). Legend as Fig S.8.
References
1 Chen WW, Niepel M & Sorger PK (2010) Classic and contemporary approaches to
modeling biochemical reactions. Genes Dev. 24, 1861–75.
2 Tzafriri AR (2003) Michaelis-Menten kinetics at high enzyme concentrations. Bull. Math.
Biol. 65, 1111–29.