Additional file 1
I. Derivation of model equations for various conditions with Michaelis Menten kinetics (K1)
A.
Enzyme kinetics for one enzyme one substrate condition
k
kcat
f

E+ S 
 P+ E ,
 [E.S] 
kb
k f  rate of complex formation, k b  rate of complex dissociation,k cat  rate of product formation.
For [E.S]=constant(steady state assumption)
[E].[S]
[S]
[E.S]
[E.S]
[S]
Km
Km

=
=
=
=
ETotal E+[E.S] E+ [E].[S] 1+ [S] Km+[S]
Km
Km
ETotal .[S]
[E.S]
[S]

=
 [E.S] =
ETotal Km+[S]
Km+[S]
[S]
Km
[S]
1+
Km
Here, k cat .[E.S]= v,the flux ofproduct formation
k .E .[S]
 k cat .[E.S]= cat Total
=
Km+[S]
k cat .ETotal .
[S]
Km     [a]
v=
[S]
1+
Km
In our models when E Total =Sig,S=MKKK , equation [a] gives
k cat .ETotal .
[MKKK]
Km
 v1=
    [b]
[MKKK]
1+
Km
Fordephosphorylation of MKKK  P to MKKK by phosphatase Phos1: ETotal =Phos1,S=MKKK - P.
k cat .Sig.
The flux of MKKK  P dephosphorylation is given as ,
 v2=
[MKKK - P]
Km
    [c]
[MKKK - P]
1+
Km
k cat .Phos1.
Dynamics of MKKK  P is given as
d[MKKK  P]
 v1  v2
    [d]
dt
The conservation equation states,
MKKK Total  MKKK  MKKK  P
 MKKK  MKKK Total  MKKK  P
   [e]
Hence at any point of time, MKKK-P concentration is determined from equation [d], and
concentration of MKKK at that time could be calculated using equation [e].
B.
For two step processes (double phosphorylation-dephosphorylation) such as MKK and MK
phosphorylation-dephosphorylation cycles, competition between two substrates for their common
enzyme arises.
Here,
k
k
k2a
k-1a
kcat2
kcat1
-2a
1a 


P2+ E 
 [E.S2] 
P1+ E
 S2 + E+ S1 
 [E.S1] 
P1,P2 are the products and S1,S2are respective substrates. E is the enzyme.
Under such conditions,
[E].[S1]
[S1]
k cat1 .ETotal .
Km1 
Km1     [f]
v S1 
[E]  [E.S1]  [E.S2] 1  [S1]  [S2]
Km1 Km2
and similalrly,
[S2]
k cat2 .ETotal .
Km2     [g]
v S2 
[S1] [S2]
1

Km1 Km2
For the phosphorylation of MKK to MKK  P and MKK  P to MKK  PP
ETotal  MKKK  P,S1  MKK,S2  MKK  P.
For the dephosphorylation of MKK-PP to MKK  P and MKK  P to MKK
k cat1 .ETotal .
ETotal  Phos2,S1  MKK  PP,S2  MKK  P.
For the phosphorylation of MK to MK  P and MK  P to MK  PP
ETotal  MKK  PP,S1  MK,S2  MK  P.
For the dephosphorylation of MK-PP to MK  P and MK  P to MK
ETotal  Phos3,S1  MK  PP,S2  MK  P.
DynamicsofMKK  P is givenas
d[MKK  P]
 v3  v5  v4  v6     [h]
dt
DynamicsofMKK  PP is givenas
d[MKK  PP]
 v4  v5     [i]
dt
DynamicsofMK  P is givenas
d[MK  P]
 v7  v9  v8  v10     [j]
dt
DynamicsofMK  PP is givenas
d[MK  PP]
 v8  v9     [k]
dt
The conservation equation states,
MKK Total  MKK  MKK  P  MKK  PP     [l]
MK Total  MK  MK  P  MK  PP       [m]
The dynamics of M1, M2, M3, and M4 are thus captured in general by the same set of differential
equations. The differential equations capturing the complete dynamics of the MAPK cascade are,
d[MKKK - P]
= v1 - v2
- - -[n]
dt
d[MKK - P]
= v3 - v4 + v5- v6 - - -[o]
dt
d[MKK - PP]
= v4 - v5
- - - -[p]
dt
d[MK - P]
= v7 - v8 + v9 - v10 - - - - [q]
dt
d[MK - PP]
= v8 - v9
- - - - [r]
dt
And the mass conservation equations of the complete system are
[MKKK]Total = [MKKK]+[MKKK - P]
[MKK]Total = [MKK]+[MKK - P]+[MKK - PP]
[MK]Total = [MK]+[MK - P]+[MK - PP]
The equations [o]-[r] are same as equations [h]-[k] as described above. Here, depending upon the
values of flux equations (v1-v10), dynamics of each system is distinctly shaped.
C.
Derivation of flux equations for M1-M4 in unsequestrated conditions with K1
M1
Here the phosphatases are specific to each layer of the three layer cascade. The flux v1-v10 could
be easily derived from the above equations. MKKK phosphorylation and dephosphorylation are
given by equation [b] and [c] respectively. MKK and MK layer phosphorylation dephosphorylation
can be easily derived from equations [f] and [g] above.
(Equations from now onwards are numbered identical to the equation number in the main text)
d[MKKK - P]
=
dt
d[MKK - PP]
=
dt
d[MK - PP]
=
dt
- - - [1]
k3.MKKK - P.MKK
k4.MKKK - P.MKK - P
K3
K4
MKK
MKK - P
MKK
MKK - P
1 +
+
1 +
+
K3
K4
K3
K4
k5.Phos2.MKK - PP
k6.Phos2.MKK - P
K5
K6
+
MKK - PP
MKK - P
MKK - PP
MKK - P
1 +
+
1 +
+
K5
K6
K5
K6
d[MKK - P]
=
dt
d[MK - P]
=
dt
Sig.k1.MKKK
Phos1.k2.MKKK - P
K1
K2
MKKK
MKKK - P
1+
1+
K1
K2
k4.MKKK - P.MKK - P
K4
MKK
MKK - P
1 +
+
K3
K4
k5.Phos2.MKK - PP
K5
MKK - PP
MKK - P
1 +
+
K5
K6
- - - [2]
- - - [3]
k8.MKK - PP.MK - P
k7.MKK - PP.MK
K8
K7
MK
MK - P
MK
MK - P
1 +
+
1 +
+
K7
K8
K7
K8
k9.Phos3.MK - PP
k10.Phos3.MK - P
K9
K10
+
- - - [4]
MK - PP
MK - P
MK - PP
MK - P
1 +
+
1 +
+
K9
K10
K9
K10
k8.MKK - PP.MK - P
K8
MK
MK - P
1 +
+
K7
K8
k9.Phos3.MK - PP
K9
MK - PP
MK - P
1 +
+
K9
K10
- - - [5]
In the equations [1] – [5], ki is the catalytic rate of the ith reactions and K i is the Km of the ith
reaction. ‘Sig’ in equation [1] represents the incoming signal that activates the cascade.
M2
Here Phos1 is shared between MKKK and MKK layer. The reaction schema for such condition is
given as
Phos1+MKK  [Phos1.MKK -P]  MKK -P + Phos1 + MKKK -P  [Phos1.MKKK -P]  Phos1+MKKK
+
MKK -PP
[Phos1.MKK -PP]

Phos1+MKK -P
Flux of MKKK-P dephosphorylation
k2.Phos1.MKKK -P
K2
v2 
MKKK -P MKK -PP MKK -P
1+
+
+
K2
K5
K6
Flux of MKK-PP dephosphorylation
k5.Phos1.MKK - PP
K5
v5 
MKKK - P MKK - PP MKK - P
1+
+
+
K2
K5
K6
Flux of MKK-P dephosphorylation
k6.Phos1.MKK -P
K6
v6 
MKKK -P MKK -PP MKK -P
1+
+
+
K2
K5
K6
The equations [1], [2] and [3] above are modified as
d[MKKK -P]
=
dt
Sig.k1.MKKK
k2.Phos1.MKKK -P
K1
K2
- - -[5]
MKKK
MKKK -P MKK -PP MKK -P
1+
1+
+
+
K1
K2
K5
K6
k3.MKKK -P.MKK
k4.MKKK -P.MKK -P
d[MKK -P]
K3
K4
=
MKK MKK -P
MKK MKK -P
dt
1+
+
1+
+
K3
K4
K3
K4
k5.Phos1.MKK -PP
k6.Phos1.MKK -P
K5
K6
+
- - - [6]
MKKK -P MKK -PP MKK - P
MKKK -P MKK -PP MKK -P
1+
+
+
1+
+
+
K2
K5
K6
K2
K5
K6
d[MKK -PP]
=
dt
k5.Phos1.MKK -PP
k4.MKKK -P.MKK -P
K5
K4
- - - [7]
MKK MKK -P
MKKK -P MKK -PP MKK -P
1+
+
1+
+
+
K3
K4
K2
K5
K6
M3
Here Phos2 is shared between MKK and MK layer. The reaction schema for such condition can be
given as:
Phos2 +MK

[Phos2.MK -P]
MK -P
+
Phos2+ MKK  [Phos2.MKK -P]  MKK -P + Phos2 + MKK -PP  [Phos2.MKK -PP]  Phos2+MKK -P
+
MK -PP
[Phos2.MK -PP]

Phos2 +MK -P
Here the flux of MKK-PP dephosphorylation
k5.Phos2.MKK - PP
K5
v5 
MKK - PP MKK - P MK - PP MK - P
1+
+
+
+
K5
K6
K9
K10
Flux of MKK-P dephosphorylation
k6.Phos2.MKK - P
K6
v6 
MKK - PP MKK - P MK - PP MK - P
1+
+
+
+
K5
K6
K9
K10
Flux of MK-PP dephosphorylation
k9.Phos2.MK - PP
K9
v9 
MKK - PP MKK - P MK - PP MK - P
1+
+
+
+
K5
K6
K9
K10
Flux of MK-P dephosphorylation
k10.Phos2.MK - P
K10
v10 
MKK - PP
MKK - P MK - PP
MK - P
1+
+
+
+
K5
K6
K9
K10
The equations [2], [3], [4] and [5] are modified as
k3.MKKK -P.MKK
k5.Phos2.MKK -PP
k4.MKKK -P.MKK -P
d[MKK -P]
K3
K5
K4
=

+
MKK MKK -P
MKK MKK -P
MKK -PP MKK -P MK -PP MK -P
dt
1+
+
1+
+
1+
+
+
+
K3
K4
K3
K4
K5
K6
K9
K10
k6.Phos2.MKK -P
K6

   [9]
MKK -PP MKK -P MK -PP MK -P
1+
+
+
+
K5
K6
K9
K10
d[MKK -PP]
=
dt
k5.Phos2.MKK -PP
k4.MKKK -P.MKK -P
K5
K4

- - -[10]
MKK MKK -P
MKK -PP MKK -P MK -PP MK -P
1+
+
1+
+
+
+
K3
K4
K5
K6
K9
K10
k8.MKK -PP.MK -P
k9.Phos2.MK -PP
k7.MKK -PP.MK
d[MK -P]
K8
K9
K7
=

+
MK MK -P
MK MK -P
MKK -PP MKK -P MK -PP MK -P
dt
1+
+
1+
+
1+
+
+
+
K7
K8
K7
K8
K5
K6
K9
K10
k10.Phos2.MK -P
K10

     [11]
MKK -PP MKK -P MK -PP MK -P
1+
+
+
+
K5
K6
K9
K10
d[MK -PP]
=
dt
k8.MKK -PP.MK -P
k9.Phos2.MK -PP
K8
K9

    [12]
MK MK -P
MKK -PP MKK -P MK -PP MK -P
1+
+
1+
+
+
+
K7
K8
K5
K6
K9
K10
M4
Here Phos1 is shared between MKKK and MKK layer and Phos2 is shared between MKK and MK
layer. The reaction schema for such condition can be given as:
Phos2 +MK

[Phos2.MK -P]
MK -P
+
Phos2 +MKK  [Phos2.MKK -P]  MKK -P+ Phos2 + MKK -PP  [Phos2.MKK -PP]  Phos2 +MKK -P
+
+
+
Phos1 +MKK  [Phos1 .MKK -P]  Phos1 MK -PP Phos1  [Phos1.MKK -PP]  Phos1 + MKK -P
+
Phos1 + MKKK  [Phos1 .MKKK - P]  MKKK -P [Phos2.MK -PP]

Phos2 +MK -P
For this model, differential equations capturing the dynamics of MKKK-P, MK-P and MK-PP would
be identical to equations [5], [11] and [12] respectively. Equations capturing the dynamics of MKKPP and MKK-P will change as Phos1 and Phos2 are functional in the MKK layer. The flux of
dephosphorylation corresponding to Phos1 is referred with the suffix “a” and the flux of
dephosphorylation corresponding to Phos2 is referred with the suffix “b”, in the below equations.
Flux of MKK-PP dephosphorylation by Phos1
k5a.Phos1.MKK -PP
K5a
v 5a 
MKK -PP MKK -P MKKK -P
1+
+
+
K5a
K6a
K2a
Flux of MKK-P dephosphorylation by Phos1
k6a.Phos1.MKK -P
K6a
v6a 
MKK -PP MKK -P MKKK -P
1+
+
+
K5a
K6a
K2a
Flux of MKK-PP dephosphorylation by Phos2
k5b.Phos2.MKK - PP
K5b
v 5b 
MKK - PP
MKK - P MK - PP
MK - P
1+
+
+
+
K5b
K6b
K9b
K10b
Flux of MKK-P dephosphorylation by Phos2
k6b.Phos2.MKK - P
K6b
v6b 
MKK - PP
MKK - P MK - PP
MK - P
1 +
+
+
+
K5b
K6b
K9b
K10b
Dynamics of MKK-PP and MKK-P is thus represented as
k3.MKKK - P.MKK
k5a.Phos1.MKK - PP
k4.MKKK - P.MKK - P
d[MKK - P]
K3
K5a
K4
=

+
MKK MKK - P
MKK MKK - P
MKK - PP MKK - P MKKK - P
dt
1+
+
1+
+
1+
+
+
K3
K4
K3
K4
K5a
K6a
K2a
k5b.Phos2.MKK - PP
k6a.Phos1.MKK - P
K5b
K6a
+

MKK - PP MKK - P MK - PP MK - P
MKK - PP MKK - P MKKK - P
1+
+
+
+
1+
+
+
K5b
K6b
K9b
K10b
K5a
K6a
K2a
k6b.Phos2.MKK - P
K6b

MKK - PP MKK - P MK - PP MK - P
1 +
+
+
+
K5b
K6b
K9b
K10b
d[MKK - PP]
=
dt
   [13]
k5a.Phos1.MKK - PP
k4.MKKK - P.MKK - P
K5a
K4

MKK MKK - P
MKK - PP MKK - P MKKK - P
1+
+
1+
+
+
K3
K4
K5a
K6a
K2a
k5b.Phos2.MKK - PP
K5b

MKK - PP MKK - P MK - PP MK - P
1+
+
+
+
K5b
K6b
K9b
K10b
    [14]
D.
Modification of flux equations in sequestrated conditions in K1 models
Phosphatase sequestration results in an additional step in the dephosphorylation process where the
unphosphorylated kinase can form a complex with the phosphatase before being released as the
product. The sequestration effect is considered in the final product release step.
For example in model M1, MKK layer dephosphorylation in presence of its sequestration with Phos2
is given as,
Phos2+MKK  [Phos2.MKK]  [Phos2.MKK -P]  MKK -P+ Phos2
+
MKK -PP
[Phos2.MKK -PP]

Phos2+MKK -P
Flux of MKK-PP dephosphorylation is modified as
k5.Phos2.MKK - PP
K5
v5 
MKK - PP MKK - P MKK
1+
+
+
K5
K6
Kse
And the flux of MKK-P dephosphorylation is modified as
k6.Phos2.MKK - P
K6
v6 
MKK - PP MKK - P MKK
1+
+
+
K5
K6
Kse
Where
MKK
Kinase
, or in general
captures the sequestration effect in the steady state.
Kse
Kse
Kse is defined as the affinity of the unphosphorylated kinases towards the phosphatase. For all the
sequestrated models, the flux of dephosphorylation will be updated by their respective
Kinase
.
Kse
Please refer to the SBML models 1-16 for the complete set of equations for each of the models M1M4.
II. Development of elementary mass action models (K2) of MAPK signal transduction
The mass action models were built similar to the original model by Huang and Ferrell and the
parameters were used from the model developed by Markevich et. al., where the later study
inspected the effect of sequestration, for enzyme substrate concentrations in the same order of
magnitude. The models are given as SBML models 17-32 in the additional material files. The
reactions of the four models are given in the additional table 1B.
III. Conversion of elementary mass action models (K2) to steady state models (K2_QSS).
K2_QSS were built based on the original parameters used for building K2. Here Km of a reaction was
calculated based on the forward (kf), backward (kb) and catalytic (kcat) rate of the reactions in K2.
For example in the reaction
kf
kcat

 [MKKK -P.Phos1] 
MKKK -P +Phos1 
 MKKK +Phos1

kb
Km is calculated as Km=
kb +kcat
. The kcat of each reaction in K2 was also used as the kcat of the
kf
corresponding reaction in K2_QSS.
For the sequestration condition, the enzyme and substrate forms a reversible complex as shown
below.
kf1
kf2
k2


 [MKKK -P.Phos1] 


 MKKK + Phos1
MKKK -P +Phos1 
 [MKKK.Phos1] 


kb1
kb2
The kinetic parameter Kseq which represents such sequestration effect was calculated for K2_QSS
as Kseq=
kb2
. Here, as the value of kb2 increases, the effect of sequestration also becomes
kf2
stronger, assuming rest of the parameters remain constant. The flux of phosphorylation and
dephosphorylation for simulation of all the K2_QSS models could be derived in the same lines as
explained for the K1 models, as explained in above sections.