Supplemental Fig. 1
Dynamic simulations using the constraint-based modelling approach
a, System comprising one reaction with
internal species A, B and C. b, Set of
constraints applied to the system. (1)
Dynamic
mass
balance
equation,
analogous to ODEs but written as the
product of the stoichiometry matrix
(matrix containing the stoichiometric
coefficients of all network reactions, rows
representing
species
and
columns
representing reactions) and the flux
distribution vector (vector representing
the
fluxes
of
each
reaction).
(2)
Inequations constraining V1, the flux of
reaction 1. Positive flux signifies reaction
irreversibility. (3) Regulatory rule. c, A
polyhedral flux cone representing the
space of steady state solutions computed
using static flux balance analysis (FBA)
(i.e. where S x V = 0). Each point in state
space corresponds to a flux vector whose
coordinates are the fluxes of each reaction
in the network. An optimal solution for a
given objective function, V*, is shown on an edge of the cone. d, Diagram of dynamic FBA.
First, static FBA is applied to calculate the optimal network state for the initial environment. In a
closed system, since molecular exchanges with the environment are imbalanced as a result of
network activity, network and environmental states evolve together. Under the quasi-steady-state
assumption, the environmental state is updated at each time step to reflect a steady optimal
activity of the network during the previous time interval, and the optimal state is recalculated,
using static FBA.
In the following, we provide details about the simulation (e.g. reaction network, parameters) for
each model discussed in the main text, show alternative ways to represent the Boolean model
discussed in Figure 1 b and give a detailed mathematical analysis for the model discussed in
Figure 2.
Supplemental Fig. 2
Simulation of a simple network using different mathematical formalisms
1. Representations of a simple gene network with negative feedback using a Boolean
formalism
Discrete time Boolean model with synchronous update. a. to d. are four equivalent ways of
specifying the Boolean functions for all model components (here simplified to three: B_mRNA,
Bn protein - corresponding to all its homo-multimers - and AP) a. network diagram b. finite state
machine c. truth table containing the updating rules d. wiring diagram.
2. Reaction network and parameters for the simulation of the models
Reaction Network for model simulated deterministically (c) and stochastically (e) in which B
represses
its
transcription
by
binding
to
its
own
transcription
activator
AP.
Parameters used for simulation:
k1=0.1 s-1 (B_mRNA production)
k2=0.5 s-1 (B production)
kon=1e09 M-1 s-1 (complex formation)
koff=0.01 s-1
kdeg=log(2)/80 s-1
Initial conditions: 100 AP molecules, 0 molecules for remaining species
Reaction Network for model simulated deterministically (d), stochastically (f) in the case in
which B oligomerization is considered. B octamer is now the species that binds to the
transcription factor AP, repressing B transcription.
Parameters used for simulation:
k1=0.1 s-1 (B_mRNA production)
k2=0.5 s-1 (B production)
kdeg=log(2)/80 s-1
kon=1e09 M-1 s-1 (complex formation between B8 and AP)
koff=0.01 s-1
kon2=1e06 M-1 s-1 (dimer formation)
koff2=0.01 s-1
kon4=1e06 M-1 s-1 (tetramer formation)
koff4=0.01 s-1
kon8=1e09 M-1 s-1 (octamer formation)
koff8=0.1 s-1
Initial conditions: 100 AP molecules, 0 molecules for remaining species
Supplemental Fig. 3
Example of mathematical formalism independent pitfalls in modelling
In all three cases considered, rates corresponding to reactions of the same type were given
identical values, such that the same parameter set was used each time, specifying constants for all
four reaction types (counting forward and reverse reactions).
Parameters:
Dimerization rate: kon=1e09 M-1s-1
Dissociation rate: koff=1 s-1
Monomer modification rate: ka=1e-02 s-1
Monomer demodification rate: ka=1e-02 s-1
Initial conditions: 600 M molecules, 0 molecules all remaining species.
Set of differential equations derived from the set of reactions shown in the general model
(no dependence specified between dimerization and modification reactions):
dM/dt = -ka1 * M + ki1 * MA - kon1 * M * M + 2 * koff1 * D - kon2 * M * MA + koff2 * DA
dMA/dt = ka1 * M - ki1 * MA - kon3 * MA * MA + 2 * koff3 * DAA - kon2 * M * MA + koff2 * DA
dD/dt = - ka2* D + ki2 * DA + 0.5 * kon1 * M * M - koff1 * D
dDA/dt = ka2 * D - ki2 * DA - ka3 * DA + ki3 * DAA + kon2 * M * MA - koff2 * DA
dDAA/dt = ka3 * DA - ki3 * DAA + 0.5 * kon3 * MA * MA - koff3 * DAA
Mathematical derivation of the steady state concentration of DAA for the three cases described in
Figure 2 of the main text, showing the impact of different assumptions on the interdependence of
dimerization and modification reactions, as a function of parameter values chosen for the model:
the pool size, Mo, for all monomeric and dimeric forms of the protein, and specific sets of
reaction rates.
(a): total dependence of dimerization on activity
Setting all disallowed reactions rates to zero and setting ka1=ka; ki1=ki; kon3=kon; koff3=koff,
we have, at equilibrium (indicated by the letter e after the variable name) :
(1) dMe/dt = -ka * Me + ki * MAe = 0
(2) dMAe/dt = ka * Me - ki * MAe - kon * MAe² + 2 * koff * DAAe = 0
(3) dDAAe/dt = 0.5 * kon * MAe² - koff * DAAe = 0
Expressing all variables as a function of Me :
(1)  (1') MAe = ka/ki * Me
(3)  (3') DAAe = 0.5 * kon/koff * MAe²
Equation of mass conservation :
Mo = M + MA + 2 * DAA
Substituting (1') and (3') in the previous equation gives :
Mo = Me + ka/ki * Me + kon/koff * (ka/ki * Me)²
kon/koff * (ka/ki)²
Mo
* Me² + Me =0
1 + ka/ki
1 + ka/ki
Mo
1

K * Me² + Me = 0 , where K = kon/koff * ka/ki *
1 + ka/ki
1 + ki/ka

The steady state value of M is the positive root of a second degree polynomial:
-1 + 1 + 4 * K *
Me =
Mo
1 + ka/ki
2*K
The steady state values for all species are then :
Me = MAe =
1
Mo
( 1+4*K*
-1)
2K
1 + ka/ki
DAAe = Mo/2 - Me = Mo/2 -
If ka=ki, by setting R =
1
Mo
( 1+4*K*
-1)
2K
1 + ka/ki
koff
, these expressions simplify to
Mo*kon
Me = MAe = Mo * R * ( 1+
1
 1)
R
DAAe = Mo [0.5 - R * ( 1+
1
 1)]
R
and
(b): total dependence of activation on dimerization
Setting all disallowed reaction rates to zero, we have, at equilibrium:
(1) dMe/dt = - kon1 * Me² + 2 * koff1 * De = 0
(2) dDe/dt = - ka2 * De + ki2 * DAe + 0.5 kon1 * Me² - koff1 * De = 0
(3) dDAe/dt = ka2 * De - ki2 * DAe - ka3 * DAe + ki3 * DAAe= 0
(4) dDAAe/dt = ka3 * DAe - ki3 * DAAe = 0
Expressing all variables as a function of Me, we find :
kon1
ka2*kon1
De =
* Me² ; DAe = ka2/ki2 * De =
* Me² ;
2koff1
2*ki2*koff1
ka2*ka3*kon1
DAAe = ka3/ki3 * DAe =
* Me²
2*ki2*ki3*koff1
Equation of mass conservation :
Mo = M + 2 * (D + DA + DAA)
Substituting D, DA, DAA in the previous equation, we get :
ka2ka3
Mo = Me + kon1/koff1 * Me² (1 + ka2/ki2 +
)
ki2ki3
ka2ka3
 [kon1/koff1 (1 + ka2/ki2 +
)] * Me² + Me - Mo = 0
ki2ki3

K' * Me² + Me - Mo = 0
The steady state value of M is the positive root of a second degree polynomial:
Me =
1 + 4 * K' * Mo  1
ka2ka3
, where K' = kon1/koff1 * (1 + ka2/ki2 +
)
2 * K'
ki2ki3
For DAA, we get :
kon1*ka2*ka3
* Me²
2*koff1*ki2*ki3
Substituting and simplifying gives the following expression for the steady state concentration of active dimer:
1
1
DAAe =
( Mo 1 + 4 * K' * Mo)
ki2*ki3 ki3
2K'
(

 1)
ka2*ka3 ka3
If we further consider the case where ka2=ki2 and ka3=ki3, and set koff1=koff, kon1=kon, we get:
DAAe =
kon
koff
kon
and Me =
( 1 + 12
* Mo  1)
koff
6kon
koff
koff
By setting R =
this simplifies to:
Mo * kon
Mo
12
1
R
12
Me =
* R *( 1+
- 1) and DAAe = Mo (1 - *( 1+ -1))
6
R
6
6
R
K' = 3 *
Conclusions drawn for the special case where modification/demodification rates are equal and
rate constants for reactions of the same type are given identical values are the following:
-totally dependent cases
The fraction of molecules in active dimer form at steady state is always a function of the ratio
R = Kd / Mo, where Kd is the dissociation constant of the monomer/dimer equilibrium (Kd =
koff/kon), and Mo is the size of the pool of monomeric parts (the number of monomers plus
twice the number of dimers).
When dimerization occurs only after modification of the monomers, the maximal steady state
concentration for active dimer is achieved when no monomers are left in solution, i.e. when the
dissociation constant is close to zero. If monomer modification is restricted to the dimer form
however, the concentration of active dimer cannot go higher than one third of the previous limit,
simply because in this case, all three dimeric forms (non-modified, singly modified, and dually
modified) are in equal concentrations.
N.b. Unlike the next case where the network contains cycles, because the paths are linear, each
pair of inverse reactions have balanced activities, i.e. they have equal fluxes.
-totally independent case
Here, by virtue of symmetry of the system (including identical rates), it is clear that any steady
state concentration for M will be a steady state concentration for MA, and likewise D and DAA
will have identical steady state concentrations, and no net flux of molecules can occur between M
and MA (since both modification and demodification reactions are first order reactions and their
rates are equal). However, since all dimerization reactions have the same rate, M = MA forces the
heterodimerization reaction flux to be double that of homodimerization reaction fluxes
(kon*M*M vs 0.5*kon*M*M), preventing their reverse reactions to have equal fluxes. This
establishes that the steady state concentration for DA is likely higher than that of D or DAA, and
that, in this case, the concentrations of the different forms of the protein are kept steady by having
a net flux of particles across the network, flowing from along the two symmetrical outer branches
of the network. Although no simple formula is available for the concentration of active dimer,
that of unmodified monomer is Me = Mo * R/4 (√(1+4/r) - 1).
N.b. Choosing a kon for the heterodimerization reaction half of the others allows all reaction
fluxes to be balanced at steady state and a calculation similar to the previous ones to be done, as
follows:
(c): total independence of dimerization reaction rates on activation state and vice versa
With all reaction rates of a given reaction type being identical, the degrees of freedom in this case are again four.
(1) dMe/dt = -ka * Me + ki * MAe - kon * Me² + 2 * koff * De - kon * Me * MAe + koff * DAe = 0
(2) dMAe/dt = ka * Me - ki * MAe - kon * MAe² + 2 * koff * DAAe - kon * Me * MAe + koff * DAe = 0
(3) dDe/dt = -ka * De + ki * DAe + 0.5 * kon * Me² - koff * De = 0
(4) dDAe/dt = ka * De - ki * DAe - ka * DAe + ki * DAAe + kon * Me * MAe - koff * DAe = 0
(5) dDAAe/dt = ka * DAe - ki * DAAe + 0.5 * kon * MAe² - koff * DAAe = 0
Here, we'll add the constraint that modification/demodification reaction rates are equal, and modify a previous constraint,
by imposing that the heterodimerization reaction have a rate half of the others, allowing the forward/reverse
reaction pairs to have equal flux at steady state. In such a case we have:
Me = MAe and De = DAe = DAAe
With this, equation (3) now simplifies to:
0.5* kon * Me² - koff * De = 0  De =
kon
* Me²
2*koff
Equation of mass conservation :
Mo = M + MA + 2 * D + 2 * DA + 2*DAA
Substituting all but M and D for their equilibrium values, then substituting De we get :
Mo = 2* Me + 6 * De = 2 * Me + 3 * kon/koff * Me²
kon
 3*
* Me² + 2 * Me - Mo = 0
koff
The steady state value of M is the positive root of a second degree polynomial:
-1 + 1 + K'' * Mo
, where K'' = 3 * kon/koff
K''
koff
R
3
Setting R =
:
Me = Mo* * ( 1+  1)
kon*Mo
3
R
Me =
To get the steady state concentration of active dimer, we first write:
DAAe = Mo/6 - Me/3
Substituting Me, then R and simplifying gives :
DAAe =
1
R
3
* Mo ( 0.5 * ( 1+  1))
3
3
R
Supplemental Fig. 4
Effect of localization of species on cellular processes
Reactions considered for the simulation
Parameters used for simulation:
D= 0.8 μm2 s-1
kon=1e07 M-1 s-1
koff=1 s-1
kcat=1 s-1
kpA=1 s-1 (protein A production)
kd=1e06 M-1 s-1 (proteins A and B degradation)
kpB=7e-04 M-1 s-1 (protein B production)
Initial conditions: TF=200 molecules, TFP=TFKn=TFPPh=A=0 molecules, Kn=Ph=60
molecules, B=250 molecules
Supplemental Fig. 5
Example of model dependent putative pitfalls in modelling: continuous versus discrete
concentrations
Scheme of the model: B can form a complex with its promoter prom, giving B_prom. B_prom
catalyses
the
production
of
B.
B
degradation
occurs
by
first
Parameter values used for the simulations of the HIGH and LOW cases:
Reactions constants (units)
HIGH
LOW
k_on (M-1 s-1)
108
108
k_off (s-1)
8.3
1.66
k_prod (s-1)
1
0.2
k_deg (s-1)
10-2
10-2
order
decay.