Supplementary information
Some additional information related to the figures and parameters is given below. An additional
figure (Fig. S1) is also included.
1. The reactions which are studied are either first order reactions or second order reactions. All
first order rate constants are assumed to have nominal values of 0.1s-1 unless otherwise
mentioned.
2. Figure 2 deals with detailed descriptions of binding and unbinding and complex formation.
Such a scheme is capable of resulting in simple transcritical bifurcations as well as
transcritical bifurcations and bistability. Parameters (all variables and parameters
dimensionless) for (a) All kinetic constants =1, except k3 which is equal to 3. k2 is the catalytic
rate of the forward reaction. The total phosphatase is equal to 5, and the initial condition for
X and X* are respectively 40 and 2. (b) The same as (a) except k3, which is equal to 10.
3. Figure 3 deals with an autocatalytic reaction in an irreversible loop as well as in open chains
of reversible reactions. All first order reactions have rates 0.1s-1. The total amount of
substrate in the system (a conserved quantity) is equal to 1M. The kinetic constant of the
autocatalytic reaction is the bifurcation parameter.
4. Stochastic simulations of networks. First order rate constants =0.1s-1, k/V (where k is the
autocatalytic reaction rate constant =0.01/no.s. For a volume of 1m3, k=6/sM. The total
number of molecules initially is chosen to be 40 (equidistributed between A,B,C and D).
5. Again, all reaction rate constants in the ambient network (i.e. apart from the autocatalytic
module) have the nominal value of 0.1s-1. (b) One of the kinetic constants in the ambient
network (mentioned in the diagram) is chosen to have a lower value of 0.001 (c) The same as
(b), except that the rate constants here correspond to reactions in the clockwise direction (if
we regard the autocatalyzing direction as clockwise). (d) and (e) show the effect of having all
reactions in the ambient network relatively small in magnitude. In each plot shown are the
cases where all the constants have a value 0.001s-1 (blue) along with a situation where all
kinetic constants in the ambient network in the anticlockwise direction (k-1,k-2,k-4) all have
values 0.004 (d) and 0.008 (e). The total amount of substrate in this case is 10M.
6. All first order rate constants 0.1s-1. The total amount of enzyme (part of the negative
feedback loop, inactivated by D) =100 molecules. (a) kf/V= 2 x10-2/no.s (b) kf/V=2/no.s,
where kf is the feedback constant. k1/V=2 x10-3/no.s, k3/V (k3 is the autocatalytic feedback) is
5 x10-3/no.s. The initial number of all substrate species is 30, and an equal amount of active
and inactive enzyme E are present.
7. We consider a situation where the autocatalytic module (in the bistable range) is embedded
in a closed reaction network. For this purpose, we consider the same basal conditions for the
module, as detailed above in 2(b) (and shown in Fig. 2(b)). We embed this in a closed
network, where every reaction of the closed network is described by mass action kinetics.
Here all reactions of the ambient network have rate constant =1s-1 (comparable to catalytic
reaction rate of the module), unless otherwise specified. Fig. S1 studies the effect of
variation of reaction rate constants of the ambient network.
Fig S1: We examine the effect of embedding an autocatalytic module in the bistable regime (model
M2), in a closed reversible chain as depicted. (a) shows how the basal bistable behaviour (present
alongside a zero steady state) is completely destroyed, and no threshold behaviour is observed.
(b)When the counterclockwise reaction rates, are made small (a tenth of the nominal value), one at
a time, bistable behaviour may indeed be seen. The lowering of the reaction rate (kco3 )responsible
for the reaction producing X*, leads to the broadest range of bistability. (c)On the other hand, the
lowering of reaction rates in the clockwise direction, one at a time, all lead to loss of bistability and
threshold behaviour. (d) depicts a scenario where all reaction rates of the ambient network are
reduced to a tenth of the nominal value: bistability is indeed seen in this case, though this is reduced
and destroyed by making (all) counter-clockwise reaction rates bigger. This shows how even
relatively weak surrounding networks can have a nontrivial effect. On the other hand if in (b), one of
the reaction rates kco3 is made even smaller, the bistable range may be substantially increased even
relative to basal conditions, as shown in (e). k2 is the catalytic constant of the autocatalytic reaction.