Valoración de los procedimientos para el diagnóstico del sistema de información comercial.

The simplicity of this core motif allowed me to analytically study the solutions to the steady state equations (see Appendix E). In particular, I was able to derive a set of inequalities in the kinetic parameters and total concentrations of the substrate and kinase that provide a set ofnecessary and sufficient conditions for the existence of three steady states in the system (Equation 5.3, see also Appendix E for the derivation of this equation). From these conditions I derive the following necessary condition for bistability (the indexing of the rate constants is given in Equation 5.1 inMethods: (κ3−κ6) (ηrκ9κ10−ηtκ8κ11)>0 (5.4) where, ηr= κ1 κ2+κ3 ηt= κ4 κ5+κ6

are the inverses of the Michaelis constants of the kinases Kr (the kinase at the

relaxed state) andKt (the kinase at the tense state) respectively. Analysis of this

equation reveals key features of the system that are necessary for bistability. I find that the switching reactions between the two states of the kinase, as well as between the kinase-substrate complexes are crucial for bistability. That is, both κ8 and κ9

cannot be zero, and both κ10 and κ11 cannot be zero. Thus, the structure of the

reaction system composing of a futile signalling cycle driven by a two-state kinase is crucial for enabling bistability.

Equation 5.4 provides two key features for bistability. Firstly, the two in- terconnected futile cycles between S and Sp, defined by the two kinase states,

need to operate at different catalytic rates (i.e. κ3 ̸= κ6). Secondly, the switch-

Figure 5.2: Expanded signalling networks without detailed balance. (A) Extended network obtained by adding an enzyme catalysing one of the transitions between the two states of the kinaseK in the core motif. (B) Extended network obtained by adding a protein to the core motif such that steric effects from the binding of the added protein with the enzyme makes the transitions between different states of the kinaseK irreversible. Both extensions maintain the capacity for bistability.

KtS) needs to occur at different rates, and in a way opposing the difference in

the catalytic rates. Specifically, if the futile cycle for the relaxed state of the ki- nase (i.e. Kr and KrS) has the highest catalytic activity (i.e. κ3 > κ6), then

ηrκ9κ10 needs to be larger than ηtκ8κ11. As a consequence, the clockwise inter-

changing cycle, Kr → KrS → KtS → Kt → Kr, corresponding to the product of

the rate constantsκ1κ10(κ5+κ6)κ9, needs to dominate over the anti-clockwise cycle,

Kr → Kt → KtS → KrS → Kr, corresponding to the product κ4κ11(κ2 +κ3)κ8.

Symmetrically, if Kt has higher catalytic activity than Kr (i.e. κ3 <κ6) then the

anti-clockwise cycle needs to dominate.

A further constraint on the rates governing the transitions among the four forms of the kinase might arise from thermodynamics. Particularly, these transi- tions form a local state cycle, which must follow the principle of detailed balance if we assume no additional energy input into the system [278–281]. This results in a thermodynamic constraint on the reaction kinetics such that the product of the rate constants in the clockwise direction must equal the product of the reverse rate constants (i.e. κ1κ5κ9κ10 =κ2κ4κ8κ11). It must also be noted, however, that this

constraint would be relaxed if the conformational switching between the enzyme states were directed by energy input (e.g. phosphorylation-dephosphorylation re- actions, (Figure 5.2A) or steric effects with enzyme binding with other proteins or enzymes (Figure 5.2B).

Table 5.1: Number of bistable parameter sets found by sampling parameters of the core motif. Sampling is performed under two conditions, relaxed form and under the thermodynamic constraint. The total number of sampled parameter sets is 105.

With thermodynamic constraint Without thermodynamic constraint 2787(∼2.8%) 14492(∼14%)

To determine whether these conditions on kinetic rates can be simultaneously satisfied in cellular signalling networks, I tabulated kinetic parameters from the literature (see Table 2.2 and references therein). I then sampled 105 parameter

Figure 5.3: Parameter sets that allow for bistability, sampled in a biologically feasible range. (A) Sampled parameter sets plotted in the space of κ3

κ6 vs.

ηrκ9κ10

ηtκ8κ11. The blue

dots (resp. yellow triangles) correspond to the parameter sets sampled without (resp. with) the thermodynamic constraint. In accordance with the sufficient and necessary condition (seeMethods), all sampled parameters that allow for bistability fall into the two regions that meet at (1,1). (B, C) Boxplots of the rate constants sampled without (B) and with (C) the thermodynamic constraint, shown on log10-scale. The conditioning on bistability changes the distribution of the rate constants. In the inequality for bistability (Equation 5.3 in the main text) the groups of rate constants

κ1, κ2, κ3, κ9, κ10 and κ4, κ5, κ6, κ8, κ11 appear symmetrically in the inequality

in the sense that if the two groups of parameters are swapped, the inequality is fulfilled if and only if it was so before swapping. Hence the rate constants κ1 and

κ4 follow the same distribution, κ2 and κ5 do as well, and so on. This symmetry

is visible in the boxplots. The range of each parameter generally shrinks under the thermodynamic constraint compared to without the constraint.

sets around these known kinetic parameters and checked whether the necessary and sufficient conditions for bistability were satisfied (see Methods). This analysis showed that the futile signalling cycle displays bistability in a biologically plausible parameter regime, even when thermodynamic constraints are taken into account (Figure 5.3 and Table 5.1).

Figure 5.4: Schematic of minimal signalling motif displaying bistability. Car- toon representation of the two interconnected reaction cycles constituting the core bistable system. The arrows represent reactions in the system and are labelled with the kinetic parameters from Equation 5.1. Two rectangles (dashed line) with dif- ferent background colour show the two futile cycles with Kr (green) and Kt (red)

competing for the substrate (in the intersected region of the two rectangles).

5.3.3 Bistability can be seen as arising from competition between