Conclusiones del tercer capítulo

It is interesting to note that the mathematical conditions derived in Equation 5.4 impose a specific structure onto the core motif, which can be seen as two connected reaction cycles that are driven by the two states of the kinase competing for the same substrate (Figure 5.4). Equation 5.4 shows that the flows of these two competing reaction cycles need to have a specific relationship for bistability to emerge. To

better understand these ensuing reaction fluxes, I have analysed the steady states of the system for increasing total kinase concentration, as a proxy for an increasing signal (Figure 5.5, see also Appendix E). For a fixed set of parameters in the bistable regime such that κ3 > κ6, κ9 > κ8, κ10 > κ11, and ηrκ9κ10 > ηtκ8κ11 (see Table

5.2), I find that in the low signal regime, where the total level of kinase is low, there is a large flux from KrS into KtS, resulting in the accumulation of KtS. Thus

in this low signal regime, the slow futile cycle driven by Kt (which has the lower

catalytic activity) dominates (i.e. [Kr] + [KrS] <[Kt] + [KtS]) and the system is

at low state (i.e. small [Sp]) (Figure 5.5, red dots). In the high signal regime, the

fast futile cycle driven byKr dominates (i.e. [Kr] + [KrS]>[Kt] + [KtS]) and the

system is at the high state (i.e. large [Sp]). The substrate is largely converted to the

phosphorylated form, which results into the accumulation ofKr (Figure 5.5, green

dots). Whether the Kr mediated or Kt mediated cycle dominates is primarily

determined by the condition ηrκ9κ10 > ηtκ8κ11, which relates to the inverse of

Michealis-Menten constants associated with each kinase forms and the transition rates between these forms in a free and substrate-bound state.

Table 5.2: Example parameter sets that enable bistable dynamics in the core sig- nalling motif. The table shows the parameter sets used for the generation of the bifurcation plot in Figure 5.5.

Parameter Unit Value Reaction

κ1 µM−1s−1 86.78 Kr+S →KrS κ2 s−1 3.583 KrS →Kr+S κ3 s−1 92.84 KrS →Kr+Sp κ4 µM−1s−1 1.200 Kt+S→KtS κ5 s−1 0.02626 KtS→Kt+S κ6 s−1 0.2644 KtS→Kt+Sp κ7 s−1 2.357 S →Sp κ8 s−1 0.01310 Kr →Kt κ9 s−1 0.7842 Kt→Kr κ10 s−1 1.041 KrS →KtS κ11 s−1 0.008057 KtS→KrS [Stot] µM 9.994 — [Ktot] µM 0∼3 —

This analysis derived from the necessary parameter conditions leads to an intuitive view, in which the bistability in the system is understood as a result of the two futile cycles driven by the two forms of the kinase competing for the substrate. Furthermore, the competing kinase forms need to have opposite dominance in terms of being able to bind the substrate and their catalytic activity, such that the form dominating catalytically (κ3 >κ6) needs to be weaker in terms of substrate binding

kinetics (i.e. assumingκ9 =κ10=κ8 =κ11, we need to haveηr >ηt).

Figure 5.5: Bifurcation plot of core bistable signalling motif. The solid line corre- sponds to the stable steady state levels of [Sp] with increasing signal given by the

total concentration of kinase [Ktot]. The dashed line corresponds to the unstable

steady states. The parameter values used to generate the bifurcation plot are listed in Table 5.2. The four little cartoons, drawn as inset, are showing the allocation of all species concentration and corresponding reaction fluxes at the different levels of [Ktot], as indicated by the coloured dots. Within each cartoon, the size of each

blue box stands for the relative amount of species (logarithmically scaled), while the thickness of the arrows stands for the relative levels of the reaction fluxes (loga- rithmically scaled) calculated with mass-action kinetics, namelyκ1[Kr][S],κ2[KrS],

5.3.4 Increasing the number of kinase states in the signalling cycle leads to unbounded multistationarity

Recognising that bistability in the core motif is linked to the competition between the two futile cycles, it is intriguing to consider whether adding more competing cycles increases the number of steady states. To expand from the simplest motif towards more complicated systems, one way of increasing competing cycles is to increase the number of two-state kinases, while the other is to increase the number of states of a single kinase. I find that both expansions of the minimal system result in an increase of the number of steady states.

Firstly, I considered the case of multiple kinases with two states (Figure 5.6A). In this case, multiple two-state kinases in a futile cycle lead to multistation- arity (Figure 5.6A). With the number of kinasesnincreasing, the number of steady states linearly scales with n. We prove that the system can admit at most 2n+ 1 steady states and further thatnof them are unstable (see Appendix E). The other

n+ 1 steady states are presumably stable. Secondly, multistability can be achieved by one kinase with multiple states (Figure 5.6B). When the kinase has 3 distinct states, the system can have 3 steady states at most, but a four-state kinase results in the possibility of 5 steady states at most (Figure 5.6B, see Appendix E). The general scenario with ann-state kinase is too complex mathematically, and does not admit the approach used to analyse systems with multiple two-state kinases. However, we conjecture that the number of positive steady states grows linearly withn as well, such that the system admits at mostn+ 1 positive steady states ifn is even andn

positive steady states ifnis odd.