General model assumptions

A central premise of systems biology is that a model is only as good as its assumptions. Therefore assumptions represent compromises that can be the strength or Achille’s heel of any given framework. The model developed here will focus on the receptor-ligand interac- tion and downstream G protein cycle. The majority of these assumptions will be centred on the initial conditions, particularly species concentrations. This is crucial given that ODE models are subject to mass action kinetics where the rate of reaction is a product of an intrinsic rate constant and the concentration of the reacting components.

While the receptor and G protein concentrations have already been determined for the Sc. cerevisiae pheromone response, all strains used in this study have been genetically modified. In particular the endogenous GPCR, STE2, has been deleted and replaced by the A1R. Moreover, the A1R is under the control of a constitutive promoter at the URA3

locus. In contrast the STE2 receptor is under the control of a pheromone-responsive pro- moter. Consequently, A1R expression levels could vary from that of STE2 in an unmodified

strain. However, in the absence of any data regarding receptor number, we are forced to assumed an A1R concentration of 160nM, the experimentally determined endogenous STE2

concentration.

The Gβγ dimer, and the relevant loci, have been unmodified in this strain. Therefore a Gβγ concentration of 160nM was assumed. The GPA1 gene, however, has been deleted. GPA1, and GPA1 transplants have been integrated at the TRP1 locus under the control of the endogenousGPA1 promoter. This leads to the assumption that GPA1 expression in the transplant system is consistent with that of unmodified strains, i.e. 160nM. Credibility is led to this assumption by the Western blots of Brown et al. (2000), who developed this system, that indicate equal GPA1 expression in all strains.

The rate parameters determined by Yi et al. (2003) will be used as initial conditions for model fitting. These parameters were determined using a FRET reporter system in which GPA1 and STE18 have been deleted and fluorescently-modified variants integrated into their respective loci under the control of their endogenous promoters. The fluorophores are cyan fluorescent protein (CFP) and yellow fluorescent protein (YFP), derivatives of GFP, and can interfere with protein-protein interactions thus influencing the results. Given that

these parameters have been used to build more comprehensive models, and that we will only be using these rates as initial conditions for fitting, these constants will be used to directly inform the model. However, in the absence of an RGS, G protein-mediated GTP hydrolysis is under the sole control of the RGS-fold of the Gα subunit. We will therefore assume that GTP-hydrolysis and heterotrimeric G protein reformation are not influenced by the receptor. Consequently, GTP-hydrolysis will be constrained to 14.4 nM−1 hour−1, as determined by Yi et al. (2003) and implemented by Kofahl and Klipp (2004), Smith et al. (2009) and Croft et al. (2013).

A further experimental consideration of the mathematical model being developed here is is one of structural identifiability. This aspect of dynamic modelling is concerned with uniqueness of solutions. Put simply, this means a given model output, fitted to experi- mental data, can only be a consequence of a unique combination of initial conditions and parameters (Chis et al., 2011b). If a model structure fits this description it is termed glob- ally and structurally identifiable (Lockley et al., 2015). However, in the absence of global identifiability, local identifiability can be achieved in which one can isolate the neighbour- hood a parameter resides in. While not ideal for model fitting, local identifiability can be useful (Raue et al., 2009, 2010). While mathematically intensive, there are a number of user-friendly tools to perform structural identifiability analysis (Chis et al., 2011a; Ogung- benro and Aarons, 2011; Maiwald et al., 2012). Here, structural identifiably was performed as a the first step of model development. The GenSSI toolbox for Matlab was used due to its availability and relative ease of use (Chis et al., 2011a).

There are ways to increase the identifiability of a mathematical model. The parameters or species concentrations that can be measured directly are called observables. The more observables available to inform a mathematical model, the greater its identifiability and the more accurate its predictions (Anguelova et al., 2012). At present, theSc. cerevisiae trans- plant strains contain two potential measures of transcription, growth and β-galactosidase activity. This study exploits the latter. But with a single observable, global and struc- turally identifiability can be difficult to achieve. Minimising the model to its most basic components can compensate for this. For example, no significant basal signal is produced by the A1R in yeast. Therefore constitutive receptor and G protein activation can be sac-

rificed for the sake of simplicity. Further, not all parameters need to be fitted. Parameters that are unlikely to vary as a function of ligand or G protein, such as those governing downstream signalling and transcription, can be constrained to increase the likelihood of global and structural identifiability.