6.4.1
Model fitting and discarding model variants
Fitting the models to the data generated in chapter 4 was done first for the original model A1B1C1D1E1. The model showed good agreement with the data; however, some non-identifiabilities remained since it was not possible to secure measure- ments for all species as originally intended. Having a model that was not fully constrained by the data, the question of model discrimination was posed more rig- orously through an indistinguishability analysis, to determine if the model could be distinguished from alternative variants with the measurements provided. Since the gene expression data obtained for ste11 suggested that only a small increase was taking place in response to pheromone, it was desired to establish if a model without positive feedback would provide a better explanation of the data. The indistinguishability analysis confirmed that the measurements given could be sufficient to make the discrimination, while comparing the AIC scores showed that the standard model including a positive feedback was indeed the superior option; however, the comparison was not fully satisfactory because the positive feedback was only affected by the data indirectly through other model reactions.
Measurements that directly affected the positive feedback were available from the expression data of ste11, so the analysis was repeated using the equivalent A2 model, which was necessary to allow the fitting of mRNA dynamics. This model would also be unidentifiable, but again allowed the distinction to be made between models with or without a positive feedback loop. The model including a positive feedback was found to be decidedly better suited to explain the data, reinforcing the previous result.
During the transition from a model with one-step process for gene expression to the model with a two-step process, the parameter for transcription factor degradation kdp was fixed based on the estimates obtained with the simpler model. This
would decrease the degrees of freedom during the estimation procedure, and could alter the results that would be obtained otherwise; however, since the estimated value was in very good agreement with previous experimental reports (Kjærulff
et al., 2007), this increased the confidence that this rate constant had been well determined and could be fixed at the estimated value.
Further indistinguishability analyses were attempted to resolve the best option for each model variant; however, the results were inconclusive as the computa- tional complexity escalated too quickly before a positive answer could be found. Nonetheless, there were still grounds to resolve one last option. The model vari- ants of type B considered whether it was necessary to introduce a delay between pheromone detection and transcription factor activation, and one of the parame- ters that was found to be identifiable was the activation rate constant ka. Based
on the estimated value of ka, and on the simulation analyses that would follow,
this process occurs extremely fast, which allows the model including a time delay to be eliminated with confidence, even when the indistinguishability analysis was inconclusive.
6.4.2
Final model iteration and the measurements required
for full identifiability
To determine the extent of uncertainty that remained in the model due to uniden- tifiable parameters, the parameter vectors along each PL were used to simulate the model and visualise the spread between the resulting trajectories. This uncer- tainty can be exploited to determine the most informative experiments that could
be performed to minimise the spread of the trajectories, and thus the uncertainty in the model.
Several experimental designs were tested but none managed to constrain the model enough to achieve full identifiability, even if all species could be measured. This suggested that the model was too complex given the information content provided by the noisy data. Since a reduction of noise in the data did not seem like a viable option, the discrepancy can be resolved through model reduction.
A reduced version of the A2 model (A2R), was shown to become fully identifiable with the addition of measurements of total Ste11, suggesting an immediate avenue for further progress along this line of investigation. Since the model reduction was performed based on insensitive parameters, the quality of the fits to data obtained previously were not affected, and model A2R became the final version of the model developed in this work.
6.4.3
Discovering roles for feedback control in
S. pombe
pheromone signalling
Although full identifiability of the model was not obtained with the available data sets, the remaining uncertainty in the model was completely localised to the concentration scale of transcription factor, which did not preclude the possibility of drawing conclusions based on other aspects of the model. To gain a sense of the nominal behaviour of the system, the model was simulated using a range of pheromone doses that cover a standard dose-response experiment in S. pombe. Comparing the normal model behaviour to simulations lacking either one of the feedback loops revealed roles for both of them in the discrimination of elevated doses of pheromone.
The proposed mechanism assumes the existence of thresholds that define the point at which the cell commits to the sexual development programme. A concentration threshold dictates the minimum amount of active transcription factor that must be present in the cell to fully launch the transcriptional network, while a time threshold is defined by the period of time over which full activation must be sus- tained for the commitment to become irreversible. Perturbation of the parameters responsible for feedback then shows that the strength of the feedback loops deter- mines the dose of pheromone that is required to satisfy both thresholds, with each feedback having exclusive control over one of the thresholds. The positive feed- back controlling the concentration threshold, while the negative feedback controls the time threshold. These roles for feedback control go beyond simply enhancing expression or terminating signalling, and can only be understood from a systems level perspective.
The existence of these thresholds in cell fate determination have both experi- mental as well as theoretical support. For example, simulations have shown that the number of downstream genes activated by a transcription factor determines the concentration it must have in order for all genes to be transcribed robustly (Del Vecchio et al., 2008). Since Ste11 has at least 78 genes under its control (Mata and B¨ahler, 2006), this provides the basis to postulate a concentration threshold. Evidence for a time threshold comes from observing the need for Ste11 to remain active for the induction of both early and late genes in the network (Mata et al., 2007; Harigaya and Yamamoto, 2007), and supports the idea that converting a pheromone input dose to the duration of Ste11 activation is one of the mechanisms employed byS. pombe for cellular decision making.
The idea of dose to duration encoding in cell signalling has been demonstrated experimentally from an input-output point of view (Behar et al., 2008; Hao and O’Shea, 2012), but the explanation of how it arises in a specific pathway has not been shown. The model developed in this work provides a new tool to explore
this behaviour with direct correspondence to specific molecular species. In ad- dition, the availability of strains without Sxa2 activity provides the opportunity for a number of model predictions to be tested immediately through similar mea- surements to those performed in this work, simply by using additional pheromone doses, for example the prediction of saturation in the transcription of sxa2 above 10−7M pheromone. Finally, the model can be used to explore the behaviour of pheromone signalling in scenarios that are not yet available experimentally, such as the importance of the positive feedback in the absence of high basal transcription factor production.