Model-driven experimental design identifies counter-acting feedback regulation in the osmotic stress response of yeast

Cells rely on mitogen-activated protein kinases (MAPKs) to survive environmental stress. In yeast, activation of the MAPK Hog1 is known to mediate the response to high osmotic conditions. Recent studies of Hog1 revealed that its temporal activity is subject to both negative and positive feedback regulation, yet the mechanisms of feedback remain unclear. By designing mathematical models of increasing complexity for the Hog1 MAPK cascade, we identified pathway circuitry sufficient to capture Hog1 dynamics observed in vivo. We used these models to optimize experimental designs for distinguishing potential feedback loops. Performing experiments based on these models revealed mutual inhibition between Hog1 and its phosphatases as the likely positive feedback mechanism underlying switch-like, dose-dependent MAPK activation. Importantly, our findings reveal a new signaling function for MAPK phosphatases. More broadly, they demonstrate the value using mathematical models to infer targets of feedback regulation in signaling pathways.

We next measured the dynamics of another upstream signaling component, at multiple 175 salt concentrations, with and without Hog1 kinase inhibition. Our rationale was that these 176 experiments would provide important additional data for informing our models and identifying 177 targets of feedback regulation. We chose the MAP2K Pbs2 because it is more abundant than any one of the MAP3Ks, is common to both input branches of the pathway and is 179 phosphorylated when activated (Tatebayashi et al., 2020). Thus, we used the Phos-tag western 180 blotting technique described above to measure the dose-and Hog1-kinase dependency of Pbs2 181 phosphorylation dynamics. As shown in Figure 2D, osmotic stress stimulation caused a mobility 182 shift of Pbs2 that was (7) fast and partial and also (8) transient. This behavior mirrored two of 183 the wildtype properties, but unlike Hog1, Pbs2 did not become fully phosphorylated at 150 mM 184 KCl. When Hog1 was kinase-inhibited, Pbs2 phosphorylation was also (9) fast and partial, but

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Details of our parameter optimization method are provided in the Methods section.

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We fit Model I to our data presented in Figure 2. We simulated wildtype behavior with the 226 full system and simulated kinase-inhibited behavior by removing the Hog1-dependent negative feedback loop. As shown in Figure 3C ( Figure 3E). To test the model, we exposed cells, with and without Hog1 254 activity, with 350 mM KCl and measured glycerol accumulation over time ( Figure 3E). While 255 glycerol exhibited a higher-than predicted increase (four-fold vs two-fold), the dynamics were 256 similar to the model prediction ( Figure 3F). The discrepancy in abundance is likely due to Hog1-257 independent glycerol production. We observed 1-to 2-fold increase of glycerol accumulation in  where Hog1-mediated positive feedback ( • 1) increases its own activation.

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We used the same procedure as described above to train the models. A summary of each 295 model's fit to the data is provided in Figure 4A.  Figure 4E (left)). These models also followed 302 similar Hog1-as dynamics as the previously published though the previously published data is 303 slightly higher than that seen in our data ( Figure 4E, right compared to Figure 4D, center). Even 304 with this small discrepancy, these data suggest Hog1 phosphorylates a pathway component at 305 or below that of the MAP2K Pbs2, forming a positive feedback loop.

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To complete our systematic screen of potential circuitries, we also added positive

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Following this strategy, we computationally generated 1000 random input profiles of 324 increasing salt concentrations and predicted Hog1 response to each input profile using Models 325 IIb and IIc. These step profiles were designed so that they could be experimentally tested in 326 vivo. We ranked the resulting input profiles based on which generated the largest differences in the Hog1 response ( Figure 5B). For example, Figure 5C shows three selected inputs ("Step") 328 that correspond to the Hog1 dynamics predicted by the two models in Figure 5D.
Step #100 329 generated similar predictions among the models while Step #990 resulted in distinct Hog1 330 behaviors.
Step #550 also predicted model-dependent dynamics, but the differences were too 331 small to be experimentally decipherable. Generally, the input profiles that produced the greatest 332 difference between the Hog1 behaviors were those that allowed Hog1 to adapt to an initial step 333 of KCl before introducing a second step (shaded area in Figure 5B). For Step #990, Model IIb 334 predicted that Hog1 would show a diminished response to the second step of stimulus, but 335 Model IIc predicted full Hog1 phosphorylation in response to this second step ( Figure 5D right 336 column). These results indicated that Step #990 would discriminate between the two models.

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We then measured the biological Hog1 response to Step #990. We exposed cells to the

Positive feedback is independent of feedback phosphorylation
Our modeling results suggested that positive feedback amplifies the signal at the level of 356 Hog1. There are two ways in which feedback phosphorylation could activate the MAPK: 357 increase its phosphorylation rate ( Figure 6) or decrease its dephosphorylation rate (Figure 7).
Since positive feedback must happen quickly, it seemed likely that the target of feedback where 1 is phosphatase-driven Hog1 suppression and 2 is Hog1-driven phosphatase 390 suppression. Here, the total phosphatase concentration is conserved.

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We determined whether Model III could perform equal to or outperform Model IIc. We  setting their concentration to 0 ( Figure 7B). Fitting to these additional data, we found that Model particularly in the wildtype strain ( Figure 7C, bottom). Model III could also correctly predict the ptp2Δptp3Δ strain to a single step of 350 mM KCl ( Figure 7D,