 6A and S6) (Hohmann et al., 2007;Mattison and Ota, 2000;O'Rourke and Herskowitz, 1998;Westfall and Thorner, 2006;Young et al., 2002). Sensitivity analysis of the wild-type (WT) model that was trained with diverse kinetics allowed us to categorize model parameters into two main groups of insensitive (sensitivity = 0) and sensitive (sensitivity = 1) parameters that we then use to predict putative mutations ( Figures 6A, 6B, and S6).
To determine whether diverse kinetic cell stimulations can identify biological mechanisms, we computationally introduced six mutations in our true model (Transparent Methods). These include three knockout mutants that remove basal deactivators on x 1 (Db 1, blue cross), x 2 (Db 2, red cross), or x 3 (Db 3, teal cross). Mutants Db 1 and Db 2 can be interpreted as removing the constitutive deactivation of SLN1 or SHO1 branches, respectively, which could change the half-life of their active states. Mutant Db 3 could represent regulation of Pbs2 through deletion of Ptc1/2/3 phosphatases. The node x 4 can be mutated by removing kinase activity of x 4 (DK, purple cross, e.g., kinase dead or kinase activity inhibited) and can be regulated through overexpression (b 4, OE, orange), and underexpression (b 4, UE, brown) of the basal deactivator on x 4 such as the Ptp2/3 phosphatases ( Figure 6A).
We simulated corresponding synthetic pathway activations from all six mutants upon all 54 kinetic stimulations (Transparent Methods). Pathway activations upon a representative kinetic stimulation input of t 9 are shown for all six mutants compared with the WT in Figure 6C. To quantify the differences in signaling dynamics between normal and mutant cells, we define severity as the difference in the activation dynamics of a mutant compared with that of the WT ( Figure 6D and Transparent Methods). We observed that pathway activation in some mutant strains shows no difference from WT under all kinetic inputs (Db 1 and Db 3, mutation ''severity'' = 0), whereas other mutants with non-zero severity (Db 2, DK, UE, and OE) are different from WT ( Figure 6C). Comparing the sensitivity with respect to specific parameters in the model ( Figure 6B) with the severity of the corresponding mutants' effect on signaling dynamics ( Figure 6E) highlights that sensitive parameters are indicators of how much specific mutations will affect signaling under different kinetics of specific types ( Figure S6). Furthermore, constraining the parameters of WT model on its synthetic data under diverse kinetic stimulations enabled us to accurately predict the activation responses of all the six mutants over time under all 54 kinetic inputs tested (Figures 7 and S7). These results are quantitatively summarized in Figure 7E, showing that the prediction errors for simulated mutations are comparable to prediction errors of non-mutated WT cells.

DISCUSSION
Our experimental and simulation results demonstrate that different kinetic cell stimulations of a pathway give rise to distinct signaling activation dynamics (Figure 1). When compared with the same amount of  Figure 5B) on its signaling dynamics upon diverse kinetics (red in Figure 5J) reveals insights into the response of several mutated models under all tested kinetic inputs. (A) The six mutants of the WT model correspond to deletions of basal regulators (e.g., phosphatases) on x 1 (Db 1, blue), x 2 (Db 2, red), or x 3 (Db 3, teal); removal of the kinase activity (e.g., kinase dead or inhibited MAPK) of x 4 (DK, purple); and under-or overexpressing b 4 (e.g. phosphatase) that regulates x 4 (UE in orange and OE in brown). (B) Sensitivity analysis of WT model with respect to model parameters around their best value from the fits predicts insensitive (e.g., l 3 and l 4 corresponding to Db 1 ) and sensitive (l 15 and l 16 corresponding to DK) mutants (see Figure S6). (C) Comparing the activation dynamics of mutants (simulated using L 0 in Table S1)  iScience Article any type of homogeneous kinetics, kinetically diverse cell stimulations perform much better to constrain complex model parameter sets and result in substantially reduced predictions errors (Figures 3 and 4). The FIM analysis approach provides a rigorous and clear mathematical interpretation of this effect. Specifically, the eigenvector of the FIM corresponding to the greatest eigenvalue indicates the parameter combination that is most likely to be accurately identified using a given kinetic stimulation ( Figure S4O). By comparing eigenvectors corresponding to large FIM eigenvalues, it is easy to see which kinetic type may be most effective to constrain specific parameters of a complex regulatory network. Similarly, by examining eigenvectors corresponding to small FIM eigenvalues, it becomes apparent which parameter combinations cannot be precisely identified using a specific kinetic input. By choosing diverse and complementary input kinetics, such that the full parameter space is spanned by high-eigenvalue FIM eigenvectors from one or multiple kinetic inputs, it becomes possible to constrain the entire parameter set ( Figures 4F, 4G, and S4L-S4O).
Better constrained parameters make it easier to identify predictive models of signal transduction (Figures 5  and S5). This also enables improved predictions of pathway activation dynamics for protein mutant strains upon kinetic stimulations (Figures 6, S6, and S7). To illustrate these predictions, we revisited the HOG pathway as an example ( Figure 6). We predicted that mutations that alter the SLN1 branch (b 1 ) will not impact Hog1 signaling dynamics, which is consistent with previously published observation that changes in Ypd1 do not alter Hog1 signaling (Mukherji, 2010). Next we focused on the node x 2 that describes the SHO1 branch in which mutating b 2 could result in an increase or decrease in Hog1 signaling. We predicted that Db 2 results in increased Hog1 signaling amplitude. In addition, the SHO1 branch can be altered in its activity by the feedback regulation from Hog1 kinase. In our model, we predicted that removing Hog1 kinase activity or inhibiting Hog1 kinase activity (DK) results in increased and prolonged Hog1 activation, which is consistent with previous published experimental work on small molecule inhibition of Hog1 activity (Muzzey et al., 2009;Westfall and Thorner, 2006). Next, we focused on the node x 3, which represents Pbs2. Our model predicted that deletions of individual phosphatases such as Ptc1/2/3 would not alter Hog1  Figure 1H) for each of the six mutants ( Figure 6A) compared with WT. These predictions are made using parameters constrained from five independent fits of WT model tarined on its signaling dynamics upon six diverse kinetics in Figure 5J. See also Figure S7.

OPEN ACCESS
iScience 23, 101565, October 23, 2020 11 iScience Article signaling dynamics through changed Pbs2 activity (Db 3 ) (Mapes and Ota, 2004;Warmka et al., 2001). Last, x 4 represents the terminal kinase Hog1 that can be deactivated through phosphatases Ptc1/2/3 and Ptp2/3 (b 4 ) (Martín et al., 2005;Murakami et al., 2008;Warmka et al., 2001;Young et al., 2002). The model predicted that changing the value of b 4 through over-or underexpression has a strong impact on Hog1 signaling intensity. The experimental observations of the effect of above-mentioned mutants on the HOG pathway dynamics under step profiles qualitatively aligns with our simulated data for all mutants validating our modeling approach for mutants ( Figure 6) (O'Rourke and Herskowitz, 1998;Saito and Posas, 2012;Tatebayashi et al., 2015;Warmka et al., 2001). Using simulated data, we verified our mutant findings by quantitatively comparing predictions from each mutant to their corresponding synthetic data (Figure 7).
These results demonstrate that kinetic cell stimulations are ideally suited to discover novel regulatory interactions, reveal key functional proteins, identify predictive and biologically meaningful models, and help to gain novel insights to cell function or dysfunction. The potential for generality of kinetic stimulations is supported by emerging studies in many pathways and in different cell types. Examples are differential cellular responses Given the complexity of signal transduction networks (Akhurst and Hata, 2012;Cildir et al., 2016;Hanahan and Weinberg, 2000;Hata and Chen, 2016;Hotamisligil and Davis, 2016) and their limited response bandwidths (Hersen et al., 2008), steps or pulsatile stimulations of even varying intensities or frequencies may not provide enough kinetics to efficiently probe the rich dynamics underlying these networks (Billings, 2013;Isermann and Mü nchhof, 2011). Our approach on the other hand is widely applicable to many biological pathways that respond to a kinetic cell stimulation with a dynamic signaling response, and our approach may have far-reaching implications for predicting pathway response upon specific mutations or drugs (Akhurst and Hata, 2012;Cildir et al., 2016;Hanahan and Weinberg, 2000;Handly et al., 2016;Hata and Chen, 2016;Hotamisligil and Davis, 2016;Janes and Lauffenburger, 2013;Rowland et al., 2011;Vanhaelen et al., 2017;Lim and Bruce Mayer, 2017). Being able to predict the pathway activation dynamics upon mutations or upon environmental changes may help design better drug treatment regimes. In addition, these results could benefit our understanding of human biology, particularly in areas such as optogenetics, gene regulatory networks, or synthetic biology, where predictive understanding of the system behavior with respect to extracellular kinetics or intercellular genetic perturbations is of immense interest (