Toward a Comprehensive Model of a Yeast Stress Response Pathway

TOWARD A COMPREHENSIVE MODEL OF A YEAST STRESS RESPONSE PATHWAY

Patrick Charles McCarter

A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Curriculum in Bioinformatics and Computational Biology.

Chapel Hill 2017

ii ABSTRACT

Patrick Charles McCarter: Toward a comprehensive model of a yeast stress response pathway (Under the direction of Timothy Elston)

Cells rely on cyto-protection programs to survive exposure to external stressors. In eukaryotes, many of these programs are regulated by Mitogen-Activated Protein Kinases (MAPK). Aberrant signaling in MAPK pathways is associated with pathologies including cancer and neurodegenerative disease. Thus understanding how MAPKs are dynamically regulated is critical for human health. We leverage the simplicity and genetic tractability of S. cerevisiae (yeast) to experimentally investigate and mathematically model MAPK activity in the prototypical High Osmolarity Glycerol (HOG) pathway. The HOG pathway transmits osmostress to Hog1, a MAPK which is homologous to the p38 and JNK kinases, via two distinct signaling branches (Sho1, Sln1). Once phosphorylated,

Hog1 induces activation of the hyper-osmotic stress adaptation program. Our goal was to identify and characterize the feedback network that regulates Hog1 activity during hyper-osmotic stress.

We previously demonstrated that Hog1 phosphorylation is encoded via positive feedback, and in agreement with previous studies, we showed that Hog1 dephosphorylation is encoded via negative feedback. We used mathematical modeling to define simplified MAPK feedback networks that featured various combinations of negative and positive feedback loops. We then used a modified

Approximate Bayesian Computation (ABC) parameter estimation and model selection algorithm to rank each feedback network based on its ability to reproduce the Hog1 phosphorylation training data.

We then used the top models to design a set of experiments that could be used to further differentiate the likelihood of each model to be selected as the consensus model. Our simulations suggested that the likelihood of each model could be further differentiated if yeast were exposed to dynamic hyper-osmotic stress conditions. Our simulations also suggested that the HOG pathway would be most sensitive to hyper-osmotic stress if the positive feedback operates at the level of the MAPK. To test our model predictions, we used live-cell ﬂuorescence microscopy and a microﬂuidic device to measure nuclear accumulation of a Hog1-GFP fusion protein as a function of the dynamic hyper-

iii osmotic stress conditions suggested by our simulations. We observed that nuclear accumulation of Hog1-GFP was most consistent with the model that suggested positive feedback operates at the level of the MAPK. These findings further suggest that positive feedback directly amplifies Hog1 phosphorylation. Furthermore our study exemplifies the power of integrating mathematical models into an existing quantitative experimental framework.

iv This work is dedicated to every person who has sacriﬁced a small part of their life, so that I could

live the whole of my own.

v ACKNOWLEDGEMENTS

I would like to ﬁrst acknowledge Dr. Timothy Elston, who served as my primary research mentor. I would also like to acknowledge Dr. Henrik Dohlman and Dr. Beverly Errede who served as co-advisors. It was an honor to train under the direction of three excellent scientists who each brought a unique perspective to collaborative research. I would also like to acknowledge all members of the Elston, Errede, and Dohlman labs, with special consideration given to lab members who served as collaborators on research projects. Dr. Denis Tsygankov, who served as my rotation mentor in the

Elston Lab; Dr. Michal Nagiec, who I collaborated with as a co-ﬁrst author on the research project presented in chapter 2; Dr. Justin English, who I collaborated on a research project that provided a strong foundation to my dissertation project. I’d also like to acknowledge Dr. Matthew Martz, and

Lior Vered who served as the main collaborators on my dissertation project.

I would also like to acknowledge several faculty members from The University of North Carolina at Chapel Hill whom I was also able to collaborate with on projects outside of the yeast focus of the Elston lab. In particular, I’d like to acknowledge Dr. Robert Duronio and Dr. Christina

Lone-Swanson for the collaboration studying cell cycle proteins in Drosophilia melanogaster. I’d like to acknowledge Dr. Alan Jones, Dr. Kang-Ling Liao, Dr. Meral Tunc-Ozdemir, and James

Draper for the collaborations studying signal transduction pathways in Arabidopsis thaliana.

I would like to acknowledge the Program in Molecular and Cellular Biophysics. I’d like to especially acknowledge Dr. Barry Lentz and Lisa Philippie, who ﬁrst introduced me to Biophysics while I was a M.S. student at North Carolina Agricultural and Technical State University. Barry and

Lisa were paramount to my success as a Ph.D. student. Similarly, I would like to acknowledge every member of the BBSP ofﬁce and also all student and faculty members of the Initiative for Maximizing

Student Diversity (IMSD), with special consideration given to Dr. Ashalla Magee-Freeman and Dr.

Jessica Harrell for their support through every step of the Ph.D. process.

I would like to acknowledge Dr. Kenneth Flurchick, who served as M.S. advisor at North

Carolina Agricultural and Technical State University. I was perfectly happy with ﬁnishing my B.S.

vi in Physics until meeting Dr. Flurchick. He not only inspired me to continue to pursue the scientiﬁc questions that ﬁrst drove me into Physics, but has remained committed to my success through every step of the process.

I would like to acknowledge every member of my family. Each person made it a priority to ensure that I was able to complete my Ph.D. with as much support as possible. Lastly I acknowledge my wife Jocelyn, who has been my biggest supporter from the moment I told her I wanted to earn my Ph.D.

vii TABLE OF CONTENTS

LIST OF TABLES ...... xi

LIST OF FIGURES ...... xii

LIST OF ABBREVIATIONS ...... xiv

1 Dynamic precedence setting in counter-acting feedback networks ...... 1

1.1 Introduction ...... 1

1.2 Results ...... 6

1.2.1 Models M1 - positive and negative feedback acting on MAPK components . . 8

1.2.2 Models M2 - MAPK translocation...... 8

1.2.3 Parameter Estimation ...... 8

1.2.4 Model Selection ...... 9

1.2.5 Positive feedback has most signiﬁcant impact when targeted to the MAPK . . 14

2 1.2.6 Model m(1,0) is most consistent with previous experimental analysis ...... 20

2 1.2.7 Model m(1,0) is most consistent with previous mutational analysis ...... 21

2 1.2.8 Model m(1,0) is most consistent with previous dynamic analysis ...... 24 1.2.9 Models are differentiated by increases in hyper-osmolarity...... 26

2 1.2.10 Model m(1,0) predicts a variable Hog1 response under physiological conditions ...... 27

1.3 Discussion...... 30

1.4 Materials and Methods...... 31

1.4.1 Model design and equations ...... 31

viii 1.4.2 Models M1 - positive and negative feedback acting on MAPK components . . 32

1.4.3 Models M2 - MAPK translocation...... 33

1.4.4 Fitting models to experimental data...... 33

1.4.4.1 An ABC-SMC algorithm for parameter estimation ...... 34

1.4.4.2 Ranking the candidate models ...... 37

1.4.5 Strain genotype and cell growth conditions ...... 38

1.4.6 Live-cell imaging and microﬂuidics ...... 38

1.4.6.1 Segmenting DIC images using a local adaptive binarization algorithm ...... 39

1.4.6.2 Watershedding cells, and aligning nuclei to cells ...... 43

1.4.6.3 Measuring Hog1-GFP intensities and cell size ...... 44

2 Signal inhibition by a dynamically regulated pool of monophosphorylated MAPK . . . . 45

2.1 Overview...... 45

2.2 Introduction...... 46

2.3 Results ...... 48

2.3.1 The phosphorylation state of Fus3 determines mating pathway output...... 48

2.3.2 The phosphorylation state of Fus3 is dynamically regulated...... 49

2.3.3 Ste5 and Msg5 regulate the kinetics of Fus3 phosphorylation...... 53

2.3.4 Ste5 and Msg5 cooperate to maintain a pool of mono-phosphorylated Fus3...... 54

2.3.5 Loss of a phosphatase and MAPK-scaffolding confer synthetic supersensitivity to pheromone...... 56

2.3.6 Discussion: ...... 60

2.3.7 Material and Methods: ...... 62

2.3.7.1 Transcriptional reporter and halo assay ...... 64

2.3.7.2 Cell extracts and immunoblotting ...... 64

2.3.7.3 Microscopy ...... 65

2.3.7.4 Mathematical Model ...... 65

ix A Supplementary Material to Chapter 2 ...... 69

B Supplementary Material to Chapter 3 ...... 83

B.1 Introduction to Mathematical Model ...... 83

B.1.1 Fus3 phosphorylation is regulated by multiple mechanisms ...... 84

B.1.2 Fus3 phosphorylation is processive, Fus3 dephosphorylation is distributive . . 86

B.2 Computational Methods ...... 88

B.2.1 Processive models of three dynamic MAPK phosphorylation states ...... 88

B.2.2 A Distributive model of three dynamic MAPK phosphorylation states ...... 89

B.2.3 Fitting models via Markov Chain Monte Carlo simulations ...... 89

B.3 Results ...... 89

B.3.1 Fus3 undergoes processive phosphorylation and distributive dephosphorylation 89

B.3.2 Feedback phosphorylated Ste5 diminishes the Fus3 phosphorylation rate ...... 91

B.3.3 The Ptp2, Ptp3, and Msg5 phosphatases synergistically regulate Fus3 phosphorylation ...... 94

B.3.4 Fus3 regulates the sensitivity of yeast to mating pheromone through feedback phosphorylation of Ste5 and induction of Msg5 ...... 97

B.3.5 The processive model is preferred over the distributive model ...... 99

B.4 Discussion ...... 102

REFERENCES ...... 110

x LIST OF TABLES

1.1 Interpretation of Bayes factors according to [49] ...... 10

2 2 1.2 Bayes factors for model m(1,0) and m(1,2) ...... 13 1.3 Genotype of the yeast strain used for live-cell microscopy ...... 38

2.1 All strains used in this study. *Mutants are derived from BY4741 ...... 63

2.2 Plasmids used in this study ...... 63

2.3 Oligonucleotides used in this study ...... 64

B.1 Parameter values for the Combined model...... 103

B.2 Parameter values for the Limiting Case #1 model...... 105

B.3 Parameter values for the Limiting Case #2 model...... 107

B.4 Parameter values for the Distributive model...... 109

xi LIST OF FIGURES

1.1 High osmolarity leads to Hog1 phosphorylation, nuclear translocation and stress adaptation ...... 4

1.2 ABC-Model Selection suggest temporally separated feedback network ...... 17

1.3 Positive feedback most-likely operates at the level of the MAPK ...... 20

2 1.4 Model m(1,0) is most consistent with previous experimental studies ...... 25

2 1.5 Model m(1,0) is most consistent with novel experimental studies ...... 29 1.6 Yeast cells as captured by DIC images ...... 40

1.7 DIC images of yeast cells segmented with simple binarization algorithm ...... 41

1.8 DIC images of yeast cells segmented with the “LocalAdaptiveBinarize” algorithm . 43

1.9 Workﬂow for aligning nuclei to cells ...... 44

2.1 Signal Inhibition by mono-phosphorylated Fus3...... 49

2.2 Analysis of differentially phosphorylated forms of Fus3...... 52

2.3 Dynamics of differentially phosphorylated forms of Fus3...... 54

2.4 Dynamics and mathematical model of the differentially phosphorylated forms of Fus3...... 57

2.5 Synergistic activation of Fus3...... 59

2 A.1 Selected parameter histograms and correlations for Model m(1,0) ...... 69

2 A.2 Selected parameter histograms and correlations for Model m(2,0) ...... 70

2 A.3 Selected parameter histograms and correlations for Model m(3,0) ...... 70

2 A.4 The response of model m(1,0) to periodic applications of hyper-osmotic stress . . . 71

2 A.5 The response of analog sensitive model m(1,0) to periodic applications of hyper-osmotic stress ...... 72

2 A.6 The response of model m(2,0) to periodic applications of hyper-osmotic stress . . . 73

2 A.7 The response of analog sensitive model m(2,0) to periodic applications of hyper-osmotic stress ...... 74

2 A.8 The response of model m(3,0) to periodic applications of hyper-osmotic stress . . . 75

xii 2 A.9 The response of analog sensitive model m(3,0) to periodic applications of hyper-osmotic stress ...... 76

2 A.10 The response of model m(1,0) to continuously increasing hyper-osmotic stress . . . 77

2 A.11 The response of analog sensitive model m(1,0) to continuously increasing hyper-osmotic stress ...... 78

2 A.12 The response of wild type model m(2,0) to continuously increasing hyper- osmotic stress ...... 79

2 A.13 The response of analog sensitive model m(2,0) to continuously increasing hyper-osmotic stress ...... 80

2 A.14 The response of wild type model m(1,0) to continuously increasing hyper- osmotic stress ...... 81

2 A.15 The response of analog sensitive model m(3,0) to continuously increasing hyper-osmotic stress ...... 82

B.1 A Schematic of the Yeast Mating Pheromone Response MAPK cascade ...... 84

B.2 Fus3 transitions between three phosphorylation states ...... 86

B.3 MAPK phosphorylation can be distributive or processive ...... 87

B.4 Fus3 phosphorylation is regulated by multiple mechanisms ...... 88

B.5 All processive models align with the wild type Phos-tag data ...... 91

B.6 The Combined model best aligns with the ste5ND Phos-tag data ...... 93

B.7 Phosphorylated Ste5 contributes to Fus3 dephosphorylation ...... 95

B.8 Fus3 induces a negative feedback loop through Msg5 ...... 97

B.9 A processive MAPK phosphorylation model aligns with the ste5ND & msg5∆ experimental data ...... 99

B.10 A processive MAPK phosphorylation model outperforms a Distributive model . . . 101

B.11 A processive model is preferred over the Distributive model...... 102

B.12 Histograms of parameter in the Combined Model ...... 104

B.13 Histograms of parameter in the Limiting Case #1 Model ...... 106

B.14 Histograms of parameter in the Limiting Case #2 Model ...... 108

xiii LIST OF ABBREVIATIONS

S. cerevisiae Saccharomyces cerevisiae

HOG High-Osmolarity Glycerol

MAPK Mitogen-Activated Protein Kinase

MAP3K Mitogen-Activated Protein Kinase Kinase Kinase

MAP2K Mitogen-Activated Protein Kinase Kinase

Thr Threonine

Tyr Tyrosine

ABC Approximate Bayesian Computation

MCMC Markov-Chain Monte Carlo

SMC Sequential Monte Carlo

SSD Sum-of-Squared Deviations

GFP Green-Fluorescent Protein

YFP Yellow-Fluorescent Protein

SCM Synthetic Complete Media

KCl Potassium Chloride

DIC Differential Interference Contrast

xiv CHAPTER 1 Dynamic precedence setting in counter-acting feedback networks 1

1.1 Introduction

Many information transfer processes are modiﬁed in real-time by feedback. In natural systems such as cellular signal transduction pathways, the downstream response is used to amplify or diminish the upstream signal via positive or negative feedback loops. Yet, how these integrated feedback networks are coordinated is not well understood. In this work, we investigate the spatial and temporal coordination of positive and negative feedback loops within the yeast (Saccharomyces cerevisiae)

High-Osmolarity Glycerol (HOG) pathway [12, 64]. The HOG pathway is a Mitogen-Activated

Protein Kinase pathway (MAPK), which has long served as a prototype signal transduction pathway to study eukaryotic stress responses. As the name implies, activation of the HOG pathway results in an up-regulation of genes that lead to the production of glycerol and other osmolytes that allow yeast to restore osmotic homeostasis with the extra-cellular environment. While it is well-appreciated that the HOG pathway contains multiple negative feedback loops that restore osmotic balance and regulate the duration of the hyper-osmotic stress response [29, 31, 71, 90–92, 95, 115], recent biochemical analysis has suggested that positive feedback also ampliﬁes activation of the HOG pathway during hyper-osmotic stress [12, 23, 64, 112]. Many of the proteins, and protein-protein interactions that contribute to the HOG pathway have been identiﬁed, yet how this counter-acting feedback network is integrated into the core pathway to coordinate the hyper-osmotic stress response is not well understood.

Hyper-osmotic stress causes water to be expelled from the cytoplasm to the extracellular space

(Fig. 1.1A). Yeast recover cell volume and restore osmotic homeostasis through activation of the HOG

1This chapter is being drafted as part of a publication to be submitted to the journal Science. The authors are as follows: Patrick C. McCarter, Lior Vered, Matthew K. Martz, Henrik G. Dohlman, Beverly E. Errede, Timothy C. Elston

1 pathway (Fig. 1.1A and B). The HOG pathway is activated by two pressure-sensitive transmembrane proteins (Sho1 and Sln1), which sense changes in cell turgor pressure [61, 80, 87, 105, 106, 114,

116]. The Sln1 branch, (summarized in Fig. 1.1B, left branch), is controlled by Sln1, an auto- phosphorylated histidine-kinase that initiates a phospho-transfer relay via Ypd1 to Ssk1 during osmotic homeostasis [61]. Sln1 auto-phosphorylation is inhibited during hyper-osmotic conditions, resulting in Ssk1 dephosphorylation [87]. Dephosphorylated Ssk1 binds and induces phosphorylation of the Ssk2/22 MAPK Kinase Kinase (MAP3K) [87]. The phosphorylated MAP3Ks bind and phosphorylate the Pbs2 MAPK Kinase, (MAP2K) [104], a scaffold protein that additionally binds and phosphorylates Hog1, the terminal MAPK [12].

The exact mechanism that leads to activation of the Sho1 branch is unknown (Fig. 1.1B, right), but several proteins including Hkr1, Msb2, Opy2, Ste20, Ste50, and Sho1 are utilized to transmit hyper-osmotic stress to Hog1 via a separate MAPK cascade (Fig. 1.1B, right branch) [63, 78, 79, 93,

105, 106, 114, 119]. Sho1 localizes the MAPK cascade to the plasma membrane by binding an SH3 domain in the MAP2K, which is again Pbs2 [86]. The MAP3K (which is Ste11 in the Sho1 branch) also binds and phosphorylates the MAP2K (Pbs2), which again induces MAPK phosphorylation

[87, 105, 106, 114, 116]. The analogy between these two separate pathways is illustrated in Fig. 1.1B, in which all kinases are labelled simply as MAP3K, MAP2K, or just MAPK for Hog1.

Once phosphorylated, the MAPK phosphorylates proteins that mediate cell cycle arrest [18,

25, 117], as well as proteins and transcription factors that regulate glycerol efﬂux and production

[31, 57, 58, 91, 95, 103], and phosphatases that regulate Hog1 phosphorylation [48, 71, 115]. The resulting restoration of osmotic balance and negative feedback that attenuates pathway activity also aligns the duration of MAPK phosphorylation with the severity of the hyper-osmotic stress

(Fig. 1.1A-C) [6, 48, 71, 76, 115].

MAPK phosphorylation is switch-like, (not signiﬁcantly phosphorylated until a threshold stress level is reached, then becomes fully phosphorylated), and also perfectly-adaptive, (returns to within

10% of the pre-stimulus phosphorylation level in the presence of sustained hyper-osmolarity as the cell restores turgor pressure), (Fig. 1.1A-C) [23]. The MAPK mediates perfect-adaptation through a negative feedback mechanisms that control the efﬂux and production of glycerol and other osmolytes

(Fig. 1.1A-C) [23, 57, 75, 103, 112], using a MAPK mutant that could be selectively inhibited

(MAPK-inhibited), English et al. discovered that the MAPK also mediates switch-like activation of

2 the hyper-osmotic stress response through an uncharacterized positive feedback loop (Fig. 1.1C-D)

[23]. The MAPK phosphorylates multiple upstream pathway components, including the putative osmo-sensor Sho1, and Ste50, a scaffold protein that is constitutively bound to Ste11 [36, 114, 116].

However Hao et al. demonstrated that Hog1-mediated feedback phosphorylation of Sho1 is not required for switch-like Hog1 phosphorylation [36]. English et al. demonstrated that precluding

Hog1-mediated feedback phosphorylation of the Ste50 scaffold protein also does not affect the

Hog1 phosphorylation rate [23]. Moreover, mutating the MAPK phosphorylation consensus sites on these proteins delays, but does not abolish Hog1 dephosphorylation during sustained hyper-osmotic stress. Thus, the Sho1 and Ste50 feedback targets alone are not sufﬁcient to coordinate MAPK phosphorylation. Therefore our goal was to identify and characterize the MAPK-mediated feedback network that encodes switch-like MAPK phosphorylation and perfect-adaptation during sustained hyper-osmotic stress.

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Figure 1.1: High osmolarity leads to Hog1 phosphorylation, nuclear translocation and stress adaptation

(A) Cartoon demonstrating the correlations between Hog1 phosphorylation, nuclear localization, and cell size during sustained high-osmolarity. The impact of high-osmolarity on cell size, Hog1 localization (cytoplasm, white ﬁlled circles; nucleus, black ﬁlled circles). (B) Schematic of the HOG pathway that describes how osmotic stress is transmitted through the MAPK cascades controlled by the Sho1 and Sln1 branches. Signal propagates in the direction of black solid arrows.

Starred proteins indicate phosphorylated MAPK components. Gray double-sided arrows indicate scaffold interactions in the Sln1 branch. Ssk2 and Ssk22 are combined as Ssk2/22. (C and D) The percentage of dual phosphorylated MAPK

(y-axis) in wild type (left) and Hog1-inhibited, (right, a variant whose activity is inhibited by addition of an ATP analog) yeast is depicted as functions of dose (see legend) and time (x-axis). (These data were originally published by English et al.

[23]).

To characterize this feedback network, we developed a quantitative biology platform that integrated mathematical modeling, machine-learning algorithms, and live-cell ﬂuorescence microscopy.

We ﬁrst utilized a systematic mathematical modeling study to determine the minimum set of HOG pathway features required to reproduce the reported Hog1 phosphorylation dynamics under wild

4 type (Fig. 1.1C) and Hog1-inhibited conditions (Fig. 1.1D) [23]. This involved us using a modiﬁed

Approximate Bayesian Computation-Sequential Monte Carlo parameter estimation and model selection algorithm (ABC-SMC, [70, 89, 108]) to identify feedback networks that could reproduce all aspects of the Hog1 phosphorylation training data obtained by English et al. [23]. We discovered that only models that temporally separated the initiation of positive and negative feedback were able to reproduce the training data. These models indicated that positive feedback could target any MAPK cascade level (MAP3K, MAP2K, or MAPK), but the “best performing” models suggested that positive feedback most likely targets the MAPK. To understand why this network was preferred over networks that featured positive feedback at the MAP2K or MAP3K levels, we used a combination of simulations that lacked positive or negative feedback and Time-Dependent Sensitivity Analysis to quantify the impact of each feedback loop on MAPK phosphorylation [44]. We discovered that positive feedback ampliﬁes MAPK phosphorylation during weak, intermediate, and severe hyper- osmotic stress. However this behavior is only observed when the positive feedback loop is targeted to the MAPK. The Time-Dependent Sensitivity Analysis further suggested that positive feedback maximizes MAPK phosphorylation by taking early precedence over negative feedback, and that negative feedback then causes MAPK phosphorylation to return to basal levels by over-taking the initial precedence of positive feedback.

To further validate our modeling results, we compared the ability of models that featured positive feedback at the MAPK, MAP2K, or MAP3K level, to simulate Hog1 phosphorylation that aligned with Hog1 dynamics observed in previous experimental studies [73]. We again observed that the model with positive feedback targeted to the MAPK level provided the best agreement with the previous studies. However, because the competing models with positive feedback targeted to the

MAP2K and MAP3K provided marginal agreement with the previously generated data, we could not fully eliminate either model from consideration as the consensus model. To further differentiate the likelihood of each model to be selected as the consensus model, we used each model to simulate

Hog1 phosphorylation under a variety of novel hyper-osmotic stress conditions. We subjected each model to simulated periodic, or step-wise increasing, and continuously increasing hyper-osmotic conditions. We discovered that the likelihood of the models to be selected as the consensus model could not be further differentiated by periodic applications of hyper-osmotic stress. In contrast, we discovered that likelihood of each model to be selected as the consensus model could be fully

5 differentiated with step-wise increasing and continuously increasing hyper-osmotic stress conditions.

In particular, we observed that the model with positive feedback targeted to the MAPK suggested that the HOG pathway was substantially more sensitive to hyper-osmotic stress than the models with positive feedback targeted to the MAP2K and MAP3K. Furthermore, this model suggested that the pathway would only partially respond to increases in the severity of the hyper-osmotic stress.

To determine whether the HOG pathway was as sensitive to hyper-osmotic stress as suggested by the model with positive feedback targeted to the MAPK, we took advantage of a strong positive correlation between MAPK phosphorylation and MAPK nuclear localization [29, 42, 73, 76], we developed an experimental protocol by using a micro-fluidic device and live-cell fluorescence microscopy to measure nuclear translocation of a Hog1-GFP fusion protein under the experimental conditions suggested by our modeling study. We observed that Hog1-GFP accumulated in the nucleus at the hyper-osmotic stress level suggested by the mathematical model with positive feedback that targeted the level of the MAPK. This result confirmed that our model selection algorithm had correctly suggested that positive feedback most-likely operates at the level of the MAPK.

1.2 Results

Our goal was to identify pathway architectures that could produce rapid and full MAPK phosphorylation, that also perfectly adapted, (returned to within ten percent of the pre-stimulated value), during sustained hyper-osmotic stress. We simultaneously required each architecture to produce graded MAPK phosphorylation that maintained dose-dependent steady state levels when all MAPK- mediated feedback loops were inactive. We divided our modeling study into two speciﬁc aims.

First, we sought to identify mathematical models that could recapitulate the wild type MAPK phosphorylation proﬁles when the MAPK-mediated feedback loops were active, and MAPK-inhibited phosphorylation proﬁles when the MAPK-mediated feedback loops were inactive. Second, we sought to identify the “best” mathematical model to use for prediction of MAPK phosphorylation dynamics when yeast were subjected to novel hyper-osmotic stress conditions.

Mathematical models have proven powerful tools to describe how simple signaling motifs drive complex physiological responses [26, 34, 35, 54, 62, 109]. However developing an informative mathematical model is often not trivial, and the difﬁculty is magniﬁed when the model is required to

6 quantitatively describe a physiological process. If the goal of the model is to quantitatively describe a physiological process, the ﬁrst challenge is to obtain a set of quantitative measurements that can be used to “train” the model to describe the physiological process. If the training data sets are available, the next challenge is to determine which biological features are needed for the model to reproduce the training data. However care must also be taken to ensure that the model does not include so many biological features that it is over-parameterized and lacks the ability to predict the system’s response to novel experimental conditions.

Our strategy to determine which features of the HOG pathway are required to reproduce the

MAPK phosphorylation training data, and to also avoid over-parameterization was to start by testing whether a minimal model of the “core” MAPK cascade of the HOG pathway could be parameterized to reproduce the training data. Our core model was represented as a system of ordinary differential equations (ODEs) that described the kinetic behavior of components of the HOG pathway. It is known that to reestablish an osmotic balance between the interior and exterior of the cell, the phosphorylated MAPK leads to the closing of glycerol conducting channels and to the generation of internal osmolytes [57, 75, 103]. Therefore the core model contained an equation to simulate the active states of three kinase levels of a MAPK cascade (MAP3K, MAP2K, MAPK), and one equation to simulate a generic signaling component “X” that was activated by the MAPK. We assumed that

X forms a negative feedback loop that attenuates the inﬂuence of the input signal s(t). Henceforth this top-level negative feedback will be referred to as the “input attenuating” negative feedback loop.

We assumed that prior to experiencing osmotic stress each signaling component was in its inactive state. To model the initiation of signaling, the activation rate for the MAP3K was assumed to be proportional to the amount of osmotic stress experienced by the cells. In turn, the activation rate for the MAP2K was assumed to be proportional the amount of active MAP3K and the activation rate of the MAPK assumed to be proportional to the amount of active MAP2K. To capture the adaptive processes, we assumed that the activation of X was proportional to the amount of active MAPK.

Thus as the amount of active MAPK increased, the amount of active X also increased. Then, as X increased the input attenuating negative feedback loop would mediated stronger inhibition of the signal transmitting into the core pathway, resulting in an eventual loss of MAPK activation. We show a schematic representing this core model in Fig. 1.2 (A, left diagram), and present the model with equations 1.3-1.6 in the Methods section.

7 1.2.1 Models M1 - positive and negative feedback acting on MAPK components

We next expanded the core model into a set of models that included an additional MAPK- mediated positive feedback loop and/or an additional X mediated negative feedback loop. In total

1 we developed fifteen additional mathematical models m(j,k), where the superscript denoted that the model was a member of the M1 model set and the (j,k) subscript pair denoted the location of positive and negative feedback within the MAPK cascade. We assumed that positive feedback loops increased the activation rate of the specified component and negative feedback loops increased the the deactivation rate of the specified component.

1.2.2 Models M2 - MAPK translocation

Hyper-osmotic shock induces MAPK translocation from the cytoplasm to the nucleus. MAPK is explicitly involved in the transcriptional regulation of numerous genes and small molecules that coordinate the hyper-osmotic stress response. We sought to determine whether MAPK translocation could affect MAPK phosphorylation dynamics. Model set M2 extended the M1 models by assuming

∗ ∗ ∗ that the activated MAPK (MAPK ) could transition between two states (MAPKAand MAPKB). ∗ ∗ MAP2K enzymatically catalyzed the transition of MAPK to MAPKA, which then transitioned to ∗ ∗ ∗ MAPKB. We assumed that MAPKA and MAPKB transitioned between states at basal rates that ∗ were ineligible for feedback ampliﬁcation. We designated MAPKA to be uniquely responsible for ∗ ∗ positive feedback and MAPKB to be uniquely responsible for X activation. This designation created a temporal separation between the activation of positive and negative feedback. We hypothesized that temporal separation would ﬁrst allow positive feedback to amplify MAPK phosphorylation to maximally saturable levels. Then, the delayed accumulation of negative feedback (X∗) would again drive the pathway towards perfect adaptation.

1.2.3 Parameter Estimation

Our next goal was to determine if any model from among the M1 and M2 sets could reproduce the MAPK phosphorylation training data. To accomplish this goal, we used a modiﬁed ABC-SMC parameter estimation algorithm to search for parameter vectors that could minimize the sum-of- squared deviations (SSD) between a given model’s simulated MAPK phosphorylation time series

8 and the corresponding MAPK phosphorylation training data sets [70, 89, 107, 108]. Here we present a brief explanation of our modified ABC-SMC algorithm, while full details of the algorithm are given in the Methods section. Our ABC-SMC method utilized an ABC-Rejection Sampler to identify an initial set of “N” parameter vectors, (a single vector of rate parameters that describe each rate equation in a given model), for each model [89]. We stipulated that in order for the parameter vector to be retained during ABC-Rejection Sampling, the model had to only “moderately” describe the training data when simulated with the candidate parameter vector, where we define an algorithm to quantitatively determine moderate agreement with the training data in the Methods section. Once an initial set of “N” parameter vectors were identified, we used a series of eight “n-step” ABC-Markov

Chain Monte Carlo simulations to explicitly evolve each parameter vector toward optimal regions of parameter space. Consistent with all ABC-SMC implementations, each consecutive ABC-MCMC simulation was required to generate a better ﬁt to the data than the previous ABC-MCMC simulation

[70, 107, 108]. However, we additionally stipulated that parameter vectors that became “trapped” in sub-optimal parameter space regions during any ABC-MCMC simulation could be resampled, with the stipulation that the chain was discarded if it was trapped a second time. This strategy allowed us to mitigate the computational costs of over-sampling from bad regions of parameter space, while simultaneously maximizing the coverage of good parameter space regions.

1.2.4 Model Selection

Our next goal was to determine which models from the M1 and M2 candidates provided the “best” representation of the HOG pathway. In pure Bayesian approaches, candidate models can be ranked based on each model’s ability to optimize an analytic “likelihood” function L(θ|x) [70].

Likelihood functions provide a quantitative measurement of how well a candidate model, given a set of parameter vectors, is supported by the training data [70]. The caveat of pure Bayesian methods in systems biology is that it is often difficult or impossible to define an explicit likelihood function for the complex models needed to represent the biology on a systems level [70, 107, 108]. ABC algorithms overcome these difficulties by substituting a “numeric” optimization of a least-squares function, (such as a the sum-of-squared deviations, SSD), which measures how well the model output is aligned with the training data, in place of the explicit optimization of the likelihood function

[70, 89, 107, 108]. ABC algorithms then utilize Bayes factors to determine which of two candidate

9 models is better supported by the training data [49, 107, 108]. Given two models, the Bayes factor is computed by taking the ratio of the likelihood function for each model [49, 107, 108]. In ABC algorithms the explicit calculation of the likelihood function is replaced by using a least-squares metric to measure the difference between the model’s representation of the training data and the training data [70, 107, 108]. The magnitude of the Bayes factor quantifies the “evidence” that one model, given a specific set of parameter vectors, provides better support for the training data than a competing model with an analogous set of parameter vectors. Given two models, the Bayes factor is computed by taking the ratio of the likelihood function for model m1 versus m2, and is defined as

P (x|m1) P (m1|x)/P (m2|x) B1,2 ≈ ≈ , (1.1) P (x|m2) P (m1)/P (m2) where x is the training data, P(mi) is the prior probability of selecting the model and P(mi|x) is the marginal posterior distribution of model mi [49, 70, 107, 108]. In the special case where the prior probabilites of two models are uniform, equation 1.1 reduces to

P (m1|x) B1,2 = . (1.2) P (m2|x)

The magnitude of the Bayes factor quantiﬁes how strongly one model is favored over another.

According to [49], and can be interpreted as,

Bayes factor Evidence against model M2 1 − 3 very weak

3 − 20 positive

20 − 150 strong

> 150 very strong

Table 1.1: Interpretation of Bayes factors according to [49]

ABC-SMC algorithms can be implemented such that model selection is performed after or simultaneously with parameter estimation [107, 108]. Our modiﬁed ABC-SMC algorithm was designed to determine the “likelihood” that any given model would be selected as the “consensus”

10 model, relative to every other candidate model, after each round of parameter estimation. Our

ABC-SMC algorithm stipulated that a model had to identify “N” parameter vectors that could meet the SSD threshold value during the speciﬁed round of parameter estimation (see Methods), in order to progress to the next round of parameter estimation. Our ABC-SMC algorithm also stipulated that the parameter vectors would be equally and minimally-optimized after each round of parameter estimation (see Methods). Therefore, computing a Bayes factor for each model after each round of parameter estimation reduced to only taking the ratio of the number of successful parameter vectors for model mi versus model mj as described in equation 1.2 above. Thus, every model that was able to identify the required number of parameter vectors during any round of parameter estimation had equal likelihood to be selected as the consensus model after that round. However if any model failed to identify the required number of parameter vectors during any round of parameter estimation, then the likelihoods of selecting any successful model as the consensus model versus the failed model were equal.

We visualized the relative likelihood of selecting any M1 or M2 candidate model as the consensus model during the nine rounds of parameter estimation in panels A and B of Fig. 1.2. We represent the relative likelihood of selecting a given model as the consensus model using stacked histograms, where each colored histogram delineates the relative likelihood of selecting a speciﬁc candidate model after each round of parameter estimation. Histograms that describe the relative likelihood of selecting a consensus model after the ABC-Rejection Sampler are shown below the red horizontal line. The next eight stacked histograms represent the relative likelihood of selecting each model as the consensus model after the corresponding rounds of ABC-MCMC parameter estimation. The height of each histogram was determined by counting the number of parameter vectors that were able to meet the speciﬁed SSD value. Histograms of unit height indicate that the model was able to identify the 500 parameter vectors required to advance to the next round of parameter estimation.

1 In panel A of Fig. 1.2, we observed that model m(3,0) was the ﬁrst model to fail to identify the required number of parameter vectors required to advance to the next round of parameter estimation, and was thus eliminated from contention during the during the second round of ABC-MCMC parameter estimation. It is trivial to see that all other successful models had equal likelihood to be

1 selected as the consensus model over m(3,0) using the Bayes factor formulism described by equation 1.2. As we show in in panels A and B of Fig. 1.2, several models failed to identify the required

11 number of parameter vectors to successfully progress through each round of parameter estimation,

1 1 (stacked histograms between red and black lines). In total, only three M1 models (m(0,0), m(0,3), 1 2 2 2 2 2 2 2 2 2 m(2,2)), and ten M2 models (m(1,0), m(1,1), m(1,2), m(1,3), m(2,0), m(2,1), m(2,2), m(2,3), m(3,0), 2 m(3,3)) had equal likelihood to be selected as the consensus model after the nine rounds of parameter estimation (Fig. 1.2, histograms touching the black horizontal line).

Our next challenge was to determine a consensus model from among the best performing M1 and

M2 models. Our ABC-SMC parameter estimation algorithm was designed to maximize the coverage of optimal parameter space regions, while minimizing the time spent sampling from sub-optimal parameter space regions. This stipulation allowed us to limit the number of steps each candidate parameter vector could take when searching for minimally optimal parameter space regions. By limiting each parameter vector’s random walk, our ABC-SMC parameter estimation algorithm was able to explore multiple parameter space regions and identify a diverse set of “minimally-optimized” parameter vectors. However the breadth of coverage came at the cost of in-depth explorations of each minimally optimized parameter space region.

Model selection for ABC-SMC algorithms is dependent on the quantity and quality of each model’s best set of parameter vectors. Therefore we next defined an ABC-Model Selection algorithm to explicitly increase the quality of each model’s parameter vectors. To perform model selection, we reduced the SSD threshold value (progressed to fitting schedule level 10), to further restrict the allowable deviation between the simulated and training data. We then updated the ABC-MCMC parameter estimation algorithm by increasing the number of steps that each parameter vector could take during the random walk. Additionally, we reasoned that because the parameter vectors were starting each random walk in a minimally optimized region of parameter space, the increase in the number of steps for the random walk would be sufficient to determine whether each parameter vector could be further optimized to reach the SSD10 threshold value. Therefore we eliminated the resampling of stalled parameter vectors. We then used updated ABC-MCMC parameter estimation conditions to update the likelihood of each top-performing candidate model by searching for a set of

“maximally-optimized” parameter vectors. We stipulated that any model that was able to identify a parameter vector that met the SSD10 threshold value would be considered as a viable candidate to be selected as the consensus model. However the model was eliminated if it failed to identify at least one “maximally-optimized” parameter vector.

12 Interestingly, our ABC-Model Selection algorithm did not identify a parameter vector from any

M1 model that could meet the reduced SSD threshold value (Fig. 1.2A, absence of histograms above the black horizontal line). Thus all M1 models were eliminated from contention to be selected as the consensus model. In contrast, all ten M2 models were able to identify parameter vectors that met the reduced SSD threshold value (Fig. 1.2B, histograms above the black horizontal line). To determine which M2 models were best supported by the training data, we again used equation 1.2 to determine the relative likelihood of selecting each M2 model as the consensus model. As we show in table

2 2 1.1 models m(1,0) and m(1,2) had the highest likelihoods to be selected as the consensus models, 2 2 with model m(1,0) having a negligibly higher likelihood than model m(1,2). Using the Bayes factor interpretation framework deﬁned in table 1.1, we concluded that there was very weak evidence that

2 2 2 model m(1,0) was favored over model m(1,2), and slightly more evidence that m(1,0) was favored 2 2 2 2 over m(1,1) and m(1,3). However there was positive evidence m(1,0) and m(1,2) were signiﬁcantly favored over all other M2 candidate models.

2 2 2 2 2 2 2 2 2 2 Model m(1,0) m(1,1) m(1,2) m(1,3) m(2,0) m(2,2) m(2,3) m(3,1) m(3,0) m(3,3) Parameters (#) 289 163 283 180 87 75 9 79 107 81

2 m(1,0) 1. 1.77 1.02 1.61 3.32 3.85 32.11 3.65 2.71 3.56 2 m(1,2) .97 1.73 1. 1.57 3.25 3.73 31.44 3.58 2.64 3.49

2 2 Table 1.2: Bayes factors for model m(1,0) and m(1,2)

Row 1) The ten M2 models that were able to identify parameter vectors during the ﬁrst round of model selection. Row 2)

2 The number of parameter vectors identiﬁed for each model. Rows 3 and 4) Bayes factors for models m(1,0) (row 3) and

2 m(1,2) (row 4) are shown versus every other M2 model. The Bayes factor was computed by taking the ratio of the evidence for model 1 over model 2, and interpreted according to [49].

2 While the preference for m(1,0) was negligible, the favorability of this model over the more complex version of the model demonstrated that our ABC-Model Selection did not favor complex models over simple models. Additionally, our ABC-Model Selection algorithm effectively “penalized”

2 2 2 2 the more complex versions of model m(1,0),(m(1,1), m(1,2), and m(1,3)) during model selection. The 2 2 continuation of this trend for models m(2,0) and m(3,0) when compared to the more complex versions

13 of each model, conﬁrmed that our ABC-Model Selection was not biased to favor complex models over simpler models.

1.2.5 Positive feedback has most signiﬁcant impact when targeted to the MAPK

Given that any model that was able to identify at least one parameter vector during the ﬁrst round of model selection was considered a successful model, we sought to understand why model

2 2 m(1,0), (and by extension all three complex versions of m(1,0)), was selected as the consensus model 2 2 over m(2,0) and m(3,0). To accomplish this goal, we first compared the ability of each model, when simulated with the parameter vectors identified during the first round of model selection, to simulate

MAPK phosphorylation dynamics that were aligned with the Hog1 phosphorylation training data sets.

To understand why all M1 models ultimately failed to reproduce the Hog1 phosphorylation

1 training data, we visualized the MAPK phosphorylation time series for model m(2,2) versus the Hog1 phosphorylation training data (Fig. 1.2C and D). Under wild type conditions, we observed that

1 model m(2,2) did not simulate full (100%) MAPK phosphorylation at any level of hyper-osmotic 1 stress (weak, intermediate, or severe) (Fig. 1.2C). However model m(2,2) did simulate MAPK phosphorylation that persisted at sub-maximal levels, where the persistence was encoded as a function of the hyper-osmotic stress severity. Under Hog1-inhibited conditions, we observed that

1 1 model m(2,2) did simulate graded MAPK phosphorylation. However model m(2,2) was unable to simulated dose-dependent steady state MAPK phosphorylation levels. These results demonstrated

1 that model m(2,2) was insufﬁcient to reproduce the Hog1 phosphorylation training data under wild type and Hog1-inhibited conditions.

We next visualized the MAPK phosphorylation time series for the characteristic M2 models,

2 2 2 (m(1,0), m(2,0), and m(3,0)), when simulated under wild type conditions (Fig. 1.2E-F). As expected, 2 model m(1,0) simulated MAPK phosphorylation that was most consistent with the wild type Hog1 2 phosphorylation training data sets (Fig. 1.2E, row 1). Although model m(1,0) simulated less MAPK phosphorylation (70%) during weak hyper-osmotic stress than what was observed for the training

2 data, m(1,0) was able to simulate maximal MAPK phosphorylation at the intermediate and severe 2 hyper-osmotic stress levels. Consistent with the Hog1 phosphorylation training data, m(1,0) simulated MAPK phosphorylation that returned to basal levels shortly after reaching the 70% phosphorylated

14 level during weak hyper-osmotic stress. Also consistent with the Hog1 phosphorylation training data,

2 m(1,0) simulated MAPK phosphorylation that persisted at maximal percentages during intermediate and severe hyper-osmotic stress, where the duration of the persistence was dependent on the severity of the hyper-osmotic stress (dose-to-duration encoded).

2 2 In contrast, neither model m(2,0) or m(3,0) simulated MAPK phosphorylation that reached the maximal levels observed at any hyper-osmotic stress level in the wild type Hog1 phosphorylation

2 2 training data sets (Fig. 1.2E, rows 2 and 3). Although models m(2,0) and m(3,0) did simulate MAPK phosphorylation that was dose-to-duration encoded, each model simulated Hog1 dephosphorylation kinetics that were uniquely misaligned with different aspects of the corresponding training data sets.

2 In particular, model m(2,0) simulated MAPK phosphorylation that persisted at sub-maximal levels for a longer duration than what was observed in the each wild type Hog1 phosphorylation training data set (Fig. 1.2E, row 2). The model also simulated switch-like MAPK dephosphorylation kinetics that returned MAPK phosphorylation to basal levels at a faster rate than observed in the wild type

2 data sets. In contrast, model m(3,0) simulated MAPK phosphorylation that persisted at sub-maximal levels, however the duration of the persistence was signiﬁcantly shorter than what was observed in

2 2 the wild type Hog1 phosphorylation training data sets. Similar to model m(2,0), model m(3,0) then simulated switch-like MAPK dephosphorylation kinetics that returned MAPK phosphorylation to basal levels (Fig. 1.2E, row 3).

We next visualized the average MAPK phosphorylation time series for each model when simu-

2 lated under Hog1-inhibited conditions. Under these conditions, model m(1,0) again simulated MAPK phosphorylation that was most consistent with the corresponding MAPK phosphorylation training data sets (Fig. 1.2F, row 1). This model was also able to simulate graded MAPK phosphorylation at each hyper-osmotic stress level, where the simulated steady state MAPK phosphorylation percentage aligned with the steady state MAPK phosphorylation percentage observed in the corresponding Hog1 training data sets.

2 2 Alternatively, models m(2,0) and m(3,0) only simulated graded MAPK phosphorylation at the weakest hyper-osmotic stress level (Fig. 1.2F, rows 2 and 3). Both models then simulated switch-like

MAPK phosphorylation at the intermediate and severe hyper-osmotic stress levels that also reached steady state levels earlier in the time course than what was observed in the corresponding Hog1 phosphorylation training data sets. Interestingly, we noted that these models established steady state

15 MAPK phosphorylation at the maximal MAPK phosphorylation levels observed in the wild type simulations.

These results suggested that MAPK phosphorylation dynamics were sensitive to the location

2 of the positive feedback loop. Because model m(1,0) was the only model that simulated a greater percentage of MAPK phosphorylation at the intermediate and severe hyper-osmotic stress levels under wild type conditions than Hog1-inhibited conditions, we hypothesized that positive feedback

2 had a greater impact on the magnitude of MAPK phosphorylation for model m(1,0) than models 2 2 2 m(2,0) and m(3,0). As a result, the positive feedback ensured that model m(1,0) attained full MAPK phosphorylation during intermediate and severe hyper-osmotic stress.

Thus, through a systematic search of model space we determined that the set of model features required to reproduce all aspects of the Hog1 phosphorylation training data sets were: (i) positive feedback and (ii) a delayed negative feedback loop that attenuated the input signal. These results suggest that the temporal separation of positive and negative feedback encodes full MAPK phosphorylation, and dose-to-duration encoding in the presence of sustained hyper-osmotic stress. We then used an objective model selection algorithm to select a consensus model from with the top performing candidate models. Our ABC-Model Selection algorithm suggested that the uncharacterized positive feedback most likely impacts the MAPK. Because our model selection algorithm also favored the

2 2 2 simplest model within each positive feedback grouping (m(1,0), m(2,0), and m(3,0)), we decided to use these three characteristic models to understand the impact of positive feedback at each MAPK cascade level, under a variety of different experimental conditions.

16 A B Hyper-Osmotic Hyper-Osmotic Stress Stress MAP3K MAP3K MAP2K MAP2K MAPK MAPK ; A MAPK ; % C D Hyper-Osmotic Hyper-Osmotic P P Hyper-Osmotic Stress Stress

Stress Weak MAP3K Intermediate MAP3K Severe 0$3. 0$3.

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MAPKA MAPKA MAPK MAPK X B X B

Hyper-Osmotic Hyper-Osmotic Stress P Stress P MAP3K MAP3K

MAP2K 0$3. MAP2K 0$3.

MAPK MAPK A A X MAPK X MAPK B B Figure 1.2: ABC-Model Selection suggest temporally separated feedback network

(A and B) The relative likelihood of selecting any model from the M1 (panel A) or M2 (panel B) model sets are shown using stacked histograms after each round of parameter estimation and one round of model selection. The likelihood of each model after the ABC-Rejection Sampler is shown with histograms under the red horizontal line. The likelihood of each model after each round of ABC-MCMC parameter estimation is shown with histograms below the black horizontal line. The likelihood of each model after one round of parameter estimation is shown with histograms above the black

1 horizontal line. (C) Wild type simulation results for model m(2,2). Average pp-Hog1(%) simulation curves are generated from the ﬁnal set of parameter vectors (y-axis), and are shown versus the wild type data (data points) and time (x-axis).

(D) Same as C, except simulations compare the Hog1-inihibited model conditions versus the Hog1-inhibited data. (E and

2 F) Same as C and D, except wild type and Hog1-inhibited simulations are shown versus the data for models m(1,0) (row

2 2 1), m(2,0) (row 2), and m(3,0) (row 3).

17 Our next goal was to understand the impact of positive feedback on each model. To test the hypothesis that positive feedback has the most signiﬁcant impact when targeted to the MAPK, we disrupted positive feedback in each model, and then re-simulated each model using the previously described hyper-osmotic stress conditions.

Consistent with our hypothesis that positive feedback had the greatest impact on the magni-

2 tude of MAPK phosphorylation when targeted to the MAPK, model m(1,0) displayed the greatest reduction in the magnitude of MAPK phosphorylation under these conditions, (Fig. 1.3A, row

1). Although the loss of positive feedback resulted in a signiﬁcant reduction in the magnitude of

MAPK phosphorylation at each hyper-osmotic stress level, we did not observe a signiﬁcant change in the persistence of MAPK phosphorylation, or in the MAPK dephosphorylation kinetics for this

2 2 model. Interestingly, models m(2,0) and m(3,0) displayed a reduction in the magnitude of MAPK phosphorylation at the weak hyper-osmotic stress level (Fig. 1.3A, rows 2 and 3). However these models did not simulate a reduction in the magnitude of MAPK phosphorylation at the intermediate

2 2 and severe hyper-osmotic stress levels. Unlike model m(1,0), model m(2,0) displayed a signiﬁcant reduction the persistence of MAPK phosphorylation and in the MAPK dephosphorylation rate at the

2 intermediate and severe hyper-osmotic stress levels (Fig. 1.3A, row 2). In contrast, model m(3,0) did not display a signiﬁcant change in the persistence of MAPK phosphorylation or in the MAPK dephosphorylation rate (Fig. 1.3A, row 3).

These results suggested that positive feedback impacted each model dynamically, and that the impact of positive feedback was dependent on the hyper-osmotic stress level and time. Therefore, our next objective was to quantify the time points at which feedback signiﬁcantly impacted Hog1 activity. To accomplish this goal, we used time-dependent sensitivity analysis to determine how small perturbation in the magnitude of each feedback parameter would affect MAPK phosphorylation dynamics, when each model was simulated at weak, intermediate, and severe hyper-osmotic stress

[44]. To perform this analysis, we computed the partial derivative of the time-series with respect to a small change in the rate parameters that determined the magnitude of positive feedback (α) and input-attenuating negative feedback (β)[44]. We then used color-coded shading to visualize the time-dependent impact of the small perturbation in each parameter value on a characteristic MAPKA time-series, when simulated at each hyper-osmotic stress level (Fig. 1.3B-D).

18 2 Our sensitivity analysis revealed that model m(1,0) was the only model continuously sensitive to positive feedback at all three hyper-osmotic stress levels 1.3B-D, row 1, blue shading). We observed that a reduction in the positive feedback parameter resulted in less MAPKA phosphorylation at every time point of the time series (blue shading below the black curve). Similarly, an increase in the positive feedback parameter resulted in an increase in MAPKA phosphorylation at every time point

2 of the time series (blue shading above the black curve). In contrast, m(1,0) was only sensitive to negative feedback during the MAPKA dephosphorylation phase at any hyper-osmotic stress level (Fig. 1.3B, row 1, pink shading). We observed that a decrease in the negative feedback parameter resulted in increased MAPKA phosphorylation only during the latter time-points in the time series, (pink shading above the black curve). Similarly, an increase in the negative feedback rate resulted in less MAPKA phosphorylation at the latter time points in the time series, (pink shading below

2 2 the black curve). Alternatively, MAPKA phosphorylation for models m(2,0) and m(3,0) was only initially sensitive to positive feedback during weak hyper-osmotic stress, and was also most sensitive to positive feedback during the dephosphorylation phase for all hyper-osmotic stress levels (Fig.

1.3B-D, rows 2 and 3, blue shading). Interestingly, all models were also simultaneously sensitive to positive and negative feedback during the MAPKA dephosphorylation phase during intermediate and severe hyper-osmotic stress (Fig. 1.3C and D, purple shading).

Collectively, these results demonstrated that the maximum proportion and persistence of MAPK phosphorylation was sensitive to the location of positive feedback. Positive feedback had the most signiﬁcant impact on the magnitude of MAPK phosphorylation when it operated at the level of the MAPK. However, Positive feedback had the most signiﬁcant impact on the duration of MAPK phosphorylation when it operated at the level of the MAP2K. Additionally, the location of the positive feedback loop impacted the MAPK dephosphorylation rate. Models with positive feedback acting at the MAPK displayed approximately linear dephosphorylation kinetics. Alternatively, models with positive feedback acting above the MAPK displayed switch-like dephosphorylation kinetics.

Our time-dependent sensitivity analysis then revealed that positive feedback initially operated independently of negative feedback, and that positive feedback ampliﬁed MAPK phosphorylation

2 early in the time series for each model (albeit only at weak hyper-osmotic stress for models m(2,0) 2 and m(3,0). However, the magnitude of the negative feedback eventually became strong enough to overcome positive feedback, and drove the system towards perfect adaptation. Thus these results

19 further demonstrate that the temporal separation of positive and negative feedback encode rapid and maximal MAPK phosphorylation and dose-to-duration encoding when yeast are exposed to sustained hyper-osmotic stress that does not change in magnitude.

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Figure 1.3: Positive feedback most-likely operates at the level of the MAPK

2 2 2 (A) Simulation results for models m(1,0), m(2,0), and m(3,0) are shown when simulated absent positive feedback. Solid curves are generated from the mean result of each model when simulated with the parameters identiﬁed during model selection. (B-D) Time-Dependent Sensitive Analysis is performed for a characteristic MAPKA time series for each model, at weak (B), intermediate (C), and severe (D), hyper-osmotic stress. The black solid line is the unperturbed MAPKA time series, blue shading shows the impact of the change in the positive feedback parameter α, red shading shows the impact of the change in the input-attenuating negative feedback parameter β, purple shading shows time points where positive and negative feedback have simultaneously impact.

2 1.2.6 Model m(1,0) is most consistent with previous experimental analysis

Model selection algorithms are powerful tools to help determine consensus models. However model selection alone is often not sufﬁcient to conﬁrm that a consensus model is the best representation of the experimental system. Our strategy to further determine the validity of our model selection algorithm was to compare the ability of each characteristic model to predict the response

20 of the MAPK to experimental conditions that were not included in the training data sets. If our

2 model selection algorithm was correct, then model m(1,0) would simulate MAPK phosphorylation dynamics that better aligned with the additional experimental Hog1 nuclear translocation data sets

2 2 than models m(2,0), and m(3,0). The HOG pathway is one of the most extensively studied signal transduction pathways available. Thus there were a substantial amount of pre-existing data sets from which we could compare the predictive ability of each model. Of particular interest, are data sets generated from live-cell microscopy that measure nuclear translocation of a Hog1-(Fluorescent

Protein) construct. Ferrigno et al. initially demonstrated that dual-phosphorylated Hog1 translocates from the cytoplasm to the nucleus [29]. It is also well-accepted that the duration of Hog1 nuclear localization is positively correlated with the duration of Hog1 phosphorylation [29, 71, 76]. Because of the strong correlation between Hog1 phosphorylation and Hog1 translocation, we ﬁrst tested whether each model could simulate MAPK phosphorylation that was qualitatively consistent with

Hog1 nuclear localization under a variety of previously conducted experimental conditions.

2 1.2.7 Model m(1,0) is most consistent with previous mutational analysis

Many previous studies have used live-cell ﬂuorescence microscopy to measure nuclear localization of a Hog1-Fluorescent Protein (Hog1-GFP, Hog1-YFP) construct, when yeast are exposed to various hyper-osmotic stress conditions [42, 73, 76]. These studies have demonstrated that unlike the all-or-nothing response of Hog1 phosphorylation, the amount of Hog1 that is localized in the nucleus at any moment of time is saturable [42, 73, 76]. Although the entire pool of phosphorylated

Hog1 does not localize to the nucleus, the duration of Hog1 nuclear localization is consistent (strong positive correlation) with the duration of Hog1 phosphorylation. Many studies have also shown that the amount of Hog1 localized to the nucleus during hyper-osmotic stress is sensitive to Pbs2 abundance [73, 76]. Mettetal et al. demonstrated that a reduction in the abundance of Pbs2 resulted in a smaller percentage of Hog1 localized to the nucleus during intermediate hyper-osmotic stress compared to wild type [73]. In addition, the Hog1 nuclear import rate was reduced from switch-like to linear, and the persistence of Hog1 nuclear localization at the reduced maximal level was eliminated.

Hog1 also gradually translocated out of the nucleus immediately after reaching the reduced maximal level. Thus these results also demonstrated that the switch-like Hog1 nuclear import and export kinetics were dependent on the abundance of Pbs2.

21 An accurate model of the HOG pathway must be able to reproduce these results when simulated under conditions of Pbs2 depletion. To determine whether each model could reproduce these results, we systematically reduced the amount of MAP2K in each model by increments of 25%, and then simulated MAPK phosphorylation at the intermediate hyper-osmotic stress level for 60

2 minutes. We observed that model m(1,0) was the only model that simulated MAPK phosphorylation dynamics that were qualitatively consistent with Hog1 nuclear localization under conditions of Pbs2 depletion (Fig. 1.4, compare to Fig. 2D of Mettetal et al. [73]). In order for this model to simulate

MAPK phosphorylation that was consistent with Hog1 nuclear localization under conditions of Pbs2 depletion, the availability of Pbs2 had to be reduced to 25% of the “wild type” model value. Model

2 m(1,0) simulated a MAPK phosphorylation rate that was reduced from switch-like to graded, and a maximal level of MAPK phosphorylation that was reduced by 85% when compared to wild type

(100% to 40%). Additionally this model did not simulate MAPK phosphorylation that persisted at the reduced maximal level. Instead and consisted with the Hog1 nuclear localization data MAPK dephosphorylation began immediately after reach the reduced maximal value (Fig. 1.4A, row 1).

2 2 Although models m(2,0) and m(3,0) reproduced the reduction in the maximum level of Hog1 phosphorylation, these models did not reproduce the reduction in the MAPK phosphorylation rate from switch-like to graded (Fig. 1.4A, rows 2 and 3). Furthermore, these models simulated

2 MAPK phosphorylation that persisted at the reduced maximal level. Model m(2,0) did simulate a reduced MAPK dephosphorylation rate, However the MAPK dephosphorylation rate for model

2 m(3,0) remained switch-like MAPK dephosphorylation kinetics. 2 The simulations under conditions of MAP2K depletion further suggested that model m(1,0) was most representative of the HOG pathway. We postulated that the linear MAPK phosphorylation kinetics exhibited by this model were the result of positive feedback directly amplifying the diminished signal ﬂowing into the MAPK. Similarly, we postulated that the slow and linear Hog1 dephosphorylation kinetics resulted from the slower accumulation of negative feedback. To test these hypotheses, we again used time-dependent sensitivity analysis to determine the time points at which a characteristic MAPKA time series was most sensitive to positive and negative feedback when the

MAP2K abundance was reduced. Our sensitivity analysis revealed that similar to wild type, MAPKA phosphorylation was most sensitive to positive feedback during the phosphorylation phase (Fig.

1.4A, inset. row 1). However unlike wild type, MAPKA phosphorylation also became sensitive to

22 negative feedback immediately after reaching its maximum phosphorylation level. The overlapping sensitivity between positive and negative feedback continued until MAPKA phosphorylation returned

2 2 to the basal level. Similar to the wild type sensitivity analysis, neither model m(2,0) or m(3,0) were sensitive to positive feedback at the early time points of the time series. However unlike the wild type sensitivity analysis, these models were also only sensitive to negative feedback during the later time points of the time series (Fig. 1.4A, insets, rows 2 and 3).

This round of sensitivity analysis suggested that because the positive feedback loop targeted

2 the MAPK, only model m(1,0) was able to compensate for reduced signal transmission caused by MAP2K under-expression. We postulated that the under-expression of the MAP2K caused the pathway to become more sensitive to positive feedback. We then demonstrated that the increased sensitivity to positive feedback allowed the MAPK to linearly accumulate phosphorylation in the presence of the diminished signal. This linear accumulation also made the pathway sensitivity to negative feedback earlier in the time course, but did not increase the magnitude of the sensitivity to negative feedback. Instead the sensitivity to negative feedback was balanced by the sensitivity to positive feedback. As a result the pathway failed to achieve maximal MAPK phosphorylation, but also maintained a steady-state level of MAPK phosphorylation after the initial dephosphorylation

2 phase. Because the positive feedback operated at or above the MAP2K for models m(2,0) and 2 m(3,0), there was no downstream mechanism to compensate for the loss of signal caused by MAP2K under-expression. Thus the magnitude of MAPKA phosphorylation was reduced, but the kinetics of

MAPKA phosphorylation remained unchanged. Considered together, the simulations under conditions of MAP2K depletion and the sensitivity

2 analysis provided another set of qualitative evidence that model m(1,0) was most representative of the HOG pathway. Our next goal was to identify additional experimental conditions that could further differentiate the qualitative behaviors of each model. To accomplish this goal, we used the wild

2 2 2 type version of models m(1,0), m(2,0), and m(3,0) to simulate MAPK phosphorylation under dynamic hyper-osmotic stress conditions. We then again compared the simulated MAPK phosphorylation dynamics with pre-existing Hog1 nuclear localization data sets.

23 2 1.2.8 Model m(1,0) is most consistent with previous dynamic analysis

Mettetal et al. also employed an engineering approach to characterize the temporal dynamics of

Hog1 nuclear localization [73]. Their study demonstrated that Hog1 could translocate into and out of the nucleus on the same time scale as a square wave of intermediate hyper-osmotic stress, with a four minute period (see Fig. S2 [73]). An accurate model of the HOG pathway must also be able to reproduce these results. To determine whether each model could simulate MAPK phosphorylation that was consistent with these data, we updated the stress input function “s(t)” to be a square wave function of intermediate amplitude, with a period of four-minutes. We then simulated each model for 15 minutes, and compared the simulated MAPK phosphorylation dynamics with the previously reported Hog1 nuclear localization dynamics [73].

2 When simulated with the square wave input function, model m(1,0) simulated MAPK phosphorylation dynamics that was most consistent with the data presented by Mettetal et. al. [73]

(Fig. 1.4B, row 1). Each model suggested that the MAPK would be maximally phosphorylated

2 during the ﬁrst and second pulses of hyper-osmotic stress. However, model m(1,0) was the only model that suggested that the MAPK would be minimally dephosphorylated when the hyper-osmotic stress was removed after the ﬁrst and second hyper-osmotic stress pulses (Fig. 1.4B, row 1). We hypothesized that positive feedback was only capable of sustaining MAPK phosphorylation during

2 the dephosphorylation phase for model m(1,0). To test this hypothesis we re-simulated each model in 2 the absence of positive feedback. We observed that compared to wild type, model m(1,0) simulated diminished MAPK phosphorylation during each hyper-osmotic stress pulse, and increased MAPK dephosphorylation when the hyper-osmotic stress was removed (Fig. 1.4C, row 1). Alternatively,

2 2 when the positive feedback was removed, models m(2,0) and m(3,0) simulated MAPK phosphorylation that was consistent with the wild type simulations (Fig. 1.4C, rows 2 and 3). These results

2 conﬁrmed that positive feedback was only able to enhance MAPK phosphorylation for model m(1,0) when the hyper-osmotic stress was present, and that positive feedback was able to diminish MAPK dephosphorylation when the hyper-osmotic stress was removed.

24 A Pbs2 Knockdown B Periodic C No Positive Feedback m10 m10 Hyper-Osmotic 100 100 Stress Time Series Positive FB

Negative FB

A Overlap FB 80 MAP3K 80

60 60 MAP2K

40 40 MAPK A 20 20

0 X* MAPK 0 B 0 5 10 15 0 5 10 15

Time (Minutes) Time (Minutes) m20 m20 Hyper-Osmotic Stress 100 100

A 80

MAP3K 80

60 60 MAP2K

40 40 MAPK A 20 20

0 X* MAPK 0 B 0 5 10 15 0 5 10 15

Time (Minutes) Time (Minutes) m30 m30 Hyper-Osmotic 100 100 Stress

A 80 80 MAP3K

60 60

MAP2K 40 40

20 20 MAPKA 0 0 0 5 10 15 X* MAPKB 0 5 10 15

Time (Minutes) Time (Minutes)

2 Figure 1.4: Model m(1,0) is most consistent with previous experimental studies

2 2 2 (A) MAPK phosphorylation for models m(1,0) (row 1) and m(2,0) (row 2), and m(3,0) (row 3) is shown for each model when simulated under conditions of Pbs2 depletion (MAP2K availability reduced to 25% of wild type). The insets in each row of (A) are Time-Dependent Sensitivity Analysis of a characteristic MAPKA time series when simulated under conditions of Pbs2 depletion. (B) Same as A, except the models are simulated using a square wave input function “s(t)”.

These comparisons with the Hog1 nuclear localization data presented by Mettetal et al. further

2 suggested that model m(1,0) was most representative of the HOG pathway. Additionally, the simulation of each model using a dynamic input function demonstrated that the location of the positive

2 feedback loop influenced the dynamic behavior of the HOG pathway. Model m(1,0) suggested that positive feedback confers “memory” to the HOG pathway when yeast are exposed to rapidly fluctuating hyper-osmotic stress conditions. However, because the negative feedback eventually overtakes the impact of positive feedback, the HOG pathway eventually adapts to the fluctuating environment.

Interestingly, we noted that the qualitative differences in each model’s MAPK phosphorylation dynamics were less pronounced when simulated with faster and slower square wave frequencies

25 (Supplemental Materials). These results suggested that the temporal dynamics of the hyper-osmotic stress could be further exploited to determine a consensus model.

1.2.9 Models are differentiated by increases in hyper-osmolarity

We next tested whether each model would respond differently to discrete increases in the severity of the hyper-osmotic stress. We simulated each model using a stress input function that increased from 0mM KCl to 350mM KCl (intermediate hyper-osmotic stress) in discrete steps at twenty-minute

2 intervals (Fig. 1.5A). We observed that model m(1,0) was the only model that generated discernible MAPK phosphorylation during the ﬁrst step of hyper-osmotic stress (Fig. 1.5A, row 1). For this step, MAPK phosphorylation achieved a maximum of approximately 40%, and then returned to near basal levels. The next step increase of hyper-osmotic stress generated MAPK phosphorylation that achieved a maximum of 20%. However, MAPK phosphorylation did not return to basal levels during the duration of the second step. Each subsequent step increase in hyper-osmotic stress elicited approximately the same percentage of MAPK phosphorylation as the second step, however MAPK phosphorylation did not fully return to basal levels during any remaining step.

2 2 Alternatively, models m(2,0) and m(3,0) generated only minimal MAPK phosphorylation during the ﬁrst twenty-minute step of hyper-osmotic stress (Fig. 1.5A, rows 2 and 3). The next step elicited

2 2 a signiﬁcant increase in MAPK phosphorylation to 50% for model m(2,0) and 35% for model m(3,0). 2 Similar to model m(1,0), MAPK phosphorylation then returned to basal levels for both models. 2 However, unlike model m(1,0), which elicited the same amount of MAPK phosphorylation for the 2 2 remaining three step increases, both m(2,0) and m(3,0) simulated less MAPK phosphorylation during the third step than the second step, more MAPK phosphorylation during the fourth step than the third step, and then less MAPK phosphorylation during the ﬁfth step than the fourth step.

Our next goal was to determine whether any model had accurately predicted the response of the HOG pathway to step-wise increasing applications of hyper-osmotic stress. To accomplish this goal, we combined live-cell microscopy, a previously described microﬂuidic device [39] to measure nuclear localization of a Hog1-GFP (green ﬂuorescent protein) construct [4] as a function of step-wise increases salt induce hyper-osmotic stress. We then used a robotic control system (GradStudent

Dial-A-Wave v3, gravity-ﬂow system) to programmatically increase the KCl concentration in the microﬂuidic chamber from 0mM KCl to 350mM KCl in 70mM steps, at twenty-minute intervals. We

26 observed Hog1-GFP localized to the nucleus during the ﬁrst step of incremental increase in osmotic stress, (Fig. 1.5C). Hog1-GFP was localized to the nucleus for approximately three-minutes, and then returned the cytoplasm. Consistent with the model predictions, each step-wise increase in KCl elicited nuclear localization of Hog1-GFP, that again returned to the cytoplasm during the duration

2 of the step. These results revealed that model m(1,0) had accurately predicted that the HOG pathway was sensitive to 70mM KCl. The analysis of the remainder of the time course revealed that this model had also correctly predicted the dynamics of Hog1 nuclear translocation during the remaining step-wise increases in the severity of the hyper-osmotic stress.

2 1.2.10 Model m(1,0) predicts a variable Hog1 response under physiological conditions

2 Thus far the model m(1,0) had proven to be an effective tool to understand how Hog1 activity under a variety of laboratory conditions, which featured, designable mutations and discrete changes in external osmolarity that were sustained or transient. However we lacked intuition about how the model and the HOG pathway would respond to more physiologically relevant changes in osmolarity.

2 To determine whether model m(1,0) would be the most appropriate model to predict Hog1 activity under physiological conditions, we updated the stress input function “s(t)” to be a function that continuously increased from 0mM KCl to 600mM KCl at a rate of 6mM KCl per minute. We also deﬁned the function such that the input signal would cease to increase when it reached a maximum value equivalent to 600mM KCl. The function would instead maintain this maximum value for an additional twenty minutes. To be consistent with our previous simulations, we simulated each of the characteristic models under these hyper-osmotic stress conditions.

When the hyper-osmotic stress increased from 0mM KCl to 600mM KCl at a rate of 6mM KCl

2 per minute, model m(1,0) suggested that Hog1 would begin to be phosphorylated within five minutes 2 of the initial hyper-osmotic stress exposure (Fig. 1.5B, row 1, left inset). Alternatively, models m(2,0) 2 and m(3,0) did not simulate significant MAPK phosphorylation until fifteen minutes after the intial 2 hyper-osmotic stress exposure. Thus model m(1,0) was most sensitive to weak hyper-osmotic stress concentrations.

Interestingly, we noted that Hog1 phosphorylation never surpassed 40% at any point during this simulation. This was surprising given that the ramp simulated a ﬁnal hyper-osmotic stress

27 concentration of 600mM KCl (severe hyper-osmotic stress). One explanation for this outcome is that the negative feedback loop suppresses the maximal MAPK phosphorylation response under these conditions.

Interestingly, we observed that while the mean MAPK phosphorylation time series was smooth,

2 2 there were signiﬁcant ﬂuctuations in the standard deviation of models m(1,0) and m(2,0). We hypothesized that the positive feedback loop was driving the early sensitivity to hyper-osmotic stress

2 for model m(1,0). We additionally hypothesized that positive and negative feedback operated on the same time scale under these physiological conditons. To test these hypotheses, we again conducted Time-Dependent Sensitivity Analysis on a characteristic MAPKA time series for each model.

2 Consistent with our hypothesis, we observed that model m(1,0) displayed the earliest sensitivity to positive feedback under these continously increasing hyper-osmotic stress conditions (Fig. 1.5B, row

1, right inset). Interestingly, we noted that this early sensitivity to positive feedback was coupled with a corresponding early sensitivity to negative feedback. We then visualized the sensitivity to

2 2 positive and negative feedback for models m(2,0) and m(3,0). Although these models did not display early sensitivity to positive feedback, the later sensitivity to positive feedback was also coupled with simultaneous sensitivity to negative feedback.

These results suggested that the temporal separation of the feedback network was dependent on the dynamics of the input signal, and that only discrete increases in external osmolarity would result in a temporally separated feedback network. To test this hypothesis, we again used our robotic control system (GradStudent Dial-A-Wave v3, gravity-flow system) to continuously increase the KCl concentration in the microfluidic chamber from 0mM KCl to 550mM KCl at a rate of 5.5mM KCl per minute. We then measured nuclear localization of a Hog1-GFP (green fluorescent protein) construct at each minute of the time series. Consistent with our model predictions, we observed that Hog1-GFP accumulated in the nucleus within ten minutes after the initial exposure to hyper-osmotic stress, (Fig.

1.5D). Furthermore, we observed that while the overall proportion of Hog1-GFP localized to the nucleus increased with the hyper-osmotic stress concentration, the time series appeared to oscillate about the mean response. Consistent with our model predictions, when the hyper-osmotic stress ceased to increase 100 minutes after the initial exposure, the proportion of Hog1-GFP localized to the nucleus also ceased to increase. Thus this live-cell microscopy experiment conﬁrmed that model

2 m(1,0) had accurately predicted the response of Hog1 to physiological changes in hyper-osmolarity.

28 ) 350 600 A 450 B mM

( 150 300 150

KCl 0 0 0 20 40 60 80 100 0 20 40 60 80 100 120

Hyper-Osmotic m10 m10 Stress 100 100 40 SS0$3. $ 30 MAP3K 80 80 20 (%) 10 60 60 0 0 5 10 15 20 25 30 MAP2K

Hog1 40 40 MAPK - A 20 20 pp 0 0 X* MAPK B 0 20 40 60 80 100 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes)

Hyper-Osmotic m20 m20

Stress 100 100 40 30 MAP3K 80 80 20 (%) 10 0 60 60 0 5 10 15 20 25 30 MAP2K

Hog1 40 40 - MAPK A 20 20 pp 0 0 MAPK X* B 0 20 40 60 80 100 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes)

Hyper-Osmotic m30 m30

Stress 100 100 40 30 MAP3K 80 80 20 (%) 10 60 60 0 MAP2K 0 5 10 15 20 25 30

Hog1 40 40 - MAPK A 20 20 pp 0 0 MAPK X* B 0 20 40 60 80 100 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) ' (

7RWDO 1XFOHXV

+RJ 7LPH 0LQXWHV 7LPH 0LQXWHV

2 Figure 1.5: Model m(1,0) is most consistent with novel experimental studies

2 2 2 (A) MAPK phosphorylation for models m(1,0) (row 1) and m(2,0) (row 2), and m(3,0) (row 3) is shown when simulated using a step-wise increasing input function “s(t)”. (B) Same as A, except the models are simulated using a continuously increasing input function “s(t)”. (B, insets) Time-Dependent Sensitive Analysis is performed for a characteristic MAPKA time series for each model. (C) Experimentally generated Hog1-GFP nuclear translocation time series. Yeast were exposed to step-wise increases in the concentration of KCl. Each step increased the concentration of KCl in the microﬂudic chamber by 70mM. (D) Experimentally generated Hog1-GFP nuclear translocation time series. Yeast were exposed to hyper-osmotic stress that continuously increased from 0mM KCl to 550mM KCl at a rate of 5.5mM per minute. (Error bars are calculated using the Standard Error of the Mean from all single cell measurements.)

29 1.3 Discussion

In this work we used an integrated quantitative biology platform to characterize the impact of positive and negative feedback on the Hog1 MAPK. We ﬁrst used a modifed Approximate

Bayesian Computation - Sequential Monte Carlo algorithm to identify feedback networks that could explain Hog1 phosphorylation in the presence and absence of Hog1 activity. Our algorithm ﬁrst suggested that two features were needed to explain the Hog1 phosphorylation dyanmics : 1) Positive feedback, and 2) a delayed negative feedback that serves a mechanisms to mitigate the inﬂuence of hyper-osmotic stress environments over time.

Given that we identified multiple models that satisfied the above criteria, our next task was to determine which of the models could be predict the response of Hog1 to novel hyper-osmotic stress conditions. To this end we implemented an ABC-Model Selection algorithm that was able to rank each model, based on its “evidence” in support of the training data. We then divided the models into two classes, and further analyzed the simpler of the two groups. We then showed that the influence of positive feedback on Hog1 phosphorylation was dependent on both location of the positive feedback loop and the dynamics of the hyper-osmotic stress environment. Using the training data, along with simulated mutational analysis, we were able to determine that positive feedback had the greatest impact on Hog1 phosphorylation when it targeted the MAPK directly. Furthermore using the models as a guide, we designed a novel set of dynamic hyper-osmotic stress conditions to determine that the positive feedback most-likely targets the MAPK directly.

To confirm our model selection, we tested the model by using it to predict the response of Hog1 to these experimental conditions. We then showed that not only did our consensus model align with the newly generated Hog1 phosphorylation data. The model additionally predicted that the response of Hog1 would be significantly more variable under physiological hyper-osmotic stress conditions. In summary, using our quantitative biology platform, we were able to: 1) identify a set of mathematical models that could explain Hog1 phosphorylation dynamics in wild type yeast, and in yeast in which the activity of dual-phosphorylated Hog1 had been inhibited via mutation (Hog1-inhibited), 2) use an objective model selection algorithm to rank the favorability of each model, 3) confirm our model selection by determining the ability of each ranked model to predict the response of Hog1 to novel hyper-osmotic stress conditions, 4) execute a subset of the novel experimental conditions using

30 live-cell microscopy and our microfluidic chamber to confirm that our objective model selection algorithm had indeed selected the consensus model from among the candidate model set, 5) confirm that our consensus mathematical model had accurately predicted that variability of the Hog1 response was partially encoded by the hyper-osmotic stress environment, and by the intrinsic relationships between the strength and temporal dynamics of positive and negative feedback.

Mitogen-Activate Protein Kinases are the master regulators of many cellular response programs.

Perhaps the most-intriguing of all our ﬁndings relates the dynamics of the input signal to the variability in the pathway. This result is particularly striking when we consider that many chemotherapies and anti-cancer treatments target MAPK pathways. Given that we have shown that the dynamics of the input signal (hyper-osmotic stress) affects the uniformity of the MAPK response, we propose that the key to optimizing anti-cancer therapies for individual patients may be to optimize the dynamics of the anti-cancer treatment to maximize the affect on the diverse population of cells and cell-type involve in the malignacy. While this is undoubtedly a daunting task, there is a large pool of failed anti-cancer therapies that initially showed promising results during early testing. Perhaps the utility of these agents could be realized, if one considers the dynamics of their administration in the clinical environment.

1.4 Materials and Methods

1.4.1 Model design and equations

All models were represented as systems of ordinary differential equations (ODEs) that described the kinetic behavior of components of a generic MAPK cascade (MAP3K, MAP2K, MAPK). We assumed that prior to experiencing osmotic stress each signaling component was in its inactive state, and that the activation and deactivation kinetics of each kinase followed Michaelis-Menten kinetics.

We assumed that the activation rate for the MAP3K was proportional to the amount of osmotic stress experienced by the cells. Similarly, the activation rates for the MAP2K and MAPK were proportional to the amounts of active MAP3K and MAP2K, respectively. Once activated, the MAPK activated an additional component “X” that modeled the restoration of osmotic homeostasis through a negative feedback loop that inhibited the input signal “s(t)” [57, 75, 103]. This core model is fully described by equations 1.3-1.6.

31 d[MAP 3K∗] k s(t) (MAP 3K − [MAP 3K∗]) = 1 T otal [X∗] ∗ dt (1 + ) K1M + (MAP 3KT otal − [MAP 3K ]) βs (1.3)

∗ k2[MAP 3K ] − ∗ K2M + [MAP 3K ]

∗ ∗ ∗ ∗ d[MAP 2K ] k3[MAP 3K ](MAP 2KT otal − [MAP 2K ]) k4[MAP 2K ] = ∗ − ∗ (1.4) dt K3M + (MAP 2KT otal − [MAP 2K ]) K4M + [MAP 2K ]

∗ ∗ ∗ ∗ d[MAP K ] k5[MAP 2K ](MAP KT otal − [MAP K ]) k6[MAP K ] = ∗ − ∗ (1.5) dt K5M + (MAP KT otal − [MAP K ]) K6M + [MAP K ]

∗ ∗ ∗ ∗ d[X ] k7[MAP K ](XT otal − [X ]) k8[X ] = ∗ − ∗ (1.6) dt K7M + (XT otal − [X ]) K8M + [X ]

1.4.2 Models M1 - positive and negative feedback acting on MAPK components

1 To create a given model m(i,j), we assumed that positive feedback loops increased the vmax of the activation rates and negative feedback loops increased the vmax of the deactivation rates. For

1 example, the equation that simulated the MAP2K in model m(2,2) became

∗ ∗ ∗ ∗ d[MAP 2K ] (k3[MAP 3K ] + α[MAP K ])(MAP 2KT otal − [MAP 2K ]) = ∗ dt K3M + (MAP 2KT otal − [MAP 2K ]) (1.7) ∗ ∗ (k4 + β[X ])[MAP 2K ] − ∗ , K4M + [MAP 2K ] where α and β are proportionality constants that determined the magnitude of positive and negative feedback ampliﬁcation.

32 1.4.3 Models M2 - MAPK translocation

2 To create a given model m(i,j), we replaced equations (1.5 and 1.6) with equations (1.8, 1.9, and

1.10) below. We then again assumed that positive feedback loops increased the vmax of the activation rates and negative feedback loops increased the vmax of the deactivation rates. Then any M2 model could be constructed from model equations (1.3, 1.4, 1.8, 1.9, and 1.10).

∗ ∗ ∗ ∗ d[MAP KA] k5[MAP 2K ](MAP KT otal − [MAP KA] − [MAP KB]) = ∗ ∗ dt K5M + (MAP KT otal − [MAP KA] − [MAP KB]) (1.8) ∗ k6[MAP KA] ∗ ∗ − ∗ − kn[MAP KA] + kc[MAP KB] K6M + [MAP KA]

d[MAP K∗ ] B = k [MAP K∗ ] − k [MAP K∗ ] (1.9) dt n A c B

∗ ∗ ∗ ∗ d[X ] k7[MAP KB](XT otal − [X ]) k8[X ] = ∗ − ∗ (1.10) dt K7M + (XT otal − [X ]) K8M + [X ]

1.4.4 Fitting models to experimental data

The successful fitting of a mathematical model to a set of training data implicitly depends on the appropriateness of the model for the data, and parametrically depends on the identification of model parameters that cause the model simulations to align with the training data. Here I present the modified version of the Approximate Bayesian Computation-Sequential Monte Carlo (ABC-SMC) parameter estimation algorithm that was used to identify mathematical models and parameter vectors that could minimize the sum of squared deviations (SSD) between a simulated and training data set.

The original algorithm was designed by Toni et al. [108]. Full details of their algorithm and the theory behind ABC-SMC algorithms can be found in [70, 107, 108].

33 1.4.4.1 An ABC-SMC algorithm for parameter estimation

Consistent with all ABC-SMC algorithms, our modified ABC-SMC parameter estimation algorithm filtered an initial set of parameter vectors for each model through a series of intermediate posterior parameter distributions [108]. The first posterior distribution required the model to only marginally explain the training data, while the last distribution required strict agreement with the training data. To guide parameter estimation, we used the experimental Hog1 phosphorylation training data sets to develop a “fitting criteria schedule” that consisted of a discrete set of SSD threshold values to define each posterior distribution. Each SSD threshold value was determined using a random sampling algorithm (described below), which simulated a pre-determined amount of artificial variability in each time point, of each experimental Hog1 phosphorylation time series. For brevity, I describe the schedule generating algorithm for one time series, however it is straighforward to extend the algorithm to include all experimental data sets.

2 To construct the fitting criteria schedule, we first computed the mean (µj) and variance (σj ) of each data point in the time series, from the three biological replicate experiments. We then defined a normal distribution for each data point, that was centered on the mean value, with the experimental

2 variance, N(µj, σj ). To generate the first element of the fitting schedule, we scaled the variance of the normal distribution for each data point by a factor “α”, and then created an “artificial” normal

2 2 distribution for each data point, N(µj, σj ± σj × α), where α was initially equal to .95. We then 2 2 drew a random variate from each N(µj, σj ± σj × α), and stored the collection of random variates in an “artiﬁcial” time series. Next, we computed a SSD value between the artiﬁcial and corresponding experimental time series. We then repeated this process until 10,000 SSD values had been generated.

We then set the first element of the fitting criteria schedule to be equal to the mean of the 10,000 simulated SSD values. To generate the remaining elements of the fitting schedule, we reduced α according to an exponentially decaying function, such that the decrease in consecutive α values was greater for the early schedule elements than the later schedule elements. In total, we created eleven fitting schedule elements, where α for the last element of the fitting criteria schedule scaled the variance by approximately 18%.

Our ABC-SMC parameter estimation algorithm was divided into three phases. The ﬁrst phase utilized an ABC-Rejection Sampler to identify an initial set of “N” parameter vectors θm, (a single

34 vector of model speciﬁc rate parameters), for each model m. To initialize the ABC-Rejection

Sampler for model m, we deﬁned a prior distribution, P (θm), for every rate parameter in model m. To construct a parameter vector for model P (θm), we sampled a random variate from the prior distribution of every parameter in m. We then used the model, given the parameter vector, to simulate the Hog1 phosphorylation training data under wild type and Hog1-inhibited conditions. To determine whether or not the parameter vector would be accepted, we used a “squared euclidean distance” function to compare the simulated MAPK phosphorylation time series, D∗, with the corresponding

∗ ∗ experimental time series D0, d(D0,D ). If d(D0,D ) ≤ SSD1 (the ﬁrst SSD threshold value), the parameter vector θm and its SSD value were stored as the posterior distribution of model m

SSD ∗ ∗ P (θm |d(D0,D ) ≤ SSD1). If d(D0,D ) > SSD1, the parameter vector θm was rejected. We stipulated that in order to advance to phase two of our ABC-SMC algorithm, each model was required to identify “N” parameter vectors that generated SSD values ≤ SSD1, where N = 500. We allowed each model to continue ABC-Rejection Sampling until the 500 required parameter vectors were identiﬁed, or until 10 million parameter vectors were sampled. If the model failed to identify the required number of parameter vectors after 10 million parameter vectors were sampled, then we updated the initial prior distribution of each parameter in the model, and conducted a fresh round of

ABC-Rejection Sampling. If the model failed to identify the required number of parameters after the second round of ABC-Rejection Sampling, the model was discarded.

The second phase of our ABC-SMC parameter estimation algorithm consisted of “ﬁltering” the

SSD ∗ parameter vectors identified by the ABC-Rejection Sampler, P (θm |d(D0,D ) ≤ SSD1), through SSD ∗ a series of K intermediate posterior distributions, P (θm |d(D0,D ) ≤ SSD2),..., SSD ∗ P (θm |d(D0,D ) ≤ SSDK ), defined by the fitting criteria schedule [108]. To filter the parameter SSD ∗ vectors, we used each parameter vector stored in P (θm |d(D0,D ) ≤ SSDk−1) as the starting point of an “n-step” ABC-Markov Chain Monte Carlo (ABC-MCMC) simulation that was used to guide the parameter vector towards optimal regions of parameter space. Each ABC-MCMC simulation was guided by a Metropolis-Hastings acceptance criteria [40, 72]. The Metropolis-

Hastings acceptance criteria stipulates that a proposed random step will be accepted with a probability

P = 1, if SSD∗, the SSD value generated by the proposed step is less than the SSD value of the previous step, (SSD∗ ≤ SSD∗−1). Alternatively, the random step will be accepted with a probability

0 < P < 1, if SSD∗ ≥ SSD∗−1. By stipulating that the “bad” steps in a MCMC simulation have

35 some probability P of being accepted, the Metropolis-Hastings acceptance criteria provides a mechanism to allow a MCMC simulation to avoid permanent entrapment in sub-optimal regions of parameter space. If the Markov Chain was able to identify parameter vectors that could meet or surpass the threshold value deﬁned by SSDk, we again collected the best scoring parameter vector from the chain to be used as a starting point of an ABC-MCMC simulation for the next ﬁtting

SSD ∗ schedule element, P (θm |d(D0,D ) ≤ SSDK ). Furthermore, consistent with our ABC-Rejection Sampling criteria, we again stipulated that a model could only advance to the next ﬁtting criteria schedule if it was able to identify 500 parameter vectors that met the current SSD threshold value,

SSD ∗ P (θm |d(D0,D ) ≤ SSDk). Although the Metropolis-Hastings criteria allowed a MCMC simulation to escape sub-optimal parameter space regions, the probability of escape was partially dependent on the number of remaining steps in the MCMC simulation. Given that our ABC-SMC parameter estimation algorithm stipulated that each parameter vector was only allowed to take “n-steps”, we provided an emergency mechanism to increase the probability that each ABC-MCMC would avoid getting permanently trapped in sub-optimal parameter space regions. Our emergency mechanism was to simply resample parameter vectors that became “trapped” in sub-optimal parameter space regions. To differentiate between the “free” and trapped parameter vectors, we created two temporary posterior distributions,.

Parameter vectors that were able to meet the SSDk during the ABC-MCMC simulation were stored

SSD ∗ in P (θm |d(D0,D ) ≤ SSDk), while parameter vectors that failed to meet SSDk during the ABC- SSD ∗ MCMC simulation were stored in P (θm |d(D0,D ) ≥ SSDk) [107]. After all parameter vectors SSD ∗ had been sample once, we resampled the failed parameter vectors stored in P (θm |d(D0,D ) ≥ SSD ∗ SSDk) from the starting point of the parameter as deﬁned by P (θm |d(D0,D ) ≤ SSDk−1). SSD ∗ The parameter vector was added to P (θm |d(D0,D ) ≤ SSDk) if it was able to meet SSDk during the second ABC-MCMC simulation. However, we again added the parameter vector to

SSD ∗ P (θm |d(D0,D ) ≥ SSDk) if it failed to reach the SSDk threshold value after the second ABC-MCMC simulation. After every initially trapped parameter vector was resampled, we dis-

SSD ∗ carded all parameters that were stored in P (θm |d(D0,D ) ≥ SSDk). We then resampled the SSD ∗ parameter vectors stored in P (θm |d(D0,D ) ≤ SSDk) from the parameter vectors deﬁned SSD ∗ by P (θm |d(D0,D ) ≤ SSDk−1). We also stipulated that any parameter vector that failed to meet the SSDk threshold value after the second round of ABC-MCMC simulations was stored in

36 SSD ∗ P (θm |d(D0,D ) ≥ SSDk), and then discarded. This resampling strategy continued until the 500 SSD ∗ required parameter vectors P (θm |d(D0,D ) ≤ SSDk) were identiﬁed, or until all parameter vectors had been discarded (model failure). This parameter estimation strategy allowed us to mitigate the computational costs of over-sampling from bad regions of parameter space and bad models, while simultaneously maximizing the coverage of good parameter space regions for viable models. If a model

SSD ∗ was able to identify the 500 required parameter vectors, we updated P (θm |d(D0,D ) ≤ SSDk), and progressed to the next round of ABC-MCMC parameter estimation for SSDk+1. Our ABC-Parameter Estimation algorithm assumed that each parameter vector was “equally and minimally-optimized” after the conclusion of each round of parameter estimation. Therefore we could assume that the prior probabilities of selecting each model, given the “equally and minimally- optimized” set of parameter vectors, were uniform. Under this assumption, computing a Bayes factor for each model reduced to taking the ratio of the number of parameter vectors that met the SSDk threshold value, after each round of parameter estimation.

P (m1|x) number of parameter vectors for m1 B1,2 = = . (1.11) P (m2|x) number of parameter vectors for m2

1.4.4.2 Ranking the candidate models

Model selection in systems biology is challenging. Models are often over-parameterized and the experimental data used to train models is often insufﬁcient to distinguish competing regulatory mechanisms. As a result, many mathematical models that accurately reproduce training data sets fail to predict the system’s response to novel experimental conditions. Alternatively if multiple models are sufﬁcient to reproduce the training data sets, an important task is to rank the candidate models, so that experiments capable of distinguishing competing models can be designed and prioritized.

Our ABC-SMC parameter estimation algorithm was designed to maximize the coverage of optimal parameter space regions, while minimizing the time spent sampling from sub-optimal parameter space regions. This stipulation allowed us to limit the number of steps each candidate parameter vector could take when searching for minimally optimal parameter space regions. By limiting each parameter vector’s random walk, our ABC-SMC parameter estimation algorithm was able to explore multiple parameter space regions and identify a diverse set of “minimally-optimized”

37 parameter vectors. However the breadth of coverage came at the cost of in-depth explorations of each minimally optimized parameter space region. To increase the quality of each model’s parameter vectors, we reduced the SSD threshold value to further restrict the allowable deviation between the simulated and training data. We also increased the number of steps that each parameter vector could take during the random walk. We reasoned that because the parameter vectors were starting each random walk in a minimally optimal parameter space region, the increase in the number of steps in the random walk would be sufﬁcient to determine whether each parameter vector could be further optimized. Therefore we also eliminated resampling stalled parameter vectors. We then used these updated ABC-SMC parameter estimation conditions to update the likelihood of each top-performing candidate model by further ﬁtting the candidate model to the training data for model selection.

1.4.5 Strain genotype and cell growth conditions

All live-cell ﬂuorescence microscopy experiments were conducted with S. cerevisiae (yeast) strain BY4741-207 in which the loci encoding Hog1 and Nrd1 are replaced with Hog1-GFP-

HIS3MX6 and Nrd1-mCherry-hphNT1, respectively [4]. MAT a, his3∆1, leu2∆0, met15∆0, ura3∆0,HOG1 − GF P − HIS3MX6,NRD1 − mCherry − hphNT 1. Cells were grown overnight in liquid synthetic complete medium (SCM), and then diluted to 5.0 ∗ 105cells/ml.

Cells were then allowed to complete two divisions so that the loading density was between 2.0 ∗

106cells/ml and 3.0 ∗ 106cells/ml.

Strain Description Source

BY4741-207 MATa-HOG1-GFP-HIS3MX6::NRD1-mCherry-hphNT1 [4]

Table 1.3: Genotype of the yeast strain used for live-cell microscopy

1.4.6 Live-cell imaging and microﬂuidics

Live-cell imaging experiments were conducted in a previously described microﬂuidics device

[39]. Media ﬂow into the microﬂuidics chamber was gravity controlled and determined by the height of four coupled 5 mL input syringes. The four coupled syringes were paired such that there was a pair of syringes with SCM media and a pair of syringes with SCM + KCl + Cascade Blue dye.

38 A Dial-A-Wave (GradStudent DAW v3) robotic system controlled the height of each syringe pair such that the SCM + KCl + Cascade Blue dye media flowed into the microfluidics chamber in a programmable fashion. The microfluidics device was mounted on a Prior automatic stage controlled using MetaMorph software (Molecular Devices, Sunnyvale, CA). Or alternatively mounted on a

Nikon motorized stage controlled using NIS Elements (Advanced Research) Microscope Imaging

Software. All images were acquired using a Nikon Eclipse (Ti-E Eclipse) Inverted Microscope with ﬂuorescence excitation provided by a X-Cite XLED1 LED light source. The GFP (488-nm) and mCherry (594-nm) ﬂuorescence channels were imaged using 100ms of exposure at one-minute intervals. Differential Inference Contrast (DIC) and DAPI (461-nm) images were acquired using

20ms of exposure at one-minute intervals.

Image registration was performed using ImageJ (National Institute of Health, Bethesda, MD) [97].

DIC images were registered ﬁrst via the “Descriptor-based series registration (2d/3d+t)” algorithm.

All ﬂuorescence images were then aligned to the registered DIC images. Image segmentation was then performed using Wolfram Mathematica.

1.4.6.1 Segmenting DIC images using a local adaptive binarization algorithm

Our task was to measure the ratio of nuclear localized Hog1-GFP versus total cell Hog1-GFP at each time-point in a multi-channel, (DIC, GFP, mCherry, DAPI), live-cell microscopy experiment. In order to accomplish this goal, we needed to create a “mask” (binary image) of each nucleus and cell in an image. The creation of a nucleus mask for each cell was straightforward. The Nrd1-mCherry signal intensity was localized to a speciﬁc region within the cell (nucleus), and was reliably more intense than the background signal. Thus to create a mask of the nucleus required only a simple binarization algorithm that excluded pixel groupings with a mean mCherry signal intensity below a threshold value.

Alternatively, creating a mask of each cell was not straightforward. The Hog1-GFP intensity was initially evenly distributed throughout the cell. However the Hog1-GFP intensity became more localized (at the nucleus) when the media in the microﬂuidics chamber was switched from SCM to

SCM + KCl. As a result the Hog1-GFP intensity could not be relied on to create a mask of the total cell. In anticipation of this challenge, we collected a DIC image at each time point. The DIC images had a number of distinctive features that suggested that they would be ideal for cell segmentation. In

39 particular, the image of each cell appeared three-dimensional, see Fig. 1.6 below for an example of a cell captured by DIC. This phenomenon was caused by the visual contrast between the dark edge of each cell and the transparency of the cell interior. The second advantage was that the size of each cell mask was not dependent on an internal marker with an intracellular distribution that was impacted by the experimental conditions.

Figure 1.6: Yeast cells as captured by DIC images

Three cells as captured by DIC are shown at the indicated time points of a time-course experiment.

Although the darkened edges of each cell were easy to distinguish by eye, cells in DIC images could not be reliably segmented with simple binarization algorithms. Traditional binarization algorithms rely on numeric differences in signal intensity to distinguish a region of interest (ROI) from the image background. Figures 1.7 demonstrates that the cell interior in the DIC image could not be fully distinguished from the image background with a simple binarization algorithm. As a result, it was impossible to obtain an accurate measure of the size of each cell.

40 Figure 1.7: DIC images of yeast cells segmented with simple binarization algorithm

Row 1) Three cells as captured by DIC are shown at three time points. Row 2) A simple binarization algorithm failed to identify each cell at the ﬁrst time point. The same algorithm and parameters over-segemented each cell at later time points.

Figure 1.7 also demonstrated a second problem with creating cell masks from DIC images. DIC images are generated from the interference pattern of two separate images captured with transmitted light. The interference pattern was highly sensitive to the environmental conditions of the experiment.

Any change in the environmental conditions during a time-course experiment was likely to be reﬂected in DIC images captured at different time points. While the source of the environmental change in the experiment shown in Fig. 1.7 was unknown, the altered environmental conditions were reﬂected in the background intensity of panels (b, d, and f). As a result the segmentation parameters used to distinguish the cells from the image background in Fig. 1.7 panel (d), completely eliminated the image background and a portion of each cell in Fig. 1.7 panels (d and f). These results demonstrated that a simple binarization algorithm would not be appropriate to generate a mask of each cell captured by DIC in a time course experiment.

41 One of the challenges of using DIC images to create cell mask was caused by non-uniformity in the image background intensity. Simple binarization algorithms use a ﬁxed threshold value for all pixels in the image, and are therefore unlikely to identify all ROIs if there are signiﬁcant non- uniformities in the image background. This resulted in only partial segmentation of each cell when segmented with simple binarization algorithms in Fig. 1.7. Alternatively, adaptive thresholding algorithms utilize local information about the intensity of each pixel when compared to its pixel

“neighborhood” to dynamically set a threshold value across the different intensity regions of the image. The neighborhood in this sense refers to a set number of pixels in proximity to the pixel of interest, and can be deﬁned in one, two, or three dimensions.

Because the edge of each cell in a DIC image was signiﬁcantly darker than the image background and cell interior, we hypothesized that an adaptive thresholding algorithm could be utilized to create a mask of the edge of each cell. We then conﬁrmed this hypothesis by using the built in “LocalAdaptiveThreshold” function to create a mask of the darkened edge of each cell for all time points in the previously described time course experiment. Figure 1.8 demonstrates that the

LocalAdaptiveBinarize algorithm was able to reliably create masks of the edge of each cell over the duration of the time-course experiment. However, as is shown in panel (f), the algorithm was also susceptible to over-segmentation of darkened regions within the cells. This over-segmentation occured frequently in cells with large and distinguishable vacoules. Thus the ﬁnal step in image segmentation was to manually remove these interior segmented regions from each cell edge mask.

42 Figure 1.8: DIC images of yeast cells segmented with the “LocalAdaptiveBinarize” algorithm

Row 1) Three cells as captured by DIC are shown at three time points. Row 2) The mask of the edge of each cell was created using the LocalAdaptiveBinarize algorithm at the three time points shown in Row 1. The algorithm was also able to differentiate and segment the vacuole of the bottom cell in panels (e and f).

1.4.6.2 Watershedding cells, and aligning nuclei to cells

Cells and nuclei were aligned using Wolfram Mathematica version 11 via a workflow that is fully described in figure 1.9. The first step of the workflow was to use a watershedding algorithm to create a masked image of the total cell from the cell edgemask created during image segmentation.

Step 2 required an “anchor point” to be set for each cell (white disks). Then the distance between each anchor point and an arbitrary pixel located at (0,0) (red disk), in the image was computed. Each anchor was then assigned an integer tag based on proximity to the arbitrary pixel. Next the centroid of each cell was determined using a built-in function. The distance between the centroid of each cell and the anchor point was then computed using a distance metric. Each cell was labeled with the integer tag of the anchor point in closest proximity to the centroid of the cell. In step 3 the nuclei

43 masks were imported, and the centroid of each nuclear mask was computed using a built-in function.

The distances between the centroid of each nuclei and the centroid of the newly labeled cells were then computed. In step 4 each nucleus was then labeled with the integer tag of the cell with centroid in closest proximity to the centroid of the nucleus.

Figure 1.9: Workﬂow for aligning nuclei to cells

1) A watershedding algorithm was used to create a mask of each cell. 2) Each cell was then tagged with the integer identiﬁer and the closest anchor point. 3) The distance from the nuclei to the cell tags were computed. 4) The nuclei were assigne the tag of the cell in closest proximity.

1.4.6.3 Measuring Hog1-GFP intensities and cell size

Hog1-GFP intensities were measured for each cell and nucleus at every time point. The nuclear to total Hog1-GFP intensity ratio was computed for each cell by dividing the mean of the ten most intense pixels in the nucleus by the mean intensity of the corresponding cell [76]. The nuclear to total cell Hog1-GFP intensity ratio at each time point was then normalized by the mean ratio of the four time-points that preceded the introduction of KCl media. The area of each cell was computed using a built-in function. The relative change in cell size was computed by normalizing the area of each cell at each time point by the mean area of the four time-points that preceded the introduction of SCM + KCl + Cascade Blue dye.

44 CHAPTER 2 Signal inhibition by a dynamically regulated pool of monophosphorylated MAPK 1

2.1 Overview

Protein kinases regulate a broad array of cellular processes and do so through the phosphorylation of one or more sites within a given substrate. Many protein kinases are themselves regulated through multi-site phosphorylation, and the addition or removal phosphates can occur in a sequential

(processive) or a step-wise (distributive) manner. Here we measured the relative abundance of the mono- and dual-phosphorylated forms of Fus3, a member of the mitogen-activated protein kinase (MAPK) family in yeast. We found that upon activation with pheromone, a substantial proportion of Fus3 accumulates in the mono-phosphorylated state. Introduction of an additional copy of Fus3 lacking either phosphorylation site leads to dampened signaling. Conversely, cells lacking the dual specificity phosphatase (msg5∆) or that are deficient in docking to the MAPK- scaffold (Ste5ND) accumulate a greater proportion of dual-phosphorylated Fus3. The double mutant exhibits a synergistic, or “synthetic”, supersensitivity to pheromone. Finally, we present a predictive computational model, which combines MAPK scaffold and phosphatase activities, that is sufficient to account for the observed MAPK profiles. These results indicate that the mono- and dual-phosphorylated forms of the MAPK act in opposition to one another. Moreover, they reveal a new mechanism by which the MAPK scaffold acts dynamically to regulate signaling.

1This chapter was drafted as part of a publication in press int he journal Molecular Biology of the Cell. The authors are as follows: Michal J. Nagiec, Patrick C. McCarter, Joshua B. Kelley, Gauri Dixit, Timothy C. Elston, Henrik G. Dohlman

45 2.2 Introduction

All cells must detect, interpret, and respond to a broad array of environmental signals. Most signal transduction pathways depend on the phosphorylation of cellular proteins by protein kinases.

Among the best known are the mitogen activated protein kinases (MAPKs), which are phosphorylated and activated by a MAPK kinase (MAP2K), which are in turn phosphorylated and activated by a

MAP2K kinase (MAP3K). These cascades transmit responses to a variety of stress conditions and secreted hormones, and are conserved in organisms ranging from yeast to humans.

The budding yeast Saccharomyces cerevisiae employs a typical MAPK signaling pathway to initiate the mating response. Haploid cells of the opposite mating type (a- or α-cells) secrete peptide pheromones that bind to cell surface receptors. These receptors activate a G protein and a protein kinase cascade comprised of Ste11 (MAP3K), Ste7 (MAP2K) and Fus3 (MAPK). All three of these kinases interact with the scaffold protein Ste5, which recruits the constituent kinases to the activated G protein at the plasma membrane. In addition to Fus3, a second MAPK Kss1 is activated by Ste7, but does not interact directly with Ste5. Thus while Ste5 is required for activation of Fus3 [2, 11, 30, 56, 67], it binds poorly to Kss1 and is not required for Kss1-mediated processes [17, 56, 88]. Once activated, Fus3 goes on to phosphorylate proteins necessary for new gene transcription, cell division arrest, polarized cell expansion (chemotropic growth and shmoo formation) and cell-cell fusion (Wang and Dohlman, 2004). In the absence of Fus3, Kss1 can mediate the pheromone-dependent gene induction program [11, 96], while other responses such as chemotropic growth are abrogated [39]. Thus, while Fus3 and Kss1 are activated by the same upstream protein kinases, they have distinct cellular functions. Here our focus is on the function of

Fus3 and its scaffold Ste5.

Based on the speciﬁcity of binding, it was long assumed that Ste5 acts to direct signaling toward

Fus3 and away from Kss1. While clearly required for Fus3 activation however, Ste5 does not prevent the activation of Kss1 by pheromone [2, 11, 17, 30, 56, 88]. It was later proposed that Ste5 is an allosteric regulator of Fus3. In support of that model, a fragment of Ste5 was demonstrated to promote Fus3 auto-phosphorylation at Tyr-182 [7], one of two sites normally phosphorylated upon full activation of the kinase [33]. The mono-phosphorylated form of Fus3 was partially (20-25%) activated and competent to phosphorylate a variety of substrates in vitro [7]. If Ste5 were found

46 to activate Fus3 in vivo, even partially, it would raise a host of new questions. What proportion of Fus3 is mono-phosphorylated in this manner? How does the pool of mono-phosphorylated kinase affect downstream signaling? Does auto-phosphorylation on one site facilitate, or hinder, subsequent phosphorylation at the second site? If auto-phosphorylation on one site were followed by a dissociation event and trans-phosphorylation on the second site, the scaffold might act to delay signaling. Alternatively, if the scaffold stabilizes the interaction of Ste7 and Fus3, phosphorylation might occur in a processive fashion, without substrate dissociation and reassociation. This would have the effect of minimizing the mono-phosphorylated species and accelerating full activation of the

MAPK [27, 60]. Thus scaffold proteins could either speed or slow signal transduction, depending on the mechanism employed. Apart from altering the kinetics of MAPK activation, a processive or distributive mechanism could inﬂuence the dose-response relationship for the pathway. Theoretical and biochemical studies indicate that processive phosphorylation should favor a graded output, one that aligns MAPK activity with the graded input stimulus. Accordingly, scaffold proteins might convert an inherently switch-like MAPK to one that is more graded [27, 60]. Conversely, distributive phosphorylation is thought to produce an all-or-none (switch-like) output [45]. In support of this second model, a number of functional activity assays reveal that Ste5 is necessary for slow and ultrasensitive activation, as is observed for Fus3 but not Kss1 [39, 68]. Thus the distinction between processive and distributive phosphorylation has important implications for signal encoding [85].

Considering the importance of how MAPKs are turned on and off, relatively little is known about how this occurs in cells. For example it is still not established whether Ste5, or scaffolds in general, dictate a processive vs. distributive phosphorylation mechanism in vivo. To address this question, we employ a new method to quantify the abundance of mono- and dual-phosphorylated

Fus3, in the cell, over time, and in the presence or absence of Fus3 regulators; these regulators include the scaffold Ste5 as well as the MAPK phosphatases Ptp2, Ptp3 and Msg5. We show that

Ste5 and Msg5 in particular act to limit full activation of Fus3 and that nearly half of the Fus3 pool remains non-phosphorylated after pathway stimulation. Of the protein that is phosphorylated, nearly half is in the mono-phosphorylated state. Whereas phosphorylation appears to be processive, de-phosphorylation appears to be distributive. Finally, we show that mono-phosphorylated Fus3 does not activate, but rather impedes, the pheromone response in vivo. Together these ﬁndings

47 indicate that the mono-phosphorylated species acts in a dynamic and dominant manner to limit signal transduction.

2.3 Results

2.3.1 The phosphorylation state of Fus3 determines mating pathway output.

It has long been recognized that MAP kinases must be phosphorylated on two residues in order to achieve full catalytic activity. This was originally documented for ERK2, where the protein is phosphorylated on Tyr-185 and, subsequently, on the neighboring Thr-183 [28, 41]. This dual phosphorylation alters the conformation of the protein, thereby enabling ATP to bind to the catalytic site [13]. These “activation loop” residues are conserved in the yeast ERK2 homolog Fus3, and mass spectrometry analysis indicates the same order of events occur [47]. Our goal here was to determine the relative abundance, and potential function, of the mono-phosphorylated MAPK species.

We began by constructing genomically-integrated mutant forms of Fus3 that cannot be phosphorylated at either of the activation loop residues: Tyr-182, Thr-180 or both. We then monitored the transcription of a mating-speciﬁc gene reporter. As shown in Fig. 2.1A , cells in which the endogenous FUS3 gene had been deleted produced a maximal response to pheromone resembling the response of wild-type cells. In the absence of Fus3, pathway induction is preserved due to the action of a homologous MAPK, Kss1, as reported previously [5]. In contrast, cells expressing a variant of Fus3 that can only be mono-phosphorylated (Fus3T 180A or Fus3Y 182F ) exhibited a substantially diminished response. This reduction in signal activity is particularly striking, given the near-normal response seen in the complete absence of Fus3 expression.

To determine if the Fus3 mutants limit signaling via the primary (Fus3-mediated) pathway, we repeated these experiments, but this time in cells that co-express endogenous wild-type Fus3 and either an additional copy of the MAPK or one of the phospho-site mutants. As shown in Fig.2.1B, signaling was diminished most substantially in cells where the second copy of Fus3 was mutated

(Fus3T 180A or Fus3Y 182F ). Thus the mono-phosphorylated Fus3 dampens the mating response, and does so in a genetically dominant manner. These ﬁndings suggest that the partially phosphorylated form of Fus3 acts to inhibit signaling.

48 Figure 1

A 100 WT fus3¨ 80 fus3T180A Y182F 60 fus3 fus3T180A/Y182F 40

maximial response maximial 20 (percent of wild-type) of (percent

B 100 WT+ empty vector WT+ FUS3WT 80 WT+ fus3T180A Y182F 60 WT+ fus3 WT+ fus3T180A/Y182F 40

maximial response maximial 20 (percent of wild-type) of (percent

120 WT+ empty vector 100 WT+ FUS3WT T180A 80 WT+ fus3 WT+ fus3Y182F 60 WT+ fus3T180A/Y182F 40 dose response dose

(percent of wild-type) of (percent 20

0 0 -9 -8 -7 -6 -5 -4 [ factor], M

Figure 2.1: Signal Inhibition by mono-phosphorylated Fus3.

(A) Transcription reporter (FUS1-lacZ) activity in wild-type (WT), Fus3-deﬁcient or Fus3 activation loop mutant strains

(Fus3T 180A, Fus3Y 182F , and combined Fus3T 180A/Y 182F ) stimulated with 10 µM α-factor. (B) Same analysis in wild- type cells bearing a single copy plasmid with no insert (vector), wild-type Fus3 (2xFus3), or mutations in the Fus3 activation loop. FUS1-lacZ data are presented as a percentage of maximum activity in wild-type cells (panel A and panel

B, top) and as a full dose-response curve (panel B, bottom). ± SEM (n = 3) is reported for each data point but is not visible in every case. *, statistically signiﬁcant student t-test of pair-wise comparisons for wild-type and the individual mutants.

2.3.2 The phosphorylation state of Fus3 is dynamically regulated.

Having shown that mono-phosphorylated Fus3 inhibits the mating transcription response, we next sought to measure the relative abundance of the mono-phosphorylated protein in cells. To that end we employed the Phos-tag reagent. Phos-tag is a metal-coordinating small molecule with a high afﬁnity for phosphorylated serine, threonine, and tyrosine [53]. Addition of Phos-tag to

49 acrylamide gels slows the migration of polypeptides, and does so in proportion to the number of phosphorylations on the molecule. As shown in Fig. 2.2A, the Phos-tag method revealed three species of Fus3 (corresponding to the dual-, mono-, and non-phosphorylated forms of the protein) following stimulation with 10 µM pheromone, a dose which produces full pathway activation (see

Fig. 2.1B). As expected, the dual-phosphorylated form of Fus3 was enriched following pheromone treatment, but mono-phosphorylated Fus3 was still readily detectable. As a control, we established that the individual phospho-site mutants yielded just two bands each, corresponding to the mono- and non-phosphorylated species (2.2B). While the pool of mono-phosphorylated species was enriched in these mutants, the proportion of non-phosphorylated protein was unaffected (Fig. 2.2B). The

Fus3Y 182F mutant migrated anomalously, equivalent to that of the activated wild-type form of the protein. In each case only the slower-migrating phosphorylated species was evident using p44/p42 antibodies, raised against a phosphorylated MAPK peptide. None of the phosphorylated species were detected in the absence of the upstream MAPK kinase Ste11 (Fig. 2.2A). No bands were detected in a FUS3-deﬁcient strain (Fig. 2.2B).

Ste5 is thought to act in part by stimulating the auto-phosphorylation of Fus3 on Tyr-182

[7]. Support for this model comes from in vitro studies using a peptide fragment of Ste5 [7].

To determine if Fus3 is also auto-phosphorylated in vivo, we compared the mobility of wild- type Fus3 and a catalytically inactive form of the protein (Fus3K42R). As shown in Fig. 2.2C,

Fus3K54R is mono-phosphorylated despite the lack of catalytic activity; as with the wild-type protein, the mono-phosphorylated species was evident in unstimulated conditions, and increased further upon pheromone stimulation. Thus auto-phosphorylation is not the only means by which Fus3 becomes mono-phosphorylated and activated, at least in vivo. To determine the contribution of the scaffold Ste5, we monitored Fus3 in the absence of Ste5-Fus3 interaction. Deletion of STE5 blocks signaling altogether, so as an alternative we used a mutant form of Ste5 in which Fus3 docking is disrupted (“nondocking” allele, Ste5ND). This mutant binds poorly to Fus3 [7, 65] yet produces a transcription-induction response that is actually enhanced [7]. Once again we were able to detect mono-phosphorylated Fus3, and at levels comparable to that of wild-type (STE5+) cells. However the relative abundance of dual-phosphorylated Fus3 was increased (≈ 50%) in the Ste5ND strain, as compared to wild-type (Fig. 2.2C). This conﬁrms previous observations that Ste5 does not activate, but rather limits the full activation of Fus3 in vivo [7, 39]. We infer that Ste5 acts in two opposing

50 ways to regulate Fus3 activity. First, Ste5 binds to Fus3 indirectly via Ste7, thereby promoting activation. This is based on the observation that Fus3 binds poorly to Ste5ND but is nevertheless activated by Ste7. Second, Ste5 binds to Fus3 directly and inhibits activation. This is based on the enhanced signaling observed in Ste5ND cells.

While Fus3 is phosphorylated by Ste7, Fus3 is also regulated by protein phosphatases. Ac- cordingly, the cellular pool of mono-phosphorylated Fus3 is likely to be regulated through the combined action of kinases and phosphatases. To test this we examined Fus3 in a panel of mutants lacking one or more of the known MAPK phosphatases: Ptp2 and Ptp3, which are Tyr-specific phosphatases, and Msg5, which is a dual-specificity (Tyr; Ser or Thr) phosphatase [20, 94, 118]. As shown in Fig. 2.2D, cells lacking the dual-specificity phosphatase (msg5∆) exhibited a decrease in mono-phosphorylated Fus3. Cells lacking the Tyr-specific phosphatases exhibited an increase in basal levels of dual-phosphorylated Fus3, but there was little or no effect of these mutants in pheromone-stimulated cells (Fig. 2.2D). These data suggest that Fus3 is dephosphorylated by Msg5 to enrich the cellular pool of mono-phosphorylated Fus3.

51 Figure 2

WT ste11 ∆ A B C ppFus3 0 215 0 2 15 min 100

12% PAGE pFus3 Fus3 3∆ 80 npFus3 WT T180A Y182F T180A/Y182Ffus IB:Fus3 60 ppKss1 ppFus3 ppFus3 pFus3 40 IB:phospho-p44/p42 50 npFus3 10% PAGE 20 ฀

ppFus3 M IB:Fus3

Fus3 (% total of lane) phostag 0 pFus3 0’ 15’ 0’ 15’ 0’ 15’ ppKss1 KR ND npFus3 ppFus3 WT Fus3 Ste5 pFus3 IB:Fus3 ppFus3 IB:phospho-p44/p42 G6PDH G6PDH pFus3 npFus3

D ptp 2∆ ptp3 ∆ WT ste7 ∆ ptp2 ∆ ptp3 ∆ msg5∆ msg5∆ 0 15 0 15 0 15 0 15 0 15 0 15 min

ppFus3 pFus3 npFus3

G6PDH

Figure 2.2: Analysis of differentially phosphorylated forms of Fus3.

(A) Wild-type or ste11∆ (MAPKK) mutant cells untreated (0) or treated for 2 or 15 min with 10 µM α-factor were lysed and resolved by immunoblotting after SDS-PAGE and probed with Fus3 antibodies (top), p44/p42 antibodies (top middle), with Phos-tag reagent and probed with Fus3 antibodies (bottom middle), or G6PDH load control antibodies (bottom).

Bands represent the dual-phosphorylated (ppKss1, ppFus3), mono-phosphorylated (pFus3), non-phosphorylated (npFus3), and total (Fus3) protein. Except for the topmost panel, this and all subsequent experiments were done using Phos-tag.

(B) Wild-type, Fus3 activation loop (T180A, Y182F) and fus3∆ mutants were resolved by Phos-tag SDS-PAGE and immunoblotting with Fus3 antibodies (top), p44/p42 antibodies (middle), or G6PDH load control antibodies (bottom).

Arrowheads indicate the bands of interest (i.e. those recognized by p44/42 antibodies). (C) Wild-type, fus3KR (catalytically inactive) and ste5ND (non-docking) mutants, untreated or treated for 15 min with 10 µM α-factor were resolved by

Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Representative data are shown below. Band intensity was quantiﬁed as a percentage of total Fus3 in each lane, ± SEM (n ≥ 3). (D) Wild-type, ste7∆ and phosphatase-deﬁcient ptp2∆, ptp3∆, and msg5∆ mutants, alone or in combination, were resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies (top) or G6PDH load control antibodies (bottom).

52 2.3.3 Ste5 and Msg5 regulate the kinetics of Fus3 phosphorylation.

To expand our analysis we next examined how the relative abundance of the mono- and dual- phosphorylated forms of Fus3 change over time. As shown in Fig. 2.3A, pheromone-activated cells exhibited a substantial (≈ 60%) drop in non-phosphorylated kinase within 5 min, accompanied by a corresponding increase in both the mono-phosphorylated and dual-phosphorylated forms of the protein. While the pool of dual-phosphorylated Fus3 rose rapidly and showed an initial peak at approximately 2-3 min, the pool of mono-phosphorylated Fus3 rose in a more gradual manner. Most strikingly, the relative abundance of mono- and dual-phosphorylated Fus3 was nearly equivalent following the initial peak of activation (from 5 to 60 min). Taken together, these data establish that a substantial proportion of Fus3 exists in the mono-phosphorylated state. Together with the data presented above (Fig. 2.1), these ﬁndings suggest that stimulated cells produce a combination of Fus3 that is fully phosphorylated (activated), mono-phosphorylated and non-phosphorylated (inhibitory).

To further establish the long-term consequences of differential phosphorylation, we monitored

Fus3 in cells lacking the protein phosphatases or that express mutant forms of the scaffold Ste5.

As noted above, Ste5ND cannot dock to Fus3, but nevertheless allows activation of Fus3 by the upstream kinase Ste7. As compared to wild-type, the Ste5ND mutant cells exhibited higher overall accumulation of dual-phosphorylated Fus3 (Fig. 2.3B). In addition we tested another variant of the scaffold (Ste5FB), one that cannot undergo feedback phosphorylation and binds poorly to

Fus3 [68]. As with the Ste5ND mutant, the Ste5FB mutant exhibited higher accumulation of dual- phosphorylated Fus3 (Fig. 2.3B), and this accumulation occurred more rapidly than in either the wild-type or Ste5ND strains. Thus binding to Ste5, or feedback phosphorylation of Ste5, limits the activation of Fus3 [7, 68]. When binding or feedback is blocked, a greater proportion of Fus3 is fully activated. Given our goal of understanding the scaffolding function, all subsequent experiments were done with Ste5ND.

There are at least two ways by which Ste5 could impede Fus3 activation. One possibility is that Ste5 slows phosphorylation of Fus3 by the upstream MAPK kinase, Ste7. In support of this model, Ste5 was shown to impart some distinctive time- and dose-dependent behaviors on Fus3.

Compared with Kss1, Fus3 is phosphorylated more slowly and in a more ultrasensitive fashion

[39, 68]. In the absence of docking to Ste5, Fus3 behaves like Kss1 [39]. Another possibility is that

53 Ste5 accelerates de-phosphorylation, perhaps by recruiting one or more MAPK phosphatases. As compared to wild-type, cells lacking the dual-speciﬁcity phosphatase (msg5∆) exhibited more rapid accumulation of dual-phosphorylated Fus3, although the differences narrowed after 60 min (Fig.

2.3C). In contrast, there was no effect of deleting PTP2 and PTP3. We infer that both Ste5 and Msg5 slow the accumulation of dual-phosphorylated Fus3.

Figure 3

A 100 ppFus3 pFus3 min 0 0.5 1 23 5 10 15 20 30 60 80 npFus3 ppFus3 60 pFus3 npFus3 40 G6PDH 20 Fus3 ( rel. to 15 min) 15 to rel. ( Fus3 0 0 1 2 3 4 5 10 20 30 40 50 60 time (min) B C 100 100 WT WT 80 ste5 FB 80 msg5 ste5 ND ptp2/3 60 60

40 40

20 20 ppFus3 (rel. to 15 min WT) ppFus3 ppFus3 (rel. to 15 min WT) ppFus3 0 0 0 1 2 3 4 5 10 20 30 40 50 60 0 1 2 3 4 5 10 20 30 40 50 60 time (min) time (min)

Figure 2.3: Dynamics of differentially phosphorylated forms of Fus3.

(A) Left, wild-type cells were treated for the indicated times with 10 µM ∆-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies (top), or G6PDH load control antibodies (bottom). Right, dual-phosphorylated

(ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3), quantiﬁed as a percentage of total Fus3 at

15 min. ± SEM (n ≥ 3). (B) Wild-type (WT), ste5FB (feedback phosphorylation deﬁcient), ste5ND (non-docking), ptp2∆/ ptp3∆, and msg5∆ (phosphatase-deﬁcient) mutants treated and resolved by Phos-tag SDS-PAGE, as above.

Dual-phosphorylated Fus3 is quantiﬁed as a percentage of total Fus3 at 15 min. ± SEM (n ≥ 3).

2.3.4 Ste5 and Msg5 cooperate to maintain a pool of mono-phosphorylated Fus3.

To further investigate the role of Ste5 in limiting Fus3 activity, we measured the accumulation of mono- vs. dual-phosphorylated Fus3 in wild-type cells, in cells where the interaction between

Fus3 and Ste5 had been disrupted (ste5ND) and in cells lacking the phosphatase Msg5 (msg5∆).

54 For these experiments we focused on early times and monitored the relative kinetics of Fus3 mono and dual-phosphorylation. As compared to wild-type, the peak of dual-phosphorylated Fus3 was more pronounced in the ste5ND mutant, but the mono-phosphorylated species remained roughly unchanged (Figs. 2.4A, compare top left and right panels). In the msg5∆ strain, dual-phosphorylated

Fus3 was again elevated and, in this case, the mono-phosphorylated pool of Fus3 was signiﬁcantly diminished (Figs. 2.4A, bottom left panel). These results are consistent with the view that both Ste5 and Msg5 limit the activity of Fus3. In contrast, a ptp2/3∆ double deletion strain showed only small changes in the kinetics of mono- or dual-phosphorylated Fus3 (Fig. 2.4A, bottom right panel).

In every case, the appearance of the dual-phosphorylated form of Fus3 preceded that of the mono- phosphorylated form. Although we can envision several mechanisms to account for this behavior, the simplest invokes processive phosphorylation and distributive de-phosphorylation. To determine if a distributive de-phosphorylation mechanism can account for our experimental results, and to quantify the relative roles of Ste5 and Msg5 in limiting MAPK activity, we developed three mathematical models to describe the kinetics of Fus3 activation and inactivation (Fig. 2.4B). All of our models started with the pheromone-dependent recruitment of Ste5 to the plasma membrane. We did not explicitly model the activation of the kinases Ste11 and Ste7, but rather assumed a single rate-limiting step in the activation (dual phosphorylation) and recruitment of Fus3. De-phosphorylation of Fus3 was assumed to occur in a distributive manner. The abundance of Ste5, Ptp2 and Ptp3 were assumed to remain constant, whereas the production of Fus3 and Msg5 were assumed to increase as a function of Fus3 activity (feedback through gene induction).

While our results and previous studies demonstrate that Ste5 dampens Fus3 activity, they do not establish a mechanism for this effect. Therefore, we developed three models to test various scenarios for the mechanism by which feedback phosphorylation of Ste5 by Fus3 limits Fus3 activity. In the ﬁrst scenario, feedback phosphorylation of Ste5 reduces the rate at which Fus3 is phosphorylated. In the second scenario, feedback phosphorylation of Ste5 increases the rate at which

Fus3 is dephosphorylated (perhaps by recruiting a phosphatase). We then tested each mechanism alone or in combination (third scenario).

The model equations (see Methods) were simulated using the NDSolve function in Wolfram

Mathematica 9.0. To perform parameter estimations, we implemented a Markov Chain Monte

Carlo algorithm [37] to search parameter space for sets of model parameter values that minimize

55 the “aggregated” sum of squared deviations (SSD) between experimental time courses for non- phosphorylated (npFus3), mono-phosphorylated (pFus3), and dual-phosphorylated (ppFus3) protein, as measured in wild-type, ste5ND, msg5∆, and ptp2/3∆ cells. Surprisingly, models that include increased Fus3 de-phosphorylation generated the best ﬁts to the experimental data (Fig. 2.4B, top right panel) and accurately captured the individual contributions of both Ste5 and Msg5 to signal dampening (Fig. 2.4A). Thus the best performing model suggests that Ste5 limits Fus3 activity in two ways, by inhibiting the rate of Fus3 phosphorylation and increasing the rate of de-phosphorylation.

2.3.5 Loss of a phosphatase and MAPK-scaffolding confer synthetic supersensitivity to pheromone.

To test the validity of the models, we used each model to predict Fus3 activity in a strain containing the non-docking Ste5 mutant and also lacking Msg5 (ste5ND/msg5∆). Interestingly, the models in which Fus3 de-phosphorylation is increased by feedback phosphorylation to Ste5 predicted that the double mutant strain would accumulate substantially more dual-phosphorylated Fus3 then the single mutants alone (Fig. 2.4B, bottom right panel). Experimental measurements made in the double mutant showed excellent agreement with these model predictions (Fig. 2.4B, bottom right panel, and Fig. 2.5A). At the level of transcription, the effects of the double mutant were similarly magniﬁed (Fig. 2.5B). Whereas the individual mutants conferred a ≈2-fold decrease in the EC50 for pheromone, combining the two mutations resulted in a ≥ 8 fold decrease in EC50. This increase in pheromone sensitivity is in marked contrast to the reduction in transcription activation presented in

Fig. 2.1, where accumulation of mono-phosphorylated Fus3 dampened the transcription response by up to 60%, depending on the genetic background. These opposing effects on activity are qualitatively similar to those exhibited by the ?benchmark? pathway regulator, the GTPase activating protein

Sst2. Whereas deletion of SST2 confers a decrease in the EC50, two-fold overexpression of SST2 dampens the maximum response [38]. Collectively, these data support the predictions of the model, and indicate that Ste5 and Msg5 cooperate to limit the accumulation of fully phosphorylated Fus3.

56 Figure 4

ND A Wildtype ste5 ppFus3 100 100 pFus3 ææ æ npFus3 80 80 æ L L % % æ 60 60 H æ H æ æ æ æ æ æ æ 40 ææ 40 æ æ

Fus3 æ Fus3 æ æ æ æ æ æ æ ææ æ æ 20 æ 20 æ æ æ ææ ææ æ 0 0 0 5 10 15 0 5 10 15 time HminL time HminL

msg5 D ptp2 3D 100 100 ææ 80 80 ææ L æ L æ

% æ 60 % H 60 æ æ H ææ æ æ æ æ 40 æ 40 æ æ æ Fus3 æ æ Fus3 æ æ æ æ æ 20 20 ææ ææ æ æ æ æ æ æ æ ææææ æææ 0 0 0 5 10 15 0 5 10 15 time HminL time HminL

Combined Model B 25 Increased Fus3 De-Phosphorylation Decreased Fus3 Phosphorylation Pheromone 20 15

SSD 10 Ste5c Ste5m p-Ste5m 5 0 Final 40K iterations ste5 ND & msg5 D 2 np-Fus3 pp-Fus3 100 æ 80 æ

L æ æ % 60 æ æ p-Fus3 H æ æ 40 æ Fus3 æ æ æ æ æ 20 æ æ æ 0 Msg5 æææææ æ 0 0 5 10 15 time HminL Figure 2.4: Dynamics and mathematical model of the differentially phosphorylated forms of Fus3.

(A) Time series for wild-type (top left), ste5ND (non-docking) mutant (top right), msg5∆ (bottom left), and ptp2∆/ptp3∆

(bottom right) cells treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) quantiﬁed as a percentage of total Fus3, ± SEM (n ≥ 3). Circles are experimental results. Lines are simulation results of the mathematical model shown in B. (B) Diagram of the mathematical models (left panel). Black lines represent pathway components present in all models. Green lines indicate Ste5m feedback phosphorylation by p-Fus3 and pp-Fus3. The blue line is included in the model in which feedback-phosphorylated Ste5 (pSte5) limits Fus3 phosphorylation (activation) and the red lines are included in the model in which feedback-phosphorylated Ste5 increases Fus3 de-phosphorylation

(deactivation). The combined model includes both effects. The performance of all three models is shown in the top right panel. This graph shows values for the sum of squared deviations (SSD) for the 40,000 best parameter sets found by the Monte Carlo algorithm for parameter estimation. Model predictions for the combined model and corresponding experimental results for the ste5ND msg5∆ double mutant are shown in the bottom right panel.

57 Finally we considered the role of Ste5 and Msg5 in the mating response. When exposed to high concentrations of mating pheromone, yeast cells will arrest in G1 and expand in a polarized manner. Cells lacking Sst2 arrest at concentrations much lower than that needed by wild-type cells

[14, 15], and do not polarize properly [98]. In addition, we found that cells bearing the msg5∆ and ste5ND mutations exhibited a greater sensitivity to pheromone in the growth arrest plate assay (Fig.

2.5C). To monitor cell polarization we used a custom-designed microﬂuidic device, and tracked the distribution of Bem1-GFP as a marker of the polar cap [19, 50]. Whereas wild-type and msg5∆ cells polarized quickly and expanded in a single direction, ste5ND and msg5∆ ste5ND strains turned frequently and failed to orient in any one direction (Fig. 2.5D). Thus the msg5∆/ste5ND strain exhibits transcription and polarization defects similar to those reported previously for sst2∆.

Whereas Sst2 limits activation of both Fus3 and Kss1, Ste5 and Msg5 act speciﬁcally to limit the accumulation of dual phosphorylated Fus3.

58 Figure 5

A 350 B WT 160 WT ste5 ND 140 ste5 ND msg5 120 msg5 200 ste5 ND & msg5 100 ste5 ND & msg5 80 100 60 80 40 percent of wildtype of percent 20 60 0

ppFus3 (rel. to 15 min WT) ppFus3 40 -10 -9 -8 -7 -6 -5 -4 [ factor], M 20 WT ste5 ND msg5 ste5 ND & msg5 0 0 1 2 3 4 5 10 20 30 40 50 60 HillSlope 0.9689 0.8528 0.7065 0.7282 EC50 6.092e-007 3.198e-007 2.635e-007 7.399e-008 time (min) C WT ste5 ND msg5 ¨ 1.0

0.8 1.5 µg 1.5 µg

inches 0.6

0.4 Halo Size 0.2 5 µg 15 µg 5 µg 15 µg

0.0 ¨ ¨ ¨ ND ND WT sst2 ste5 ste5 msg5 msg5

D 0’ 50’ 100’ 150’ 200’ WT ¨ msg5 ND ste5 ¨ msg5 ND ste5

Figure 2.5: Synergistic activation of Fus3.

(A) Dual-phosphorylated Fus3 in wild-type, ste5ND (non-docking), msg5∆ (phosphatase-deficient) mutants re-plotted from Fig. 3 for comparison with the ste5ND msg5∆, mutant. Dual-phosphorylated Fus3 is quantified as a percentage of total Fus3 at 15 min. ± SEM (n ≥ 3). (B) Transcription reporter data in the same strains as in A, as a percentage of maximum activity in wild-type. Inset, Hill slope and EC50 for each strain. SEM (n = 3). (C) Pheromone-induced growth arrest for cells treated with α-factor. Halo diameters are quantified for all strains at 5 µg (left), ± SEM (n = 4).

Also shown are representative halo assays for the wild-type and ste5ND msg5∆ strains at 1.5, 5 and 15 µg (right). (D)

Polarized growth in cells treated with 300 nM α-factor. Shown are representative images for wild-type, msg5∆, ste5ND, and ste5ND msg5∆ cells, each bearing the integrated polar cap marker Bem1-GFP (arrow), at the indicated times. The path of the polar cap was quantiﬁed for ten cells 45-95 min after pheromone addition. Data are representative of 2 or more experiments. Scale bar, 10 µM.

59 2.3.6 Discussion:

It is well known that MAPKs are dually phosphorylated and activated in response to an extracellular signal. Here we demonstrate that Fus3 is both mono- and dually-phosphorylated in response to pheromone stimulation. The dual-phosphorylated species accumulates rapidly and precedes the appearance of the mono-phosphorylated species, suggesting a processive mechanism of phosphorylation and a distributive mechanism of de-phosphorylation. The scaffold Ste5 and the phosphatase Msg5 act together to maintain a substantial pool of mono-phosphorylated Fus3.

Mono-phosphorylated Fus3 appears to be a functionally important protein species, one that inhibits signal transmission and does so in a functionally dominant manner.

Previous studies demonstrated that Ste5 and Ste7 compete for binding with Fus3 and, in such case, there would have to be a dissociation event between steps of auto-phosphorylation (Ste5-Fus3) and trans-phosphorylation (Ste7-Fus3) [56]. Thus, based on the literature, we had expected to observe distributive phosphorylation of Fus3. Moreover, a distributive mechanism had already been demonstrated for ERK in vitro [28, 41]. However a more recent analysis revealed that ERK is phosphorylated processively in cells [3]. As with Fus3, the mono-phosphorylated species of ERK does not precede, but rather accompanies, the appearance of the dual-phosphorylated species [3].

These differences suggest that there are additional proteins or processes in the cell that modulate

MAPK phosphorylation in vivo. According to one model, the effects of molecular crowding, as exists in the cellular milieu, act to accelerate MAPK activation so as to appear processive in vivo.

In support of that model, investigators showed that the addition of an osmolyte (to displace water and mimic the effects of molecular crowding) conferred a processive rate of activation in vitro [3].

Molecular crowding could just as easily be achieved by binding to scaffolds such as Ste5 or Pbs2 in yeast, and KSR (kinase suppressor of Ras) or JIP (c-Jun N-terminal kinase inhibitory protein) in animals [10]. Pbs2 serves as both a scaffold and a MAPKK for the p38 ortholog Hog1, which like Fus3 is phosphorylated in a processive manner in vivo [23]. In this way, scaffolds might act to increase the thermodynamic activity, and thereby promote phosphorylation, of the kinase.

Another consideration is the effects of protein localization. Whereas activated MAPKs typically translocate into the nucleus where they phosphorylate transcription regulators, other substrates are located at the plasma membrane or in the cytoplasm. For example, Fus3 directly phosphorylates

60 the nuclear transcription factor Ste12 [10, 22, 46], as well as the cytoplasmic protein Sst2 [32, 82].

Moreover, subcellular localization is highly dynamic. Under conditions where Fus3 translocates from the plasma membrane to the nucleus, Ste5 shuttles from the nucleus to the plasma membrane

[65, 66, 111]. These movements are likely to affect local concentrations of enzyme, scaffold and substrate, and consequently affect signal output. For instance, when Ste5 is permanently tethered to the plasma membrane, the transcriptional output is more graded than it is in wild-type cells [102].

Thus changes in scaffold binding, concentration, and localization can all have potentially important effects on signal output.

Given the difﬁculty of reconciling MAPK activities in vivo and in vitro, we took an alternative approach “in silico”. Speciﬁcally, we built a computational model where Fus3 is phosphorylated processively and where de-phosphorylation depends on feedback interactions with the scaffold and phosphatases. Our model also posits that feedback phosphorylation of Ste5 slows the rate of downstream signaling and protects against signal saturation. We have shown experimentally that

Ste5 cooperates with Msg5 to maintain the pool of mono-phosphorylated Fus3. Msg5 could be acting directly on Fus3 or indirectly through Ste5 or a Ste5-binding partner [94]. Ste5ND on its own has little effect on the abundance of mono-phosphorylated Fus3. When interactions with both Ste5 and

Msg5 are disrupted however, nearly all of the Fus3 protein is dually phosphorylated. Fus3 and Ste5 also regulate the abundance of Msg5 through gene induction, thereby providing another mechanism to ﬁne-tune MAPK activity over time. Independently, these mechanisms dynamically regulate Fus3 phosphorylation; Collectively, these mechanisms generate a dynamic self-regulatory network. While

Fus3 is partially redundant with the MAPK Kss1, Kss1 does not bind to Ste5 and was not considered in this analysis.

Our ﬁndings add support to the proposal that mono-phosphorylated Fus3 down regulates pathway activity, and does so through phosphorylation of Ste5 [7, 39, 68]. Further, our data provides evidence that auto-phosphorylation is a minor contributor to the pool of mono-phosphorylated Fus3. While our data points to the importance of Msg5 and de-phosphorylation in this process, other (as yet unidentiﬁed) mechanisms may further contribute to the production of mono-phosphorylated Fus3 in vivo.

Collectively, our ﬁndings reveal a substantial pool of non- and mono-phosphorylated Fus3 in pheromone-stimulated cells, and these have an inhibitory effect on signaling. This could account

61 for the ability of a second wild-type copy of Fus3 to dampen the activity of the mating pathway.

While we do not know the mechanism of inhibition in this case, there is growing evidence that even catalytically inactive kinases can have important cellular functions. Approximately 10% of all human kinase genes lack one of three key residues necessary for activity, and are therefore likely to be catalytically inert. Nevertheless, many of these “pseudokinase” proteins are expressed and have been demonstrated to perform functions that do not involve substrate phosphorylation [59](Leslie,

2013). For example, a pseudo-MAPK in yeast (Mlp1) was shown to regulate transcription [51](Kim et al., 2008) and to bind Msg5 through an unusual docking domain [81]. Here we have shown that an incompletely phosphorylated form of Fus3 exists in the absence of catalytic activity, and that this mono-phosphorylated form of the protein acts in opposition to the fully phosphorylated kinase. Given the evolutionary conservation of structurally-similar MAPKs, we anticipate ﬁnding functionally-similar, differentially-phosphorylated MAPKs in other systems. A deeper understanding of MAPK regulation should point to new mechanisms and new therapeutic strategies to inﬂuence stimulus-response behaviors in humans.

2.3.7 Material and Methods:

Experimental Methods

Strains, plasmids, and growth conditions

Standard methods were used throughout for growth, maintenance, and transformation of yeast and bacterial cultures as well as for the manipulation of DNA. Strains and plasmids used in this study are described in Tables 1 and 2. Mutagenesis was performed using the QuikChange site-directed mutagenesis kit (Life Technologies). Yeast strains carrying integrated mutations were constructed using the Delitto Perfetto method [101]. Oligonucleotides used for strain construction and DNA ampliﬁcation of FUS3 T180 and Y182 mutants are provided in Table 3. Cells were grown in synthetic complete medium, absent speciﬁc nutrients to maintain plasmid selection, and containing 2% (w/v) dextrose.

62 Strain Description Source

BY4741* MATa leu2∆ met15∆ his3-1 ura3∆ [9]

fus3∆ MATa fus3::kanMX Invitrogen

msg5∆ MATa msg5::kanMX Invitrogen

ptp2/3∆ MATa ptp2::URA3 ptp3::kanMX Nan Hao

ptp2/3∆ msg5∆ MATa ptp2::URA3 ptp3::LEU2 msg5::kanMX Nan Hao

ste5ND MATa ste5ND [39]

ste5FB MATa ste5FB This study

ste5ND msg5∆ MATa ste5ND msg5::kanMX This study

ste11∆ MATa ste11∆ Invitrogen

ste7∆ MATa ste7∆ Invitrogen

fus3T 180A MATa fus3T 180A This study

fus3Y 182F MATa fus3Y 182F This study

fus3T 180A/Y 182F MATa fus3T 180A/Y 182F This study

Table 2.1: All strains used in this study. *Mutants are derived from BY4741

Fig Plasmid Name Description Source

1 pRS316 CEN URA3 vector [100]

1 pRS316-FUS3 CEN URA3 FUS3 [37]

1 pRS316-FUS3-T180A CEN URA3 FUS3T 180A This Study

1 pRS316-FUS3-Y182F CEN URA3 FUS3T 182F This Study

1 pRS316-FUS3-T180A/Y182F CEN URA3 FUS3T 180A/T 182F This Study

1 pRS425-FUS1-lacZ 2µ LEU2 PFUS1-lacZ [43]

Table 2.2: Plasmids used in this study

63 Fig Plasmid Name

P1 Fus3-CORE 60up TTA GCA AGA ATC ATT GAC GAG TCA GCC GCG

GAC AAT TCA GAG CCC ACA GGT CAG CAA AGC

GGC GAG CTC GTT TTC GAC ACT GG

P2 Fus3-CORE 60dn CAC GTC CAT GGC CCT TGA GTA TTT GGC AGA

GGT TAA CAT CAC CTC TGG CGC CCT GTA CCA

ACG TCC TTA CCA TTA AGT TGA TC

P1 Fus3 activation-loop IRO CCA GAT GCT GAG TGA CGA

P2 Fus3 activation-loop IRO GCA GGG TAC ATG GGA AGC

Table 2.3: Oligonucleotides used in this study

2.3.7.1 Transcriptional reporter and halo assay

Growth arrest (halo) and FUS1-LacZ levels were measured after treatment with mating pheromone,

α-factor, as described previously [43]. Cells grown to A600nm ≈0.8 were stimulated for 90 min, then lysed and β-galactosidase activity was measured spectrophotometrically after 60 min. Results are from 2-3 independent experiments of 3-4 colonies per strain read in quadruplicate.

2.3.7.2 Cell extracts and immunoblotting

Protein extracts were produced by glass bead lysis in TCA as previously described [36]. Protein concentration was determined by Dc protein assay (Bio-Rad Laboratories). Protein extracts were resolved by standard SDS-PAGE or 50 µM Mn2+ -Phos-tag in 10% acrylamide SDS-PAGE according to manufacturer’s instructions (Wako Chemicals). Proteins were detected by immunoblotting with phospho-MAPK p44/42 antibodies (9101, Cell Signaling Technology) at 1:500, Fus3 antibodies (sc-

6773, Santa Cruz Biotechnology, Inc.) at 1:500, and glucose-6-phosphate dehydrogenase (G6PDH) antibodies (A9521, Sigma-Aldrich) at 1:50,000. Immunoreactive species were visualized by ﬂuo- rescence detection (Typhoon Trio+ Imager, GE Healthcare) of horseradish peroxidase-conjugated antibodies (sc-2006, Santa Cruz Biotechnology, Inc.) at 1:10,000 using ECL-plus reagent (Pierce).

Band intensity was quantiﬁed by scanning densitometry using ImageJ (National Institutes of Health).

Fus3 intensity values were ﬁrst normalized to G6PDH loading control and values for non-, mono-

64 and dual-phosphorylated bands were calculated as a percent of the total for each lane. To capture changes in Fus3 protein induction over time and/or between different genetic backgrounds, samples were normalized to the 15 min time point. Brieﬂy, the value of total Fus3 in the 15 min lane served as the denominator for calculating values of the three Fus3 forms at all time points. When comparing wild-type and mutant strains all lanes were normalized to the 15 min time point of wild-type.

2.3.7.3 Microscopy

Cells expressing Bem1-GFP were examined in a microﬂuidic device as previously described

[50](Kelley et al., 2015) in the presence of 300 nM pheromone. This was carried out on an Olympus

IX83 with an Andor Revolution XD spinning disk unit. Images were taken every 5 min as a z-series of 1 µm step size, 5 µM around the focal plane. These images were aligned in FIJI (http://fiji.sc/Fiji) using the “Descriptor-based series registration (2D/3D + t)” plugin. A single pixel Gaussian blur was applied to the fluorescence images to remove camera noise, and a maximum intensity projection was created of the stacks. To plot the polar cap path, the Bem1-GFP images were thresholded to determine the centroid of the polar cap. The centroids were recorded for each timepoint, for every cell. Using MATLAB, the Cartesian coordinates of the polar cap centroids were converted to polar coordinates, and each path was rotated such that the final 50 minutes of time points had an average angle of 0. The first 45 min of experimental data was not included in the analysis to exclude data from mitotic events. The normalized paths were converted back into Cartesian coordinates and smoothed with the “smooth” function in MATLAB, using a window size of 7 time points. Ten individual paths were plotted for each strain.

2.3.7.4 Mathematical Model

We tested three different processive models for the time-dependent behavior of Fus3 activity.

The models differ in the mechanism by which Ste5 negatively regulates Fus3 activity. The ﬁrst model assumes feedback phosphorylated Ste5 (pSte5m) has no effect on the Fus3 phosphorylation rate. The second model assumes pSte5m has no effect on the Fus3 de-phosphorylation rate. The third model combines the effect of pSte5m on phosphorylation of npFus3 and de-phosphorylation of ppFus3 and pFus3, and contains the other two models as limiting cases. We describe this model in detail and point out how the other two were obtained from it. We explicitly modeled the temporal evolution of

65 three species of Ste5: 1) free and inactive (Ste5c), 2) plasma membrane bound and active (Ste5m), and 3) feedback-phosphorylated, plasma membrane bound and less active (pSte5m). We assumed that the total population of Ste5 is conserved and that Ste5 is recruited to the plasma membrane upon stimulation with mating pheromone. The model assumed that Fus3 is dually phosphorylated in a processive manner. To test if feedback phosphorylation of Ste5 affects the activation of Fus3, we assumed the rate of Fus3 phosphorylation depends on the form of Ste5 (Ste5m or pSte5m) with which

Fus3 is associated. We assumed that Fus3 is induced and degraded, and that mono-phosphorylated

(pFus3) or dual-phosphorylated Fus3 (ppFus3) feedback-phosphorylates Ste5m. We modeled Fus3 de-phosphorylation as a distributive process. We assumed that the dephosphorylation rate depends on the current level of the induced dual-speciﬁcity phosphatase Msg5. Because the mechanism by which Ste5 limits Fus3 activity is not known, we also included the possibility that dephosphorylation rate depends on the phosphorylated Ste5 species (pSte5m).

The equations that describe the evolution of the Ste5 species are as follows:

d[Ste5m] = k1s(t)[Ste5c] + k4[pSte5m] − k2[Ste5m] dt (2.1) 0 − k3[Ste5m][ppF us3] − k3[Ste5m][pF us3]

d[Ste5c] = −k s(t)[Ste5c] + k [Ste5m] + k0 [pSte5m] (2.2) dt 1 2 2

d[pSte5m] 0 = k3[Ste5m][ppF us3] + k3[Ste5m][pF us3] dt (2.3)

− k2[Ste5m] − k4[pSte5m] where the first term in equation 2.1 describes the rate at which mating pheromone “s(t)” induces the recruitment of Ste5 to the plasma membrane. The second term defines the dephosphorylation rate for pSte5m. The third term defines the rate at which Ste5 transitions back to the cytosol. The fourth and

ﬁfth terms deﬁne rates of feedback phosphorylation of Ste5m by dual-phosphorylated Fus3 (ppFus3),

66 and mono-phosphorylated Fus3 (pFus3), respectively. Additionally the second term in 2.2 deﬁnes the rate at which pSte5m transitions back to the returns to Ste5c.

The equation that describes the evolution of the non-phosphorylated Fus3 (npFus3) concentration is

0 d[npF us3] k5[ppF us3] k5[pF us3] = k10 + + dt K1 + [ppF us3] K2 + [pF us3]

+ (kms[Msg5] + k9[pSte5m] + kb)[pF us3] (2.4) k [Ste5m][npF us3] k0 [pSte5m][npF us3] − 7 − 7 K3 + [npF us3] K4 + [npF us3]

− k6[npF us3]

The first term on the right-hand-side of equation 2.4 accounts for basal induction of Fus3. The second and third terms correspond to the positive feedback resulting from induction of Fus3 via ppFus3 and pFus3, respectively. We use Michaelis-Menten kinetics to model positive feedback. The fourth term describes the rate at which mono-phosphorylated Fus3 (pFus3) is dephosphorylated. We have assumed three mechanisms for dephosphorylation. The first depends on the phosphatase Msg5, which is induced by dual-phosphorylated Fus3 (ppFus3). The second is independent of Msg5, and is included to test the possibility that feedback phosphorylated Ste5 (pSte5m) limits Fus3 activity by increasing the dephosphorylation rate (perhaps by recruiting another phosphatase). Finally, we include a basal pFus3 dephosphorylation rate that is independent of Msg5 and Ste5. The fifth and sixth terms define rates of dual-phosphorylation mediated by Ste5m and pSte5m, respectively. These terms allow us to test the possibility that feedback phosphorylated Ste5 (pSte5m) limits Fus3 activity

0 by decreasing the phosphorylation rate (i.e. requiring k7 to be less than k7). The last term deﬁnes the degradation rate for Fus3.

The rate equations that describe dual- and mono-phosphorylated Fus3 are

d[ppF us3] k [Ste5m][npF us3] k0 [pSte5m][npF us3] = 7 − 7 dt K + [npF us3] K 3 4 (2.5)

− (kms[Msg5] + k8[pSte5m] + kb + k6)[ppF us3]

67 d[pF us3] = (kms[Msg5] + k8[pSte5m] + kb)[ppF us3] dt (2.6)

− (kms[Msg5] + k9[pSte5m] + kb + k6)[pF us3] where terms describing the phosphorylation, dephosphorylation and degradation rates of ppFus3 and pFus3 are written in the same form as those in equation 2.4.

The rate equation that describes evolution of the Msg5 concentration is

d[Msg5] kamsg5[ppF us3] = k11 + − k12[Msg5] (2.7) dt Kmsg5 where the ﬁrst and second terms are the basal and ppFus3-dependent synthesis rates, respectively.

The third term deﬁnes the rate of Msg5 degradation.

Equations 2.1-2.7 deﬁne the full processive model that allows for the possibility that Ste5 impacts both the phosphorylation and dephosphorylation rates of Fus3. The model in which pSte5m only decreases the Fus3 phosphorylation rate is obtained by setting the parameters k8 and k9 equal to zero. While the model in which pSte5m only increases the Fus3 dephosphorylation rate is achieved by

0 setting k7 equal to k7. To confirm our assumption that phosophorylation occurs through a processive mechanism, whereas dephosphorylation follows distributive kinetics, we tested a model that allowed for the possibility of distributive (sequential) mechanism for Fus3 phosphorylation. However, to generate a good fit to the data this model required that the second phosphorylation event occur substantially faster (by three orders of magnitude) than the first phosphorylation event. Thus, this model effectively reduced to a processive (single step) mechanism, justifying our use of processive phosphorylation model.

68 Supplementary Material to Chapter 2

Hyper-Osmotic k1p k1p k1p k1p Stress 80 5 5 5 4 4 60 4 3 3 3 40 MAP3K 2 2 pfb 2 k6p nfbs 20 1 1 1

0 0 0 0 MAP2K 0 1 2 3 4 5 6 0.00 0.02 0.04 0.06 0.08 0.10 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

k1p MAPKA pfb pfb pfb 6 0.12 0.12 200 5 0.10 0.10 4 150 0.08 0.08 X* MAPK 3 100 0.06 0.06 B pfb

k6p 0.04 0.04

2 nfbs 50 1 0.02 0.02 0 0 0.00 0.00 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

k1p pfb k6p k6p 5 6 100 0.4 5 80 4 4 0.3 60 3 3 0.2 2

k6p 40 k6p

2 nfbs 0.1 1 20 1 0 0.0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

k1p pfb k6p nfbs 6 6 100 0.4 5 5 80 0.3 4 4 60 3 3 0.2 40 nfbs nfbs 2 nfbs 2 0.1 1 1 20 0 0.0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

2 Figure A.1: Selected parameter histograms and correlations for Model m(1,0)

69 Hyper-Osmotic k1p k1p k1p k1p 5 5 Stress 80 5 4 4 4 60 3 3 3 MAP3K 40 2 pfb 2 2 k6p nfbs 20 1 1 1 MAP2K 0 0 0 0 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

MAPKA k1p pfb pfb pfb 1.0 1.0 5 250 0.8 0.8 X* MAPK 4 200 B 0.6 0.6 3 150 0.4 0.4 pfb 2 100 k6p nfbs 1 50 0.2 0.2 0 0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

k1p pfb k6p k6p 0.7 5 5 100 0.6 4 4 0.5 80 3 3 0.4 60 0.3 2 k6p k6p 2 40 0.2 nfbs 1 0.1 20 1 0 0.0 0 0 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

k1p pfb k6p nfbs 6 5 0.7 70 0.6 5 60 4 0.5 4 50 3 0.4 3 40 0.3 2 30 nfbs

nfbs nfbs 2 0.2 20 1 0.1 1 10 0 0.0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

2 Figure A.2: Selected parameter histograms and correlations for Model m(2,0)

Hyper-Osmotic k1p k1p k1p k1p Stress 80 5 5 5 60 4 4 4 3 3 3 MAP3K 40 2 pfb 2 2 k6p 20 nfbs 1 1 1 MAP2K 0 0 0 0 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

MAPKA k1p pfb pfb pfb 1.0 1.0 5 200 0.8 0.8 4 X* MAPK 150 B 0.6 0.6 3 100 0.4 0.4 pfb 2 k6p nfbs 1 50 0.2 0.2 0 0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

k1p pfb k6p k6p 120 5 5 0.4 100 4 4 80 0.3 3 3 60 0.2 2 k6p k6p 2

40 nfbs 0.1 1 20 1 0 0.0 0 0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

k1p k6p nfbs pfb 6 5 80 0.4 5 4 4 60 0.3 3 3 0.2 40 2

nfbs 2 nfbs nfbs 0.1 20 1 1 0 0.0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

2 Figure A.3: Selected parameter histograms and correlations for Model m(3,0)

.&O P0 P +\SHU2VPRWLF 6WUHVV 0$3. 0$3. 0$3. SS+RJ $ 7LPH 0LQXWHV 7LPH 0LQXWHV ; 0$3. %

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV

2 Figure A.4: The response of model m(1,0) to periodic applications of hyper-osmotic stress m10 Ramps under wild type conditions

71 ) 350 350 mM ( 0 0 KCl P 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Hyper-Osmotic Stress 100 100 MAP3K 80 80 (%) 60 60 MAP2K Hog1 40 40 - 20 20 MAPK pp A 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30

X* MAPKB Time (Minutes) Time (Minutes) 350 350 350

0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes) 350

0 0 20 40 60 80 100 120

100 80 60 40 20 0 0 20 40 60 80 100 120 Time (Minutes)

2 Figure A.5: The response of analog sensitive model m(1,0) to periodic applications of hyper-osmotic stress m10 Ramps under analog sensitive conditions

.&O P0 P +\SHU2VPRWLF 6WUHVV 0$3. 0$3. 0$3. SS+RJ $ 7LPH 0LQXWHV 7LPH 0LQXWHV ; 0$3. %

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV

2 Figure A.6: The response of model m(2,0) to periodic applications of hyper-osmotic stress m20 Ramps under wild type conditions

73 350 350

0 0 2 KCl (mM) m (2,0) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Hyper-Osmotic Stress 100 100 MAP3K 80 80 60 60 MAP2K 40 40 20 20 MAPK pp-Hog1(%) A 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (Minutes) Time (Minutes) X* MAPK B 350 350 350

0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes) 350

0 0 20 40 60 80 100 120

100 80 60 40 20 0 0 20 40 60 80 100 120 Time (Minutes)

2 Figure A.7: The response of analog sensitive model m(2,0) to periodic applications of hyper-osmotic stress m20 Ramps under analog sensitive conditions

.&O P0 P +\SHU2VPRWLF 6WUHVV 0$3. 0$3. 0$3. SS+RJ $ 7LPH 0LQXWHV 7LPH 0LQXWHV ; 0$3. %

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV

2 Figure A.8: The response of model m(3,0) to periodic applications of hyper-osmotic stress m30 Ramps under wild type conditions

75 350 350

0 0 2 KCl (mM) m (3,0) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Hyper-Osmotic Stress 100 100 MAP3K 80 80 60 60 MAP2K 40 40 20 20 MAPK pp-Hog1(%) A 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (Minutes) Time (Minutes) X* MAPK B 350 350 350

0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes) 350

0 0 20 40 60 80 100 120

100 80 60 40 20 0 0 20 40 60 80 100 120 Time (Minutes)

2 Figure A.9: The response of analog sensitive model m(3,0) to periodic applications of hyper-osmotic stress m30 Ramps under analog sensitive conditions

76 .&O P0 2 P (1,0) +\SHU2VPRWLF 6WUHVV 0$3. 0$3.

SS+RJ 0$3. $ ; 0$3. % 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV 2 Figure A.10: The response of model m(1,0) to continuously increasing hyper-osmotic stress m10 Ramps under wild type conditions

.&O P0 2 P (1,0)

+\SHU2VPRWLF 6WUHVV 0$3. 0$3.

0$3. $ SS+RJ ; 0$3. % 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

2 Figure A.11: The response of analog sensitive model m(1,0) to continuously increasing hyper-osmotic stress m10 Ramps under analog sensitive conditions

.&O P0 P +\SHU2VPRWLF 6WUHVV 0$3. 0$3.

SS+RJ 0$3. $

; 0$3. % 7LPH 0LQXWHV 7LPH 0LQXWHV

7LPH 0LQXWHV 7LPH 0LQXWHV 7LPH 0LQXWHV

2 Figure A.12: The response of wild type model m(2,0) to continuously increasing hyper-osmotic stress m20 Ramps under wild type conditions

79 600 600 450 450 300 300 150 150

KCl(mM) 0 0 P 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Hyper-Osmotic Stress 100 100 MAP3K 80 80 60 60 MAP2K 40 40 20 20 MAPK pp-Hog1(%) A 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120

X* MAPK B Time (Minutes) Time (Minutes) 600 600 600 450 450 450 300 300 300 150 150 150 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes) 600 600 600 450 450 450 300 300 300 150 150 150 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes)

2 Figure A.13: The response of analog sensitive model m(2,0) to continuously increasing hyper-osmotic stress m20 Ramps under analog sensitive conditions

80 600 600 450 450 300 300 150 150

KCl(mM) 0 0 2 m (3,0) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Hyper-Osmotic Stress 100 100 MAP3K 80 80 60 60 MAP2K 40 40 20 20 MAPK pp-Hog1(%) A 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120

X* MAPK B Time (Minutes) Time (Minutes) 600 600 600 450 450 450 300 300 300 150 150 150 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes)

2 Figure A.14: The response of wild type model m(1,0) to continuously increasing hyper-osmotic stress m30 Ramps under wild type conditions

81 600 600 450 450 300 300 150 150

KCl(mM) 0 0 2 m (3,0) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Hyper-Osmotic Stress 100 100 MAP3K 80 80 60 60 MAP2K 40 40 20 20 MAPK pp-Hog1(%) A 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120

X* MAPK B Time (Minutes) Time (Minutes) 600 600 600 450 450 450 300 300 300 150 150 150 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120

100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Minutes) Time (Minutes) Time (Minutes)

2 Figure A.15: The response of analog sensitive model m(3,0) to continuously increasing hyper-osmotic stress m10 Ramps under analog sensitive conditions

82 APPENDIX B

Supplementary Material to Chapter 3

B.1 Introduction to Mathematical Model

Baker’s yeast Saccharomyces cerevisiae stably propagates as either diploid or haploids, (a-type or α-type). Each haploid secretes a mating pheromone (a-factor for a-type, α-factor for α-type) that attracts the opposite haploid. If the distance between yeast of the opposite mating type is large, then the concentration of mating pheromone sensed by each haploid will be low, and each haploid will elongate towards a mating partner via chemotropic growth. If yeast of the opposite mating type are in close proximity to each other, then the concentration of mating pheromone sensed by each haploid will be high, and each yeast will form a mating projection (shmoo) in the direction of the mating partner. The shmoos then fuse upon contact to form an a-α diploid yeast. The decision to transition from chemotropic growth to the mating response is dependent on the concentration of mating pheromone sensed by the cells and also on the phosphorylation state of the Fus3 Mitogen-Activated Protein Kinase. Figure B.1 describes how pheromone is transmitted from the Mating Pheromone Receptor in a-type cells through a Mitogen-Activated Protein Kinase cascade to Fus3, the terminal MAPK. In a-type cells, α-factor binds to Ste2, a transmembrane G-Protein Coupled Receptor (GPCR) that is initially in complex with a heterotrimeric G-Protein consisting of Gpa1 (Gα-subunit) [74], Ste4 (Gβ-subuint), and Ste18 (Gγ-subunit) [113]. Gα, which is inactive in its GDP bound state, associates with the Gβγ subunits. The binding of α-factor to Ste2 induces a conformation change in the GPCR that subsequently causes Gα to exchange GDP for GTP. GTP bound Gα dissociates from the Gβγ subunits. The free Gβγ subunits then bind to Ste5, a protein scaffold that is trafficked to the plasma membrane [1, 17]. Ste5 localizes the Mitogen- Activated Protein Kinase cascade to the plasma membrane by binding the MAP3K (Ste11), MAP2K (Ste7), and MAPK (Fus3) [55, 69, 88]. Signaling through the MAPK cascade is initiated when Ste11 becomes phosphorylated by Ste20, a p21-activated Kinase [110]. Ste11 then phosphorylates Ste7 [? ], which phosphorylates Fus3 on the tyrosine182 and threonine180 phosphorylation loops [33]. Phosphorylated Fus3 dissociates from Ste5 and translocates to the nucleus [8]. Fus3 then phosphorylates Ste12, a transcription factor that leads to transcription of genes involved in the mating response [21, 22, 24], and Far1, a cyclin-dependent kinase that leads to cell cycle arrest [16, 83, 84, 99]. Although dual-phosphorylated Fus3 drives the pheromone response, recent biochemical analysis has revealed that mono-phosphorylated Fus3 down-regulates the pheromone response [77]. In the preliminary work that motivates this chapter, Nagiec et al. used a spectroscopically measured transcriptional reporter assay (Fus1-LacZ), to show that mono-phosphorylated Fus3 yielded a significantly weaker mating pheromone response than dual-phosphorylated Fus3 [77]. The authors additionally showed that mono-phosphorylated yielded a significantly weaker mating pheromone response than cells that lacked Fus3 entirely. The result that cells lacking Fus3 produced a transcriptional response was not surprising, given that Kss1 (MAPK) is partially redundant to Fus3. However this result in combination with the diminished transcriptional response produced by mono- phosphorylated Fus3 suggested that the mating pheromone response was differentially regulated by the Fus3 phosphorylation state. The authors then tested this hypothesis by repeating the transcriptional reporter assay with an exogenous copy of the MAPK, a copy the MAPK in which each

83 phosphorylation loop had been precluded from phosphorylation (fus3T 180A, fus3Y 180F ), and a copy of the MAPK in which both phosphorylation loops were inhibited (fus3T 180A & fus3Y 180F ). Consis- tent with the hypothesis that mono-phosphorylated Fus3 inhibited the mating pheromone response, the authors observed the least amount of transcriptional activity in cells with exogenous copies of mono-phosphorylated Fus3. These results conﬁrmed that the mating pheromone response was up-regulated by dual-phosphorylated Fus3, and down-regulated by mono-phosphorylated Fus3 (see Fig. B.1).

Figure B.1: A Schematic of the Yeast Mating Pheromone Response MAPK cascade Schematic of the yeast Mating Pheromone Response Pathway after α-factor (green star) has bound to the Ste2 G-Protein Coupled Receptor (GPCR). The βγ G-protein subunits are shown after separation from the α subunit and bound to Ste5. The MAPK cascade is scaffolded by Ste5 and localized to the plasma membrane. Ste20, a p21-Activated Kinase, is shown phosphorylating Ste11, which then phosphorylates Ste7, which then phosphorylates Fus3. Here, we represented phosphorylated MAPK cascade components with two gold stars. Non-phosphorylated Fus3 (npFus3) is shown as a neutral contributor to the Mating Pheromone response. Mono-phosphorylated Fus3 (pFus3) is shown as a negative regulator of the Mating Pheromone response. Dual-phosphorylated Fus3 (ppFus3) is shown as a positive regulator of the Mating Pheromone response.

B.1.1 Fus3 phosphorylation is regulated by multiple mechanisms Given that mono-phosphorylated and dual-phosphorylated Fus3 differentially regulate the mating response, our next goal was to understand the how each Fus3 phosphorylation state was regulated over time. To this end, we stimulated a-type yeast with 10 µM α-factor, a dose that produces full pathway activation, and then used Phos-tag reagent, (a metal coordinating small molecule that has high afﬁnity for phosphorylated serine, threonine, and tyrosine [3, 52]), to measure the relative abundance of each Fus3 phosphorylation state over a ﬁfteen minute time course. Addition of Phos-tag to acrylamide gels slows the migration of polypeptides, where the reduction in migration speed is proportional to the number of phosphorylations on the molecule. As we show in Fig. B.2 A, our Phos-tag analysis revealed that only non-phosphorylated Fus3, (low band), was present at the moment of α-factor addition. We observed that only minimal amounts of mono-phosphorylated (middle band)

84 and dual-phosphorylated Fus3 (highest band) were present at the thirty second time point. However all three Fus3 phosphorylation states were present at the one minute time point. Interestingly, we noted that the highest band was more intense than the middle band, and less intense than the lowest band a this time point. The intensity of the lowest band decreased at the two minute time point, and was accompanied by an increase in the intensities of the middle and highests bands, relative to the one minute time point. The intensity of the lowest band remained unchanged at the three minute time point. Although the intensity of the highest band decreased at the three minute time point, the band was more intense than the middle band. The intensity of the lowest band continued to increase at the five and ten minute time point, but then slightly decreased at the fifteen minute time point. Interestingly, we noted that the intensity of the highest band decreased at the five minute time point and was equal to the intensity of the middle band. We then noted that the intensity of these two bands remained steady and equal at the ten and fifteen minute time points. To determine the relative abundance of each Fus3 phosphorylation state, we quantified the band intensities at each time point. We then normalized the band intensities relative to the intensity of dual-phosphorylated Fus3 at the fifteen minute time point. We then visualized the relative abundance of each Fus3 phosphorylation state in Fig. B.2 B. We observed that as the proportion of non-phosphorylated Fus3 decreased, the proportion of dual phosphorylated Fus3 increased at a faster rate than mono-phosphorylated Fus3. The proportion of dual-phosphorylated Fus3 reached its maximum value at the two minute time point. However as the proportion of dual-phosphorylated Fus3 decreased at the three minute time point, the proportion of mono-phosphorylated Fus3 increased. The proportion of dual-phosphorylated Fus3 decreased again at the five minute point, however the proportion of mono-phosphorylated Fus3 remained unchanged. Interestingly, we noted that the proportion of non-phosphorylated Fus3 increased at this time point. The proportion of each Fus3 phosphorylation state remained relatively constant throughout the remainder of the time course, with the proportions of mono-phosphorylated and dual-phosphorylated Fus3 also being equivalent.

85 A

Fus3 B pFus3 ppFus3

Figure B.2: Fus3 transitions between three phosphorylation states (A) A Phos-Tag western blot showing three Fus3 phosphorylation states at the indicated time-points (top row). (B) The proportion of each Fus3 phosphorylation state is measured relative to the total of amount of Fus3 (y-axis) present in the sample at each time point (x-axis).

B.1.2 Fus3 phosphorylation is processive, Fus3 dephosphorylation is distributive Our next goal was to understand how each Fus3 phosphorylation state was regulated in time. Given that the proportion of dual-phosphorylated Fus3 accumulaed at a faster rate than mono- phosphorylated Fus3, our first task was to determine the mechanism of Fus3 phosphorylation. MAPKs can accumulate activating phosphorylations via a distributive or processive phosphorylation mechanism. Distributive phosphorylation occurs when a single MAPK is phosphorylated once during one collision event, and then phosphorylated again during a second collision event (Fig B.3 Left), [26, 45]. If Fus3 was subject to distributive phosphorylation, then we would expect to see an early accumulation of mono-phosphorylated Fus3, followed by a subsequent accumulation of dual- phosphorylated Fus3 (Fig B.3 Left). In contrast, processive phosphorylation occurs when the second phosphorylation occurs immediately after the first phosphorylation event, such that the second event is indistinguishable from the first event (Fig B.3 Right), [28, 45]. If Fus3 was subject to processive phosphorylation, then we would expect to see an early accumulation of dual-phosphorylated Fus3. Then, in order for mono-phosphorylated Fus3 to accumulate, dual-phosphorylated Fus3 would have to be dephosphorylated in a distributive manner (Fig B.3 Right).

86 Figure B.3: MAPK phosphorylation can be distributive or processive Cartoon representations of Fus3 undergoing distributive (Left) or processive (Right) phosphorylation. Representative time series for the proportion of each Fus3 phospho-form (y-axis) are shown underneath the corresponding phosphorylation mechanism. Green arrows represent phosphorylation events catalyzed by Ste7 (MAP2K). Red arrows represent dephosphorylation events.

87 Figure B.4: Fus3 phosphorylation is regulated by multiple mechanisms The schematic of the Yeast Mating Response Pathway has been updated to describe the proposed mechanisms of Fus3 phosphorylation and dephosphorylation. The legend describes the impact of the added phosphorylation - dephosphorylation cycle, and indicates the Ste5 is feedback phosphorylated by phosphorylated Fus3.

B.2 Computational Methods

B.2.1 Processive models of three dynamic MAPK phosphorylation states We tested three different processive models for the time-dependent behavior of Fus3 activity. The models differ in the mechanism by which Ste5 negatively regulates Fus3 activity. The ﬁrst model assumes feedback phosphorylated Ste5 (pSte5m) has no effect on the Fus3 phosphorylation rate. The second model assumes pSte5m has no effect on the Fus3 de-phosphorylation rate. The third model combines the effect of pSte5m on phosphorylation of npFus3 and de-phosphorylation of ppFus3 and pFus3, and contains the other two models as limiting cases. We describe this model in detail and point out how the other two were obtained from it. We explicitly modeled the temporal evolution of three species of Ste5: 1) free and inactive (Ste5c), 2) plasma membrane bound and active (Ste5m), and 3) feedback-phosphorylated, plasma membrane bound and less active (pSte5m). We assumed that the total population of Ste5 is conserved and that Ste5 is recruited to the plasma membrane upon stimulation with mating pheromone. The model assumed that Fus3 is dually phosphorylated in a processive manner. To test if feedback phosphorylation of Ste5 affects the activation of Fus3, we assumed the rate of Fus3 phosphorylation depends on the form of Ste5 (Ste5m or pSte5m) with which Fus3 is associated. We assumed that Fus3 is induced and degraded, and that mono-phosphorylated (pFus3) or dual-phosphorylated Fus3 (ppFus3) feedback-phosphorylates Ste5m. We modeled Fus3 de-phosphorylation as a distributive process. We assumed that the dephosphorylation rate depends on the current level of the induced dual-speciﬁcity phosphatase Msg5. Because the mechanism by which Ste5 limits Fus3 activity is not known, we also included the possibility that dephosphorylation rate depends on the phosphorylated Ste5 species (pSte5m). The equations that describe the processive

88 models are given in the main text. Here we show the performance of the Combined model versus all other candidate models.

B.2.2 A Distributive model of three dynamic MAPK phosphorylation states Processive MAPK phosphorylation can be thought of as a special case of distributive MAPK phosphorylation, where the second phosphorylation event occurs immediately after the ﬁrst phosphorylation event. To further determine whether Fus3 is processively phosphorylated, we tested whether a distributive model of Fus3 phosphorylation would better align with the Fus3 phosphorylation training data (see below). To create the distributive model, we replaced equations 2.5 and 2.6 with equations 8 and 9 below. Then distributive model was then fully described by equations 2.1 - 2.4, 8, 9, and 2.7.

d[pF us3] k [Ste5m][npF us3] k0 [pSte5m][npF us3] k00[[pF us3] = 7 − 7 − 7 dt K3 + [npF us3] K4 + [npF us3] K7 + [pF us3] (8) + (kms[Msg5] + k8[pSte5m] + kb)[ppF us3]

− (kms[Msg5] + k9[pSte5m] + kb + k6)[pF us3]

00 d[ppF us3] k7 [[pF us3] = − (kms[Msg5] + k8[pSte5m] + kb + k6)[ppF us3] (9) dt K7 + [pF us3]

B.2.3 Fitting models via Markov Chain Monte Carlo simulations All models were simulated using the NDSolve function in Wolfram Mathematica 9.0. To perform parameter estimation, we implemented a Markov Chain Monte Carlo (MCMC) algorithm to search for parameter vectors that minimized the “aggregated” sum of squared deviations (SSD) between simulated and experimental time courses [37]. We compared each model’s simulated proportions of non-phosphorylated (npFus3), mono-phosphorylated (pFus3), and dual-phosphorylated (ppFus3), to the corresponding experimental proportions as measured via Phos-Tag in wild type cells (Wildtype), in cells in which the binding of Fus3 to Ste5 was inhibited (ste5ND), in cells in which the Msg5 phosphatase was deleted (msg5∆), and in cells in which the Ptp2 and Ptp3 phosphatases were deleted (ptp2/3∆).

B.3 Results

B.3.1 Fus3 undergoes processive phosphorylation and distributive dephosphorylation We visualized each model’s best ﬁt, (as determined by the SSD value for the given parameter sets), to each training data set. Under wild type conditions, we observed that all processive MAPK phosphorylation models were able to simulate Fus3 phosphorylation dynamics that aligned with the corresponding proportion of each Fus3 phosphorylation state as measured by Phos-tag analysis, (Wildtype, Fig. B.5). Although each model generated an optimal ﬁt to this data set, there were differences in Fus3 phosphorylation dynamics for the Combined model when compared to the Limit- ing Case models. The Combined model suggested that the proportion of dual-phosphorylated Fus3 (ppFus3) would never surpass the proportion of non-phosphorylated Fus3 (npFus3). Alternatively,

89 the Limiting Case models suggested that the proportion of ppFus3 would surpass the proportion of npFus3 within two minutes after the addition of 10 µM α-factor. However, consistent with the Combined model, the Limiting Case models suggested that the proportion of npFus3 would surpass the proportion of ppFus3 at the ﬁve minute time point. The Combined and Limiting Case Model #2 also suggested that ppFus3 would undergo more dephosphorylation than Limiting Case Model #1 during the remainder of the time series. The Limiting Case Model #1 suggested that rate of Fus3 phosphorylation was independent of the Ste5 phosphorylation state. However this model simulated a greater proportion of ppFus3 than the Combined model and Limiting Case Model #2, which only assumed that pSte5m did not impact of the Fus3 dephosphorylation rate. These results suggested that the Ste5 phosphorylation state impacted the rate at with Fus3 became phosphorylated. Therefore we concluded that feedback phosphorylation of Ste5 by phosphorylated Fus3 served as a mechanism to regulate the proportion of ppFus3 over time.

90 AWildtype B Wildtype Pheromone Combined 100 Fus3 pFus3 80 ppFus3 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes ) C D Pheromone Limiting Case #1 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes ) E F Pheromone Limiting Case #2 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes )

Figure B.5: All processive models align with the wild type Phos-tag data (A, C, E) Diagrams of the Combined, Limiting Case #1, and Limiting Case #2 processive MAPK phosphorylation models. (B, D, F) Time series for Wildtype cells treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non- phosphorylated (npFus3) are quantiﬁed as a percentage of total Fus3, (ﬁlled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Each model’s simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

B.3.2 Feedback phosphorylated Ste5 diminishes the Fus3 phosphorylation rate We next visualized each model’s best ﬁt to Fus3 phosphorylation data generated in cells in which the binding of Fus3 to Ste5 had been inhibited, (ste5ND, Fig. B.6). Under these conditions,

91 our Phos-tag analysis revealed that the proportion of ppFus3 was maximal at the three minute time point, and that the maximum proportion of ppFus3 had increased from approximately 40% to 60% compared to wild type. Similar to wild type, the proportion of ppFus3 then decreased at the five and ten minute time points. However unlike wild type, the proportion of ppFus3 again increased at the fifteen minute time point. The increase in the proportion of ppFus3 at three minutes was accompanied by a decrease in the proportion of npFus3 from approximately 40% to 20% compared to wild type. Similar to wild type, the proportion of npFus3 increased at the five and ten minute time points, and then decreased at the fifteen minute time point. Interestingly, we did not observe a significant change in the proportion of pFus3 when compared to the wild type experiment. To simulate these conditions in each model, we reduced the rate at which phosphorylated Fus3 (pFus3, and ppFus3) feedback phosphorylated Ste5 by 90%. Under these conditions, we observed that only the Combined model simulated Fus3 phosphorylation dynamics that aligned with all aspects of the ste5ND training data, (Fig. B.6, A and B). Alternatively, the Limiting Case models were unable to simulate Fus3 phosphorylation that was significantly different from the wild type simulation of each model, at any time point, (Fig. B.6, C - D). The success of the Combined model further suggested that Ste5 had secondary impacts on Fus3 phosphorylation dynamics. In particular when the ability of phosphorylated Fus3 (pFus3 and ppFus3) to feedback phosphorylate Ste5 was diminished, the ppFus3 phosphorylation rate was increased. This loss of negative regulation resulted in an increase in the proportion of ppFus3 at the two minute time point. Alternatively, this loss of negative regulation had no appreciable impact on the pFus3 and ppFus3 dephosphorylation rates. Thus the distributive Fus3 dephosphorylation kinetics maintained the proportion of pFus3 at wild type levels. In addition, because of the faster ppFus3 phosphorylation rate, the proportion of ppFus3 remained higher than the proportion of npFus3 over the remainder of the time course.

92 Aste5 ND B ste5 ND Pheromone Combined 100 Fus3 pFus3 80 ppFus3 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes ) C D Pheromone Limiting Case #1 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes ) E F Pheromone Limiting Case #2 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes )

Figure B.6: The Combined model best aligns with the ste5ND Phos-tag data (A, C, E) Diagrams of the Combined, Limiting Case #1, and Limiting Case #2 processive MAPK phosphorylation models. (B, D, F) Fus3 phosphorylation time series for cells in which the binding of Fus3 to Ste5 has been significantly inhibited ste5ND. Cells were treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) are quantified as a percentage of total Fus3, (filled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Model simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

93 B.3.3 The Ptp2, Ptp3, and Msg5 phosphatases synergistically regulate Fus3 phosphorylation We next visualized each model’s best fit to Fus3 phosphorylation data generated in cells in which the Ptp2 and Ptp3 phosphatases were deleted, (ptp2/3∆, Fig. B.7). Under these conditions, our Phos-tag analysis revealed that the proportion of ppFus3 was again maximal at the three minute time point, and that the maximum proportion of ppFus3 increased from approximately 40% to 50% compared to wild type. Similar to wild type, the proportion of ppFus3 then decreased at the five, ten and fifteen minute time points. The increase in the proportion of ppFus3 at three minutes was accompanied by a decrease in the proportion of npFus3 from approximately 40% to 25% compared to wild type. Similar to wild type, the proportion of npFus3 increased at the five and ten minute time points, and then decreased at the fifteen minute time point. Interestingly, we again did not observe a significant change in the proportion of pFus3 when compared to the wild type experiment. While the proportion of ppFus3 was increased when compared to pFus3 at the ten and fifteen minute time points, we noted that unlike ste5ND, the proportion of ppFus3 was not greater than the proportion of npFus3 at these time points in ptp2/3∆ cells. To simulate these conditions in each model, we set the basal dephosphorylation rate (kb = 0). Under this condition, we observed that all models failed to simulate the increase in the maximum proportion of ppFus3 at the three minute time point (see the simulation curves in Fig. B.7, B, D, and F). However, the Limiting Case models simulated a greater increase in the maximum proportion of ppFus3 at this time point than the Combined model. The Combined model was only able to simulate the qualitative changes in the proportion of each Fus3 phosphorylation state at the later time points. Alternatively, the Limiting Case #1 model was able to simulate quantitative changes in the proportion of each Fus3 phosphorylation state that better aligned to the corresponding experimental data than the Combined and Limiting Case #2 models. The failure of all models to reproduce the increase in the maximum proportion of ppFus3 at the three minute time point suggested that distributive Fus3 dephosphorylation had a partial positive impact on ppFus3 phosphorylation after early exposure to mating pheromone. We hypothesize that this positive impact is caused by rapid dephosphorylation of pFus3, which serves as a negative regulator of ppFus3. The Combined model accounts for the negative impact of ppFus3 and pFus3 by including differential rates of ppFus3 phosphorylation for Ste5 and pSte5m. However the model does not account for differences in the affinity of each Fus3 phosphorylation state to phosphorylate Ste5. Therefore we concluded that because the Combined model did not account for these differences, the model was unable to reproduce the differences in each Fus3 phosphorylation state in ptp2/3∆ cells. Alternatively, the Limiting Case #1 model assumed that phosphorylated Ste5 phosphorylated npFus3 at the same rate as Ste5. Because this model was able to capture all aspects of the wild type Fus3 phosphorylation time course data, (refer to panel D of Fig. B.5), we hypothesized that the basal dephosphorylation rate compensated for the loss of negative regulation conferred by phosphorylated Ste5m in wild type cells. As a result, the loss of this basal dephosphorylation mechanism resulted in an increase in the proportion of ppFus3 at the early time points under the ptp2/3∆ conditions. Similarly, the Limiting Case #2 model assumed that phosphorylated Ste5m had no impact on the ppFus3 or pFus3 dephosphorylation rate. As a result Fus3 dephosphorylation was dependent on the basal dephosporylation rate (kb) and Msg5. Because this model was also able to capture all aspects of the wild type Fus3 phosphorylation time course data, (refer to panel F of Fig. B.5), we hypothesized that the basal dephosphorylation rate compensated for the loss of increased ppFus3 and pFus3 dephosphorylation that was dependent on phosphorylated Ste5 in wild type cells. As a result, the loss of this basal dephosphorylation mechanism again resulted in an increase in the proportion of ppFus3 at the early time points under the ptp2/3∆ conditions.

94 Aptp2/3 B ptp2/3 Pheromone Combined 100 Fus3 80 pFus3 Ste5c Ste5m p-Ste5m ppFus3 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Msg5 Time (Minutes ) C D Pheromone Limiting Case #1 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 Time Minutes 0 Msg5 ( ) E F Pheromone Limiting Case #2 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 Time Minutes 0 Msg5 ( ) Figure B.7: Phosphorylated Ste5 contributes to Fus3 dephosphorylation (A, C, E) Diagrams of the Combined, Limiting Case #1, and Limiting Case #2 processive MAPK phosphorylation models. (B, D, F) Fus3 phosphorylation time series for cells in which the Ptp2 and Ptp3 phosphatases were deleted ptp2/3∆. Cells were treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) are quantiﬁed as a percentage of total Fus3, (ﬁlled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Model simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

We next visualized each model’s best ﬁt to Fus3 phosphorylation data generated in cells in which the Msg5 phosphatase was deleted, (msg5∆, Fig. B.8). Under these conditions, our Phos-tag analysis

95 revealed that the proportion of ppFus3 was again maximal at the three minute time point, and that the maximum proportion of ppFus3 increased from approximately 40% to 60% compared to wild type. Similar to wild type, the proportion of ppFus3 then decreased at the five and ten minute time points. However unlike wild type, the proportion of ppFus3 increased at the fifteen minute time point. The increase in the proportion of ppFus3 at three minutes was accompanied by a decrease in the proportion of npFus3 from approximately 40% to 20% compared to wild type. Similar to wild type, the proportion of npFus3 increased at the five and ten minute time points, and then decreased at the fifteen minute time point. Interestingly, we again did not observe a significant change in the proportion of pFus3 when compared to the wild type experiment. While the proportion of ppFus3 was increased when compared to pFus3 at the ten and fifteen minute time points, we noted that similar to Fus3 phosphorylation dynamics under the ste5ND conditions, the proportion of ppFus3 was not greater than the proportion of npFus3 at ten and fifteen minute time points in msg5∆ cells. To simulate the deletion of Msg5 in each model, we eliminated the influence of Msg5 dependent ppFus3 and pFus3 dephosphorylation by setting the Msg5 dependent dephosphorylation rate parameter, kms = 0. Under this condition, we observed that all models reproduced the increase in the maximum proportion of ppFus3 at the three minute time point (see the simulation curves in Fig. B.8, B, D, and F). However, the Combined model simulated a greater increase in the maximum proportion of ppFus3 at the three minute time point than the Limiting Case models. The Combined and Limiting Case #1 models were also able to simulate the qualitative changes in the proportion of each Fus3 phosphorylation state at the later time points. Alternatively, the Limiting Case #2 model was not able to simulate a greater proportion of npFus3 than ppFus3 at the ten minute time point. The success of all models to reproduce Fus3 phosphorylation dynamics in msg5∆ cells, and the failure of the models to reproduce Fus3 phosphorylation dynamics in ptp2/3∆ further demonstrated that the Fus3 phosphorylation state was regulated by multiple phosphatases. However, because the loss of Msg5 in cells and in the simulations resulted in a greater increase in the proportion of ppFus3, and because the increase persisted throughout the time course, we concluded that Msg5 was a stronger regulator of the Fus3 phosphorylation state than the Ptp2 and Ptp3 combination.

96 Amsg5 B msg5 Pheromone Combined 100 Fus3 pFus3 80 ppFus3 Ste5c Ste5m p-Ste5m 60 (%) 40

np-Fus3 pp-Fus3 Fus3 20 0 p-Fus3 0 5 10 15 0 Time (Minutes ) C D Pheromone Limiting Case #1 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40

np-Fus3 pp-Fus3 Fus3 20 0 p-Fus3 0 5 10 15 0 Time (Minutes ) E F Pheromone Limiting Case #2 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40

np-Fus3 pp-Fus3 Fus3 20 0 p-Fus3 0 5 10 15 0 Time (Minutes ) Figure B.8: Fus3 induces a negative feedback loop through Msg5 (A, C, E) Diagrams of the Combined, Limiting Case #1, and Limiting Case #2 processive MAPK phosphorylation models. (B, D, F) Fus3 phosphorylation time series for cells in which the Msg5 phosphatase was deleted msg5∆. Cells were treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) are quantiﬁed as a percentage of total Fus3, (ﬁlled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Model simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

B.3.4 Fus3 regulates the sensitivity of yeast to mating pheromone through feedback phosphorylation of Ste5 and induction of Msg5 To test the validity of the models, we used each model to predict Fus3 phosphorylation dynamics in a strain that contained the nondocking Ste5 mutant and that also lacked the Msg5 phosphatase (ste5ND & msg5∆, ﬁgure B.9). Under these conditions, our Phos-tag analysis revealed that the

97 proportion of ppFus3 was greater than wild type at all time points, and that the proportion of ppFus3 was now maximal at the five minute time point. Additionally, we observed that the proportion of ppFus3 at the five minute time point increased from approximately 25% to 60% compared to wild type cells. While the proportion of ppFus3 decreased at the five, and ten minute time points in wild type cells, we observed only a minimal reduction in the proportion of ppFus3 at the five minute time point, and no change in the proportion of ppFus3 at the ten and fifteen minute time points. The increase in the proportion of ppFus3 over the time course was accompanied by a decrease in the proportion of npFus3 when compared to wild type. Unlike wild type, the proportion of npFus3 did not increase at the five, ten, and fifteen minute time points. Additionally, we observed a reduction in the proportion of pFus3 at the two, three, and five minute time points when compared to the wild type proportions. However the proportion of pFus3 increased at the ten and fifteen minute time points. To simulate these conditions in each model, we reduced the rate at which phosphorylated Fus3 (pFus3, and ppFus3) feedback phosphorylated Ste5 by 90%, and also eliminated the influence of Msg5 dependent ppFus3 and pFus3 dephosphorylation by setting the Msg5 dependent dephosphorylation rate parameter, kms = 0. Under these conditions, we observed that all models reproduced the increase in the maximum proportion of ppFus3 at the three and five minute time points (see the simulation curves in figure B.9, B, D, and F). All models then simulated a reduction in the proportion of ppFus3 at the the ten minute time point. However, only the Combined model reproduced was able to simulate an increase in the proportion of ppFus3 at the fifteen minute time point. The success of the Combined model to predict the temporal dynamics of each Fus3 phosphorylation state generated in cells that contained the nondocking Ste5 mutant and that also lacked the Msg5 phosphatase, (ste5ND & msg5∆), provided the final evidence that Combined model was preferred over each Limiting Case model. Our next goal was to determine whether this processive MAPK phosphorylation model would be preferred over a “true” distributive MAPK phosphorylation model.

98 Aste5 ND & msg5 B ste5 ND & msg5 Pheromone &RPELQHG 100 Fus3 pFus3 80 ppFus3 Ste5c Ste5m p-Ste5m 60 (%) 40

np-Fus3 pp-Fus3 Fus3 20 p-Fus3 0 0 5 10 15 0 Time (Minutes ) C D Pheromone 100 /LPLWLQJ&DVH 80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 0 p-Fus3 0 5 10 15 0 Time (Minutes ) E F Pheromone /LPLWLQJ&DVH 100

80 Ste5c Ste5m p-Ste5m 60 (%) 40 np-Fus3 pp-Fus3 Fus3 20 0 p-Fus3 0 5 10 15 0 Time (Minutes ) Figure B.9: A processive MAPK phosphorylation model aligns with the ste5ND & msg5∆ experimental data (A, C, E) Diagrams of the Combined, Limiting Case #1, and Limiting Case #2 processive MAPK phosphorylation models. (B, D, F) Fus3 phosphorylation time series for cells in which the binding of Fus3 to Ste5 was inhibited and in which the Msg5 phosphatase was deleted ste5ND & msg5∆. Cells were treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) are quantiﬁed as a percentage of total Fus3, (ﬁlled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Model simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

B.3.5 The processive model is preferred over the distributive model To conﬁrm our assumptions that Fus3 phosphorylation occurs through a processive mechanism, whereas dephosphorylation follows distributive kinetics, we tested a model that allowed for the possibility of distributive (sequential) mechanism for Fus3 phosphorylation. We observed that this model was equally able to simulate Fus3 phosphorylation dynamics as the Combined model under

99 the wild type, ste5ND, and msg5∆ conditions, (figure , B, C, and E). However similar to all other models, the distributive model was unable to account for Fus3 phosphorylation dynamics when Ptp2 and Ptp3 were deleted. We then used the distributive model to simulate Fus3 phosphorylation under the ste5ND & msg5∆ conditions. We observed that the Distributive model over-estimated the increase in the proportion of ppFus3, and under-estimated the proportion of npFus3 over the duration of the time course (figure , F). The failure of the Distributive model to predict Fus3 phosphorylation dynamics under the ste5ND & msg5∆ conditions confirmed that the Combined processive model was the preferred model.

100 AB Pheromone Wildtype 100 Fus3 pFus3 80 ppFus3 Ste5c Ste5m p-Ste5m 60 (%) 40

np-Fus3 p-Fus3 pp-Fus3 Fus3 20 0 0 0 5 10 15 Time (Minutes ) CD ste5 ND ptp2 /3 100 100 80 80 60 60 (%) (%) 40 40 Fus3 Fus3 20 20 0 0 0 5 10 15 0 5 10 15 Time (Minutes ) Time (Minutes ) EF msg5 ste5 ND & msg5 100 100 80 80 60 60 (%) (%) 40 40 Fus3 Fus3 20 20 0 0 0 5 10 15 0 5 10 15 Time (Minutes ) Time (Minutes ) Figure B.10: A processive MAPK phosphorylation model outperforms a Distributive model (A) Diagram of the Distirbutive MAPK phosphorylation models. (B Fus3 phosphorylation time series for wild type cells, (C) cells in which the binding of Fus3 to Ste5 was inhibited ste5ND, (D) cells in which the Ptp2 and Ptp3 phosphatases were deleted ptp2/3∆, (E) cells in which the Msg5 phosphatase was deleted msg5∆ msg5∆, (F) cells in which the binding of Fus3 to Ste5 was inhibited and in which the Msg5 phosphatase was deleted ste5ND & msg5∆. Cells were treated with 10 µM α-factor and resolved by Phos-tag SDS-PAGE and immunoblotted with Fus3 antibodies. Dual-phosphorylated (ppFus3), mono-phosphorylated (pFus3) and non-phosphorylated (npFus3) are quantiﬁed as a percentage of total Fus3, (ﬁlled circles, see legend). Error bars are ± Standard Error of the Mean, where n ≥ 3. Model simulations for the indicated Fus3 phosphorylation state are shown via solid curves versus the corresponding experimental data points. (This data was originally published in [77]).

We next, investigated the Fus3 phosphorylation rates identified that allowed the Distributive model to reproduce the Fus3 phosphorylation training data sets. We observed that in order to generate a good fit to the data, this model required that the second phosphorylation event to occur substantially faster (by three orders of magnitude) than the first phosphorylation event. Thus, this model effectively

101 reduced to a processive (single step) mechanism, justifying our use of processive phosphorylation model.

Combined 25 Limiting Case #1 Limiting Case #2 Distributive 20 15

SSD 10 5 0 Final 40K iterations Figure B.11: A processive model is preferred over the Distributive model. The performance of all three processive models and the distributive model as measured by the sum of squared deviations (SSD, y-axis, value× 103). Each line shows the SSD value for each model over the ﬁnal 40,000 iterations of the Markov Chain Monte Carlo parameter estimation algorithm.

B.4 Discussion

In this work, our goal was to understand how three MAPK phosphorylation states were regulated over time. We coupled a novel biochemical assay with mathematical models to quantitiatively investigate the impact various regulatory mechanisms on the phosphorylation state of Fus3 after pheromone induced activation. Our study was motivated in part, by the apparent contradictory roles of mono-phosphoyrlated Fus3 versus dual-phosphorylated Fus3. Complete ablation of Fus3 appeared to have no impact on the ﬁdelity of the mating pheromone response, as measured by a downstream transcriptional reporter. However, exogenous addition of a Fus3 mutant that could only be activated on either phosphorylation loop by Ste7 had an antagonistic affect on the mating pheromone response. Similarly, exogenous addition of a Fus3 mutant that could not be phosphorylated resulted in a more diminished trasncriptional response than complete ablation of Fus3. Given these results, we used Phos-tag analysis [3] to measure the dynamics of each Fus3 phosphorylation state, during the early time points of the mating pheromone response. Interesting, we observed that dual-phosphorylated Fus3 accumulated at a fast rate than mono-phosphorylated Fus3. We used mathematical models to determine which of two general phosphorylation mechanisms best explained this phenomena. We discovered that Fus3 phosphorylation was be described by a processive phosphorylation mechanism. We also tested several variations of the processive model to determine which other regulatory mechanism were required to explain the dynamics of each Fus3 phosphorylation state. We discovered that the best model also included explicit mechanism of regulation conferred by Ste5 and the Msg5 phosphatase. Collectively, our mathematical model analysis, and biochemical analyses demonstrated that the Fus3 phosphorylation state was dynamically regulated by several synergistic mechanisms. The scaffold Ste5 down-regulated the Fus3 dual-phosphorylated rate, Msg5 conferred substantial down- regulation of the Fus3 phosphorylated. While Fus3 itself up-regulated its dual-phosphorylation rate via positive feedback, and down-regulated its dual phosphorylation through induction of Msg5.

102 MAPK are ubiquitous in all eukaryotes. Therefore we propose that these regulatory mechanisms are likely also conserved across eukaryotes. If these regulatory mechanisms are conserved, then propose that each has the potential to be further developed to up- or down-regulate aberrantly signaling MAPK pathways in cancer, and MAPK-mediated pathologies.

Parameter Value (Combined) Description k1 1.15 × 101 [Ste5c] Activation k2 1.25 × 10−4 [Ste5m] Deactivation k2p 1.66 × 10−2 [pSte5m] Deactivation k3 2.73 × 10−4, 4.09 × 10−5 [Ste5m] Feedback Phosphorylation k3p 4.81 × 10−2, 7.21 × 10−3 [Ste5m] Feedback Phosphorylation kb 1.29 × 10−4 Fus3 Dephosphorylation k4 4.36 × 10−1 [pSte5m] Dephosphorylation k5 1.15 [Fus3] Synthesis k5p 1.7 × 10−4 [Fus3] Synthesis k6 2.64 × 10−3 [Fus3] Degradation k7 1.43 × 10−1 [ppFus3] Phosphorylation k7p 9.13 × 10−2 [ppFus3] Feedback Phosphorylation k8 6.01 × 10−4 [ppFus3] Basal Dephosphorylation k9 1.16 × 10−3 [pFus3] Basal Dephosphorylation k10 5.19 × 10−4 [Fus3] Synthesis k11 5.16 × 10−3 [Msg5] Synthesis k12 1.09 × 10−4 [Msg5] Degradation kms 2.04 × 10−4, 0. [Msg5] Mediated Dephosphorylation kamsg5 1.06 × 10−4 [Msg5] Activation K1 2.5 × 101 Michaelis Constant K2 3.1 Michaelis Constant K3 4.12 Michaelis Constant K4 1.07 Michaelis Constant Kmsg5 1.62 Michaelis Constant

Table B.1: Parameter values for the Combined model.

The name, numerical value and function of each rate parameter in the Combined model. The rate parameters k3 and k30 are shown for the wild type (left), ste5ND (right), and ste5ND & msg5∆ (right) models.

103 Model Score k1 k2 1540 350 1520 150 300 250 1500 100 200 1480 score 150 2 Count 1460 Count Χ 50 100 1440 50 1420 0 0 0 10 20 30 40 17.5 18.0 18.5 19.0 19.5 20.0 0.000100.000150.000200.000250.000300.00035 Parameter Iterations H105L Parameter Value Parameter Value

k3 k3p kb 400 150 600 300 100 400 200 Count Count Count 50 200 100

0 0 0 0.1 0.2 0.3 0.4 0.0005 0.0010 0.0015 0.0020 0.00010 0.00015 0.00020 0.00025 0.00030 Parameter Value Parameter Value Parameter Value

k4 k5 k5p 600 150 150 500 400 100 100 300 Count Count Count 50 50 200 100 0 0 0 2 4 6 8 10 12 1.15 1.20 1.25 1.30 1.35 1.40 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 Parameter Value Parameter Value Parameter Value

k6 k7 k7p 150 200 100 80 150 100 60 100 Count Count 40 Count 50 50 20 0 0 0 0.00210.00220.00230.00240.00250.00260.00270.0028 0.080 0.085 0.090 0.095 0.022 0.024 0.026 0.028 Parameter Value Parameter Value Parameter Value

k8 k9 k10 140 100 500 120 400 100 80 80 60 300 60 Count Count 40 Count 200 40 100 20 20 0 0 0 0.00032 0.00034 0.00036 0.00038 0.00040 0.00065 0.00070 0.00075 0.00080 0.00085 0.00090 0.000 0.001 0.002 0.003 0.004 0.005 0.006 Parameter Value Parameter Value Parameter Value

k11 k12 kms 350 140 200 300 120 150 250 100 200 80 100 150 60 Count Count Count 100 40 50 50 20 0 0 0 0.000 0.001 0.002 0.003 0.004 0.005 0.0005 0.0010 0.0015 0.002 0.004 0.006 0.008 0.010 Parameter Value Parameter Value Parameter Value

kamsg5 K1 K2 250 500 200 200 400 150 150 300 100 100 200 Count Count Count 50 50 100

0 0 0 0.005 0.010 0.015 0.020 22.0 22.5 23.0 23.5 24.0 24.5 25.0 2 4 6 8 10 12 14 16 Parameter Value Parameter Value Parameter Value

K3 K4 Kmsg5 350 400 300 300 250 250 300 200 200 200 150 150 Count Count Count 100 100 100 50 50 0 0 0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5 5 10 15 Parameter Value Parameter Value Parameter Value

Figure B.12: Histograms of parameter in the Combined Model

104 Parameter Value (Limiting Case #1) Description k1 2.04 × 101 [Ste5c] Activation k2 5.89 × 10−3 [Ste5m] Deactivation k3 1.42 × 10−5, 2.14 × 10−6 [Ste5m] Feedback Phosphorylation k3p 4.5 × 10−4, 6.76 × 10−5 [Ste5m] Feedback Phosphorylation kb 1.58 × 10−3 Fus3 Dephosphorylation k4 5.69 × 101 [pSte5m] Dephosphorylation k5 1.56 [Fus3] Synthesis k5p 0. [Fus3] Synthesis k6 2.9 × 10−3 [Fus3] Degradation k7 6.94 × 10−2 [ppFus3] Phosphorylation k7p 6.94 × 10−2 [ppFus3] Feedback Phosphorylation k8 4.61 × 10−2 [ppFus3] Basal Dephosphorylation k9 9.47 × 10−2 [pFus3] Basal Dephosphorylation k10 2.01 × 10−4 [Fus3] Synthesis k11 2.4 × 10−3 [msg5] Synthesis k12 7.25 × 10−4 [msg5] Degradation kms {1.8 × 10−3., 0.} [msg5] Mediated Dephosphorylation kamsg5 1.55 × 10−4 [msg5] Activation K1 2.39 × 101 Michaelis Constant K2 1.19 Michaelis Constant K3 1.11 Michaelis Constant K4 1.95 Michaelis Constant Kmsg5 1.98 Michaelis Constant

Table B.2: Parameter values for the Limiting Case #1 model.

105 Model Score Limiting Case ð1 k1 k2 350 350 300 2600 300 250 250 200 200 2550 150 score 150 Count Count 2

Χ 100 100 50 50 2500 0 0 17 18 19 20 21 22 0.00450.00500.00550.00600.00650.00700.00750.0080 0 10 20 30 40 50 Parameter Value Parameter Value Parameter Iterations H105L

k3 k3p kb 500 600 500 400 500 400 400 300 300 300 200 Count Count 200 Count 200 100 100 100 0 0 0 0.00010 0.00015 0.00020 0.00025 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 Parameter Value Parameter Value Parameter Value

k4 k5 k5p 500 800 300 400 600

300 200 400

Count 200 Count Count 100 200 100

0 0 0 0 20 40 60 80 100 1.4 1.5 1.6 1.7 0.0001 0.0002 0.0003 0.0004 0.0005 Parameter Value Parameter Value Parameter Value

k6 k7 k7p 600 400 400 500 400 300 300

300 200 200 Count 200 Count Count 100 100 100 0 0 0 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 0.0036 0.060 0.065 0.070 0.075 0.080 0.085 0.060 0.065 0.070 0.075 0.080 0.085 Parameter Value Parameter Value Parameter Value

k8 k9 k10 800 300 300 600 200 200 400 Count Count Count 100 100 200

0 0 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.000 0.001 0.002 0.003 0.004 0.005 0.006 Parameter Value Parameter Value Parameter Value

k11 k12 kms 1000 1000 500

800 800 400

600 600 300

Count 400 Count 400 Count 200

200 200 100

0 0 0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.000 0.002 0.004 0.006 0.008 0.010 Parameter Value Parameter Value Parameter Value

kamsg5 K1 K2 700 600 600 600 500 500 400 400 400 300 300 Count Count Count 200 200 200 100 100 0 0 0 0.00000.00020.00040.00060.00080.00100.00120.0014 20 21 22 23 24 25 0 5 10 15 Parameter Value Parameter Value Parameter Value

K3 K4 Kmsg5 1000 600 1000 500 800 800 400 600 600 300 400 Count Count Count 400 200 100 200 200 0 0 0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0 5 10 15 20 25 0 5 10 15 20 Parameter Value Parameter Value Parameter Value

Figure B.13: Histograms of parameter in the Limiting Case #1 Model k7p was set equal to k7 during parameter estimation and ﬁgure generation.

106 Parameter Value Description k1 2.29 × 101 [Ste5c] Activation k2 1.72 × 10−2 [Ste5m] Deactivation k3 1.01 × 10−4, 5.07 × 10−6 [Ste5m] Feedback Phosphorylation k3p 3.29 × 10−4, 1.65 × 10−5 [Ste5m] Feedback Phosphorylation kb 2.87 × 10−3 Fus3 Dephosphorylation k4 4.16 × 101 [pSte5m] Dephosphorylation k5 1.23 [Fus3] Synthesis k5p 1.63 × 10−4 [Fus3] Synthesis k6 2.47 × 10−3 [Fus3] Degradation k7 8.05 × 10−2 [ppFus3] Phosphorylation k7p 7.67 × 10−2 [ppFus3] Feedback Phosphorylation k8 0. [ppFus3] Basal Dephosphorylation k9 0. [pFus3] Basal Dephosphorylation k10 1.08 × 10−4 [Fus3] Synthesis k11 6.27 × 10−4 [msg5] Synthesis k12 1.17 × 10−4 [msg5] Degradation kms 1.37 × 10−3, 0. [msg5] Mediated Dephosphorylation kamsg5 1.21 × 10−4 [msg5] Activation K1 2.49 × 101 Michaelis Constant K2 3.69 Michaelis Constant K3 1.03 Michaelis Constant K4 6.9 Michaelis Constant Kmsg5 2.26 Michaelis Constant

Table B.3: Parameter values for the Limiting Case #2 model.

107 Model Score Limiting Case ð2 k1 k2 2640 350 250 2620 300 200 250 2600 150 200 2580

score 150 Count Count

2 100

Χ 2560 100 50 50 2540 0 0 2520 20 21 22 23 24 25 26 0.014 0.015 0.016 0.017 0.018 0.019 0.020 0.021 0 10 20 30 Parameter Value Parameter Value Parameter Iterations H105L

k3 k3p kb 500 600 600 500 400 400 300 400 300 200 Count Count Count 200 200 100 100 0 0 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.00000.00020.00040.00060.00080.00100.00120.0014 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 0.0036 Parameter Value Parameter Value Parameter Value

k4 k5 k5p 300 600 250 400 500 200 300 400 150 300 200 Count Count Count 100 200 100 50 100 0 0 0 0 20 40 60 80 100 1.10 1.15 1.20 1.25 1.30 1.35 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 Parameter Value Parameter Value Parameter Value

k6 k7 k7p 350 400 400 300 250 300 300 200 200 200 150 Count Count Count 100 100 100 50 0 0 0 0.00220.00240.00260.00280.00300.00320.00340.0036 0.070 0.075 0.080 0.085 0.090 0.095 0.065 0.070 0.075 0.080 0.085 0.090 Parameter Value Parameter Value Parameter Value

k8 k9 k10 800 1000 1000

800 800 600

600 600 400

Count 400 Count 400 Count 200 200 200

0 0 0 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 0.000 0.001 0.002 0.003 0.004 0.005 Parameter Value Parameter Value Parameter Value

k11 k12 kms 800 800 400 600 600 300 400 400 200 Count Count Count 200 200 100

0 0 0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0002 0.0004 0.0006 0.0008 0.000 0.002 0.004 0.006 0.008 0.010 Parameter Value Parameter Value Parameter Value

kamsg5 K1 K2 1000 600 350 300 800 500 250 400 600 200 300 400 150 Count Count Count 200 100 200 100 50 0 0 0 0.000 0.001 0.002 0.003 0.004 24.0 24.2 24.4 24.6 24.8 25.0 0 5 10 15 Parameter Value Parameter Value Parameter Value

K3 K4 Kmsg5 500 350 800 400 300 250 600 300 200 400 200 150 Count Count Count 100 100 200 50 0 0 0 1.0 1.1 1.2 1.3 1.4 0 5 10 15 0 5 10 15 20 25 Parameter Value Parameter Value Parameter Value

Figure B.14: Histograms of parameter in the Limiting Case #2 Model

108 Parameter Value Description k1 3.25 × 101 [Ste5c] Activation k2 1.12 × 10−4 [Ste5m] Deactivation −4 k2p 1.11 × 10 [p-Ste5]m Deactivation k3 3.52 × 10−3, 1.76 × 10−4 [Ste5m] Feedback Phosphorylation k3p 1.31 × 10−4, 6.55 × 10−6 [Ste5m] Feedback Phosphorylation kb 1.31 × 10−4 Fus3 Dephosphorylation k4 2.04 × 10−1 [pSte5m] Dephosphorylation k5 7.63 × 10−1 [Fus3] Synthesis k5p 0. [Fus3] Synthesis k6 2.12 × 10−3 [Fus3] Degradation k7 8.04 × 10−2 [ppFus3] Phosphorylation k7p 2.73 × 10−2 [ppFus3] Feedback Phosphorylation k8 8.58 × 10−4 [ppFus3] Basal Dephosphorylation k9 1.4 × 10−3 [pFus3] Basal Dephosphorylation k10 2.77 × 10−4 [Fus3] Synthesis k11 1.72 × 10−3 [msg5] Synthesis k12 2.79 × 10−4 [msg5] Degradation kms 9.87 × 10−3, 0. [msg5] Mediated Dephosphorylation kamsg5 1.36 × 10−3 [msg5] Activation K1 2.1 × 101 Michaelis Constant K2 2.1 Michaelis Constant K3 1.5 Michaelis Constant K4 2.24 × 101 Michaelis Constant Kmsg5 2.27 Michaelis Constant k7pp 1.95 × 101 Distributive Phosphorylation k7ppm 9.84 × 101 Michaelis Constant Table B.4: Parameter values for the Distributive model.

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