# Resolving Early Signaling Events in T-Cell Activation Leading to IL-2 and FOXP3 Transcription

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Mathematical Model: Overview and Scope

**Figure 1.**Reaction network of the T-cell activation model. Solid arrows denote reactions for which the forward and reverse directions are indicated; dashed arrows connect reactions (either forward or reverse) with their catalysts. Colored arrows denote reactions that are catalyzed by specific species as indicated. Symbols and reactions are described in Appendix 1.

#### 2.2. Global Sensitivity Analysis

#### 2.3. Parameter Identification

#### 2.4. Experimental Datasets for Model Development

Species | Variable(s) | Experiment Type | Cell Type | Figure | Source(s) |
---|---|---|---|---|---|

TCR | αCD3 | Dynamic dose response | Jurkat | 2 | [45,46] |

Zap70 | αCD3 | Dose response | Jurkat | 4 | [40] |

Erk | Sanguinarine, U0126 | Dynamic dose response | Jurkat | 5 | [41] |

Ca${}^{2+}$ | αCD3 | Dynamics | Jurkat | 6 | Appendix 3 |

IκBα | αCD3+αCD28 | Dynamics | Jurkat | 7 | Appendix 4 |

Akt, PKCθ | αCD3, αCD28 | Input response | CD4+ T-cells (murine) | 8 | [32] |

## 3. Model Development

#### 3.1. Modeling TCR Trafficking

**Figure 2.**Resting and ligand-induced (

**A**) surface TCR expression and internalization and (

**B**) degradation activity in response to varying doses of αCD3 stimulation (data shown in [45]).

#### 3.2. Tuning the Roles of CD45 and SHP1

**Figure 3.**Representation of the phosphatase CD45 in the model. The model states $CD{45}^{p}$ and $CD{45}_{n}$ are represented by the regions outlined in blue and red, respectively. $CD{45}_{n}^{p}$, which is the intersection between the two modeled states outlined in purple, has both positive and negative roles, but is not explicitly modeled. $CD{45}_{tr}$ represents completely inactive CD45 that is translocated away from its substrates and is also not explicitly modeled.

**Figure 4.**Zap70-Y319 phosphorylation in response to various doses of αCD3 stimulation in the absence of Erk feedback. Jurkat cells were incubated in the presence of a Mek1/2 inhibitor (2 μg/mL U0126) and stimulated with αCD3 at the indicated concentrations. Samples were taken 5 min post-stimulation and analyzed by western blot. The data shown are the means and standard errors from at least three independent experiments (data source: [40]).

#### 3.3. Tuning Erk Signaling

**Figure 5.**Erk activation in response to various doses of (

**A**) MKP inhibitor sanguinarine and (

**B**) Mek1/2 inhibitor U0126. Doses of sanguinarine and U0126 were administered at 15 and 6 min post-stimulation (10 μg/mL αCD3), respectively. Samples of phospho-Erk were taken at indicated times relative to the inhibitor doses and measured via western blot. The data shown are the means and standard errors from at least three independent experiments (data source: [41]).

#### 3.4. Modeling Calcium Signaling

**Figure 6.**Intracellular calcium release in response to stimulation with 10 μg/mL αCD3. The data shown are the means and standard errors from 12 independent experiments.

#### 3.5. Modeling CD28 Costimulation

#### 3.6. Modeling NFκB Signaling

**Figure 7.**IκBα phosphorylation following 2 μg/mL αCD3 and αCD28 costimulation. The data shown are quantified western blot data from one experiment.

#### 3.7. Modeling mTOR Signaling

**Figure 8.**T-cell signaling in response to doses of αCD3 and αCD28 as measured by (

**A**) P-Akt and (

**B**) P-PKCθ. CD4+ T-cells were stimulated (40 min) with 0.5 mg/mL plate-bound αCD3, 2.5 mg/mL of soluble αCD28 or both. Bar graphs quantify phosphorylation of Akt and PKC, with each sample normalized to the level of unphosphorylated protein in one experiment representative of three replicates (data source: [32]).

## 4. Results and Discussion

#### 4.1. Model Corroboration

#### 4.1.1. Experimental Datasets for Model Corroboration

Species | Variable(s) | Experiment Type | Cell Type | Figure | Source |
---|---|---|---|---|---|

Ca${}^{2+}$ | αCD3, αCD28, PMA, ionomycin | Dynamic input response | Jurkat | 9 | [27] |

NFAT | Cyclosporin A | Dose response | Jurkat | 9 | [53] |

NFAT, AP1, NFκB | αCD3, αCD28, PMA | Input response | Jurkat | 10 | [27] |

Lck, Erk | CD45 | Knockdown | Murine DPthymocytes | 11 | [55] |

Ca${}^{2+}$ | CD45 | Knockout | Murine DP thymocytes | 11 | [56] |

IL-2 | SHP1, PMA, aTCR, ionomycin | Input response, Knockdown | Jurkat | 12 | [49] |

#### 4.1.2. Corroboration of Signaling Events and Transcription Factor Activation

**Figure 9.**Intracellular calcium and NFAT signaling in response to various combinations of stimuli. (

**A**) Jurkat cells were stimulated as indicated (αCD3, αCD28, ionomycin: 1 μg/mL; PMA: 10 ng/mL), and intracellular Ca${}^{2+}$ release was monitored over time (data sampled from [27]). Solid lines represent corresponding model simulations. (

**B**) NFAT activity in response to 10 μg/mL αCD3 stimulation and varying doses of calcineurin inhibitor cyclosporin A (CsA). The data are measured at 30 min post-stimulation (data sampled from [53]).

**Figure 10.**Activity of transcription factors (

**A**) NFAT, (

**B**) NFκB and (

**C**) AP1, as well as (

**D**) synthesis of IL-2 in response to various combinations of stimuli (αCD3, αCD28: 1 μg/mL; PMA: 10 ng/mL). Activity was measured 15 min post-stimulation (data sampled from [27]). Left and right axes correspond to model simulations and measured relative absorbance values, respectively.

#### 4.1.3. Corroboration of CD45 Activity

**Figure 11.**Downstream TCR signaling in response to CD45 knockdown. (

**A**) Model simulations of Lck phosphorylation at the negative regulatory residue Y505 as a function of CD45 activity (data sampled from [55]). (

**B**) Model simulations of Erk phosphorylation 3 min after stimulation with 0 or 50 μg/mL αCD3 as a function of CD45 activity (data sampled from [55]). (

**C**) Model simulations of intracellular calcium release over time in wild-type (WT) and CD45 knockdown mutant (5% of WT activity) stimulated with 10 μg/mL αCD3 at 90 s as a function of CD45 activity (data sampled from [56]).

#### 4.1.4. Corroboration of SHP1 Activity

#### 4.2. Weak CD28 Costimulation Predicted to Elevate FOXP3 Transcription

**Figure 12.**IL-2 reporter stimulation in response to SHP1 knockdown. The model simulates the wild-type and SHP1 knockdown mutant (C453S mutation, resulting in catalytically inactive SHP1) stimulated with 10 μg/mL αCD3. IL-2 reporter stimulation was measured 2 h post-stimulation (data sampled from [49]).

**Figure 13.**(

**A**) IL-2 and (

**B**) FOXP3 transcription in response to stimulation by combinations of αCD3 and αCD28. Results show outputs 30 min after stimulation.

#### 4.3. Reduced CD45 Activity Predicted to Elevate FOXP3 Transcription

**Figure 14.**Model response to various CD45- and SHP1-knockdown scenarios. (

**A**) Intracellular calcium; (

**B**) Erk; (

**C**) PKCθ; (

**D**) Akt; (

**E**) IL-2; and (

**F**) FOXP3 transcription factor activation in systems with various levels of CD45 and SHP1 downregulation. Note that downregulation is shown as percentages of wild-type activity (i.e., 100% corresponds to normal function, 0% corresponds to full knockout). Results show outputs 30 min after stimulation by 10 μg/mL αCD3 and αCD28.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix 1: Model Equations

State | Rate Equation | Biological Meaning | Initial | Total | Source(s) |
---|---|---|---|---|---|

TCR${}_{b}$ | ${R}_{TC{R}_{lig}}-{R}_{TC{R}_{p}}-{R}_{iTC{R}_{b}}$ | Ligand-bound TCR | 0 | $2\times {10}^{5}$ | [40] |

TCR${}_{p}$ | ${R}_{TC{R}_{p}}-{R}_{iTC{R}_{p}}$ | Phosphorylated TCR-ζ chain | 0 | $2\times {10}^{5}$ | [40] |

TCR${}_{i}$ | ${R}_{iTC{R}_{b}}+{R}_{iTC{R}_{p}}+{R}_{iTC{R}_{f}}-{R}_{TC{R}_{exo}}-{R}_{TC{R}_{deg}}$ | Internalized TCR | $2\times {10}^{4}$ | $2\times {10}^{5}$ | [40,46] |

TCR${}_{deg}$ | ${R}_{TC{R}_{deg}}-{R}_{TC{R}_{synth}}$ | Degraded TCR | $2\times {10}^{4}$ | $2\times {10}^{5}$ | [40,46] |

Zapb | ${R}_{1}-{R}_{8}$ | Protein tyrosine kinase Zap70 bound to the phosphorylated TCR-ζ chain | 0 | $9.3\times {10}^{4}$ | [40] |

Zap* | ${R}_{8}-{R}_{9}$ | Activated Zap70 (phosphorylated at Y493 in the activation loop) | 0 | $9.3\times {10}^{4}$ | [40] |

Zapp | ${R}_{9}-{R}_{4}+{R}_{5}$ | Doubly phosphorylated Zap70 (at Y493 and Y319) | $9\times {10}^{3}$ | $9.3\times {10}^{4}$ | [40] |

SFKdp | ${R}_{2}-{R}_{3}-{R}_{4}-{R}_{20}$ | Src family kinases (including Lck and Fyn) with dephosphorylated inhibitory site (Y505 on Lck) | 100 | $1\times {10}^{5}$ | [40] |

SFKdp-Zapp | ${R}_{4}-{R}_{4a}$ | Dephosphorylated SFK bound to pY319 of Zap70 | 0 | $9.3\times {10}^{4}$ | [40] |

SFKdpS59p | ${R}_{20}-{R}_{22}$ | Dephosphorylated SFK phosphorylated at serine-59 by activated Erk | 0 | $1\times {10}^{5}$ | [40] |

SFK* | ${R}_{3}+{R}_{5}-{R}_{21}$ | Free fully activated SFK | 0 | $1\times {10}^{5}$ | [40] |

SFK*-Zapp | ${R}_{4a}-{R}_{5}$ | Fully activated SFK bound to pY319 of Zap70 | 0 | $9.3\times {10}^{4}$ | [40] |

SFK*S59p | ${R}_{21}+{R}_{22}$ | Fully activated SFK phosphorylated at serine-59 by activated Erk | 0 | $1\times {10}^{5}$ | [40] |

CD45 ${}^{p}$ | $-{R}_{23}$ | Positive regulatory role of transmembrane tyrosine phosphatase CD45 | $1\times {10}^{5}$ | $1\times {10}^{5}$ | [40] |

CD45 ${}_{n}$* | ${R}_{23a}$ | Negative regulatory role of CD45 | 0 | $1\times {10}^{5}$ | [40] |

Cbpp | ${R}_{6}-{R}_{7}$ | Phosphorylated transmembrane scaffold protein Cbp (also known as PAG) | 50 | $5\times {10}^{4}$ | [40] |

Csk* | ${R}_{7}$ | Membrane-localized protein tyrosine kinase Csk recruited by Cbpp | $2.5\times {10}^{3}$ | $5\times {10}^{4}$ | [40] |

SHP1* | ${R}_{10}$ | Tyrosine phosphatase SHP1 recruited to the membrane and activated | 0 | $1\times {10}^{6}$ | [40] |

LATp | ${R}_{11}-{R}_{12}-{R}_{19}$ | Phosphorylated transmembrane protein LAT at tyrosine residues | 0 | $5\times {10}^{4}$ | [40] |

SOSb | ${R}_{19}$ | LATp-bound scaffold protein Grb2 and guanine nucleotide exchange factor SOS | 0 | $5\times {10}^{4}$ | [40] |

PLCγp | ${R}_{12}$ | Activated phospholipase Cγ and bound to LATp | 0 | $5\times {10}^{4}$ | [40] |

DAG | ${R}_{13}-{R}_{13a}-{R}_{14}$ | Diacylglycerol | 0 | $1\times {10}^{7}$ | [40] |

IP ${}_{3}$ | ${R}_{13}-{R}_{13b}$ | Inositol 1,4,5-triphosphate | 0 | $1\times {10}^{7}$ | [40] |

RasGRP* | ${R}_{14}$ | Activated Ras guanine nucleotide releasing protein (RasGRP) | 0 | $1\times {10}^{5}$ | [40] |

RasGTP | ${R}_{15}$ | Guanine triphosphate (GTP)-bound Ras protein | 0 | $1\times {10}^{7}$ | [40] |

Raf* | ${R}_{16}$ | Phosphorylated and activated mitogen-activated protein (MAP) kinase kinase kinase Raf | 0 | $4\times {10}^{4}$ | [40] |

Mek* | ${R}_{17}$ | Phosphorylated and activated MAP kinase kinase Mek | 0 | $2\times {10}^{7}$ | [40] |

Erk* | ${R}_{18}$ | Phosphorylated and activated MAP kinase (MAPK) Erk | $2.0960\times {10}^{6}$ | $2\times {10}^{7}$ | [40] |

AP1* | ${R}_{26}$ | Activated transcription factor activator protein 1 | 0 | $2\times {10}^{7}$ | Derived |

Ca ${}^{2+}$ | ${R}_{27}-{R}_{28}$ | Cytoplasmic calcium ions released from intracellular stores (endoplasmic reticulum) | $3.011\times {10}^{4}$ | $1\times {10}^{8}$ | [58] |

CaM* | ${R}_{28}$ | Calcium-binding protein calmodulin bound to calcium | 0 | $1\times {10}^{6}$ | Derived |

CN* | ${R}_{29}$ | Activated calcium-dependent serine-threonine phosphatase calcineurin | 0 | $1\times {10}^{6}$ | Derived |

NFATn | ${R}_{30}$ | Dephosphorylated NFAT with unobstructed nuclear localization signal | 0 | $1\times {10}^{6}$ | Derived |

CD28* | ${R}_{32}$ | Ligand-bound and activated CD28 coreceptor | 0 | $2\times {10}^{5}$ | Derived |

PI3K* | ${R}_{33}$ | Activated phosphoinositide 3-kinase related kinase (PI3K) | 0 | $1\times {10}^{4}$ | Derived |

PIP ${}_{3}$ | ${R}_{34}$ | Phosphatidylinositol 3,4,5-trisphosphate | 0 | $1\times {10}^{7}$ | Derived |

PDK1* | ${R}_{35}$ | Activated 3-phosphoinositide-dependent kinase-1 (PDK1) | 0 | $1\times {10}^{4}$ | Derived |

PKCθ* | ${R}_{36}+{R}_{14a}$ | Active protein kinase C-θ | 0 | $1\times {10}^{4}$ | Derived |

IKK* | ${R}_{37}$ | Activated IκB kinase | 0 | $1\times {10}^{4}$ | Derived |

IκBαp | ${R}_{38}$ | Phosphorylated IκB marked for proteasomal degradation | 0 | $1\times {10}^{4}$ | Derived |

NFκBn | ${R}_{39}$ | Nuclear NFκB | 0 | $1\times {10}^{4}$ | Derived |

AKT* | ${R}_{40}$ | Activated serine-threonine kinase Akt, also known as protein kinase B (PKB) | 0 | $1\times {10}^{4}$ | Derived |

TSC1-TSC2 | $-{R}_{41}$ | GTPase-activating protein (GAP) consisting of tuberous sclerosis complex 1 (TSC1) and TSC2 | $1\times {10}^{4}$ | $1\times {10}^{4}$ | Derived |

RhebGTP | $-{R}_{42}$ | GTP-bound Ras homolog enriched in brain (Rheb) GTPase | $5\times {10}^{3}$ | $1\times {10}^{4}$ | Derived |

mTORC1* | ${R}_{43}$ | Activated mammalian target of rapamycin (mTOR) complex 1 | 0 | $1\times {10}^{4}$ | Derived |

mTORC2* | ${R}_{44}$ | Activated mTORC2 | 0 | $1\times {10}^{4}$ | Derived |

PTEN* | ${R}_{45}$ | Activated phosphatase and tensin homolog (PTEN) | $1\times {10}^{4}$ | $1\times {10}^{4}$ | Derived |

IL2 | ${R}_{46}$ | Interleukin-2, a cytokine marking T-cell activation | 0 | $1\times {10}^{4}$ | Derived |

FOXP3 | ${R}_{47}$ | Forkhead box P3, regulator of regulatory T-cell development and function | 0 | $1\times {10}^{4}$ | Derived |

Reaction | Equation | Biological Meaning | |
---|---|---|---|

R ${}_{TC{R}_{lig}}$ | = | ${k}_{f,{r}_{00}}*TC{R}_{lig}*TC{R}_{f}-{k}_{r,{r}_{00}}*TC{R}_{b}$ | Association/dissociation of ligand and TCR complex |

${R}_{TC{R}_{p}}$ | = | $({k}_{f1,{r}_{0}}*(SFKact+SFKactS59p)+{k}_{f2,{r}_{0}}*SFKactZapp)*TC{R}_{b}-({k}_{r1,{r}_{0}}*SHP1act+{k}_{r3,{r}_{0}}*CD{45}_{n}+{k}_{r2,{r}_{0}})TC{R}_{p}$ | SFK-mediated phosphorylation and SHP1/CD45-mediated dephosphorylation of ligand-bound TCR complex |

${R}_{iTC{R}_{f}}$ | = | ${k}_{int}*TC{R}_{f}$ | Internalization of free TCR |

${R}_{iTC{R}_{b}}$ | = | ${k}_{int}*TC{R}_{b}$ | Internalization of ligand-bound TCR |

${R}_{iTC{R}_{p}}$ | = | ${k}_{int}*TC{R}_{p}$ | Internalization of phosphorylated TCR |

${R}_{TC{R}_{exo}}$ | = | ${k}_{exo}*TC{R}_{i}$ | Exocytosis of internalized TCR |

${R}_{TC{R}_{deg}}$ | = | ${k}_{deg}*TC{R}_{i}$ | Degradation of internalized TCR |

${R}_{TC{R}_{synth}}$ | = | ${k}_{synth}*TC{R}_{deg}$ | Synthesis of new TCR |

${R}_{1}$ | = | ${k}_{f,{r}_{1}}*(2*TCRp-Zapb-Zapact-(Zapp-Zap{p}_{0})-SrcbactZapp-SrcdpZapp)*Zap-{k}_{r,{r}_{1}}*Zapb$ | Association/dissociation of phosphorylated TCR complex and Zap70 |

${R}_{2}$ | = | ${k}_{f,{r}_{2}}*CD{45}^{p}*SFK-{k}_{r,{r}_{2}}*Cskact*SFKdp$ | SFK dephosphorylation by CD45 and re-phosphorylation by Csk* at the inhibitory site (Y505 in Lck, Y528 in Fyn) |

${R}_{3}$ | = | $({k}_{f,{r}_{3}}*(TC{R}_{b}+TC{R}_{p}))*SFKdp-({k}_{r1,{r}_{3}}*SHP1act+{k}_{r3,{r}_{3}}*CD{45}_{n}+{k}_{r2,{r}_{3}})*SFKact$ | SFK phosphorylation at the activation loop (Y394 in Lck, Y417 in Fyn) by autophosphorylation (or by another kinase) and dephosphorylation by SHP1 |

${R}_{4}$ | = | ${k}_{f,{r}_{4}}*(Zapp-Zap{p}_{0})*SFKdp-{k}_{r,{r}_{4}}*SFKdpZapp$ | Association/dissociation of Zapp and SFKdp |

${R}_{4a}$ | = | ${k}_{f,{r}_{4}}*(Zapp-Zap{p}_{0})*SFKdp-{k}_{r,{r}_{4}}*SFKdpZapp$ | TCR-mediated phosphorylation of Zapp-bound SFKdp |

${R}_{5}$ | = | ${k}_{f,{r}_{5}}*SFKactZapp-{k}_{r,{r}_{5}}*SFKact*(Zapp-Zap{p}_{0})$ | Dissociation/association of activated SFK and Zapp |

${R}_{6}$ | = | $({k}_{f1,{r}_{6}}*(SFKact+SFKactS59p)+{k}_{f2,{r}_{6}})*Cbp-{k}_{r,{r}_{6}}*CD{45}^{p}*Cbpp$ | Cbp phosphorylation by activated SFK (or other kinases) and dephosphorylation by CD45 |

${R}_{7}$ | = | ${k}_{f,{r}_{7}}*Cbpp*Csk-{k}_{r,{r}_{7}}*Cskact$ | Cbpp-mediated activation of Csk |

${R}_{8}$ | = | $({k}_{f1,{r}_{8}}*(SFKact+SFKactS59p)+{k}_{f2,{r}_{8}}*SFKactZapp)*Zapb-({k}_{r1,{r}_{8}}*SHP1act+{k}_{r2,{r}_{8}})*Zapact$ | Zap70 phosphorylation at the activation loop (Y493) by activated SFK (SFK* and SFK*-Zapp) and dephosphorylation by PTPs including SHP1 |

${R}_{9}$ | = | $({k}_{f1,{r}_{9}}*(Zapact+(Zapp-Zap{o}_{0})+SFKactZapp+SFKdpZapp)+{k}_{f2,{r}_{9}}*(SFKact+SFKactS59p)+{k}_{f3,{r}_{9}}*SFKactZapp)*Zapact-({k}_{r1,{r}_{9}}*SHP1act+{k}_{r2,{r}_{9}})*(Zapp-Zap{p}_{0})$ | Additional Zap70 phosphorylation at Y319 by activated SFK and Zap70 and dephosphorylation by PTPs including SHP1 |

${R}_{10}$ | = | ${k}_{f,{r}_{10}}*SFKact*SHP1-{k}_{r,{r}_{10}}*SHP1act$ | SHP1 activation (by SFK*) and deactivation |

${R}_{11}$ | = | ${k}_{f,{r}_{11}}*(Zapact+(Zapp-Zap{p}_{0})+SFKactZapp+SFKdpZapp)*LAT-({k}_{r1,{r}_{11}}*SHP1act+{k}_{r2,{r}_{11}})*LATp$ | LAT phosphorylation by activated Zap70 and dephosphorylation SHP1 |

${R}_{12}$ | = | ${k}_{f,{r}_{12}}*LATp*PLCg-{k}_{r,{r}_{12}}*PLCgp$ | PLCγ phosphorylation by LATp |

${R}_{13}$ | = | ${k}_{f,{r}_{13}}*PLCgp*PIP2$ | PLCγ-mediated hydrolysis of PIP ${}_{2}$ to IP${}_{3}$ and DAG |

${R}_{13a}$ | = | ${k}_{r,{r}_{13a}}*DAG$ | Degradation of DAG |

${R}_{13b}$ | = | ${k}_{r,{r}_{13b}}*IP3$ | Degradation of IP ${}_{3}$ |

${R}_{14}$ | = | ${k}_{f,{r}_{14}}*DAG*RasGRP-{k}_{r,{r}_{14}}*RasGRPact$ | DAG-mediated activation of RasGRP |

${R}_{14a}$ | = | ${k}_{f,{r}_{14a}}*DAG*PKC\theta $ | DAG-mediated activation of PKCθ |

${R}_{15}$ | = | $({k}_{f1,{r}_{15}}*RasGRPact+{k}_{f2,{r}_{15}}*SOSb)*RasGDP-{k}_{r,{r}_{15}}*RasGTP$ | RasGRP- and Grb2SOS-mediated activation of Ras |

${R}_{16}$ | = | ${k}_{f,{r}_{16}}*RasGTP*\left(Raf\right)-{k}_{r,{r}_{16}}*Rafp$ | Ras-mediated activation of Raf |

${R}_{17}$ | = | ${k}_{f,{r}_{17}}*Rafp*Mek-{k}_{r,{r}_{17}}*Mekp$ | Rafp-mediated activation of Mek |

${R}_{18}$ | = | ${k}_{f,{r}_{18}}*Mekp*Erk-{k}_{r,{r}_{18}}*(Erkp-Erk{p}_{0})$ | Mekp-mediated activation of Erk |

${R}_{19}$ | = | ${k}_{f,{r}_{19}}*LATp*Grb2SOS-{k}_{r,{r}_{19}}*SOSb$ | LATp-mediated association and activation of the Grb2-SOS complex |

${R}_{20}$ | = | ${k}_{f,{r}_{20}}*SFKdp*(Erkp-Erk{p}_{0})-{k}_{r,{r}_{20}}*SFKdpS59p$ | Erkp-mediated phosphorylation of SFKdp at serine-59 |

${R}_{21}$ | = | ${k}_{f,{r}_{21}}*SFKact*(Erkp-Erk{p}_{0})-{k}_{r,{r}_{21}}*SFKactS59p$ | Erkp-mediated phosphorylation of SFKact at serine-59 |

${R}_{22}$ | = | $({k}_{f,{r}_{3}}*(TC{R}_{b}+TC{R}_{p}))*SFKdpS59p-({k}_{r1,{r}_{3}}*SHP1act+{k}_{r3,{r}_{3}}*CD{45}_{n}+{k}_{r2,{r}_{3}})*SFKactS59p$ | TCR-mediated activation and SHP1-mediated deactivation of SFK-s59p |

${R}_{23}$ | = | ${k}_{f,{r}_{23}}*(TCRb+TCRp)*CD{45}^{p}-{k}_{r,{r}_{23}}*(CD{45}_{tot}-CD{45}^{p})$ | Positive regulatory role of CD45 and translocation caused by receptor cluster formation |

${R}_{23a}$ | = | ${k}_{f,{r}_{23a}}*(SFKact+SFKactZapp+SFKactS59p)*CD{45}_{n}-{k}_{r,{r}_{23a}}*(CD{45}_{tot}-CD{45}_{n})$ | Negative regulatory role of CD45 and recruitment to receptor cluster |

${R}_{26}$ | = | ${k}_{f,{r}_{26}}*(Erkp-Erk{p}_{0})*PKC\theta act*AP1-{k}_{r,{r}_{26}}*AP1act$ | Erkp- and PKCθ-mediated activation of AP1 |

${R}_{27}$ | = | ${k}_{f,{r}_{27}}*IP3*C{a}_{s}-{k}_{r,{r}_{27}}*(Ca-C{a}_{0})$ | IP ${}_{3}$-induced calcium release into the cytoplasm |

${R}_{28}$ | = | ${k}_{f,{r}_{28}}*(Ca-C{a}_{0})*CaM-{k}_{r,{r}_{28}}*CaMact$ | Association/dissociation of calcium and calmodulin |

${R}_{29}$ | = | ${k}_{f,{r}_{29}}*CaMact*CN-{k}_{r,{r}_{29}}*CNact$ | Calmodulin-mediated activation of calcineurin |

${R}_{30}$ | = | ${k}_{f,{r}_{30}}*CNact*NFATp-{k}_{r,{r}_{30}}*NFATn$ | Calcineurin-mediated dephosphorylation and nuclear translocation of NFAT |

${R}_{32}$ | = | ${k}_{f,{r}_{32}}*CD{28}_{lig}*CD{28}_{f}-{k}_{r,{r}_{32}}*CD28act$ | Association/dissociation of ligand and CD28 |

${R}_{33}$ | = | ${k}_{f,{r}_{33}}*CD28act*(Zapact+(Zapp-Zapp0)+SFKdpZapp+SFKactZapp)*PI3K-{k}_{r,{r}_{33}}*PI3Kact$ | PI3K activation by CD28 and Zap70 and deactivation |

${R}_{34}$ | = | ${k}_{f,{r}_{34}}*PI3Kact*PIP2-({k}_{r1,{r}_{34}}*PTENact+{k}_{r2,{r}_{34}})*PIP3$ | PI3K-mediated phosphorylation of PIP ${}_{2}$ and PTEN-mediated dephosphorylation of PIP${}_{3}$ |

${R}_{35}$ | = | ${k}_{f,{r}_{35}}*PIP3*PDK1-{k}_{r,{r}_{35}}*PDK1act$ | PIP ${}_{3}$-mediated activation of PDK1 |

${R}_{36}$ | = | $({k}_{f1,{r}_{36}}*PDK1act+{k}_{f2,{r}_{36}}*mTORC2act)*PKC\theta -{k}_{r,{r}_{36}}*PKC\theta act$ | Activation of PKCθ mediated by PDK1, DAG, and mTORC2 |

${R}_{37}$ | = | $({k}_{f1,{r}_{37}}*PKC\theta act+{k}_{f2,{r}_{37}}*Aktp)*IKK-{k}_{r,{r}_{37}}*IKKact$ | PKCθ- and AKT-mediated activation of IKK |

${R}_{38}$ | = | ${k}_{f,{r}_{38}}*IKKact-{k}_{r1,{r}_{38}}*IkBp-{k}_{r2,{r}_{38}}*NFkBn$ | IKK-mediated phosphorylation and 26S proteasome-mediated degradation of IκBα; NFκB-induced synthesis of new IκBα |

${R}_{39}$ | = | ${k}_{f,{r}_{39}}*IkBp-{k}_{r,{r}_{39}}*NFkBn$ | Activation and nuclear translocation of NFκB |

${R}_{40}$ | = | $({k}_{f1,{r}_{40}}*PDK1act+{k}_{f2,{r}_{40}}*mTORC2act+{k}_{f3,{r}_{40}}*PDK1act*mTORC2act)*Akt-{k}_{r,{r}_{40}}*Aktp$ | PDK1- and mTORC2-mediated phosphorylation of AKT |

${R}_{41}$ | = | ${k}_{f,{r}_{41}}*\frac{Akt{p}^{{n}_{{r}_{41}}}}{Akt{p}^{{n}_{{r}_{41}}}+{k}_{{r}_{41}}^{{n}_{{r}_{41}}}}*TSC-{k}_{r,{r}_{41}}*TSC2p$ | AKT-mediated phosphorylation, dissociation, and deactivation of TSC |

${R}_{42}$ | = | ${k}_{f,{r}_{42}}*TSC*RhebGTP-{k}_{r,{r}_{42}}*RhebGDP$ | GAP activity of TSC on Rheb |

${R}_{43}$ | = | ${k}_{f,{r}_{43}}*(RhebGTP-RhebGT{P}_{0})*mTORC1-{k}_{r,{r}_{43}}*mTORC1act$ | RhebGTP-mediated activation of mTORC1 |

${R}_{44}$ | = | ${k}_{f,{r}_{44}}*PI3Kact*mTORC2-({k}_{r1,{r}_{44}}*mTORC1act+{k}_{r2,{r}_{44}})*mTORC2act$ | PI3K-mediated activation and mTORC1-mediated inhibition of mTORC2 |

${R}_{45}$ | = | $({k}_{f1,{r}_{45}}*FOXP3+{k}_{f2,{r}_{45}})*PTEN-{k}_{r,{r}_{45}}*\frac{{(TC{R}_{b}+TC{R}_{p})}^{{n}_{{r}_{45}}}}{{(TC{R}_{b}+TC{R}_{p})}^{{n}_{{r}_{45}}}+{k}_{{r}_{45}}^{{n}_{{r}_{45}}}}*PTENact$ | TCR-mediated inhibition and FOXP3-mediated activation of PTEN |

${R}_{46}$ | = | ${k}_{f,{r}_{46}}*\frac{AP1ac{t}^{{n}_{1,{r}_{46}}}}{AP1ac{t}^{{n}_{1,{r}_{46}}}+{k}_{1,{r}_{46}}^{{n}_{1,{r}_{46}}}}*\frac{NFAT{n}^{{n}_{2,{r}_{46}}}}{NFAT{n}^{{n}_{2,{r}_{46}}}+{k}_{2,{r}_{46}}^{{n}_{2,{r}_{46}}}}*\frac{NFkB{n}^{{n}_{3,{r}_{46}}}}{NFkB{n}^{{n}_{3,{r}_{46}}}+{k}_{3,{r}_{46}}^{{n}_{3,{r}_{46}}}}-({k}_{r1,{r}_{46}}*FOXP3+{k}_{r2,{r}_{46}})*IL2$ | AP1, NFAT, and NFkB regulate transcription of IL-2; FOXP3-mediated inhibition of IL-2 |

${R}_{47}$ | = | ${k}_{f,{r}_{47}}*\frac{AP1ac{t}^{{n}_{1,{r}_{47}}}}{AP1ac{t}^{{n}_{1,{r}_{47}}}+{k}_{1,{r}_{47}}^{{n}_{1,{r}_{47}}}}*\frac{NFAT{n}^{{n}_{2,{r}_{47}}}}{NFAT{n}^{{n}_{2,{r}_{47}}}+{k}_{2,{r}_{47}}^{{n}_{2,{r}_{47}}}}-({k}_{r1,{r}_{47}}*(mTORC1*mTORC2)+{k}_{r2,{r}_{47}})*FOXP3$ | AP1 and NFAT regulate transcription of FOXP3; mTOR-mediated inhibition of FOXP3 |

Parameter | Biological Meaning | Value | 95% CI | Units | Source |
---|---|---|---|---|---|

${k}_{f,{r}_{00}}$ | Association rate of ligand and TCR complex | 0.0900 | [0.0558, 0.1231] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{00}}$ | Dissociation rate of ligand and TCR complex | $3\times {10}^{-4}$ | [$1.2346\times {10}^{-4}$, $5.5632\times {10}^{-4}$] | min${}^{-1}$ | Fitted |

${k}_{f1,{r}_{0}}$ | Phosphorylation rate of ligand-bound TCR mediated by SFK* and SFK*-S59p | 0.3000 | [0.0475, 3.8961] | (mol·min)${}^{-1}$ | [40] |

${k}_{f2,{r}_{0}}$ | Phosphorylation rate of ligand-bound TCR mediated by SFK*-Zapp | 1.13$\xb7{k}_{f1,{r}_{0}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r1,{r}_{0}}$ | Dephosphorylation rate of ligand-bound TCR mediated by activated SHP1 | 0.0022 | [0.0015, 0.0221] | (mol·min)${}^{-1}$ | [40] |

${k}_{r2,{r}_{0}}$ | Constitutive dephosphorylation rate of ligand-bound TCR | 16.1100 | [1.0731, 61.7493] | min${}^{-1}$ | [40] |

${k}_{r3,{r}_{0}}$ | Dephosphorylation rate of ligand-bound TCR mediated by activated CD45 | 0.0300 | [$8.1854\times {10}^{-4}$, 0.1574] | (mol·min)${}^{-1}$ | Fitted |

${k}_{in{t}_{min}}$ | Resting TCR internalization rate | 0.0100 | min${}^{-1}$ | [46] | |

${k}_{in{t}_{max}}$ | Maximum induced TCR internalization rate | 0.0380 | min${}^{-1}$ | [46] | |

${n}_{int}$ | TCR internalization Hill coefficient | 2 | unitless | Derived | |

${K}_{int}$ | Enzyme quantity producing half-maximum TCR internalization rate | 0.05·[PKCθ]${}_{total}$ | mol | Derived | |

${k}_{exo}$ | Constitutive TCR exocytosis rate | 0.0789 | min${}^{-1}$ | Derived | |

${k}_{de{g}_{min}}$ | Resting TCR degradation rate | 0.0011 | min${}^{-1}$ | [45] | |

${k}_{de{g}_{max}}$ | Maximum induced TCR degradation rate | 0.0033 | min${}^{-1}$ | [45] | |

${n}_{deg}$ | TCR degradation Hill coefficient | 2 | unitless | Derived | |

${K}_{deg}$ | Enzyme quantity producing half-maximum TCR internalization rate | 0.05·[SFK]${}_{total}$ | mol | Derived | |

${k}_{synth}$ | TCR synthesis rate | ${k}_{de{g}_{min}}$ | min${}^{-1}$ | Derived | |

${k}_{f,{r}_{1}}$ | Association rate of Zap70 to phosphorylated TCRζ-chain | $6\times {10}^{-4}$ | [$5.4721\times {10}^{-5}$, $9.6525\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{1}}$ | Dissociation rate of Zap70 to phosphorylated TCRζ-chain | 1.2600 | [0.4913, 41.7225] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{2}}$ | Dephosphorylation rate of SFK at the inhibitory site by CD45 | $3\times {10}^{-6}$ | [$2.0755\times {10}^{-6}$, $3.6068\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{2}}$ | Phosphorylation rate of SFK at the inhibitory site by Csk | 0.1199 | (mol·min)${}^{-1}$ | Derived | |

${k}_{f,{r}_{3}}$ | Phosphorylation rate of SFKdp at the activation site mediated by TCRb and TCRp | 13.7700 | [1.3177, 75.4525] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r1,{r}_{3}}$ | Dephosphorylation rate of SFK* at the activation site by activated SHP1 | ${k}_{r1,{r}_{0}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r2,{r}_{3}}$ | Constitutive dephosphorylation rate of SFK* at the activation site | ${k}_{r2,{r}_{0}}$ | min${}^{-1}$ | [40] | |

${k}_{r3,{r}_{3}}$ | Dephosphorylation rate of SFK* at the activation site by activated CD45 | ${k}_{r3,{r}_{0}}$ | (mol·min)${}^{-1}$ | Derived | |

${k}_{f,{r}_{4}}$ | Association rate of SFKdp to Zapp | 0.0217 | [$2.169\times {10}^{-4}$, 2.1690] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{4}}$ | Dissociation rate of SFKdp to Zapp | 0.0025 | [$2.415\times {10}^{-4}$, 0.0151] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{4a}}$ | Phosphorylation rate of Zapp-bound SFKdp at the activation site mediated by TCRb and TCRp | ${k}_{f,{r}_{3}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r1,{r}_{4a}}$ | Dephosphorylation rate of Zapp-bound SFK* at the activation site by activated SHP1 | 0.0068 | [0.0012, 0.0641] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{4a}}$ | Constitutive dephosphorylation rate of Zapp-bound SFK* at the activation site | 13.4040 | [1.3409, 84.0744] | min${}^{-1}$ | Fitted |

${k}_{r3,{r}_{4a}}$ | Dephosphorylation rate of Zapp-bound SFK* at the activation site by activated CD45 | 0.0300 | [0.0019, 0.3542] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f,{r}_{5}}$ | Dissociation rate of SFK* to Zapp | 70.7880 | [68.7880, 102.3198] | min${}^{-1}$ | Fitted |

${k}_{r,{r}_{5}}$ | Association rate of SFK* to Zapp | ${k}_{f,{r}_{4}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{f1,{r}_{6}}$ | Phosphorylation rate of Cbp by SFK* | $1.788\times {10}^{-6}$ | [$1.0338\times {10}^{-6}$, $1.788\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{6}}$ | Constitutive phosphorylation rate of Cbp | 0.0207 | [0.0127, 0.1200] | min${}^{-1}$ | Fitted |

${k}_{r,{r}_{6}}$ | Dephosphorylation rate of Cbpp by CD45 | $1.9644\times {10}^{-4}$ | (mol·min)${}^{-1}$ | Derived | |

${k}_{f,{r}_{7}}$ | Association rate of Csk to Cbpp | $6.984\times {10}^{-4}$ | [$1.2137\times {10}^{-5}$, 0.0070] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{7}}$ | Dissociation rate of Csk to Cbpp | 0.6635 | min${}^{-1}$ | Derived | |

${k}_{f1,{r}_{8}}$ | Phosphorylation rate of bound Zap by SFK* and SFK*S59p | 0.0021 | [0.0015, 0.0022] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{8}}$ | Phosphorylation rate of bound Zap by SFK*-Zapp | 1.13$\xb7{k}_{f1,{r}_{8}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r1,{r}_{8}}$ | Dephosphorylation rate of Zap* by activated SHP1 | ${k}_{r1,{r}_{0}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r2,{r}_{8}}$ | Constitutive dephosphorylation rate of Zap* | ${k}_{r2,{r}_{0}}$ | min${}^{-1}$ | [40] | |

${k}_{f1,{r}_{9}}$ | Phosphorylation rate of Zap* by free and bound Zapp | $3\times {10}^{-4}$ | [$3\times {10}^{-6}$, $3.6068\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{9}}$ | Phosphorylation rate of Zap* by SFK* and SFK*S59p | ${k}_{f1,{r}_{8}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{f3,{r}_{9}}$ | Phosphorylation rate of Zap* by SFK*-Zapp | 1.13$\xb7{k}_{f1,{r}_{8}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r1,{r}_{9}}$ | Dephosphorylation rate of Zapp by activated SHP1 | ${k}_{r1,{r}_{0}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r2,{r}_{9}}$ | Constitutive dephosphorylation rate of Zapp | ${k}_{r2,{r}_{0}}$ | min${}^{-1}$ | [40] | |

${k}_{f,{r}_{10}}$ | Activation rate of SHP1 by SFK* | $8.19\times {10}^{-6}$ | [$8.1121\times {10}^{-6}$, $8.2192\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{10}}$ | Deactivation rate of SHP1 | 0.3660 | [0.3450, 0.3861] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{11}}$ | Phosphorylation rate of LAT by activated Zap | $3\times {10}^{-4}$ | [$2.5835\times {10}^{-4}$, $3.9477\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r1,{r}_{11}}$ | Dephosphorylation rate of LAT by activated SHP1 | 0.0020 | [0.0017, 0.0025] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{11}}$ | Constitutive dephosphorylation rate of SHP1 | 90 | [87.9332, 130.0896] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{12}}$ | Association rate of PLCγ and LATp (PLCγ is immediately phosphorylated) | 0.0030 | [0.0021, 0.0033] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{12}}$ | Lumped rate of dephosphorylation of PLCγ and its dissociation from LATp | 300 | [295.1992, 302.0020] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{13}}$ | PIP ${}_{2}$ hydrolysis rate catalyzed by PLCγp | $3\times {10}^{-8}$ | [$2.9199\times {10}^{-8}$, $3.0011\times {10}^{-8}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{13a}}$ | DAG degradation rate | 2.4311 | [1.3990, 2.9228] | min${}^{-1}$ | Fitted |

${k}_{r,{r}_{13b}}$ | IP ${}_{3}$ degradation rate | ${k}_{r,{r}_{13a}}$ | min${}^{-1}$ | Derived | |

${k}_{f,{r}_{14}}$ | Activation rate of RasGRP by DAG | 0.0030 | [0.0025, 0.0052] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{14}}$ | Inactivation rate of RasGRP | 30 | [17.2632, 36.0679] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{14a}}$ | Activation rate of PKCθ by DAG | $3\times {10}^{-4}$ | [$2.7665\times {10}^{-5}$, $3.7263\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f1,{r}_{15}}$ | Rate of Ras guanine nucleotide exchange catalyzed by activated RasGRP | $1.2\times {10}^{-5}$ | [$8.3020\times {10}^{-6}$, $1.7345\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{15}}$ | Rate of Ras guanine nucleotide exchange catalyzed by recruited SOS | $1.2\times {10}^{-6}$ | [$1.2111\times {10}^{-7}$, $8.3020\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{15}}$ | Constitutive rate of RasGTP hydrolysis to RasGDP | 30 | [24.9529, 43.3632] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{16}}$ | Activation rate of Raf by RasGTP | $2.4\times {10}^{-4}$ | [$1.6604\times {10}^{-4}$, $2.8854\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{16}}$ | Constitutive rate of Raf inactivation by phosphatase | 30 | [24.9529, 43.3632] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{17}}$ | Activation rate of Mek by activated Raf | 0.0030 | [0.0025, 0.0036] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{17}}$ | Constitutive rate of Mek inactivation by phosphatase | 30 | [24.9529, 36.0679] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{18}}$ | Activation rate of Erk by activated Mek | $3\times {10}^{-6}$ | [$2.4953\times {10}^{-6}$, $3.6068\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{18}}$ | Constitutive rate of Erk inactivation by phosphatase | 30 | [24.9529, 36.0679] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{19}}$ | Association rate of Grb-SOS complex to LATp | $6\times {10}^{-4}$ | [$5.7393\times {10}^{-6}$, $9.151\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{19}}$ | Dissociation rate of Grb-SOS complex from LATp | 30 | [3.3632, 100] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{20}}$ | Phosphorylation rate of SFKdp at S59 by activated Erk | $3\times {10}^{-5}$ | [$6.3433\times {10}^{-7}$, 0.0011] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{20}}$ | Constitutive dephosphorylation rate of SFKdp at S59 | 30 | [4.5543, 80.0101] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{21}}$ | Phosphorylation rate of SFK* at S59 by activated Erk | ${k}_{f,{r}_{20}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r,{r}_{21}}$ | Constitutive dephosphorylation rate of SFK* at S59 | ${k}_{r,{r}_{20}}$ | min${}^{-1}$ | [40] | |

${k}_{f,{r}_{22}}$ | Phosphorylation rate of SFKdpS59p at the activation site mediated by TCRb and TCRp | ${k}_{f,{r}_{3}}$ | (mol·min)${}^{-1}$ | Derived | |

${k}_{r1,{r}_{22}}$ | Dephosphorylation rate of SFK*S59p at the activation site by activated SHP1 | ${k}_{r1,{r}_{3}}$ | (mol·min)${}^{-1}$ | [40] | |

${k}_{r2,{r}_{22}}$ | Constitutive dephosphorylation rate of SFK*S59p at the activation site | ${k}_{r2,{r}_{3}}$ | min${}^{-1}$ | [40] | |

${k}_{r3,{r}_{22}}$ | Dephosphorylation rate of SFK*S59p at the activation site by activated CD45 | ${k}_{r3,{r}_{3}}$ | (mol·min)${}^{-1}$ | Derived | |

${k}_{f,{r}_{23}}$ | Translocation rate of CD45 mediated by receptor complex | $3\times {10}^{-7}$ | [$6.5488\times {10}^{-8}$, $6.2679\times {10}^{-7}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{23}}$ | Constitutive return rate of CD45 | 0.0030 | [$9.1223\times {10}^{-5}$, 0.1009] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{23a}}$ | Activation rate of CD45 negative regulator by SFK* | $6\times {10}^{-7}$ | [$4.151\times {10}^{-7}$, $3.98\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{23a}}$ | Constitutive deactivation rate of CD45 negative regulator | 0.0030 | [$9.2531\times {10}^{-5}$, 0.0961] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{26}}$ | Activation rate of AP1 by Erk* and PKCθ* | $3\times {10}^{-9}$ | [$3.3439\times {10}^{-10}$, $7.9843\times {10}^{-8}$] | (mol${}^{2}\xb7$min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{26}}$ | Constitutive deactivation rate of AP1* | 30 | [23.1193, 43.8583] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{27}}$ | Release rate of calcium stored in the endoplasmic reticulum by IP ${}_{3}$ | 0.0300 | [0.0212, 0.0380] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{27}}$ | Constitutive calcium uptake rate | 30 | [19.0031, 54.1565] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{28}}$ | Association rate of calmodulin to calcium | $1.5\times {10}^{-8}$ | [$1.5267\times {10}^{-9}$, $1.0877\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{28}}$ | Dissociation rate of calmodulin from calcium | 0.3000 | [0.0182, 3.2856] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{29}}$ | Activation rate of calcineurin by calmodulin* | $1.5\times {10}^{-7}$ | [$5.8843\times {10}^{-8}$, $8.6424\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{29}}$ | Constitutive deactivation rate of calcineurin* | 0.3000 | [0.0943, 19.1092] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{30}}$ | Activation rate of NFAT (NFAT immediately translocates to nucleus) | $1.5\times {10}^{-6}$ | [$1.0869\times {10}^{-6}$, $1.9341\times {10}^{-6}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{30}}$ | Constitutive deactivation rate of NFAT | 0.3000 | [0.0210, 0.3433] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{32}}$ | Association rate of ligand to CD28 coreceptor | 0.0300 | [0.0199, 0.0360] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{32}}$ | Dissociation rate of ligand from CD28 coreceptor | $6\times {10}^{-5}$ | [$1.8795\times {10}^{-5}$, $7.4673\times {10}^{-5}$] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{33}}$ | Activation rate of PI3K by ligand-bound CD28 and activated Zap | $3\times {10}^{-9}$ | [$3.9548\times {10}^{-10}$, $5.1943\times {10}^{-9}$] | (mol${}^{2}\xb7$min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{33}}$ | Constitutive deactivation rate of PI3K | 30 | [15.3560, 37.5731] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{34}}$ | Phosphorylation rate of PIP ${}_{2}$ by PI3K* | 3 | [0.3923, 5.7161] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r1,{r}_{34}}$ | Dephosphorylation rate of PIP ${}_{3}$ by PTEN* | 30 | [17.4422, 65.3412] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{34}}$ | Constitutive dephosphorylation rate of PIP ${}_{3}$ | $1\times {10}^{-10}$ | [$5.1834\times {10}^{-11}$, $2.7164\times {10}^{-10}$] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{35}}$ | Activation rate of PDK1 by PIP ${}_{3}$ | $3\times {10}^{-5}$ | [$9.3939\times {10}^{-6}$, $5.7164\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{35}}$ | Constitutive deactivation rate of PDK1* | 30 | [5.4422, 40.4623] | min${}^{-1}$ | Fitted |

${k}_{f1,{r}_{36}}$ | Activation rate of PKCθ by PDK1* | $3\times {10}^{-6}$ | [$8.4379\times {10}^{-7}$, $1.0892\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{36}}$ | Activation rate of PKCθ by mTORC2* | $3\times {10}^{-5}$ | [$8.0982\times {10}^{-7}$, $9.9339\times {10}^{-5}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{36}}$ | Constitutive deactivation rate of PKCθ* | 0.3000 | [0.1754, 1.3217] | min${}^{-1}$ | Fitted |

${k}_{f1,{r}_{37}}$ | Activation rate of IKK by PKCθ* | 0.0015 | [$9.0321\times {10}^{-4}$, 0.0065] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{37}}$ | Activation rate of IKK by Aktp | 0.0030 | [0.0019, 0.0046] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{37}}$ | Constitutive deactivation rate of IKK* | 15 | [5.3394, 48.7211] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{38}}$ | Phosphorylation rate of IκBα by IKK* | 0.4500 | [0.4109, 0.5512] | min${}^{-1}$ | Fitted |

${k}_{r1,{r}_{38}}$ | Proteasomal degradation rate of pIκBα | 0.1500 | [0.1093, 0.1978] | min${}^{-1}$ | Fitted |

${k}_{r2,{r}_{38}}$ | Deactivation rate of IκBα by NFκB | 0.1500 | [0.0936, 0.2380] | min${}^{-1}$ | Fitted |

${k}_{f,{r}_{39}}$ | Activation rate of NFκB by IκBα deactivation | 0.1500 | [0.0994, 0.2121] | min${}^{-1}$ | Fitted |

${k}_{r,{r}_{39}}$ | Constitutive deactivation rate of NFκB | 0.0150 | [0.0124, 0.0272] | min${}^{-1}$ | Fitted |

${k}_{f1,{r}_{40}}$ | Phosphorylation rate of Akt by PDK1* | $1\times {10}^{-10}$ | [$7.9346\times {10}^{-11}$, $2.7832\times {10}^{-10}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f2,{r}_{40}}$ | Phosphorylation rate of Akt by mTORC2* | $1\times {10}^{-10}$ | [$9.3976\times {10}^{-11}$, $1.7832\times {10}^{-10}$] | (mol·min)${}^{-1}$ | Fitted |

${k}_{f3,{r}_{40}}$ | Phosphorylation rate of Akt by PDK1* and mTORC2* | $1\times {10}^{-8}$ | [$6.4853\times {10}^{-9}$, $5.7164\times {10}^{-8}$] | (mol${}^{2}\xb7$min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{40}}$ | Dephosphorylation rate of Aktp by phosphatases | 1 | [0.5422, 7.0427] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{41}}$ | Phosphorylation rate of the TSC2 subunit of the TSC1-TSC2 complex (complex immediately dissociates) | 10 | [6.8726, 30.3895] | min${}^{-1}$ | Fitted |

${k}_{r,{r}_{41}}$ | Dephosphorylation rate of TSC2p and its association with TSC1 | 1 | [0.3783, 8.2627] | min${}^{-1}$ | Derived |

${n}_{{r}_{41}}$ | TSC2 phosphorylation Hill coefficient | 5 | unitless | Derived | |

${k}_{{r}_{41}}$ | Enzyme quantity producing half-maximum TSC2 phosphorylation rate | 0.15·[Akt]${}_{total}$ | mol | Fitted | |

${k}_{f,{r}_{42}}$ | Rate of Rheb guanine nucleotide exchange catalyzed by TSC1-TSC2 complex | $1\times {10}^{-4}$ | [$3.6763\times {10}^{-5}$, $9.9339\times {10}^{-4}$] | (mol·min)${}^{-1}$ | Derived |

${k}_{r,{r}_{42}}$ | Rate of RhebGDP to RhebGTP exchange | 1 | [0.3020, 10.5309] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{43}}$ | Activation rate of mTORC1 by RhebGTP | $1\times {10}^{-4}$ | [$6.3984\times {10}^{-5}$, 0.0047] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r,{r}_{43}}$ | Constitutive deactivation rate of mTORC1* | 1 | [0.2653, 4.7164] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{44}}$ | Activation rate of mTORC2 by PI3K* | 0.0030 | [2.7360e-4, 0.0993] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r1,{r}_{44}}$ | Deactivation rate of mTORC2* by mTORC1 | 0.0030 | [0.0023, 0.0102] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{44}}$ | Constitutive deactivation rate of mTORC2* | 3 | [1.3631, 3.8319] | min${}^{-1}$ | Fitted |

${k}_{f1,{r}_{45}}$ | Activation rate of PTEN by FOXP3* | $1\times {10}^{-4}$ | [$1.8647\times {10}^{-5}$, 0.0010] | (mol·min)${}^{-1}$ | Derived |

${k}_{f2,{r}_{45}}$ | Constitutive activation of PTEN | 1 | [0.1925, 18.9529] | min${}^{-1}$ | Derived |

${k}_{r,{r}_{45}}$ | Deactivation of PTEN* by TCR complex | 20 | [11.4372, 106.1915] | min${}^{-1}$ | Fitted |

${n}_{{r}_{45}}$ | PTEN* deactivation Hill coefficient | 10 | unitless | Derived | |

${k}_{{r}_{45}}$ | Ligand-bound TCR quantity producing half-maximum PTEN* deactivation | 0.75·[TCR]${}_{total}$ | mol | Derived | |

${k}_{f,{r}_{46}}$ | Activation rate of IL-2 activity | $1\times {10}^{4}$ | [574.7247, $1.8543\times {10}^{5}$] | mol·min${}^{-1}$ | Fitted |

${n}_{1,{r}_{46}}$ | Hill coefficient for AP1-induced IL-2 activity | 2 | unitless | Derived | |

${n}_{2,{r}_{46}}$ | Hill coefficient for NFAT-induced IL-2 activity | 2 | unitless | Derived | |

${n}_{3,{r}_{46}}$ | Hill coefficient for NFκB-induced IL-2 activity | 2 | unitless | Derived | |

${k}_{1,{r}_{46}}$ | AP1 quantity producing half-maximum IL-2 activity | 0.1·[AP1]${}_{total}$ | mol | Derived | |

${k}_{2,{r}_{46}}$ | NFAT quantity producing half-maximum IL-2 activity | 0.3·[NFAT]${}_{total}$ | mol | Derived | |

${k}_{3,{r}_{46}}$ | NFκB quantity producing half-maximum IL-2 activity | 0.1·[NFκB]${}_{total}$ | mol | Derived | |

${k}_{r1,{r}_{46}}$ | Deactivation rate of IL-2 activity by FOXP3 | $1\times {10}^{-4}$ | [$3.4974\times {10}^{-5}$, 0.0093] | (mol·min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{46}}$ | Constitutive deactivation rate of IL-2 | 1 | [0.1832, 8.9847] | min${}^{-1}$ | Derived |

${k}_{f,{r}_{47}}$ | Activation rate of FOXP3 activity | $1\times {10}^{4}$ | [748.5767, $2.4329\times {10}^{4}$] | mol·min${}^{-1}$ | Fitted |

${n}_{1,{r}_{47}}$ | Hill coefficient for AP1-induced FOXP3 activity | 2 | unitless | Derived | |

${n}_{2,{r}_{47}}$ | Hill coefficient for NFAT-induced FOXP3 activity | 2 | unitless | Derived | |

${k}_{1,{r}_{47}}$ | AP1 quantity producing half-maximum FOXP3 activity | 0.1·[AP1]${}_{total}$ | mol | Derived | |

${k}_{2,{r}_{47}}$ | NFAT quantity producing half-maximum FOXP3 activity | 0.1·[NFAT]${}_{total}$ | mol | Derived | |

${k}_{r1,{r}_{47}}$ | Deactivation rate of FOXP3 activity by mTOR | $2\times {10}^{-7}$ | [$7.3598\times {10}^{-8}$, $8.9275\times {10}^{-6}$] | (mol${}^{\xb7}2$min)${}^{-1}$ | Fitted |

${k}_{r2,{r}_{47}}$ | Constitutive deactivation rate of FOXP3 activity | 1 | [0.3685, 4.7328] | min${}^{-1}$ | Derived |

## Appendix 2: Parameter Sensitivity Analysis

## Appendix 3: Calcium Flux Measurements

**Figure A1.**Parameter sensitivity indices. Values are standardized by rows. Red = most sensitive; Green = least sensitive.

## Appendix 4: Phospho-IκBα Measurements

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Perley, J.P.; Mikolajczak, J.; Buzzard, G.T.; Harrison, M.L.; Rundell, A.E.
Resolving Early Signaling Events in T-Cell Activation Leading to IL-2 and FOXP3 Transcription. *Processes* **2014**, *2*, 867-900.
https://doi.org/10.3390/pr2040867

**AMA Style**

Perley JP, Mikolajczak J, Buzzard GT, Harrison ML, Rundell AE.
Resolving Early Signaling Events in T-Cell Activation Leading to IL-2 and FOXP3 Transcription. *Processes*. 2014; 2(4):867-900.
https://doi.org/10.3390/pr2040867

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Perley, Jeffrey P., Judith Mikolajczak, Gregery T. Buzzard, Marietta L. Harrison, and Ann E. Rundell.
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