Combining Theoretical and Experimental Techniques to Study ...jarciero/Publication_26.pdf · 1...

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Number of words: 6784 1 Number of figures: 7 2

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Combining Theoretical and Experimental Techniques to Study 5

Solid Organ Transplant Rejection 6

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Julia C. Arciero1,*, Andrew Maturo1, Anirudh Arun2, Byoung Chol Oh2, Gerald 8 Brandacher2, and Giorgio Raimondi2,* 9 10 1 Department of Mathematical Sciences, Indiana University-Purdue University 11 Indianapolis, Indianapolis, IN, USA 12 2 Vascularized and Composite Allotransplantation Laboratory, Department of Plastic and 13

Reconstructive Surgery, Johns Hopkins School of Medicine, Baltimore, MD, USA 14 15 *Correspondence: 16 Dr. Julia Arciero 17 [email protected] 18

19 Dr. Giorgio Raimondi 20

[email protected] 21 22 Abstract 23 The quality of life of organ transplant recipients is compromised by complications associated 24

with life-long immunosuppression, such as hypertension, diabetes, opportunistic infections, and 25

cancer. Moreover, the absence of established tolerance to the transplanted tissues causes limited 26 long-term graft survival rates. Thus, there is a great medical need to understand the complex 27 immune system interactions that lead to transplant rejection so that novel and effective strategies 28

of intervention that redirect the system toward transplant acceptance (while preserving overall 29 immune competence) can be identified. This study implements a systems biology approach in 30 which an experimentally-based mathematical model is used to predict how alterations in the 31

immune response influence the rejection of mouse heart transplants. Five stages of conventional 32 mouse heart transplantation are modeled using a system of 13 ordinary differential equations that 33 tracks populations of both innate and adaptive immunity as well as proxies for pro- and anti-34 inflammatory factors within the graft and a representative draining lymph node. The model 35 correctly predicts that in the absence of CD4+ T cells the transplant survives indefinitely, while 36

in the absence of CD8+ T cells the transplant is quickly rejected. The model also predicts that 37

decreasing the translocation rate of effector cells from the lymph node to the graft delays 38

transplant rejection. Increasing the starting number of quiescent regulatory T cells in the model 39 yields a significant but somewhat limited protective effect on graft survival. Surprisingly, the 40 model predicts that a delayed appearance of alloreactive T cells has an impact on graft survival 41 that does not correlate linearly with the time delay. This computational model represents one of 42 the first comprehensive approaches towards simulating the many interacting components of the 43 immune system. Despite some limitations, the model provides important suggestions of 44 experimental investigations that could improve the understanding of rejection. Overall, the 45

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systems biology approach used here is a first step in predicting treatments and interventions that 46 can induce transplant tolerance while preserving the capacity of the immune system to protect 47

against legitimate pathogens. 48

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Keywords 50 Mathematical model, transplant, rejection, immune response, antigen presenting cells, T cells, 51 cytokines 52 53

1. Introduction 54 Organ transplantation is a life-saving surgical procedure through which the functionality of a 55

failing organ can be restored via replacement with a functioning one. Transplants are performed 56 for a wide variety of organs, including skin, heart, kidney, liver, pancreas, spleen, and lung [1]. 57

However, without the administration of immunosuppressive drugs, the recipient’s immune 58

system recognizes the transplanted tissue as a foreign and potentially dangerous material and 59 responds with a massive immune attack that ultimately destroys the graft. This immune response 60 represents a major roadblock in the development of effective therapeutic regimens for the care of 61 patients requiring organ transplants. Current therapeutic regimens rely on chronic 62

immunosuppression. However, the quality of life of transplant recipients is compromised by 63 complications that derive from life-long immunosuppression (such as hypertension, diabetes, 64

opportunistic infections, and cancer) and by the limited long-term graft survival rates due to the 65 absence of established immune tolerance to the transplanted tissues. Ultimately, 20% or more of 66 transplanted patients die by 5 years post-transplant. Thus, there is a pressing need for a new 67

investigative approach to understand the systemic effects that arise from the dynamic interactions 68 between components of the immune system and transplanted tissues. 69

70 Previous hypothesis-driven research has provided important insight into the complex interactions 71 among the multiple components of the immune system, including T cells, antigen-presenting 72 cells, and cytokines [2-10]. Such studies have helped to determine the critical players and 73 processes in transplant rejection. For example, the rationale for “costimulation blockade” 74 therapies stems from such studies. These therapies, which target a key step of lymphocyte 75 activation, aim to control T cell activation and promote transplant survival. They have been 76 shown to be a potent strategy for promoting long-term acceptance of transplants in rodents [11-77 15] and primates [16-18]. However, their clinical translation encountered serious difficulties, 78 and ultimately costimulation blockade therapies were only approved as maintenance therapies 79 [19] since they could not promote tolerance [20]. To date, the only clinically successful avenue 80 of transplant tolerance induction has been through protocols that induce hematopoietic 81 chimerism [21, 22] (the coexistence of donor and recipient hematopoietic cells) via donor bone 82 marrow co-transplantation with the organ to be transferred. This procedure requires heavy 83 conditioning of patients and carries a significant risk of immunological complications (e.g., the 84 development of graft versus host disease). Consequently, this approach is applicable only in a 85 very restricted cohort of patients in need of a transplant. Thus, a valid and widely applicable 86 strategy to alter the reactivity of the immune system of transplant recipients in a robust and 87 reliable way is still needed. 88

89 Biological studies of rejection face various challenges. Experimental in vivo models of immune 90

rejection can elucidate precise information regarding select immune cell dynamics and the 91

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production and distribution of cytokines. However, conclusions about the system as a whole and 92 the generalizability of these conclusions to other species or types of allografts are further 93 complicated by factors such as procedural variability between models of rejection and variability 94 in parameter measurements. These factors, in combination with the complexity of the immune 95

response, motivate the use of an integrated theoretical and experimental approach to unravel the 96 inter-connected components of the immune response that contribute to transplant rejection. A 97 mathematical model of allograft rejection, refined by multiple clinical and experimental 98 observations, can help to identify variables and parameters that play a significant role in immune 99 system dynamics and yield a better understanding of the complex mechanisms of transplant 100

rejection. 101 102 Several computational models have been implemented to predict the dynamics of the immune 103 system in response to viral or bacterial infections [23-26], although only a few theoretical studies 104

have addressed transplant rejection. A recent publication used agent-based modeling (ABM) to 105 investigate solid organ transplant rejection [27]. In their study, the model provides an abstract 106

representation of the innate and adaptive immune components involved in the acute rejection 107 process of a solid organ transplant. The study does not use experimentally-based parameter 108

values, but it gives a range of possible responses to a transplant challenge without replicating a 109 specific disease process. Another recent study [28] used ordinary differential equations to model 110 the impact of the initial inflammatory response to a surgical insult on overall graft damage. 111

These studies have addressed general transplant immunology questions and have studied a very 112 specific aspect of the initiation of the transplant rejection response, but they do not offer the 113

capacity to capture the important intricacies of the rejection response in a combined experimental 114 and theoretical system that could lead to useful predictions to design new experimentations. The 115 mathematical model presented in the current study aims to provide useful theoretical predictions 116

of transplant rejection based on biologically relevant parameter values, initial conditions, and 117

cellular interactions. 118

The objectives of this study are (i) to develop a theoretical model to predict the effect of the 119

immune response dynamics on the rejection of an organ transplant based on experimental 120 measurements, and (ii) to identify new and effective strategies to promote transplant acceptance 121 that could be investigated experimentally. The model is composed of a system of ordinary 122

differential equations describing the cellular dynamics in the lymph node and graft in the context 123 of a simulated acute rejection of murine heart allograft. The model equations and parameters are 124 based on previous immune system models and are designed to incorporate key assumptions and 125

experimental observations of the immune response to transplants. The model captures the 126 known behavior of mouse heart rejection and recapitulates the effect of previously reported 127 experimental manipulations. It also underscores the relative importance of the ratios of effector 128

versus regulatory T cells on the speed of graft rejection. Importantly, the model predicts a 129 previously unappreciated behavior when altering the timing of T cell exposure to the graft, 130 providing details for the design of new experimentations that could confirm or refute these 131 findings. Ultimately, we believe this model could become an innovative tool to improve our 132

understanding of transplant rejection and significantly aid in the design of new and effective 133 strategies of immune intervention. 134

135 2. Methods 136 137

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2.1 Model Development 138 In this study, a mathematical model of murine heart transplants is developed to investigate the 139 interactions between the host immune system and transplanted graft. A compartmental model is 140 used in which all interactions are assumed to occur in either the graft or the draining lymph node. 141

A separate compartment for blood is not included, but the rates of exit and entry of the various 142 cells into the graft or lymph node are assumed to account for transit time in the blood. Table 1 143 provides the definition and description of all the variables tracked by this model. As with any 144 model, some assumptions and simplifications are necessary to address a specific question using 145 quantitative techniques. 146

147 The following list provides a summary of the assumptions made in this study: 148 149

Antigen Presenting Cells (APCs): a single population of APCs is defined in the model 150 and includes the populations of dendritic cells, macrophages, and B cells; 151

Antigen Presentation: direct, indirect, and semi-direct antigen presentation pathways 152 are grouped into a single function; no distinction is made between the origin of the APCs 153 (donor or recipient); 154

Rejection Mechanisms: only cell-mediated mechanisms of graft cell destruction are 155 included since the absence of B cells in mouse heart transplant models does not improve 156

graft survival [29]; 157

Lymphoid tissue: the activation of the immune response is restricted to an ideal lymph 158

node that drains the graft. The contribution of the response by multiple lymphoid tissues 159 is accounted for via modification of the translocation rate of activated T cells from the 160 blood to the graft; 161

Naïve T cells: the population of graft-reactive T cells is considered to be homogeneously 162

naïve. No contribution of memory T cells is considered at this stage. A continuous 163 output of newly generated T cells (from the thymus) is assumed to maintain a constant 164

number of naïve T cells in the lymph node. Of the total T cell population, 5% are 165 considered to be allo-reactive [30]; 166

Inflammatory Response: the danger signals generated by the surgical procedure of 167 transplantation (e.g., exposure to bacterial and viral components, surgical trauma, and 168 ischemia/reperfusion injury) and the inflammatory cytokines released by graft tissues and 169

immune components are all included in a single population (Cp); 170

Anti-inflammatory Response: the anti-inflammatory cytokines (e.g., IL-10, TGF-β) 171 normally produced by graft tissues and cells of the immune system as compensatory 172 mechanisms are all included in a single population (Ca); 173

Graft Cells Turnover: following destruction, the growth of heart cells is assumed to be 174 negligible for the timeframe considered in the model [31]. 175

176 With these assumptions, the dynamics between the immune system and the graft are described in 177 the following five stages (and are depicted in Figure 1): 178 179

1. Transplantation. Transplantation (introduction of the graft) occurs at Day 0 and is 180

captured by the model using the following initial conditions (listed in Table 3) for the 181 graft population (G), pro-inflammatory cytokines (Cp), and immature APCs (Aimm): G(0) 182

= 5,600,000 cells [32, 33], Cp(0) = 50 pg/ml, and Aimm = 2000 cells [34, 35]. G(0) was 183

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chosen by extrapolating the number of cells in a mouse heart based on the average mass 184 of a mouse heart and the average cell density of a human heart. Pro-inflammatory 185 cytokines are assumed to be present at time 0 since transplantation is associated with 186 surgical trauma and exposure to bacterial and viral agents. Additionally, during the 187

procedure, the reconnection to the recipient circulation initiates the process of 188 ischemia/reperfusion injury, causing rapid accumulation of inflammatory mediators [36-189 38]. Although there is a general agreement on the presence of inflammatory elements at 190 time 0, the overall amount is not known and thus an arbitrary value is chosen here. The 191 kinetics of the accumulation of Cp have been described previously [39, 40]. The presence 192

of these inflammatory cytokines leads to a rapid influx of host immature APCs into the 193 graft (representing the influx of circulating monocytes rapidly converting in the tissues 194 into APCs). 195 196

2. APC maturation and presentation of donor antigens in the lymph node. Once 197 exposed to Cp, immature APCs are activated into mature APCs in the graft (Amat

G). The 198

maturation of APCs contributes to an increased accumulation of pro-inflammatory factors 199 as well as, in a delayed fashion, to the production of anti-inflammatory factors (Ca). Once 200

mature, APCs exit the graft and travel to the draining lymphoid tissue. 201 202

3. Activation of T cells in the lymph node. In the theoretical lymph node, naïve CD8+ 203

effector T cells (TEN), naïve regulatory T cells (TRN), and naïve CD4+ helper T cells 204 (THN) that have the capacity to recognize donor antigens are assumed to be present 205

initially at background levels of 55000, 9500, and 70000 cells, respectively. Upon 206 entering the lymph node, Amat

LN facilitate the activation of T cells. As shown in Figure 1, 207 Amat

LN are necessary to promote the activation of naïve CD8+ (TELN), CD4+ (TH

LN), and 208

regulatory (TRLN) T cells in the lymph node. CD8+ T cell activation is dependent on the 209

licensing of interacting APCs by activated CD4+ T cells. Once activated, regulatory T 210 cells inhibit the activation of CD8+ and CD4+ T cells. T cell proliferation in the lymph 211 node depends on the autocrine and paracrine effects of growth factors (e.g., IL-2). 212

Regulatory T cells are unable to produce and secrete these growth factors, and thus their 213 proliferation is delayed and dependent on the presence of activated CD4+ and CD8+ T 214

cells [41]. 215 216

4. T cell infiltration of the graft. Following their activation, TELN, TH

LN, and TRLN exit the 217

lymph node and search for the inflamed tissues of the graft. It is important to note that 218 not all T cells exiting the lymph node will locate the graft. Also, the model assumes that 219 T cells originate from multiple lymph nodes, even though only one lymph node is 220 explicitly depicted and described. The translocation rate parameter value accounts for 221

these two situations. 222

223 5. Destruction of the graft. In the graft, TH

G promote the conversion of Aimm into 224 inflammatory antigen presenting cells (Ainf); this represents the activation of 225 macrophages into inflammatory cells that release cytotoxic agents (e.g., reactive oxygen 226 species) that induce death of surrounding graft cells. This process is inhibited by TR

G [42, 227 43]. The release of pro-inflammatory cytokines is promoted in the presence of Amat

G, 228 TE

G, THG, and Ainf. The release of Ca is assumed to depend on Amat

G, TR, Cp, G, and Ainf 229

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[44]. The presence of Ca inhibits the conversion of Aimm into Amat or Ainf. Ainf and TEG 230

direct the destruction of the graft, while TRG inhibit graft destruction. 231

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2.2 Model Equations. 233 The interactions described in the five stages are modeled using a system of 13 ordinary 234 differential equations that tracks cell populations and cytokine concentrations within the graft 235 and lymph node. Many of the parameter values are taken directly from literature sources, some 236 were obtained experimentally, and the remaining are estimated according to experimental 237 assumptions and observations. The initial values of all model variables are given in Table 2. 238

The model parameter values, units, and references are listed in Table 3. 239 240 The model equations describe the activation, proliferation, natural decay, destruction, and 241 inhibition of the various populations when appropriate. The interactions in the lymph node are 242

modeled using four equations, and the interactions in the graft are modeled using nine equations. 243 The superscripts LN and G denote cell populations in the lymph node and graft, respectively. 244

245 In Equation 1, the rate of change of mature APCs in the lymph node (Amat

LN) is defined. This 246

rate depends on the entrance of mature APCs from the graft at rate eA (first term) and on the 247 natural decay of Amat

LN in the lymph node (second term). 248 249

𝑑𝐴𝑚𝑎𝑡

𝐿𝑁

𝑑𝑡= 𝑒𝐴𝐴𝑚𝑎𝑡

𝐺 − 𝜇𝐴𝐴𝑚𝑎𝑡𝐿𝑁 (1) 250

251 The naïve alloreactive T cell populations (TEN, TRN, and THN) in the lymph node are assumed to 252 be constant since a background population of these cells is always present (due to thymopoiesis). 253

254

In Equation 2, the rate of change in CD8+ T cells in the lymph node is shown to depend on T cell 255

activation (term 1), decay (term 2), proliferation (term 3), and translocation (term 4). The 256 activation of TE

LN depends on the presence of both AmatLN and TH

LN while TRLN inhibit this 257

process [10, 43, 45]. Proliferation of the TELN cells depends on Amat

LN and TELN and occurs at 258

rate rE (used as a proxy for the production, secretion, and autocrine effect of IL-2) [46]. The 259 translocation of TE

LN from the lymph node is assumed to occur at rate eE. 260

261 262

𝑑𝑇𝐸

𝐿𝑁

𝑑𝑡=

𝑎𝐸𝑇𝐸𝑁𝐴𝑚𝑎𝑡𝐿𝑁 𝑇𝐻

𝐿𝑁

(𝛾1+𝐴𝑚𝑎𝑡𝐿𝑁 )(𝛼1+𝑇𝑅

𝐿𝑁)− 𝜇𝐸𝑇𝐸

𝐿𝑁 +𝑟𝐸𝑇𝐸

𝐿𝑁𝐴𝑚𝑎𝑡𝐿𝑁

𝛽1+𝐴𝑚𝑎𝑡𝐿𝑁 − 𝑒𝐸𝑇𝐸

𝐿𝑁 (2) 263

264 The rates of change of regulatory T cells in the lymph node (Equation 3) and CD4+ T cells in the 265

lymph node (Equation 4) contain the same four terms (activation, decay, proliferation, and exit) 266 as the equation for CD8+ T cells, with a few important differences: the activation of TR

LN and 267 TH

LN depends only on AmatLN, and the proliferation of TR

LN occurs only in the presence of TELN 268

or THLN. The activation of TH

LN is inhibited by TRLN [10, 42]. 269

270

𝑑𝑇𝑅

𝐿𝑁

𝑑𝑡=

𝑎𝑅𝑇𝑅𝑁𝐴𝑚𝑎𝑡𝐿𝑁

𝛾2+𝐴𝑚𝑎𝑡𝐿𝑁 − 𝜇𝑅𝑇𝑅

𝐿𝑁 +𝑟𝑅𝑇𝑅

𝐿𝑁(𝑇𝐸𝐿𝑁+𝑇𝐻

𝐿𝑁)

𝛼2+𝑇𝑅𝐿𝑁 − 𝑒𝑅𝑇𝑅

𝐿𝑁 (3) 271

272

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𝑑𝑇𝐻

𝐿𝑁

𝑑𝑡=

𝑎𝐻𝑇𝐻𝑁𝐴𝑚𝑎𝑡𝐿𝑁

(𝛾3+𝐴𝑚𝑎𝑡𝐿𝑁 )(𝛼3+𝑇𝑅

𝐿𝑁)− 𝜇𝐻𝑇𝐻

𝐿𝑁 +𝑟𝐻𝑇𝐻

𝐿𝑁𝐴𝑚𝑎𝑡𝐿𝑁

𝛾4+𝐴𝑚𝑎𝑡𝐿𝑁 − 𝑒𝐻𝑇𝐻

𝐿𝑁 (4) 273

274 The rate of change of the Aimm population in the graft is defined in Equation 5. The first term 275 represents the influx of Aimm due to the presence of the graft and pro-inflammatory cytokines. 276

The second term accounts for the natural decay of Aimm. The third term indicates the loss of 277 immature APCs once they become activated into Amat

G. The fourth term defines the THG-278

mediated activation of Aimm into Ainf. In both of these last two terms, the conversion of Aimm into 279 activated populations is inhibited by Ca and TR

G [44, 45, 47-49]. 280 281

𝑑𝐴𝑖𝑚𝑚

𝑑𝑡= 𝑘𝐶𝑝

𝐶𝑝𝐺 − 𝜇𝐴𝑖𝑚𝑚𝐴𝑖𝑚𝑚 − 𝑎𝑝1 (1 −𝐶𝑎

2

𝜂12+𝐶𝑎

2) (𝐴𝑖𝑚𝑚𝐶𝑝

𝛼4+𝑇𝑅𝐺 ) − 𝑎𝑝2 (1 −282

𝐶𝑎2

𝜂22+𝐶𝑎

2) (𝐴𝑖𝑚𝑚𝑇𝐻

𝐺

𝛼5+𝑇𝑅𝐺 ) (5) 283

284

Equation 6 describes the dynamics of mature APCs in the graft (AmatG). The first term defines 285

the activation of AmatG by Cp, which is inhibited by Ca and TR

G [44, 45, 47-49]. The second term 286 is the natural decay of Amat

G, and the last term accounts for the exit of AmatG from the graft to the 287

lymph node. 288 289

290 𝑑𝐴𝑚𝑎𝑡

𝐺

𝑑𝑡= 𝑎𝑝1 (1 −

𝐶𝑎2

𝜂12+𝐶𝑎

2) (𝐴𝑖𝑚𝑚𝐶𝑝

𝛼4+𝑇𝑅𝐺 ) − 𝜇𝐴𝐴𝑚𝑎𝑡

𝐺 − 𝑒𝐴𝐴𝑚𝑎𝑡𝐺 (6) 291

292

In Equation 7, inflammatory APCs are differentiated from Aimm in the presence of THG (term 1) 293

and are assumed to exhibit natural decay (term 2). 294

𝑑𝐴𝑖𝑛𝑓

𝑑𝑡= 𝑎𝑝2 (1 −

𝐶𝑎2

𝜂22+𝐶𝑎

2) (𝐴𝑖𝑚𝑚𝑇𝐻

𝐺

𝛼5+𝑇𝑅𝐺 ) − 𝜇𝐴𝑖𝑛𝑓𝐴𝑖𝑛𝑓 (7) 295

296

The rates of change for CD8+, regulatory, and CD4+ T cells in the graft (Equations 8, 9, and 10, 297 respectively) depend on the rate at which they enter the graft (term 1), their natural decay (term 298

2), and their proliferation (term 3). The parameter k that multiplies the exit rate of T cells from 299 the lymph node accounts for the fact that not all T cells exiting the lymph node reach the graft 300 and that T cells arrive from multiple lymph nodes. 301 302

𝑑𝑇𝐸

𝐺

𝑑𝑡= 𝑘𝑒𝐸𝑇𝐸

𝐿𝑁 − 𝜇𝐸𝑇𝐸𝐺 + 𝑟𝐸𝐺 (1 −

𝐶𝑎2

𝜂32+𝐶𝑎

2)𝑇𝐸

𝐺𝐺

𝛽2+𝐺 (8) 303

304

𝑑𝑇𝑅

𝐺

𝑑𝑡= 𝑘𝑒𝑅𝑇𝑅

𝐿𝑁 − 𝜇𝑅𝑇𝑅𝐺 +

𝑟𝑅𝐺𝑇𝑅𝐺(𝑇𝐸

𝐺+𝑇𝐻𝐺)

𝛼6+𝑇𝑅𝐺 (9) 305

306

𝑑𝑇𝐻

𝐺

𝑑𝑡= 𝑘𝑒𝐻𝑇𝐻

𝐿𝑁 − 𝜇𝐻𝑇𝐻𝐺 + 𝑟𝐻𝐺 (1 −

𝐶𝑎2

𝜂42+𝐶𝑎

2)𝑇𝐻

𝐺𝐴𝑚𝑎𝑡𝐺

𝛾5+𝐴𝑚𝑎𝑡𝐺 (10) 307

308 Equation 11 describes the dynamics of the mass of the graft. The first and second terms 309 represent the destruction of the graft due to Ainf and TE

G, respectively. TRG work to inhibit the 310

destruction of the graft. No growth of graft cells is assumed in this model as stated in our model 311 assumptions. 312

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313

𝑑𝐺

𝑑𝑡= −

𝑑𝑖𝑛𝑓𝐴𝑖𝑛𝑓𝐺

(𝛼7+𝑇𝑅𝐺)

−𝑑𝐸𝑇𝐸

𝐺𝐺

(𝛼8+𝑇𝑅𝐺)

(11) 314

315

As defined in Equation 12, the release of pro-inflammatory cytokines is triggered by AmatG, TE

G, 316 TH

G, and Ainf. The release of Cp by each of these cells is inhibited by TRG. The natural decay of 317

Cp is modeled in the last term. 318 319

𝑑𝐶𝑝

𝑑𝑡=

𝜌1𝐴𝑚𝑎𝑡𝑔

𝛼9+𝑇𝑅𝐺 +

𝜌2𝑇𝐸𝐺


𝜌3𝑇𝐻𝐺


𝜌4𝐴𝑖𝑛𝑓

𝛼12+𝑇𝑅𝐺 − 𝜇𝐶𝑝

𝐶𝑝 (12) 320

321 Equation 13 describes the release of Ca due to Amat

G , TRG , Cp , and Ainf , respectively. The last 322

term gives the natural decay of Ca. 323

324

𝑑𝐶𝑎

𝑑𝑡= 𝜉1𝐴𝑚𝑎𝑡

𝐺 + 𝜉2𝑇𝑅𝐺 + 𝜉3𝐶𝑝𝐺 + 𝜉4𝐴𝑖𝑛𝑓 − 𝜇𝐶𝑎

𝐶𝑎 (13) 325

326

2.3 Experimental Data Collection 327 Male 8 to 10 week old Balb/C (H-2d), and C57BL/6 (B6; H-2b) mice were purchased from the 328 Jackson Laboratory (Bar Harbor, ME) and housed in specific pathogen-free facilities at Johns 329 Hopkins University, Baltimore, MD, USA. All experiments were conducted according to 330

Institutional Animal Care and Use Committee-approved protocols. 331 332

Heterotopic (intra-abdominal) heart transplantation was performed from BALB/c to B10 mice, as 333 previously described [50]. On day 7 post transplantation, cells from grafts were isolated using an 334 adaptation of the technique described by Setoguchi et al. [51]. Briefly, tissues were digested at 335

37°C via 3 consecutive 15 minute incubations in PBS containing Collagenase IV (560U/ml; 336

Worthington,) DNAse I (275U/ml; Amresco), and Dispase II (0.4U/ml, Roche). Leukocytes were 337 enriched using a 24% Histodenz - (Sigma-Aldrich) based gradient separation. These preparations 338 where then used to quantify the content of CD4+, CD8+, and regulatory T cells (Treg) in the 339

rejecting hearts via flow cytometry. Cells were stained using anti-CD4+ and anti-CD8+ mAb 340 (from BD Bioscience) and anti-Foxp3 mAb (Affymetrix/eBioscience) according to the 341

manufacturer protocols; samples were acquired using a BD LSR-II flow cytometer. Data was 342 analyzed via FlowJo analysis software (FlowJo, LLC). 343 344 Table 4 summarizes the absolute counts and relative ratios of T cell subsets infiltrating a 345 rejecting heart on post-operative day (POD) 7 deriving from such analysis. From this data, the 346

biological variability observed between animals in the total number of each subset that infiltrate 347 the heart is clearly evident. Strikingly, however, the ratios among T cell subsets were maintained 348

within very narrow ranges. Consequently, we used the average number of T cells to set the scale 349 for the number of T cells in the model, and we optimized various model parameters to the 350 observed ratios of T cells. 351 352 A similar approach was used to determine the average number of each T cell subset in a typical 353

lymph node. Our data agree with a previously published data set [3]. Briefly, collection of 16 354 lymph nodes from multiple animals averaged the identification of 17e6 CD8+ T cells, 22e6 CD4+ 355 T cells, and 3e6 Treg. This renders 1.1e6 CD8+ T cells, 1.4e6 CD4+ T cells, and 0.19e6 Treg in 356

9

the average lymph node. Considering that approximately 5% of T cells are reactive against 357 donor antigens, the average lymph nodes contains (at time 0) 55,000 CD8+ T cells, 70,000 CD4+ 358 T cells, and 9,500 Treg. 359 360

2.4 Parameter Estimation. 361 The model contains 61 parameters. Many of the values of these parameters have been obtained 362 directly from experimental studies [1-6, 8-10, 26, 39, 40, 43-46, 50, 52-62] or other 363 mathematical models of the immune system [3, 23-26, 63-69]. Table 3 provides a list of all the 364 model parameter values and sources for their values when possible. A definition of “estimated” 365

in Table 3 indicates that the value was not found directly in the literature but was estimated 366 according to known relationships and ratios among cell populations in the model. For example, 367 due to the potency and cellular similarities of regulatory T cells and helper T cells, the activation 368 rate of TR

LN is assumed to be smaller than the activation rate of THLN [43]. As another example, 369

the death rate of 𝐴𝑖𝑛𝑓 is assumed to equal the death rate of AmatG. The constant values for TEN 370

and THN are obtained from experiments conducted in the present study (Table 4). According to 371

reported ratios [26, 43, 70], the TRN population should be chosen to be about one tenth of the 372 helper T cell initial populations. 373

374 Several remaining model parameters are optimized (and are defined as “optimized” in Table 3) 375

to satisfy the following experimental observations: 376 377

1) Presence of all T cells 378 a. APC conditions [34, 35] 379

i. AmatG have a peak population of approximately 18,000 cells. 380

ii. AimmG have a peak population of approximately 12,000 cells. 381

iii. AmatG and Aimm

G peak between days one and three. 382

b. Graft destruction [60] 383 i. A 75% reduction of the graft mass occurs by 12-14 days following 384

transplantation. 385

c. T cell ratios (Table 4) 386 i. The maximum TE

G value is approximately five times greater than the 387 maximum TH

G value. 388

ii. The maximum THG value is approximately seven times greater than the 389

maximum TRG value. 390

iii. The maximum number of TELN occurs at approximately 4 days post-391

transplantation. 392 iv. The maximum number of TE

G occurs at approximately 6 days post-393

transplantation. 394

2) Absence of Helper T Cells [7, 58] 395 No graft rejection. 396

3) Absence of Effector T cells [7, 58] 397 Rejection should be delayed slightly. 398

4) Absence of all T Cells [34, 35] 399 a. No damage to the graft. 400

b. APC measures 401 402

10

3. Results 403 404

3.1 Model Verification 405 The following four model simulations were used to confirm that the model results indeed reflect 406

the assumptions built into the model based on expected physiological behavior. 407 408

3.1.1 Timing of graft rejection 409 In the absence of any outside interference, cardiac transplants are rejected at approximately 12-410 14 days after transplantation. Tanaka et al. [60] performed in vivo visualization of murine 411

cardiac allograft rejection and identified the cessation of the heartbeat to occur on day 12, which 412 corresponded to a 75% reduction in the measured luminescence of donor tissue from transgenic 413 luciferase-GFP (green fluorescent protein)-modified mice. The present model uses this as an 414 approximate metric, defining graft rejection once the number of graft cells has decreased by 75% 415

of their initial number. Figure 2 (panel A) shows the time dynamics of graft rejection predicted 416 by the model. The behavior of other key populations including APCs in the graft, T cells in the 417

lymph node, T cells in the graft, and cytokines in the graft are shown in Figure 2 (panels B-F). 418 The number of T cells in the lymph node peaks around day 6-7 in the lymph node and day 7-9 in 419

the graft, which agrees with experimental observations [65]. The ratios of TE:TH and TH:TR at 420 their peaks are calculated to be 3.8 and 7.9, respectively, in the graft, which is consistent with 421 experimental values obtained in this study (Table 4). 422

423

3.1.2 Graft rejection in the absence of CD8+ T cells (TE) 424 As demonstrated experimentally [7], transplant rejection can occur even if no CD8+ cells are 425 present in the system; the time to rejection is just slightly delayed. The absence of CD8+ cells is 426 simulated in the model by modifying the initial value of naïve effector T cells to be TEN=0. As a 427

result, no effector T cells are generated in the lymph node, but graft rejection is predicted to 428

occur at day 22 (dashed curve, Figure 3A) instead of day 11 when all T cells are present (solid 429 curve, Figure 3A). Rejection is predicted to occur despite the absence of CD8+ T cells since 430 activated CD4+ T cells in the graft promote the differentiation of inflammatory APCs (Figure 431

3B) which cause graft destruction. Figure 3C serves to explain why the graft is not destroyed 432 sooner when no CD8+ T cells are present given that Ainf is much higher in their absence (Figure 433

3B). When all T cells are present, the graft is destroyed by both Ainf and CD8+ T cells (terms 1 434 and 2 in Equation 11, respectively). The contribution of each of these terms to the rate of change 435

of the graft population is plotted in Figure 3C (solid curves). Specifically, the contribution of the 436 Ainf term is shown in red, the contribution of the CD8+ T cells is shown in black, and the sum of 437 these contributions is shown in blue. The dashed curves correspond to these same cases when 438 CD8+ T cells are absent. Note that the contribution of Ainf (dashed red curve) and the sum of the 439 contributions (dashed blue curve) lie on top of each other since the contribution of the CD8+ T 440

cells is zero (black dashed curve). As can be seen in Figure 3C, the contribution of Ainf when 441 CD8+ T cells are absent exceeds the total solid blue curve until a time point between day 5 and 442

10 when the blue solid curve exceeds the dashed blue curve. This explains the steep decline in 443 graft population initially in the absence of T cells followed by a slower decay than when all T 444 cells are present. 445

446

3.1.3 Graft acceptance in the absence of all T Cells 447

11

As discussed in [10, 71], animals with no T cells (i.e., no CD4+ T cells, no CD8+ T cells, and no 448 regulatory T cells) are incapable of rejecting transplants. To simulate conditions of no T cells in 449 the model, the naïve T cell populations are set to zero: TEN = TRN = THN = 0. As a result, no T 450 cells are generated in the lymph node or graft. Although Amat

G are activated, the absence of T 451

cells or Ainf prevents any damage to the graft (Figure 4A). The APC dynamics in the graft under 452 these conditions are compared with data reported by Oberbarnscheidt et al. [34] in Figure 4B, 453 and the levels of the pro- and anti-inflammatory cytokines are shown in Figure 4C. 454

455

3.1.4 Graft acceptance in the absence of TH 456 Several studies [7, 46, 58] have demonstrated that the presence of CD4+ T cells is a necessary 457 and sufficient condition for rejection. As predicted by the model, in the absence of CD4+ T cells 458 in the lymph node (THN=0), the graft is accepted since no damage-inducing cells are activated 459 without the contribution of CD4+ T cell. 460

461

3.2 Model Simulations 462 The model is used to predict the effect of altering the number of naïve regulatory T cells 463 (adoptive transfer) and the translocation rate of T cells from the lymph node to the graft. Insight 464

from these simulations may have eventual application in designing improved therapeutic 465 strategies that promote tolerance of transplants. 466 467

3.2.1 Adoptive Transfer of regulatory T cells 468 Adoptive transfer is a technique by which T cells are obtained from an animal, stimulated in a 469

polyclonal or antigen-specific fashion, and grown in culture. The cells are then transferred back 470 into the original animal or into a separate animal with the overall goal of expanding the 471 frequency of those T cells. Ultimately, this procedure can be exploited to increase or decrease the 472

reactivity of the immune system. Adoptive transfer has been employed using Treg, aiming to 473

counter graft destruction, and is currently under active investigation for its clinical translation 474 [61, 72]. The size, frequency, and type of these transfers can vary greatly depending on the 475 system and overall treatment goal. Here, a single injection of naïve regulatory T cells into the 476

lymph node immediately prior to transplantation is simulated by varying TRN from 9500 cells to 477 3e8 cells. Figure 5 shows that the graft survival time increases nonlinearly with the injection 478

dose. However, transplant rejection is still expected [61] unless immunosuppression (which does 479 not affect Treg activity) or pre-activation of injected Tregs is used to reduce the levels of the 480

other T cells so that a large ratio of TR to T cells is maintained. 481 482

3.2.2 Translocation Rates 483 The ease with which T cells can travel between the lymph node and the graft is expected to 484 influence the destruction of the graft. For example, decreasing the rate (eE) at which TE cells 485

translocate from the lymph node to the graft should extend the survival of the graft, though not 486 indefinitely. Figure 6 depicts the effect of eE on graft survival time alone (magenta) or in 487

combination with the translocation rate of CD4+ cells (eE and eH, blue curve) and the 488 translocation rate of regulatory T cells (eE and eH and eR, black). Under normal model 489 conditions, eE = eH = eR = 0.001 day-1. If eE is increased, the graft survival time is decreased. If 490 both eE and eH are increased, the graft survival time is even more decreased. However, if eE, eH, 491 and eR are increased the survival time is longer because more Tregs are present to inhibit the 492 effects of the CD8+ and CD4+ T cells. The logic is reversed to the left of eE = 0.001 day-1. 493

12

494

3.2.3 Delayed injection of T cells 495 In Figure 7, the model is used to simulate the effect of introducing T cells into a system that 496 originally has no T cells for a fixed number of days (simulations for 10 (red), 20 (blue), 30 497

(black), 40 (magenta), and 50 (green) days are shown). This was used to assess the ability of the 498 model to predict the outcome of published experiments in which T cells were introduced into a 499 lymphopenic animal 50 days after heart transplantation. The rationale for this test was that the 500 healing process would make the graft incapable of initiating the rejection response. The reported 501 results, however, refuted that hypothesis and showed a complete rejection initiated even when T 502

cells were introduced 50 days after transplant [2]. The model presented in the current study fails 503 to predict this outcome, but provides valuable insight into the behavior of the system modeled. 504 For example, the red curve in Figure 7A shows that the model predicts graft acceptance when no 505 T cells are present and graft destruction once T cells are introduced starting at day 10. As 506

indicated by the additional curves in Figure 7A, the steady state population of graft cells (e.g., 507 the population of graft cells after 200 days) does not change monotonically with the number of 508

days till lymphocyte injection. That is, the steady state number of graft cells is higher if T cells 509 are injected at 20 days instead of 10 days, but lower if T cells are injected at 40 days instead of 510

10 days. This unexpected behavior is summarized in Figure 7B, which shows the population of 511 graft cells at 200 days as a function of the day at which T cells are injected. This graph clearly 512 shows the non-monotonic relationship between these values. 513

514

4. Discussion 515 516

In this study, a mathematical model of transplant rejection that encompasses both innate and 517 adaptive elements of the immune response is presented. The model is based on combining 518

experimentally observed ratios of different types of T cells in the lymph node and graft as well as 519

the time at which their numbers are maximum together with defined characteristics of the 520 immune response that have been reported in the literature [2-10, 26, 43-45, 50, 53, 54, 58, 60, 61, 521 70, 71, 73]. Our efforts in the development of this transplant rejection model were driven by its 522

ultimate application as a tool to provide a better understanding of the complex dynamics that 523 underlie the rejection response and to provide a novel and powerful perspective to predict new 524

methods for preventing graft rejection. Three hypothetical immune interventions are explored in 525 this study: modulation of the frequency of naïve regulatory T cells, alteration of the migration of 526

T cells to the graft, and transient depletion of the T cell pool. First, we considered a simple 527 experiment of adoptive transfer of naïve regulatory T cells simulating conditions where the 528 starting number of resting regulatory T cells in the lymph node was altered. Our model indicates 529 that a higher number of regulatory T cells causes an increase in the time to allograft rejection 530 (Figure 5). As expected, however, the impact on graft protection is modest and requires what 531

would be a non-physiological augmentation of regulatory T cell numbers to achieve a therapeutic 532 effect. As indicated below, this model is well suited to investigate which combination of 533

strategies could maximize the impact of regulatory T cell adoptive transfer [61]. For example, 534 although indirectly, the model simulations already suggest a powerful effect of activated Treg 535 migrating to the graft (see comments on third simulation below). While this paper only considers 536 a simple example, the ultimate goal of adoptive transfer is to maintain a high level of regulatory 537 T cells so that they accumulate in both the lymph node and the graft [43, 44, 50, 61]. Achieving 538 the greatest possible ratio of TR to other T cells would yield the maximum inhibitory effect on 539

13

the activation of TE, TH, Cp, Amat, and Ainf, and, as a result, provide a significant protection to the 540 graft. 541

542 Second, as shown in Figure 6, reducing the translocation rate of TE

LN has a nonlinear effect on 543

graft destruction. For example, a 50% decrease in eE yields an 82% increase in graft survival 544 time, while a 50% increase in eE decreases the graft survival time by 34%. Decreasing both the 545 translocation rate of TE

LN and the translocation rate of THLN causes an even more pronounced 546

increase in graft survival time. This protective effect is not only due to a more limited damage 547 inflicted directly by a reduced number of translocating T cells but is also due to the powerful 548

suppressive effect of regulatory T cells that localize to the graft. In fact, the concomitant 549 reduction of eR with eE and eH shows a much more limited prolongation of graft survival. This 550 behavior helps to explain why the inhibition of TE

LN translocation to the graft has a more 551 beneficial effect than their complete absence (Figure 3). This is probably due to the contribution 552

of TELN to the expansion of regulatory T cells in the lymph node that would then more efficiently 553

control the remaining immune response, a situation that would not occur in the absence of TELN. 554

Thus, the manipulation of activated T cell migration could have a more profound therapeutic 555 effect than the prevention of their activation or their deletion, as long as the migration of 556

activated regulatory T cells is not concomitantly affected. Such complex dynamics could 557 contribute to understanding the disparate therapeutic effects observed when targeting specific 558 chemokine receptors [74, 75]. Alternatively, this result highlights the importance of using 559

activated regulatory T cells rather than resting ones for adoptive transfer strategies. 560 561

Third, the theoretically predicted nonlinear and non-monotonic relationship between graft 562 survival and the delayed appearance of alloreactive T cells suggests that new experiments to 563 confirm such a relationship are needed to determine if the results suggest a new method for 564

promoting graft survival. The prediction of our model is in discordance with the experimental 565

observation that the re-introduction of T cells 50 days post-transplant causes a prompt rejection 566 response [2]. This underscores the need to adapt the theoretical model to incorporate other 567 important mechanisms that would contribute to such an outcome. At the same time, this 568

discrepancy indicates that the basic principles implemented in our model are not sufficient to 569 explain the intricate behavior of the immune system and suggest that additional scenarios need to 570

be investigated experimentally. We can speculate two plausible scenarios: a) the accumulation of 571 pro-inflammatory mediators follows a longer kinetic that supports delayed activation (though not 572

observed experimentally), or b) the phenomenon of lymphopenia induced proliferation of T cells 573 (observed when T cells are transferred into a lymphopenic mouse) causes the non-specific 574 activation of T cells that can travel directly to the graft and initiate the rejection response [76]. 575 The experimental validation of these possible hypothesis would strengthen the understanding of 576 the non-linear and non-monotonic behavior predicted in this scenario by our model. For 577

example, the model prediction of graft rejection when T cells are administered at 40 days versus 578 the model prediction of near graft acceptance when T cells are administered at 42 days warrants 579

additional investigation. This improved understanding would be essential in determining the 580 extent to which the transient elimination of T cells would be more effective and, possibly, what 581 combinatorial intervention strategy would maximize this effect. 582 583

4.1 Model Limitations. 584

14

Some of the choices and assumptions made in this study limit the capabilities of the model. 585 First, the model focuses on the interactions of T cells, APCs, and inflammatory cytokines, but 586 does not include small scale details such as cell signaling or the secretion of various factors. 587 Additionally, cytokines are grouped into 2 categories (pro- and anti-inflammatory signals) and 588

are tracked only in the graft, not in the lymph node. Considering the vast number of individual 589 cytokine molecules involved in a full immune response as well as their independent dynamics, 590 the relative strength of their effects on the overall immune system (as well as independent effects 591 on individual cell types), and their unique production and decay rates, a model that accounts for 592 each cytokine molecule individually will rapidly become complicated and cumbersome. But, 593

studies show that the overall balance of these signals and their specific varieties can significantly 594 impact graft outcome [4, 6, 40, 44]. Effects of pro- and anti-inflammatory cytokines are 595 assumed to be included in parameters such as rE, rR, and rH. The tradeoff of specificity for 596 simplicity allows the model to reproduce general behavior. 597

598 Some of the diversity and antigen-specificity of the various cell populations are generally 599

neglected in the model. B cells and memory cells are excluded, allowing the model to be 600 compared with experimental preparations only involving a naïve T cell repertoire [6, 7, 9, 54, 58, 601

73, 77]. The model does not accurately represent the accumulation of immature APC in the graft 602 when no T cells are present in the system. This limitation likely derives from the simplification 603 of incorporating multiple APC types into one variable and having Ainf and Amat

G originating from 604

the same starting population. Moreover, the effects of mechanisms of tolerization, namely the 605 induction of T cell anergy by immature APC or the conversion of T cells into regulatory T cells, 606

are not presently included but could be incorporated in the future. These processes contribute 607 significantly to the underlying anti-inflammatory processes, as they allow Aimm and TR

G to 608 inhibit activated cells in ways currently not being modeled. Modeling these factors may require 609

probabilistic considerations of co-stimulatory encounters of antigen with or without pro-610

inflammatory signals. Additionally, graft rejection experiments typically conclude upon 611 rejection and no further measurements of the graft mass are taken. Thus, any model predictions 612 post-rejection unfortunately cannot be compared to available experimental observations. 613

614 The model also assumes that the entire graft is attacked at Day 0. In reality, due to the three-615

dimensional heterogeneity of the system, sites undergoing an inflammatory response are 616 damaged more. There are also early and late inflammatory populations that could be included in 617

the model using a time delay. This would require converting the system into delay differential 618 equations, as in [25]. 619 620

4.2 Model extensions. 621 Excitingly, despite the presented limitations, multiple avenues of experimentation to understand 622

the rejection response and to assess the efficacy of therapeutic interventions are suggested by the 623 results obtained with this model. For example, the current model predicts that altering the TRN 624

population has a significant impact on graft survival, as shown in Figure 5 and 6. The size, 625 timing, and repetition of TR transfers can vary widely; many experiments have started to identify 626 appropriate combinations for maximizing graft life [10, 43-45, 50, 61, 73]. This model can be 627 used to simulate a multitude of adoptive transfer regimens that may or may not have been 628 explored experimentally. In addition, pharmaceutical immunosuppression can be simulated by 629 targeting terms in the equations that represent chemical pathways. In particular, there is much 630

15

interest in directly manipulating pro- and anti-inflammatory signals as novel immunosuppressive 631 strategies; model simulation could help to identify optimal regimens. The model can also be 632 used to assess the compatibility between the current strategy of immunosuppression and 633 experimental immune interventions and guide the identification of optimal conversion strategies. 634

Overall, the model can be used to hypothesize which pathways are viable targets for 635 pharmaceutical intervention based on parameter sensitivity analysis and model dynamics. 636 Combined with a continuous cycle of suggested experimentation and model optimization, this 637 approach has potential for valuable contributions in the quest of transplant tolerance induction. 638

639

Conflict of Interest 640 The authors declare that the research was conducted in the absence of any commercial or 641 financial relationships that could be construed as a potential conflict of interest. 642 643

Authors and Contributors 644 JA, AM, AA, BO, GB, and GR contributed to the theoretical elaboration of the computational 645

model described herein; JA and AM wrote the model script; JA, AM, and AA performed model 646 simulations; BO, GB, and GR executed all wet-lab experiments; JA, AM, AA, BO, GB, and GR 647

wrote the manuscript while providing critical feedback. 648 649

Funding 650 This work was supported by a Burroughs Wellcome Fund BWF Collaborative Research Travel 651 Grant (to JA), the IUPUI School of Science Institute for Mathematical Modeling and 652

Computational Science Grant to Enhance Interdisciplinary Research and Education (iM2CS-653 GEIRE) award (to JA), a National Institutes of Health grant number UL1TR000005 in the form 654 of a CTSI-PEIR (to GR), and grant number R21HL127355 (to GR). 655

656

Abbreviations 657 APCs: Antigen Presenting Cells 658 POD: Post-operative Day 659

660

Acknowledgements 661 We thank Dr. Yoram Vodovotz, Dr. James Faeder, and Ms. Joyeeta Dutta-Moscato for 662 invaluable feedback in the initial elaboration of the theoretical model of transplant rejection. We 663

also thank Xiaoling Zhang of the Ross Research Flow Cytometry Core (Johns Hopkins 664 University) for invaluable assistance with all flow cytometry-based experiments. 665

666

667 668

669 670

671 672 673 674 675 676

16

Figure Captions 677 678 Figure 1. Schematic of the five stages of transplant rejection defined in the theoretical 679 model. Circled numbers correspond to the stage number. 1) Transplantation is represented in 680

the model by positive initial conditions for the graft population, pro-inflammatory cytokines, and 681 immature APCs. 2) The activation of Aimm into Amat by pro-inflammatory factors, and the 682 translocation of Amat

G from the graft to the lymphoid tissue. Pro- and anti-inflammatory 683 cytokines attract Aimm to the graft. 3) The activation of TE

LN, THLN, and TR

LN in the lymph node. 684 T cell activation is facilitated (blue lines) by Amat

LN and THLN and inhibited (red lines) by TR

LN. 685

4) The translocation of T cells from the lymph node to the graft. 5) In the presence of THG, 686

Aimm are activated into AmatG or Ainf (blue lines). Graft cells are destroyed by Ainf and TE

G. Each 687 of these processes is inhibited by TR (orange lines). 688 689

Figure 2. (A) Graft rejection is predicted to occur approximately 11 days following 690 transplantation. (B) Model predicted values of immature APCs (Aimm, magenta), mature APCs 691

(AmatG, green), and inflammatory APCs (Ainf, cyan) in the graft. (C) Model predicted values of 692

regulatory T cells (black), CD4+ T cells (blue), CD8+ T cells (red) in the lymph node. (D) Model 693

predicted values of regulatory T cells (black), CD4+ T cells (blue), CD8+ T cells (red) in the 694 graft. (E) Model predicted concentration of pro-inflammatory cytokines (Cp) and anti-695 inflammatory cytokines (Ca) in the graft. (F) Ratio of Ca:Cp in the graft. 696

697 Figure 3. (A) Graft rejection is delayed by approximately ten days in the absence of CD8+ T 698

cells. Dashed curve: model prediction in the absence of effector T cells. Solid curve: model 699 prediction in the presence of all T cells. (B) Model predicted values of inflammatory APCs in 700 the absence of effector T cells (dashed curve) and in the presence of all T cells (solid curve). (C) 701

Individual contributions of Ainf (red curve) and CD8+ T cells (black curve) and combined 702

contribution (blue curve) to the rate of change of the graft population in the absence (dashed) and 703 presence (solid) of CD8+ T cells. 704 705

Figure 4. (A) Model predicted number of graft cells when no T cells are present. (B) Model 706 predicted number of mature APCs (green), immature APCs (magenta), and inflammatory APCs 707

(cyan) in the graft when no T cells are present. The model predictions are compared with data 708 reported in [34, 35] (asterisks). (C) Concentrations of pro-inflammatory cytokines (red) and 709

anti-inflammatory cytokines (blue) when no T cells are present. 710

711 Figure 5. Model predicted values of time until transplant rejection as the initial (and constant) 712 level of naïve regulatory T cells is varied between 9500 and 3e8 cells. 713 714

Figure 6. Model predicted values of transplant survival times as the translocation rate of CD8+ 715 T cells (eE, magenta) is varied alone or in combination with the translocation rate of CD4+ cells 716

(eH, blue) and the translocation of regulatory T cells (eR, black). 717 718 Figure 7. (A) Model predicted number of graft cells when no T cells are initially present and 719 then injected after 10 (red), 20 (blue), 30 (black), 40 (magenta), and 50 (green) days. (B) The 720 number of graft cells predicted by the model at day 200 when T cells are administered at 721 different time points between 10 and 60 days. 722

17

Table 1. Description of model variables 723

Variable Description Location

𝐴𝑚𝑎𝑡𝐿𝑁 Mature antigen presenting cells Lymph node

𝑇𝐸𝐿𝑁 Activated effector T cells

𝑇𝑅𝐿𝑁 Activated regulatory T cells

𝑇𝐻𝐿𝑁 Activated helper T cells

𝐴𝑚𝑎𝑡𝐺 Mature antigen presenting cells Graft

𝐴𝑖𝑚𝑚 Immature antigen presenting cells

𝐴𝑖𝑛𝑓 Inflammatory antigen presenting cells

𝑇𝐸𝐺 Activated Effector T cells

𝑇𝑅𝐺 Activated regulatory T cells

𝑇𝐻𝐺 Activated Helper T cells

𝐺 Graft cells

𝐶𝑝 Pro-inflammatory cytokine

𝐶𝑎 Anti-inflammatory cytokine

724 725 726

Table 2. Initial values for model variables 727

728

Variable Initial Value Unit

𝐴𝑚𝑎𝑡𝐿𝑁 0 cells

𝑇𝐸𝐿𝑁 0 cells

𝑇𝑅𝐿𝑁 0 cells

𝑇𝐻𝐿𝑁 0 cells

𝐴𝑚𝑎𝑡𝐺 200 cells

𝐴𝑖𝑚𝑚 2000 cells

𝐴𝑖𝑛𝑓 0 cells

𝑇𝐸𝐺 0 cells

𝑇𝑅𝐺 0 cells

𝑇𝐻𝐺 0 cells

𝐺 5.6e6 cells

𝐶𝑝 50 pg/mL

𝐶𝑎 0 pg/mL

729

730

731 732 733 734 735

736 737 738

18

739 Table 3. Names, values, units, and citations for all model parameters. 740

Equation Parameter Name Value Unit Source

1 𝑒𝐴 5.5 1/day [24]

1 𝜇𝐴 1.2 1/day [64]

2 𝑇𝐸𝑁 55000 cells Section 2.3

2 𝑎𝐸 3 1/day [25]

2 𝛾1 100 cells [25]

2 𝛼1 2500 cells estimated

2 𝜇𝐸 0.7 1/day [25]

2 𝑟𝐸 1.51 1/day [25]

2 𝛽1 5000 cells estimated

2 𝑒𝐸 0.001 1/day [24]

3 𝑇𝑅𝑁 9500 cells Section 2.3

3 𝑎𝑅 2.82e-4 1/day optimized

3 𝛾2 1000 cells estimated

3 𝜇𝑅 0.7 1/day [24, 78]

3 𝑟𝑅 0.02 1/day estimated


3 𝑒𝑅 0.001 1/day [24]

4 𝑇𝐻𝑁 70000 cells Section 2.3

4 𝑎𝐻 6018.9 cells/day optimized

4 𝛾3 100 cells [25]


4 𝜇𝐻 0.4 1/day [24, 25]

4 𝑟𝐻 1.51 1/day [25, 54]

4 𝛾4 4000 cells [25]

4 𝑒𝐻 0.001 1/day [24]

5 𝑎𝑝1 1500 cells/day/(pg/ml) optimized

5 1 10 pg/ml estimated


6 𝑘𝐶𝑃 0.005 1/day/(pg/ml) optimized

6 𝜇𝐴𝑖𝑚𝑚 60 1/day optimized

6 𝑎𝑝2 3.844 1/day optimized



7 𝜇𝐴𝑖𝑛𝑓 1.2 1/day [64]

8 𝑘 15 - estimated

8 𝑟𝐸𝐺 0.3 1/day optimized


8 𝛽2 4e6 cells estimated

9 𝑟𝑅𝐺 0.00375 1/day optimized


19

10 𝑟𝐻𝐺 0.755 1/day estimated

10 𝛾5 4000 cells estimated


11 𝑑𝑖𝑛𝑓 0.055 1/day optimized


11 𝑑𝐸 0.004 cells/day optimized


12 𝜌1 10.98 (pg/ml)/day optimized


12 𝜌2 0.024 (pg/ml)/day estimated


12 𝜌3 0.24 (pg/ml)/day estimated


12 𝜌4 10.95 (pg/ml)/day optimized


12 𝜇𝐶𝑝 0.15 1/day optimized

13 1 2.08e-4 (pg/ml)/cells/day estimated


13 3 4.2e-9 1/cells/day estimated


13 𝜇𝐶𝑎 0.05 1/day estimated

741 742

743

744 Table 4. Measures of T cells infiltrating a rejecting heart on POD 7. 745

CD8 CD4 CD8/CD4

Ratio

Treg Treg

(% of CD4)

Heart #1 2.7e6 5.4e5 5 7.4e4 13.6

Heart #2 4.3e5 8.3e4 5.3 1.2e4 14.9

Heart #3 6.5e5 1.8e5 3.7 2.5e4 14.1

Average 1.27e6 2.7e5 4.7 3.7e4 14.2

SE 6e5 1.2e5 0.4 1.5e4 0.3

746 747

748

20

Figures 749

750

Figure 1 751

752

753

754

755

756

757

758

759

760

761

762

763

21

Figure 2 764

765

766

22

Figure 3 767

768

769

770

23

Figure 4 771

772

773

774

775

Figure 5 776

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