# Modeling the Dynamics of T-Cell Development in the Thymus

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## Abstract

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## 1. Introduction

## 2. A Journey through Population Models of T-Cell Development

**Box 1.**Trajectory of murine intrathymic T-cell development.

#### 2.1. Early Steps of T-Cell Development

#### 2.2. Estimation of the Flow between Compartments at Steady State Using Larger Models

#### 2.3. Models for Thymus Involution and Shrinkage

#### 2.4. Regulations between Thymic Populations

## 3. Estimation of In Vivo Cell Proliferation in the Thymus

#### 3.1. Measuring the Number of Divisions by Label Dilution

#### 3.2. Nucleoside Analogue Incorporation during S Phase

#### 3.2.1. Direct EdU or BrdU Staining

#### 3.2.2. One-Point EdU or BrdU Pulse Followed by DNA Staining at Different Time-Points

#### 3.2.3. Dual Labeling with EdU and BrdU at Different Time-Points

#### 3.3. Future Models and Finding the Optimal Experimental Set-Up

## 4. Estimation of In Vivo Cell Death in the Thymus

## 5. Multi-Scale Considerations on Thymic Dynamics

#### 5.1. Linking the History of TCR Signaling to Cell Fate

#### 5.2. 3D Models of Thymic Development, APC Types and Antigen Spatial Compartmentalization

#### 5.3. Thymus Dynamical Models Can Help the Analysis of TCR Repertoires

#### 5.4. Future Types of Multi-Scale Models

## 6. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

APC | Antigen-Presenting Cell |

BrdU | Bromodeoxyuridine |

CLP | Common Lymphoid Progenitor |

CD4SP | Single-Positive CD4${}^{+}$CD8${}^{-}$ thymocyte |

CD8SP | Single-Positive CD4${}^{-}$CD8${}^{+}$ thymocyte |

CTV | Cell Trace Violet |

DC | Dendritic Cell |

DN | Double-Negative thymocyte |

DP | Double-Positive thymocyte |

EdU | 5-Ethynyl-2’-deoxyuridine |

ETP | Early T-lineage Progenitor |

iSP8 | Immature Single-Positive CD8${}^{+}$ thymocyte |

LMPP | Lymphoid-Primed Multipotent Progenitors |

MHC | Major Histocompatibility Complex |

ODE | Ordinary Differential Equation |

SP | Single-Positive thymocyte |

TCR | T-cell Receptor |

TRA | Tissue Restricted Antigen |

Treg | Regulatory T cell |

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**Figure 1.**Major developmental steps in the thymus as a basis for population models of T-cell development. (

**A**) Main stages annotated with their degree of expansion (y axis) and RAG1${}^{\mathit{GFP}}$ reporter expression (green levels). In such an experimental model, RAG1 gene regulatory elements drive expression of a reporter gene, such as GFP, the concentration of which depends on cell division and reporter protein half-life. Thus, reporter levels can be used as a timer and distinguish newly generated versus recirculating or long-term populations. The main bottlenecks in transition between thymocyte populations are $\beta $-selection, selecting for cells with functionally recombined TCR$\beta $, and positive and negative selection that select for cells with functional MHC reactive, but not self-reactive fully expressed TCR$\alpha \beta $. The hourglass denotes that loss of RAG expression could also come from cells being resident in the thymus for a long period of time (instead of recirculating), which is an open question. (

**B**) Gating strategies of functional sub-populations. The first lineage gating ‘lin-’ on the left discards B, NK and myeloid cells. (left) Detail of developmental stages inside the DN population, and a choice of markers to distinguish them. Progenitors inflowing from the blood are called Early T-lineage Progenitors (ETP) and refer to DN1a and DN1b. DN1 and early DN2a cells can also differentiate into B or NK cells while only late DN2bs are fully committed to the T-cell lineage [16]. When the DN4 population is only gated on CD4${}^{-}$CD8${}^{-}$CD28${}^{-}$CD44${}^{-}$, it also contains more differentiated populations containing TCR$\beta $ [17]. (right) Main developmental stages from the DN stage to fully mature CD4 and CD8 T cells and their export (open door symbols). Different gating strategies are shown for isolating DP and SP sub-populations. The death skulls refer to stages with high death. The term Tconv refers to conventional CD4${}^{+}$ SP, while CD8${}^{+}$ SP cells can also contain conventional and unconventional cells that are not described here. The relative size of each compartment is detailed in [17].

**Figure 2.**Population dynamics mathematical models of the thymus. (

**A**) Types of equations used when simulating thymic population dynamics, accounting for the dynamics of a population B fueled with progenitors coming from a population A, and further differentiating into a population C. (left) simple linear ODE with proliferation (round arrow), death (death skull) and differentiation (flat arrows). (middle) linear ODE with an additional regulated logistic growth according to a maximum carrying capacity K, and whose niche is shared with another population A (large box). The logistic growth control in thymus models has been implemented by inhibiting proliferation rather than enhancing death. (right) linear ODE-based generational models that simulate the cell numbers at each division within the population A. ${G}_{i}$ denotes the number of cells inside the generation i, i.e., that performed i divisions already. The rate of cells leaving a generation is $1/T$ where T is the half-life of a generation, and the rate of cells entering the next generation is $2(1/T-\delta )$ where $\delta $ is the death rate. This type of model assumes a generation-structured population behavior, i.e., that all cells perform a fixed number of divisions before exiting the compartment A, which can generate different dynamics than the linear ODE model on the left. It is also possible to add an outflow rate at each generation to change this behavior (not shown in the formula, see model variants in [19]). (

**B**) Published mathematical models, annotated with the equation design explained in A. Death skulls refers to a linear death rate, round arrows refer to proliferation, the large boxes represent a carrying capacity, while smaller sub-populations in circle denote a generational model. The red crosses denote neglected mechanisms in the models, and the open door refers to a linear outflow rate.

**Figure 3.**Parameters from four main studies [43,45,53,55]. (

**Top**) The size of each considered population is shown, at steady state in the models. Sometimes the model stabilizes at a different value than the experimental dataset, in which case the experimental value is given for comparison. All cell numbers are in million cells. (

**Bottom**) Detail of model parameters and cell numbers. All absolute values (cell numbers or flow between compartments) are rescaled to a total thymus size of 100 million cells, to be more easily compared. Technical details: Parameters from Sinclair et al. [43] are average values digitized from its Figure 3, under the “4+8” model, and the details of DP2/DP3 sub-populations are calculated from percentages shown in its Figure 7. The “parameter set 2” is shown for the study by Moleriu et al. [55]. *: this value was taken as a hypothesis and was not inferred from experimental data. **: we calculate residence time as 1/(output + death − proliferation), which is the half-life of the population dynamics. The authors instead calibrated the half-life of one cell (excluding its potential daughters), as 1/(output + death), to match experimental data, which ended up as a very long population residence time here. ***: This study did not show the number or percent of DN cells. We assumed a DN population size of 4% of the thymus to estimate the total thymus size and rescale the cell numbers to 100 million cells. ****: the calculated residence time diverged, probably because of digit precision on the parameters.

**Figure 4.**Experimental methods to measure proliferation in the thymus. (

**A**) Following the number of divisions of injected labeled cells by dye dilution. T cells were labeled with Cell Trace Violet (CTV), activated in vitro with anti-CD3 and anti-CD28 and measured for CTV intensity by flow cytometry at different time-points. Cells did not divide yet at 24 h. The first division can be seen at 36 h and up to 5 divisions can be seen at 72 h. By adoptive transfer of dye-labeled cells, their proliferation can be assessed at later time-points in vivo. (

**B**) Following the number of cells in the S phase by BrdU or EdU injection. A pulse of nucleoside analogue in vitro or in vivo labels the cells that are incorporating new DNA in the S phase (replication). An example is given of two cells that perform the cell-cycle phases at different time-points compared to the pulse. The cell in S phase during the pulse, gets a fraction of its DNA labeled depending on its S-phase duration and the pulse duration, while the cell in G1 phase did not get labeled. At the population level, the percent of labeled cells informs on the fraction of cells that were in the S phase during the effective pulse duration, while the percent of labeled DNA inside labeled cells indirectly informs on their S phase duration. (

**C**) Tracking of labeled cells at later time-points. A nucleoside analogue pulse (EdU or BrdU) can be followed by tracking the cell-cycle status at different time-points later, informing on the fate of cells that were in S phase during the pulse. (top): six populations can be quantified at each time-point: labeled and unlabeled, and in G0/G1, S or G2/M phases. (bottom) cell-cycle state (% of labeled cells in G0/G1 or S) of whole thymocytes over time after in vivo BrdU injection, which already gives an extrapolation of the duration of the G2/M duration (when cells start to be labeled in G1), or the S-phase duration (when labeled cells would have all left the S phase, if they would not come back into G1, by linear extrapolation). The duration of the full cycle, proposed to be when the labeled cells return to S phase, is less straightforward to identify and would need proper mathematical modeling. (

**D**) Dual-pulse labeling with EdU followed by BrdU to label cells that enter or leave the S phase in between pulses, and to later track the cycle stage of the labeled cells. (left) scheme of cells that will be labeled by either or nucleoside analogues depending on their cycle stage during the two pulses. (right) Example of visualization of the labeling by flow cytometry at a later time-point, where the DNA level can also be quantified for each population, giving a more precise glance in which stage of the S phase they currently are.

**Figure 5.**Mathematical approaches used to infer proliferation speed. (

**A**–

**C**) ODE-based models for simulating in vivo labeling of cells. Such models typically model an instant labeling of all cells in S phase, and possibly a decay of the labeling by proliferation (in (

**A**) only). In (

**B**), a two-pulse labeling is applied, and the dynamics of labeling are simulated for both labels. Assuming instant labeling of all cells in S phase, the first labeling stains the equilibrium value of such cells. Two strategies lead to different analytical formula: assuming the labeling interval t is negligible compared to the cell-cycle, cells cannot return in S; or simulating a 2-states Markov chain for the state of the cells at second labeling allows some cells to cycle multiple times. In (

**C**), the ODEs can be represented with a matrix formalism. (

**D**) From mean-field equations of growing populations, assuming a certain synchrony of the total cycle, the state of initially labeled cells over time can be predicted. (

**E**,

**F**) Age-structured stochastic models for cell proliferation with time distribution of each cycle phase under exponential growth, assuming delayed exponential distributions (

**E**) or with generic cycle and death times convenient when using gamma distributions (

**F**). (

**G**) Agent-based explicit simulation of each event at the cellular level, pre-defined from time distributions.

**Figure 6.**Different biological scales underlying thymic selection and models linking cellular interactions to signal and fate. (

**A**) TCR signaling, and thereby thymic selection fate, is mediated by the encounter with Antigen-Presenting Cells (APCs) displaying samples of self-peptides on their MHCs. TCR signaling can be induced by high affinity to an MHC (typically at each interaction), or to a cognate peptide (more rarely). Specific types of APCs express a larger scale of self-antigens (Tissue Restricted Antigens) and are compartmentalized in space (yellow box). (

**B**) Model predicting that T cells would show increasing signal over time due to increased TCR expression, and suggesting two self-adapting thresholds, for positive and negative selections. (

**C**) Experimental observations on ex vivo thymic slices, where T cells migrate and get signaling at each APC encounter. The encounter with cognate peptide leads to stop and strong signaling, while non-self-reactive interactions are shorter. (

**D**). Signal integration model. Each encounter with APCs leads to a transient increase in the integrated TCR signaling depending on the affinity (or avidity) of TCR-pMHC binding at each cell interaction. The integrated signal is translated into peak signal (Transient Signaling Level, TSL) and basal signal (Sustained Signaling Level, SSL), used by the T cells to decide their fate. Due to the correlation of SSL with MHC affinity and TSL with highest self-peptide affinity, the decision translates into Tconv with intermediate affinity to MHC while Tregs emerge with higher MHC affinity.

**Figure 7.**Crosstalk between recombination probabilities, proliferation and selection on the observed TCR frequencies in the repertoire. The progeny of one DN cell is shown as an example, starting from both non-recombined $\alpha $ and $\beta $ loci (crossed boxes on the left). Recombination events are shown in purple, and each V, D or J fragment is shown with different levels of red, yellow and blue/turquoise. In particular, since the V fragment is responsible for most of the interaction to the MHC, we have colored them to reflect their affinity towards at least one MHC. Due to proliferation in the DN stage, multiple daughter cells carrying different Tcrb gene recombination are ‘tested’ through $\beta $ selection, proliferate and recombine their Tcra, leading to multiple cells with the same Tcrb recombination but different Tcra recombinations, at the pre-selection DP stage. It is not clear whether cells proliferate equally between Tcrb recombination and the onset of selection. In this example, two cells proliferate once, two cells proliferate twice, and one cell leads to three daughters, as an example. Since the V gene is determining most of the contact interface to the MHC, we have shown an example where only TCRs with high affinity to an MHC survive positive selection (although we have discussed above that this could also be mediated by low affinity peptides). It is not clear whether TCR signaling impacts on the number of divisions, in which case cells with higher MHC affinity (or cross-reactive to multiple low affinity antigens) could proliferate more, which is annotated as a round arrow with a ‘?’. Finally, some cells die by negative selection and differentiate into CD4 Tconvs, CD4 Tregs or CD8 cytotoxic T cells. We have also included low observed proliferation at the SP stage. Altogether, this suggests that the frequencies of TCRs with a recombination scenario can be strongly modified by proliferation and selection between their rearrangement and thymic egress, asking for further mathematical investigation.

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**MDPI and ACS Style**

Robert, P.A.; Kunze-Schumacher, H.; Greiff, V.; Krueger, A.
Modeling the Dynamics of T-Cell Development in the Thymus. *Entropy* **2021**, *23*, 437.
https://doi.org/10.3390/e23040437

**AMA Style**

Robert PA, Kunze-Schumacher H, Greiff V, Krueger A.
Modeling the Dynamics of T-Cell Development in the Thymus. *Entropy*. 2021; 23(4):437.
https://doi.org/10.3390/e23040437

**Chicago/Turabian Style**

Robert, Philippe A., Heike Kunze-Schumacher, Victor Greiff, and Andreas Krueger.
2021. "Modeling the Dynamics of T-Cell Development in the Thymus" *Entropy* 23, no. 4: 437.
https://doi.org/10.3390/e23040437