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Edited by: Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

Reviewed by: Filippo Castiglione, Italian National Research Council (CNR), Italy; Yinghong Hu, Emory University, United States

This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Activation of naive CD8 T-cells can lead to the generation of multiple effector and memory subsets. Multiple parameters associated with activation conditions are involved in generating this diversity that is associated with heterogeneous molecular contents of activated cells. Although naive cell polarisation upon antigenic stimulation and the resulting asymmetric division are known to be a major source of heterogeneity and cell fate regulation, the consequences of stochastic uneven partitioning of molecular content upon subsequent divisions remain unclear yet. Here we aim at studying the impact of uneven partitioning on molecular-content heterogeneity and then on the immune response dynamics at the cellular level. To do so, we introduce a multiscale mathematical model of the CD8 T-cell immune response in the lymph node. In the model, cells are described as agents evolving and interacting in a 2D environment while a set of differential equations, embedded in each cell, models the regulation of intra and extracellular proteins involved in cell differentiation. Based on the analysis of

Following acute infection, the activation of naive CD8 T-cells by antigen presenting cells (APCs) triggers the synthesis of proteins controlling cell proliferation and differentiation up to the memory state. While CD8 T-cell population dynamics have been widely described, it is of great interest to better understand the molecular mechanisms driving the CD8 T-cell response. In particular, determining the effects of molecular events on the generation of memory cells is necessary for vaccine design improvement.

The CD8 T-cell immune response occurs through four main phases. First the activation of naive CD8 T-cells in secondary lymphoid organs such as lymph nodes (LN) or spleen by APCs through MHC class I antigenic peptide/T-cell receptor (TCR) binding, surface co-receptor/ligands interactions and soluble cytokines secretion. Once activated, CD8 T-cells proliferate quickly during the expansion phase, which expands the initial population by a factor of 10^{3} to 10^{5} (

The responding effector population is composite and two subsets with antagonistic fates have been described (^{lo}CD127^{hi}) and SLEC (KLRG1^{hi}CD127^{lo}) express effector features (cytotoxicity, proliferation) but MPEC are capable of differentiation into memory cells while SLEC are destined to die during the contraction phase (

Among the genes, transcription factors and proteins involved in the CD8 T-cell response, some seem to play key roles in the differentiation processes. Transcription factors Tbet and Eomesodermin (Eomes) appear to play critical roles in the acquisition of effector and memory phenotypes. It has been shown that the expression of Tbet induces the development of SLEC and represses the development of MPEC profiles (

Since a unique initial antigenic signal can trigger a complete response, additional mechanisms are necessary to generate the observed molecular-content heterogeneity. Arsenio et al. (

Nevertheless, the role of asymmetric division of polarised naive cells in the T-cell differentiation process appears to be controversial (

Less is known about the partitioning of molecular content in the course of subsequent cell divisions. However, several studies support the hypothesis that when a cell divides, a random, uneven partitioning of the molecular content occurs (

We emphasize that the

In a recent work (

Several works [see (

Gong et al. (

Prokopiou et al. (

The model presented in this article has been developed from the multi-scale agent-based model previously introduced in Prokopiou et al. (

In this paper, we are interested in understanding how, from the activation of naive CD8 T-cells, an antigen-independent regulation of intra-cellular molecular content can drive a complete CD8 T-cell response. We particularly focus on the role of molecular-content heterogeneity among a CD8 T-cell population in the generation of memory cells. We first verify our model's ability to reproduce

4 × 10^{5} naive CD8 T-cells from CD45.1+ F5 TCR transgenic mice (B6.SJL-Ptprc^{a}Pepc^{b}/BoyCrl-Tg(CD2-TcraF5, CD2-TcrbF5)1Kio/Jmar) recognizing the NP68 epitope were transferred intravenously in congenic CD45.2+ C57BL/6 mice (C57BL6/J). The day after recipient mice were inoculated intranasally with 2 × 10^{5} PFU (plaque forming units) of a vaccinia virus expressing the NP68 epitope (^{+}CD45.2^{−}) from host (CD45.1^{−}CD45.2^{+}) CD8 T-cells. Naive (CD44^{−} Mki67^{−} Bcl2^{+}), effector (CD44^{+} Bcl2-) and memory (CD44^{+} Mki67^{−} Bcl2^{+}) CD8 T-cells have been identified (

OT1 CD8 T cells mRNA expression data time courses come from the ImmGen project (

We aim at describing the molecular regulation within each CD8 T-cell during a response to an acute infection, and how the dynamical molecular state of a cell characterises its differentiation stage. We present on

Simplified molecular regulatory network in a CD8 T-cell. Red molecular factors dynamics are described by Equations (1–6); yellow molecular factors dynamics are described by Equation (7); black arrows: promotion or secretion; green arrows: transition between activated and non-activated form of IL2R; red dashed arrows: inhibition. The meaning of the numbered arrows is reported in

Description of the molecular signalling pathways in

^{°} |
||
---|---|---|

1 | Activated TCR induces the development of IL2 receptors | ( |

2 | Activated TCR induces the synthesis of Tbet | ( |

3 | Deactivation of activated IL2 receptors | ( |

4 | Activation of IL2 receptors | ( |

5 | Activated IL2 receptors induce the development of new IL2 receptors | ( |

6 | Activated IL2 receptors inhibit the expression of the |
( |

7 | Activated IL2 receptors induce the expression the |
( |

8 | Tbet enhances the inhibition of the |
( |

9 | Internal IL2 is secreted in extracellular environment | ( |

10 | External IL2 binds the non-activated IL2 receptors to activates them | ( |

11 | Tbet inhibits the secretion of IL2 | ( |

12 | TCR activation activates |
( |

13 | TCR activation inhibits the activation of Caspases (via Erk, Bim, Bax and Bcl2) | ( |

14 | Activated IL2 receptors inhibit the activation of Caspases (via Stat5, BAX et Bcl2) | ( |

15 | Tbet induces the expression of FasL | ( |

16 | FasL activates Fas through cell contact | ( |

17 | Activated Fas induces Caspases activation | ( |

18 | Tbet activates |
( |

19 | Eomes induces the expression of IL2 receptors | ( |

20 | Activated IL2 receptors induce the expression of |
( |

21 | Activated TCR inhibits |
( |

22 | Eomes inhibits the activation of Caspases (via Bcl2) | ( |

23 | Tbet inhibits the expression of |
( |

This MRN is initiated upon antigen presentation to a naive CD8 T-cell, through the engagement of the TCR. Antigenic stimulation triggers the synthesis of interleukine-2 (IL2) by the CD8 T-cell and the production of IL2 receptors (IL2R) on the cell membrane (

Antigenic stimulation independently stimulates Tbet synthesis (

Eomes expression, involved in the acquisition of memory phenotype (

The activation of IL2 receptors, of the TCR and the protein Eomes prevents apoptosis by inhibiting the activation of Caspases, in particular through the mediator protein Bcl2 (

Based on the above-described reactions, and from the equations used in Prokopiou et al. (^{*}]), Caspases ([

All parameters are positive. Parameters λ are associated to induction and inhibition effects, μ are associated to activation and deactivation of transmembrane proteins and

The effects of the external environment on the intracellular system (1–6) are taken into account through five variables. The variable _{APC} (Equations 1, 3, 5, 6) is equal to the number of APCs bound to the considered CD8 T-cell and accounts for TCR engagement. The variable _{c1} and λ_{E4} is not active in naive cells. The variable ^{cm}] is equal to the concentration of IL2 at the cell membrane, in the extracellular environment. Finally, [^{cm}] is defined as the sum of Tbet concentrations in effector and memory CD8 T-cells in contact with the considered CD8 T-cell and acts as a proxy for the expression of Fas in those cells.

We introduced the variable [_{E1}[

The positive feedback loop on Tbet is modeled with an order

Proposition 1 (_{APC} = 0, n>1 and _{u} < [Tb]_{s}, such that 0 and [Tb]_{s} are locally asymptotically stable and [Tb]_{u} is unstable

In the following, we will assume that the conditions

System (1–6) is embedded in every CD8 T-cell. Nevertheless, cell-cell contacts, stochastic events (cell cycle length, protein distribution at division) and external concentrations of IL2 affect the evolution of the system such that each CD8 T-cell develops a unique molecular profile based on its own history.

The secretion of IL2 by CD8 T-cells and its isotropic diffusion in the extracellular domain (with periodic boundary conditions) are modeled by the following PDE, introduced by Prokopiou et al. (

where [^{cm}] term, in (1–2), defined as the sum of [

Rules controlling cell division (including protein distribution at the division), apoptosis and differentiation are summarised in

Main rules applying to APCs and CD8 T-cells in the model.

APC | ❍ | ✔ | ❍ | ❍ | ❍ |

Naive | ❍ | ❍ | ❍ | ❍ | ❍ |

Pre-activated | ❍ | ✔ | ✔ | ❍ | ❍ |

Activated | ✔ | ✔ | ✔ | ✔ | ❍ |

Effector | ✔ | ✔ | ✔ | ✔ | ✔ |

Memory | ❍ | ✔ | ✔ | ✔ | ✔ |

We designed a set of rules based on the linear, irreversible differentiation scheme from Prokopiou et al. (

CD8 T-cell differentiation scheme. Red arrows: proliferation; black arrows: differentiation;

A naive CD8 T-cell binding an APC becomes pre-activated and maintains the contact with the APC thanks to good adhesion properties (cf. Section 2.4 and _{th}, the pre-activated CD8 T-cell becomes activated, leaves the APC, and starts to proliferate. When an activated CD8 T-cell divides, it gives birth to two CD8 T-cells whose states are determined by their respective concentrations of protein Tbet by comparison with a given threshold _{th}: activated if [_{th}, effector otherwise. Finally, if the concentration of protein Eomes is greater than the threshold _{th}, a dividing activated or effector CD8 T-cell will differentiate into memory cell and stop proliferating.

Division is considered only for activated and effector CD8 T-cells. The cell cycle length (hours) of a cell preparing its

When a CD8 T-cell divides, the molecular content of the mother cell is randomly divided between the two daughter cells. To account for protein distribution between daughter cells at each division and for each protein, let us introduce the parameter

For the sake of clarity, we emphasise that the degree of unevenness

The exact value of each daughter cell molecular content at birth is randomly chosen according to a probabilistic law, as detailed hereafter. Each protein concentration [_{i}[_{i})[_{i}, _{i}∈[0.9, 1] for _{i}∈[0, 1] so the quantity of molecular material is preserved at each division, given that the volume of each daughter cell is half the volume of the mother. Different degrees of unevenness will be considered in section 3.3.

One special case of division is the asymmetric division, and its associated unequal repartition of Tbet between daughter cells. To account for polarisation of naive cells by antigenic signalling and the consecutive asymmetric divisions, the first division of a CD8 T-cell following its activation by an APC is characterised by a very specific uneven distribution of protein Tbet only between the two daughter cells: a coefficient

CD8 T-cell apoptosis occurs as soon as Caspases concentration [_{th}. APCs are present from the beginning of the simulation and their lifetime is randomly chosen from the uniform law

At the cell population scale, we use a cellular Potts model (CPM), also known as Glazier-Graner-Hegeweg model (

In our model, based on that from Prokopiou et al. (_{e}.

Cell (including extracellular medium) size variation and displacement result from the succession of copies of index from nodes to neighbour nodes, based on the minimisation of the Hamiltonian Ω [see Equation (8)], thanks to a simulated annealing algorithm. More precisely, at each iteration, known as Monte Carlo Step (MCS), of the CPM, the following algorithm is executed

_{s} and, among its first order neighbours, a target node _{g}.

_{s} would copy its index on node _{g}, i.e., if cell σ(_{s}) incorporates the node _{g}.

_{s} copies its index σ(_{s}) on _{g}, i.e., _{g} is integrated by cell σ(_{s}). Else, the copy is accepted with probability exp(−ΔΩ/

Note that it is conventional to consider

The Hamiltonian Ω is computed using the following formula:

where _{τ1, τ2} accounts for the contact energy between two cells of types τ_{1} and τ_{2}. Thanks to the term _{σ} and _{σ} are the actual perimeter and area of cell σ, respectively, whereas _{τ(σ)} and _{τ(σ)} are the target perimeter and area, respectively, for a cell of type τ(σ) ; perimeter and area constraints then penalize the configurations where the effective perimeter and area are distant from the target ones. Parameters λ_{area} and λ_{pm} define the weights of those two constraints. The perimeter constraint has been added to the definition used in Prokopiou et al. (

The energy Δ_{motility} is defined by

where _{s})) is the weight associated to the motility energy for the cell σ(_{s}) and θ(σ(_{s}), _{s}) at time _{motility} is all the more high (and then the copy is all the more probably accepted) that the copy direction (_{g}−_{s}) aligns with (

The initial cell population is composed of 30 naive CD8 T-cells and 3 APCs. A simulation requires 30,000 iterations (MCS) corresponding to 20 days and 20 h in the real time, that is, 1 MCS represents 1 min. When a simulation starts, APCs are already present in the LN, ready to activate naive CD8 T-cells. We consider the initial time to be day 4 post-infection (D4 p.i.) since our

We assume that a node of the lattice corresponds to 4 × 4μ^{2} for biological interpretation. The target cell area is chosen to be 9 nodes (144μm^{2}) for CD8 T-cells and 140 nodes (2, 240μm^{2}) for APCs. The target perimeter for CD8 T-cells is 48μ

In section 3.4, we study the ability of our model to simulate a secondary response, also called memory response. Our model has first been calibrated in order to reproduce an

Parameters of Equations (1–9) have been calibrated on _{cell}+_{prot} where

and

with

Since pre-activated and activated cellular types are not identified in

Note that we did not perform a parameter estimation procedure, but a calibration of our model based on experimental data. Evaluation of accuracy and sensitivity of parameter values have been investigated in previous studies (_{th} is presented in section 2 (

We first briefly illustrate our model's ability to reproduce

CD8 T-cell population dynamics.

On

Molecular dynamics. Mean concentration of

In our model, each cell develops its own molecular profile, resulting in a heterogeneous cell population. Consequently, studying the mean concentration of a given protein among the population, as shown on

To study the molecular-content heterogeneity and its role in cellular dynamics, we show in _{s}≈118 mol/L) and one with low concentration of Tbet (≈0 mol/L). The unstable steady state of (3), defined in Proposition 1 and separating the stable equilibria 0 and [_{s}, is given by [_{u}≈21 mol/L. Moreover, cells expressing high levels of Tbet express high levels of Caspases and low levels of Eomes, a molecular profile associated with cell death and poor memory potential. On the contrary, cells expressing low levels of Tbet have good survival and memory differentiation properties since they express low levels of Caspases and high levels of Eomes. Progressively, cells with high concentrations of Tbet die (when their concentrations of Caspases reach the threshold _{th} ≈ 19 mol/L) and cells with low concentrations of Tbet differentiate into memory cells and stop proliferating (when the concentration of Eomes reaches _{th} = 16 mol/L). On D24 p.i. there is no cell with intermediary profile, most of the cells have differentiated into memory cells while a few effector cells with high Tbet concentrations still survive. One can observe that the molecular profiles of memory cells converge to the same state where [

Concentrations (mol/L) of Tbet (red), Eomes (blue), and Caspases (brown) in all cells, sorted (left to right) according to their Tbet concentration.

The coexistence of two sub-populations characterised by their concentrations of Tbet explains the population dynamics observed on

As discussed in the introduction, responding CD8 T-cells can be distinguished between short-lived (SLEC) and memory precursor (MPEC) effector cells based on the expression of two proteins: KLRG1 and CD127 (

A major source of heterogeneity in our model is the uneven molecular partitioning at cell division determined by the degree of unevenness

Size of the CD8 T-cell population at the peak of the response (black squares, left axis) and size of the memory CD8 T-cell population at the end of the response D25 p.i. (blue diamonds, left axis) as functions of the degree unevenness of molecular partitioning (mean ± standard deviation over 5 simulations). Red crosses (right axis) show memory cell generation efficiency, measured as the ratio between the size of the memory CD8 T-cell population D25 p.i. and the size of the CD8 T-cell population at the peak of the response (mean over 5 simulations).

First,

Second, the relation between the degree of unevenness and the size of the memory population generated at the end of the response is not monotonous: the biggest memory populations are observed when considering a moderate unevenness (10–50%).

In section 3.2, the role of Tbet concentration in determining the fate (death or memory differentiation) of an effector CD8 T-cell has been discussed. Additionally, we showed in Girel and Crauste (

On the opposite, when molecular partitioning is symmetrical (_{s} (high Tbet concentration) or to 0 mol/L (low Tbet concentration). This irreversibly leads to apoptosis (high Tbet concentration) or memory differentiation (low Tbet concentration) of the whole cell lineage.

Thus, our result clearly stresses that uneven partitioning allows the maintenance of a CD8 T-cell compartment with undetermined fate for some time, through cell fate reversibility. As long as it is maintained, this compartment is able to produce both effector cells destined to die and memory cells.

We also showed in Girel and Crauste (

To discuss the efficiency of memory cell generation, we compare on

One of the characteristics of memory cells is their capacity to mount more rapid effector response than naive cells and to generate an increased fraction of memory cells (

_{th} (see section 2.3.1) is reached sooner when starting with memory cells than with naive cells. As a result, the concentration of Tbet, up-regulated during APC binding, is lower after the activation of a memory cell than after the activation of a naive cell, and low Tbet level is associated to memory precursor fate and low cytotoxicity.

Number of

On

Number of CD8 T-cells, normalised by CD8 T-cell population size D7 p.i., during

Activation of naive CD8 T-cells triggers a primary immune response, characterised by a well-orchestrated program of cell proliferation, differentiation, death and migration. It is now well-known that the responding CD8 T-cell population is heterogeneous and that a single naive T-cell can generate differently fated cells (

With this in mind, we expanded a hybrid multi-scale model of the CD8 T-cell immune response, where cell behaviour is determined by intracellular molecular dynamics. Model parameters have been calibrated using

In addition to reproduce primary responses, our model easily produces secondary responses. Memory cells generated during the

We discussed how a deterministic description of molecular concentration dynamics combined with stochastic events, such as uneven partitioning of molecular content at division, can regulate the emergence and the maintenance of two sub-populations of CD8 T-cells. Those sub-populations, characterised by their molecular profiles, coexist but express different properties and antagonistic fates, comparable to those of SLEC and MPEC described in the literature (

In our model, cell phenotypic heterogeneity, associated with molecular-content heterogeneity, first arises upon asymmetric division of polarised naive cells. This heterogeneity is thereafter continuously regulated throughout the whole response by means of uneven partitioning of molecular-content at each division. This is in agreement with the observations of Lemaître et al. (

Polarisation of naive cells upon antigenic stimulation has been observed in CD4 T-cells (

In our study, increasing the degree of unevenness of molecular partitioning reduces the expansion size of the whole CD8 T-cell population whereas the size of the sub-population of memory cells is maximal for intermediate degrees of unevenness. As a consequence, the ratio between the number of memory cells generated and the magnitude of the response at its peak, viewed as a measure of memory generation efficiency, is maximised when considering a 50% degree of unevenness. As discussed above, molecular partitioning is not the only regulator of heterogeneity. In this regard, we can believe that our evaluation overestimates the value of this optimal degree of unevenness and rather indicates that generating a moderate heterogeneity all along the immune response leads to efficient memory generation.

In our manuscript, when the degree of unevenness is

Note that, in works dealing with the CD8 T-cell immune response, it is usual to consider that 5 to 10% of the cells present at the peak of the response survive the contraction phase and differentiate into memory cells (

In many mathematical models of the CD8 T-cell immune response, as those referenced in (

Cell cycle length depends in our model on the number of divisions the cell has undergone. It would be instructive to introduce a molecular control of cell proliferation, since the putative existence of coexisting sub-populations with disparate cycle lengths could considerably impact the cellular dynamics. One could for instance consider the transcription factor Foxo1, known to induce Eomes expression while repressing that of Tbet and inhibiting cell cycle progression (^{lo}Eomes^{hi} memory precursor cells discussed in section 3.2 might adopt a longer cycle than the Tbet^{hi} Eomes^{lo} cells.

In conclusion, our agent-based multiscale model successfully reproduced several aspects of the CD8 T-cell immune response at both molecular and cellular scales. Even though we cannot infer quantitative conclusions from this study, it suggests that uneven partitioning of molecular content at cell division, as a source of heterogeneity, can modulate cell fate decision and act as a regulator of the magnitude of the response and of the size of the memory cell pool. Actually, we did not consider intermediaries, namely DNA transcription and mRNA translation, between gene activation and protein synthesis. Consequently, our molecular model is an amalgam between gene activity and protein synthesis. Therefore, while our argumentation is based on uneven partitioning of the molecular content, it could also stand for the situation where, when a cell divides, the two daughter cells inherit different gene activity levels for each gene. All in all, our study focuses on molecular heterogeneity generation upon cell division in general, rather than the specific case of molecular partitioning. It stresses that dynamics observed at the cellular scale—including the initiation of the contraction phase and the origin of memory cells—can be explained by structural molecular-content heterogeneity, that is continuously regulated along the response, as CD8 T-cells divide.

The datasets generated for this study can be found in the Open Science Framework repository

mRNA expression data analysed in this study come from the ImmGen project (

All co-authors discussed the problem, approach and results. SG, OG, and FC designed the model. SG ran simulations and performed analysis. CA and JM conducted the experimental studies. SG wrote the paper. All the authors approved the final version.

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

We thank the Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules de Lyon (CC-IN2P3) for providing computing resources, particularly Pascal Calvat and Yonny Cardenas for their valuable help. We also thank the BioSyL Federation and the LabEx Ecofect (ANR-11-LABX-0048) of the University of Lyon for inspiring scientific events. We acknowledge the contribution of SFR Biosciences (UMS3444/CNRS, US8/Inserm, ENS de Lyon, UCBL) facilities. We acknowledge the contributions of the CELPHEDIA Infrastructure (

The Supplementary Material for this article can be found online at:

Parameter value tables and sensitivity analysis to parameter _{th}.

CompuCell3D simulation files used to generate the results presented in this manuscript.

^{+}T cell differentiation