# Modeling the Dependence of Immunodominance on T Cell Dynamics in Prime-Boost Vaccines

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. The Effect of Immunodominance in Acute Infections

#### 2.2. Immunodominance in Single Immunization Vaccines

#### 2.3. Immunodominance in Prime-Boost Vaccines

## 3. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Model of Homologous PB Vaccines

**Figure A1.**

**Hybrid automata diagram of the model of homologous PB vaccines.**Simulations start with the inoculation of antigen A. Clone 1 recognizes the target epitope with affinity ${\lambda}_{1}$, while clone 2 binds to an alternative epitope with affinity ${\lambda}_{2}>{\lambda}_{1}$. Therefore, the target clone is subdominant. A second inoculation of the same agent (boost inoculation) takes place at time $t={t}_{boost}$. This gives rise to a discrete change in the model, consisting in an instantaneous change of antigen A ($A({t}_{boost})=A({t}_{boost})+\Delta A$, with $\Delta A$ the dose of the boost inoculation). Different values for the prime and boost doses of A and different boost timings give rise to alternative vaccination protocols. The condition ${T}_{i}<0$ ($i=1,2$) determines the end of the clonal contraction of clone i.

## Appendix B. Model of Heterologous PB Vaccines

**Figure A2.**

**Hybrid automata diagram of the model of heterologous PB vaccines.**In this case we assume that prime consists in the inoculation of an antigen ${A}_{1}$, whereas the boost vector contains a different antigen ${A}_{2}$. Both antigens contain the target epitope, which is recognized by clone 1 with affinity ${\lambda}_{1}$. A second clone recognizes a different epitope of ${A}_{1}$ with affinity ${\lambda}_{2}>{\lambda}_{1}$ and a third clone binds an epitope on ${A}_{2}$ with affinity ${\lambda}_{3}>{\lambda}_{1}$. The target clone is therefore subdominant with respect to clones 2 and 3. Simulations start with the activation of clones 1 and 2 in response to the presence of antigen ${A}_{1}$. At time $t={t}_{boost}$ the second antigen (${A}_{2}$) is inoculated. This triggers a discrete change in the equations of the model. The activation of clone 3 introduces three new equations describing the dynamics of effector and memory T cells of clone 3 (${T}_{3}$ and ${M}_{3}$) and of the second antigen ${A}_{2}$. The equation of effector and memory T cells of clone 1 also change owing to the appearance of a new antigenic force ${A}_{2}$. Finally, the presence of a new clone of memory T cells (${M}_{3}$) also modifies the dynamics of the homeostatic interleukin H. These changes are marked in bold face in the figure.

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**Figure 1.**Hybrid dynamical system described by Equation (1). Equation (1) may lead to negative values for the populations of effector T cells (${T}_{i}$). To avoid this, the condition ${T}_{i}\le 0$ determines a discrete change in Equation (1). The variable ${T}_{i}$ is set to zero, which is equivalent to remove the variable from the model. In biological terms, this implies that clone i has completed clonal contraction. The condition $P<{P}_{m}$ means that the pathogen has been effectively neutralized. This is modeled by another discrete transition, in this case to remove the variable P from the model. We assume that memory T cell populations do not undergo clonal contraction and that the concentration of the homeostatic interleukin remains always greater than zero. In order to satisfy these biological constrains, the set of parameters must be chosen such that conditions ${M}_{i}(t)>0$, $M(t)>0$ and $H(t)>0$ hold for $t>0$.

**Figure 2.**Immunodominance in acute T cell immune responses. (

**A**) The light grey curve represents clonal expansion and contraction of a given clone in the absence of immunodominance. If the same clone is activated in the presence of a dominant clone (dashed line) it displays a curtailed expansion (solid black line). (

**B**) The presence of a dominant clone (dashed line) markedly reduces the number of memory T cells of the original clone (solid black line). (

**C**) The dominant clone leads to a more rapid elimination of the pathogen (black line) as compared to the less dominant clone alone (light grey line). (

**D**) Ratio between the number of memory T cells of two clones of affinities ${\lambda}_{1}$ and ${\lambda}_{2}$ respectively. Values above 1 (respectively below 1) indicate dominance of clone 1 (resp. of clone 2). More affine clones dominate when the number of naïve cells are similar (grey area in the figure), whereas less affine clones become dominant if they are present at higher frequency in the pool of naïve T cells. Parameters used in A and B: $k=150$, $c=40$, ${\lambda}_{1}=200$, ${\lambda}_{2}=150$, $\alpha =40$, $\beta =0.5$, ${y}_{0}=10$, $\mu =1$, $\phi ={10}^{6}$, ${\lambda}_{H}=10$, ${T}_{P1}(0)=10$ and ${T}_{P2}(0)=10$ (all in suitable units).

**Figure 3.**Immunodominance index. (

**A**) Each dot represents a numerical simulation of the model of immunodominance in single-dose vaccines. In this model, we assume that two clones of affinities ${\lambda}_{1}$ and ${\lambda}_{2}$ are simultaneously activated in response to the vaccine. For each set of parameter values, the ratio between the number of memory T cells formed of these clones is linearly related with the ratio of their affinities (correlation coefficient ${r}^{2}>0.999$ in all cases). Different combinations of parameters can be fitted to different lines passing through the origin. (

**B**) The slope of these lines can be used to define an immunodominance index ${I}_{\lambda}$. For a fixed value of $\lambda $, a higher slope implies greater differences in memory formation between concurrent clones. Therefore, higher values of ${I}_{\lambda}$ correspond to greater differences in memory formation between dominant and non-dominant clones. (

**C**) Immunodominance index for different vaccine doses (parameter ${P}_{0}$ in the equations) and (

**D**) different rates of antigen decay (parameter $\alpha $ in the model). These results suggest that higher antigen doses and higher rates of antigen decay intensify the effects of immunodominance.

**Figure 4.**Effect of homologous and heterologous prime-boost (PB) vaccines on the immunodominance index. Each figure represents the ratio between the immunodominance index measured for PB vaccines and for single-dose vaccines. Values above 1 represent situations in which the immunodominance index is greater in the PB strategy as compared to administering all the antigen in one single dose. Conversely, values below 1 indicate a reduction in the immunodominance index as a consequence of distributing the antigen in two doses. In the case of heterologous vaccines comparison is made with single-dose vaccines containing only the first antigen (left), or a combination of antigens present both in prime and boost agents (right). (

**A**–

**C**) correspond to three different time intervals between prime and boost. Threshold value of 1 is shown in gray for reference. The values of the parameters used these simulations are (in suitable units): $k=150$, $c=40$, ${\lambda}_{1}=10$, ${\lambda}_{2}=150$, ${\lambda}_{3}=200$, ${\alpha}_{1}={\alpha}_{2}=-5$, ${\mu}_{1}={\mu}_{2}=1$, $\phi ={10}^{6}$, ${\lambda}_{H}=10$, ${T}_{1}(0)={T}_{2}(0)={T}_{3}({t}_{boost})={M}_{1}(0)={M}_{2}(0)={M}_{3}({t}_{boost})=10$.

**Figure 5.**Effect of homologous and heterologous PB vaccines on the formation of memory T cells of non-dominant clones. (

**A**) The results presented here correspond to a particular choice of clone affinities for PB vaccine scenarios introduced in Figure 4. In particular, we have set ${\lambda}_{1}=10,{\lambda}_{2}=150,{\lambda}_{3}=200$ (left), and ${\lambda}_{1}=10,{\lambda}_{2}=100,{\lambda}_{3}=200$ (right). All the protocols of heterologous PB vaccines (white) give raise to more memory T cells of subdominant clone 1 than single-dose vaccines (gray) or homologous PB vaccines (dark-gray). (

**B**) According to the models of population mechanics, PB vaccine strategies can be used to change the relations of immunodominance that emerge from single-dose vaccines. In particular, the immunodominance index of a particular clone can be increased by using appropriate antigen doses in PB homologous vaccines. In contrast, heterologous PB protocols tend to reduce the value of this index, thus broadening T cell immune response triggered by vaccine antigens. These results suggest that T cell population dynamics can be explicitly used to design PB vaccine protocols to modulate immunodominance. These protocols can be combined with other techniques, such as artificial epitope modification, to further reduce immunodominance effects on specific target clones.

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**MDPI and ACS Style**

Arias, C.F.; Herrero, M.A.; Bertocchini, F.; Acosta, F.J.; Fernandez-Arias, C.
Modeling the Dependence of Immunodominance on T Cell Dynamics in Prime-Boost Vaccines. *Mathematics* **2021**, *9*, 28.
https://doi.org/10.3390/math9010028

**AMA Style**

Arias CF, Herrero MA, Bertocchini F, Acosta FJ, Fernandez-Arias C.
Modeling the Dependence of Immunodominance on T Cell Dynamics in Prime-Boost Vaccines. *Mathematics*. 2021; 9(1):28.
https://doi.org/10.3390/math9010028

**Chicago/Turabian Style**

Arias, Clemente Fernandez, Miguel Angel Herrero, Federica Bertocchini, Francisco Javier Acosta, and Cristina Fernandez-Arias.
2021. "Modeling the Dependence of Immunodominance on T Cell Dynamics in Prime-Boost Vaccines" *Mathematics* 9, no. 1: 28.
https://doi.org/10.3390/math9010028