Modeling the effect of boost timing in murine irradiated sporozoite prime-boost vaccines

Vaccination with radiation-attenuated sporozoites has been shown to induce CD8+ T cell-mediated protection against pre-erythrocytic stages of malaria. Empirical evidence suggests that successive inoculations often improve the efficacy of this type of vaccines. An initial dose (prime) triggers a specific cellular response, and subsequent inoculations (boost) amplify this response to create a robust CD8+ T cell memory. In this work we propose a model to analyze the effect of T cell dynamics on the performance of prime-boost vaccines. This model suggests that boost doses and timings should be selected according to the T cell response elicited by priming. Specifically, boosting during late stages of clonal contraction would maximize T cell memory production for vaccines using lower doses of irradiated sporozoites. In contrast, single-dose inoculations would be indicated for higher vaccine doses. Experimental data have been obtained that support theoretical predictions of the model.


A. Outline of population mechanics
Population mechanics is based on the following assumptions: 1. T cell populations are elastic and show inertia. This statement is motivated by the dynamics of effector T cells during immune response to acute infections, specifically by the fact that T cells continue to proliferate even after the pathogen has been effectively neutralized (see figure below).

Pathogen clearance
Onset of clonal contraction Inertia and elasticity draw clear analogies with models of Classical Mechanics, and call for the use of second order differential equations at the modeling level. From this viewpoint, the dynamics of a cell population results from the balance between opposite forces that tend to increase or decrease the number of cells. In particular, if a cell population x is subject to m different forces F 1 , · · · , F m , then: x (t) = F 1 + · · · + F m 2. Proportional distribution of forces. If n T cell populations compete for a given force F (t), then the magnitude of the force perceived by population i is given by: where x i (t) is the number of cells of population i at time t.
Models of population mechanics Based on the previous assumptions, it is possible to model the dynamics of multiple cell populations that compete for different forces. For instance, let x 1 , x 2 and x 3 three cell populations, and suppose that x 1 and x 2 compete for a given factor F a , while a second factor F b is shared between populations x 1 and x 3 . The corresponding mathematical model takes the following form: Parameters c i and k i represent the damping and elastic coefficients of population i. Parameter λ i,k describes the intensity of the force perceived by population i per unit of factor F k . Finally, the dynamics of factors F a and F b is given by certain functions f a and f b , which in turn may depend on the size of the cell populations.
Population mechanics models are of a modular nature and can be easily pieced together to account for additional populations. For instance, adding a fourth population that competes for factor F a in the previous example yields: We remark that solutions of the previous equations can eventually lead to negative values for some variables. Since negative values are meaningless in our current context, these models must be understood as hybrid dynamical systems. Such systems are characterized by alternating between a collection of discrete states, with transitions from a given state to another being governed by differential equations (see for instance (3) On the other hand, if the pathogen falls below a given threshold P m > 0 the infection is controlled, and hence the antigenic force disappears. When this occurs the only force acting on effector T cells is the intrinsic elastic force, while memory T cells continue to compete for the homeostatic force F H . (4) As consequence of the elastic force, effector T cells eventually disappear, and memory T cells eventually reach a steady state. (5) We will not consider situations in which the homeostatic interleukin or memory T cell populations go to zero. In any of these cases, the chosen set of parameters does not lead to a meaningful simulation.

SM2. Experimental data
Characterization of clonal expansion and contraction Table A: Number of effector T cells (%CD11ahi) after the inoculation of 10 4 irradiated sporozoites (see Figure 6.A in the main text).  Table B: Number of effector T cells (%CD11ahi) after the inoculation of 10 5 irradiated sporozoites (see Figure 6.B in the main text). Formation of memory T cells Table C: Memory T cell (% of Resident Memory T cells) formed in three vaccination protocols (P : single inoculation of 10 4 irradiated sporozoites, P B 1 : priming with 10 4 and boosting three days later with 4 × 10 4 irradiated sporozoites, and P B 2 : priming with 10 4 and boosting seven days later with 4 × 10 4 irradiated sporozoites). These data are shown in Figure 6.C in the main text.  Table D: Memory T cell (% of Resident Memory T cells) formed in three vaccination protocols (P : single inoculation of 10 5 irradiated sporozoites, P B 1 : priming with 10 5 and boosting three days later with 4 × 10 4 irradiated sporozoites, and P B 2 : priming with 10 5 and boosting seven days later with 4 × 10 4 irradiated sporozoites). These data are shown in Figure 6.D in the main text.