Mechanisms underlying vaccination protocols that may optimally elicit broadly neutralizing antibodies against highly mutable pathogens

Effective prophylactic vaccines usually induce the immune system to generate potent antibodies that can bind to an antigen and thus prevent it from infecting host cells. B cells produce antibodies by a Darwinian evolutionary process called affinity maturation (AM). During AM, the B cell population evolves in response to the antigen to produce antibodies that bind specifically and strongly to the antigen. Highly mutable pathogens pose a major challenge to the development of effective vaccines because antibodies that are effective against one strain of the virus may not protect against a mutant strain. Antibodies that can protect against diverse strains of a mutable pathogen have high “breadth” and are called broadly neutralizing antibodies (bnAbs). In spite of extensive studies, an effective vaccination strategy that can generate bnAbs in humans does not exist for any highly mutable pathogen. Here we study a minimal model to explore the mechanisms underlying how the selection forces imposed by antigens can be optimally chosen to guide AM to maximize the evolution of bnAbs. For logistical reasons, only a finite number of antigens can be administered in a finite number of vaccinations; that is, guiding the nonequilibrium dynamics of AM to produce bnAbs must be accomplished nonadiabatically. The time-varying Kullback-Leibler divergence (KLD) between the existing B cell population distribution and the fitness landscape imposed by antigens is a quantitative metric of the thermodynamic force acting on B cells. If this force is too small, adaptation is minimal. If the force is too large, contrary to expectations, adaptation is not faster; rather, the B cell population is extinguished for reasons that we describe. We define the conditions necessary for the force to be set optimally such that the flux of B cells from low to high breadth states is maximized. Even in this case we show why the dynamics of AM prevent perfect adaptation. If two shots of vaccination are allowed, the optimal protocol is characterized by a relatively low optimal KLD during the first shot that appropriately increases the diversity of the B cell population so that the surviving B cells have a high chance of evolving into bnAbs upon subsequently increasing the KLD during the second shot. Phylogenetic tree analysis further reveals the evolutionary pathways that lead to bnAbs. The connections between the mechanisms revealed by our analyses and recent simulation studies of bnAb evolution, the problem of generalist versus specialist evolution, and learning theory are discussed.

Figure 1

Definition of breadth space. (a) Depiction of the 1D fitness landscape. The origin denotes the point of highest breadth, and the fitness distribution imposed on the B cell population by the immunogen is Gaussian. Reducing the variance of the fitness focuses selection pressure on B cells that bind with high affinity to conserved epitopes. (b) Breadth space is binned into $K-1$ states: $K/2$ is the highest breadth state corresponding to the origin in (a), 1 and $K-1$ are states of lowest breadth, and 0 is the death state.

Figure 2

Fitness landscape and response of the B cell population. (a) Imposed fitness landscape ($f_i$) during prime (blue, dashed) and boost (orange, dash-dotted). (b) Response of B cell population distribution $p_i$ to fitness landscapes shown in (a).

Figure 3

Response to different vaccination protocols. (a) For each prime immunization choice of KL divergence, $D\left(p^0\right|\left|f^1\right)$ (colored curves), the number of bnAbs produced per GC is shown over a range of boost immunization choices, $D\left(p^1\right|\left|f^2\right)$. (b) Maximal bnAb titers for each of the curves shown in (a) versus $D\left(p^0\right|\left|f^1\right)$ shows that an optimal prime-boost protocol exists.

Figure 4

Germinal center (GC) survival (B cell population is not extinguished). (a) Fraction of GCs that survive prime immunization for a range of values of $D\left(p^0\right|\left|f^1\right)$. The fraction falls to $\sim 0.9$ at the optimal setting and drops significantly beyond this point. (b) Sample fitness landscapes at the points located on the curve shown in (a).

Figure 5

Learning inside a GC. Within surviving GC trajectories, the system eventually approaches a regime where $D(p\left(t\right)\left|\right|f^1)$ decreases. On average, the rate of decrease is slower (a) near optimal forcing $\left[D\right(p^0\left|\right|f^1)=3.15]$ and faster (b) beyond the optimum, $D\left(p^0\right|\left|f^1\right)=5.16$.

Figure 6

Optimal diversity of the B cell population. (a) For each GC, the change in relative entropy after prime immunization is recorded and graphed as a histogram for low (orange, solid), optimal (yellow, dashed), and high (green, dash-dotted) $D\left(p^0\right|\left|f^1\right)$. The GCs wherein the B cell population is extinguished experience a large increase of relative entropy. (b) Decrease in relative entropy is a consequence of B cells on average populating higher breadth states after prime immunization.

Figure 7

Characterizing optimal protocols. For the three prime protocols shown in Fig. 6, the numbers of evolutionary trajectories that exit the GCs as bnAbs after optimal boost are graphed as a function of the breadth state of the B cells that initially seed the trajectories at the beginning of boost.

Figure 8

Phylogeny of evolving B cell lineages for low $D\left(p^0\right|\left|f^1\right)$ during prime followed by the optimal boost. (a) An example of a population of B cells evolving in a germinal center. The initial B cells were sampled from the precursor distribution [black (solid), Fig. 2] subjected to $D\left(p^0\right|\left|f^1\right)=2.56$ and $D\left(p^1\right|\left|f^2\right)=72.12$. The yellow region corresponds to the prime and the pink region to the boost. (b) The top panel is an expanded view of the green box (green right arrow) in (a), which shows high breadth B cells generated at the end of prime and the beginning of boost. The bottom panel corresponds to the purple box (purple left arrow) in (a), which occurs at the end of boost. Colors of circles depicting B cells denote their breadth states, and the numbers to the right of each circle quantify the number of B cells in that state.

Figure 9

Phylogeny of evolving B cell lineages for high $D\left(p^0\right|\left|f^1\right)$ during prime followed by the optimal boost. (a) An example of a population of B cells evolving in a germinal center. The initial B cells were sampled from the precursor distribution [black (solid), Fig. 2] subjected to $D\left(p^0\right|\left|f^1\right)=3.39$ and $D\left(p^1\right|\left|f^2\right)=34.73$. The yellow region corresponds to the prime and the pink region to the boost. (b) The top panel is an expanded view of the green box (green right arrow) in (a), which shows high breadth B cells generated at the end of prime. The bottom panel corresponds to the purple box (purple left arrow) in (a), which corresponds to the end of boost. Colors of circles depicting B cells denote their breadth states, and the numbers to the right of each circle quantify the number of B cells in that state.

Figure 10

Phylogeny of evolving B cell lineages for optimal $D\left(p^0\right|\left|f^1\right)$ during prime followed by the optimal boost. (a) An example of a population of B cells evolving in a germinal center. The initial B cells were sampled from the precursor distribution [black (solid), Fig. 2] subjected to $D\left(p^0\right|\left|f^1\right)=2.76$ and $D\left(p^1\right|\left|f^2\right)=56.46$. The yellow region corresponds to the prime and the pink region to the boost. (b) The top panel is an expanded view of the green box (green right arrow) in (a), which shows high breadth B cells generated at the end of prime. The bottom panel corresponds to the purple box (purple left arrow) in (a), which corresponds to the end of boost. Colors of circles depicting B cells denote their breadth states, and the numbers to the right of each circle quantify the number of B cells in that state.

Figure 11

Initial distribution of B cells. (a) Precursor B cells are sampled from a log-normal distribution with $\mu =3.0$ and $\sigma =1.0$. (b) Normalizing with respect to K = 31 bins yields the precursor B cell distribution $p_i^0$.

Figure 12

Response to different vaccination protocols. For each prime immunization choice of KL divergence, $D\left(p^0\right|\left|f^1\right)$ (colored curves), the number of high breadth B cells produced per GC is shown over a range of boost immunization choices, $D\left(p^1\right|\left|f^2\right)$.

Figure 13

Germinal center (GC) survival (B cell population is not extinguished). (a) Fraction of GCs that survive prime immunization for a range of values of $D\left(p^0\right|\left|f^1\right)$. The fraction falls to $\sim 0.9$ at the optimal setting and drops significantly beyond this point. (b) Sample fitness landscapes at the points located on the curve shown in (a).

Figure 14

$D(p\left(t\right)\left|\right|f^1)$ of surviving GC trajectories when (a) $D\left(p^0\right|\left|f^1\right)=3.15$ and (b) $D\left(p^0\right|\left|f^1\right)=5.16$. To explore the equilibrium limit, the maximum population size is increased to $N_{\max }=500000$.

Figure 15

$D(p\left(t\right)\left|\right|f^1)$ within GC trajectories that lead to population extinction when (a) $D\left(p^0\right|\left|f^1\right)=3.15$ and (b) $D\left(p^0\right|\left|f^1\right)=5.16$.