Abstract
Adjuvanted influenza vaccines constitute a key element towards inducing neutralizing antibody responses in populations with reduced responsiveness, such as infants and elderly subjects, as well as in devising antigensparing strategies. In particular, squalenecontaining adjuvants have been observed to induce enhanced antibody responses, as well as having an influence on crossreactive immunity. To explore the effects of adjuvanted vaccine formulations on antibody response and their relation to proteinspecific immunity, we propose different mathematical models of antibody production dynamics in response to influenza vaccination. Data from ferrets immunized with commercial H1N1pdm09 vaccine antigen alone or formulated with different adjuvants was instrumental to adjust model parameters. While the affinity maturation process complexity is abridged, the proposed model is able to recapitulate the essential features of the observed dynamics. Our numerical results suggest that there exists a qualitative shift in proteinspecific antibody response, with enhanced production of antibodies targeting the NA protein in adjuvanted versus nonadjuvanted formulations, in conjunction with a proteinindependent boost that is over one order of magnitude larger for squalenecontaining adjuvants. Furthermore, simulations predict that vaccines formulated with squalenecontaining adjuvants are able to induce sustained antibody titers in a robust way, with little impact of the time interval between immunizations.
Introduction
Seasonal and pandemic influenza A virus (IAV) infections pose a serious threat to public health. Influenza readily spreads across borders, and can affect several countries simultaneously, resulting in considerable economic and social impact. Seasonal outbreaks cause millions of infected cases and about half a million deaths worldwide every year^{1,2}. Furthermore, the consequences of epidemics can be economically devastating, since they can also affect susceptible poultry and swine populations.
Vaccines represent a cornerstone of measures against influenza outbreaks; however, a variety of important limitations exist in terms of the availability, cost and effectiveness of currently licensed influenza vaccines. A comprehensive quantitative evaluation of the withinhost effects of vaccination is still lacking, and the elaboration of vaccination strategies that overcome these difficulties remains a fundamental challenge^{3}.
Influenza A viruses are classified into subtypes according to the antigenicity of their two main surface glycoproteins: hemagglutinin (HA) and neuraminidase (NA). The former is responsible for virus entry by binding to sialic acids on the surface of hosts cells and subsequent pHdependent fusion of the viral and endosomal membranes, while the latter mediates the release of newly produced virions from infected cells by removing sialic acid from their surfaces^{4,5,6}. Due to these different functions, neutralizing antibodies are primarily directed against the HA protein^{7}. The antibody response directed against NA, in turn, plays a role in decreasing viral spread by provoking the accumulation of virus on the cell surface, which reduces morbidity and mortality in mice^{8,9}.
Antibody responses against the virus drive antigenic drift, which consists in gradual changes to the surface proteins HA and NA. Occasionally, reassortment may lead to the introduction of a new HA or NA segment—also referred to as antigenic shift—resulting in the apparition of entirely novel strains, for which the population is immunologically naive, with potentially severe global consequences^{6,10}. To date, 18 HA and 11 NA subtypes have been identified, with only a few of them—H1, H2, and H3 and N1 and N2, respectively—found in human seasonal viruses^{11}. Within a given subtype, the mutation rate in NA is lower than that in HA^{12}—that is, NA is more antigenetically conserved—possibly owing to the fact that the antibody response is skewed towards HA, resulting in a greater selection pressure^{6,13}.
As a consequence of antigenic drift, the strain composition in seasonal vaccines has to be updated regularly^{14}. This is a costly endeavor that, at the same time, does not address the latent threat of further antigenic drift or a pandemic caused by a newly reassorted strain, since the vaccines are highly strainspecific. There is thus a need for immunization strategies that can elicit a broad immune response; specifically, the production of broadly crossreactive antibodies that confer protection from strains of the virus different from those present in the vaccine.
In pandemic situations, when a large number of doses is needed in a very short time^{15}, limited antigen availability is an additional major challenge. This may be addressed by antigensparing strategies in combination with adjuvants, which trigger a strong immune response at lower antigen doses than otherwise necessary^{16}. Interestingly, the addition of squalenecontaining adjuvants has also been observed positively influence the breadth of the antibody response by enhancing not only the overall titers but also the production of antibodies that target the NA protein^{17}. Since NA is more conserved, antibodies directed towards this protein may confer partial immunity against other influenza strains carrying the same NA subtype^{18}.
Mathematical models of biological processes can yield insight on their most essential features, as well as contribute in delineating experimental studies. While a variety of models exists that explore the withinhost dynamics of influenza infection, most mathematical descriptions of influenza are constructed at the population level^{19}. In particular, those incorporating vaccination are mainly concerned with the epidemiological consequences of different vaccination strategies in a given population—see, e.g., Weycker et al.^{20}, and van den Dool et al.^{21}.
Inside the host itself, antibody production is driven by a process referred to as affinity maturation (AM), whereby B cells undergo several rounds of proliferation, mutation and selection within specialized domains called germinal centers (GCs) towards increasing binding affinities to the antigen, ultimately differentiating into highaffinity antibodysecreting plasma cells and memory B cells^{22}. The dynamics inside the GCs are essential to the strength and crossreactivity of antibody responses to infection or vaccination, and have received considerable attention, both conceptually^{22,23,24} and from a modeling perspective to various degrees of detail^{25,26}. These models have also incorporated the effects of different vaccination strategies on the AM process, notably for the cases of malaria^{27} and HIV^{28,29}.
In this work, we construct a mathematical model to capture the withinhost effects of immunization with adjuvanted and nonadjuvanted influenza vaccine formulations. By means of a parsimonious description of the AM process, we predict the magnitude and proteinspecificity of the antibody response elicited by the different vaccine formulations at a coarse level. Using the data from Schmidt et al.^{17}, we fit our model parameters to virusneutralizing antibody titers, and qualitatively compare our results to the experimentally observed proteinspecific titers. Our simulation results show that this simple model is able to recapitulate the essential features of antibody production elicited by adjuvanted vaccine formulations, and may serve as a stepping stone for more complex analyses.
Methods
Experimental data
For the proposed mathematical modeling analysis, the experimental data previously published by Schmidt et al.^{17} is employed. Briefly, adult male and female ferrets, immunologically naive to circulating influenza strains, were immunized intramuscularly with adjuvanted and nonadjuvanted formulations of the inactivated pandemic H1N1pdm09 vaccine, followed by a boost 3 weeks later. The adjuvants tested were the squalenecontaining MF59 and AS03, as well as Diluvac Forte, which contains only vitamin E.
Post vaccination, antibody titers against total influenza virus, as well as specific to HA or NA, were determined for a series of different HA and NA proteins. Thus, both the magnitude and specificity of the immune response for the different vaccine formulations were measured. In our specific case, the output of the model outlined in the following section was fitted to the data for functional virusneutralizing antibodies.
Mathematical model of antibody production
We propose a twoepitope model in a Euclidean, onedimensional shape space, with the genotype of a given cell represented by its position \(x\in [0,1]\). The two epitopes correspond to the NA and HA proteins of the vaccine strain, and will be located at different, nonoverlapping positions—i.e., their antigenic distance is larger than some specified cutoff—in shape space. Without loss of generality, we take NA to be located at position x = 0, while HA sits at x = 1—see Fig. 1A.
The mathematical model draws mainly from the detailed AM simulations of Chaudhury et al.^{27}. Our interest, however, lies in the broader effects of the inclusion of adjuvants in the influenza vaccine, and not in the fine details of the AM process. For this reason, the variables we consider explicitly are only two, and correspond to the distributions of B cells and antibodies in the shape space, which we represent by B(x, t) and Ab(x, t), respectively. In other words, we do not distinguish between stimulated B cells, nonstimulated B cells, plasma and memory cells, but rather consider an effective behavior inside a GC, represented by the variable B(x, t). The choice of a simplified shape space, in turn, is inspired by the notion of ‘antibody landscape’ introduced by Fonville et al.^{30}. In this case, in contrast to the original formulation, it is considered that the points along the axis do not correspond to different virus strains, but rather to the individual virus glycoproteins, as described above.
AM is an inherently stochastic process, and it has been shown that stochasticity plays a decisive role in the selection of clonal lineages in the GC^{26}. However, we stress that our interest is not in tracking individual B cells nor in the actual number of B cell genotypes in the system, but rather the broad shape—with respect to protein specificity—and size of the resulting antibody distribution in shape space, and its comparison to the experimental data for Ab titers. Therefore, we consider the case in which there exists a onetoone correspondence between cell genotype and affinity, as shown below. Due to this, random drift does not play a role in our system—cells with the same affinity as treated as belonging to the same lineage—and the stochastic effects in our model arise predominantly from B cell mutations, which are small in magnitude. We thus turn to a deterministic description of the process, in which the positions of cells in shape space, and their corresponding affinities, are continuous variables, and we choose to represent mutations by an effective diffusion along the xaxis.
We examine two different forms for the time evolution of the distribution of B cells in shape space which we denote, respectively, by model A and model B. The former is given by
while the explicit form of model B is shown in Eqs (S5) and (S6) of the Supplementary Material. The dynamics above are illustrated in Fig. 2.
Let us start by addressing the growth terms in Eq. (1), and leaving the function G(t) aside for the time being. The ‘birth’ of B cells comprises three different types of events. First, a constant influx, at a rate σ_{N}, from a pool of lowaffinity naive cells, which we represent by a function H(x) that is only nonzero in low and zero affinity regions of the interval \([0,1]\), as shown in Fig. 1B. Binding affinity between a B cell and a protein at antigenic distance x is represented by a function Q_{0}(x) that takes values between 0 and 1. Following the model proposed by Chaudhury et al.^{27}, we take:
where d_{min} and d_{max} represent cutoffs for maximum and zero affinity, respectively. This means that d_{max} will determine the range of influence of H(x).
Apart from the influx of naive B cells, we represent the recruitment of ‘memory’ cells—unaccounted for in the model—as well as selection by allowing highaffinity B cells to further proliferate at a rate σ_{M} over the base rate; in the equation, Q(x) represents the total binding affinity of cells at position x, weighted by antigen immunogenicity. In other words,
where γ_{NA} and γ_{HA} are the immunogenicities of NA and HA, respectively, while x_{NA} and x_{HA} correspond to the antigenic distances to NA, \({x}_{{\rm{NA}}}\equiv x0=x\), and to HA, \({x}_{{\rm{HA}}}\equiv x1=1x\), respectively. Finally, affinityindependent B cell proliferation happens at a rate \(\tilde{r}\), and the position of the daughter cell in shape space will be given by that of the parent cell plus a normallydistributed random number with variance ν. In the meanfield limit, this translates into an overall diffusion coefficient \(D=\tilde{r}\nu /2\)^{31}.
The decay of B cells occurs at a rate g_{B}(x); the specific form of this function is chosen so that highaffinity B cells possess a longer lifespan, in order to account for memory cells. We formulate this term as follows:
where \(\tilde{Q}(x)\) is the unweighted total binding affinity of cells at position x, i.e., \(\tilde{Q}(x)\equiv {Q}_{0}(x)+{Q}_{0}(1x)\). The lifespan of memory B cells is a subject of debate, and there exists evidence for it ranging from months to years^{32,33}. Here, we set \({g}_{B}^{\mathrm{(1)}}\) so that B cells have a maximum possible mean life of 2 years. We note that what we have taken to represent the presence of memory B cells in the system in the equation above may be also regarded as a form of deathlimited selection; conversely, the selection term in Eq. (1) corresponds to birthlimited selection. These concepts have been recently introduced by Chakraborty et al.^{26}, who explored the influence of both types of selection in the loss of clonal diversity in the GC during the AM process.
Finally, the GC has a finite carrying capacity K, and B cells with similar genotypes will compete for space in the system^{23}. The competition will be in general nonlocal and can be written in the form
The exact form of the competition kernel, \({\mathscr{C}}(x)\), will determine the strength and range of this competition. For the sake of simplicity, we consider a Diracδ kernel; that is, the competition is made completely local and assumed to have unit strength. This yields the last term in Eq. (1).
The difference between models A and B is given by the modulation function G(t). This corresponds to a general measure of activity of the immune response, which we assume to vary between 0 and 1 for inactive and fully active, respectively; that is, G can be regarded as a function that determines whether affinity maturation is taking place. We explicitly choose not to identify this function with the concentration of free antigen in the system, due to the fact that the formation of the GC is dependent on antigen presence, but not on antigen quantity; once active, its subsequent dynamics and decay are essentially antigenindependent^{24}.
In the case of model A, the activity level governs all processes resulting in the growth of the B cell population, whereas model B assumes that only recruiting events—of naive and of ‘memory’ B cells—are affected by it, permitting the free proliferation of B cells as long as a nonzero quantity remains in the system. In other words, in Eq. (1) G encompasses the activity of the response as a whole, from antigen uptake and B cell migration into the GC to proliferation inside the GC itself. In Eq. (S5), in contrast, only antigen uptake and cell recruitment are affected by G. The activity level is assumed to decay exponentially with a rate μ, so that G takes the form
where t_{0} = 0 corresponds to moment of initial immunization and the t_{k}, k = 1, 2, …, to the subsequent boosts. In this case, the animal experiments were performed with a single immunization boost at t_{1} = t_{boost} = 21 days. Again, since G(t) does not represent the antigen concentration, its value is not affected by the presence of antibodies in the system.
The equation describing antibody dynamics, Eq. (2), is simpler, consists of an antibody production term, mediated by weighted B cell affinity to the antigen, and a constant exponential decay rate. Since not all B cells end up producing antibodies, but only the ones among them that differentiate into plasma cells, we multiply the production term by an attenuating factor δ. We choose to take this factor as the differentiation probability from Chaudhury et al.^{27}.
Given a nonnegative initial condition, B(x, 0) ≥ 0 and Ab(x, 0) ≥ 0, the solutions to Eqs (1) and (2) will remain nonnegative for all t > 0. This can be seen from the fact that both G(t) and H(x) are nonnegative for all t and all x, respectively, so that ∂_{t}B ≥ 0 for B → 0; as a consequence of this, we obtain ∂_{t}Ab ≥ 0 for Ab → 0, since Q is also nonnegative for all x. Furthermore, the nonlinear term in Eq. (1), stemming from the finite carrying capacity K, ensures that the solution for B remains bounded for all x, t, which, in turn, yields ∂_{t}Ab < 0 for large values of Ab. A more formal mathematical analysis of the character of the solutions to Eqs (1) and (2) is beyond the scope of this paper.
We note that, in absence of the decay in GC activity signified by G(t), these equations admit nonzero steadystate solutions. The solution for B is characterized by a constant nonzero value in the zeroaffinity region, with peaks corresponding to B cells that have some degree of specificity towards either of the glycoproteins. Since only nonzero affinity B cells end up producing antibodies, the solution for Ab will be nonzero only towards the edges of the interval, vanishing everywhere else. In both cases, the profiles of the left and rightmost portions of the distribution will be modulated by the affinity Q(x), with higheraffinity cells being more abundant in the steady state. In our particular case, however, the modulation introduced by G(t) has the effect of ultimately yielding solutions for both B and Ab that vanish in the entire interval as t → ∞.
Parameters
Despite the fact that we have attempted to keep the model simple, it still contains a large number of parameters. Due to this, we have decided to fix most of them and focus on those that govern the differences between vaccine formulations. The fixed parameters values in the model are primarily taken from Chaudhury et al.^{27}, and we have kept their labels whenever appropriate. In the cases when a single event in our model encompasses multiple events in the original one, we have assumed that the time it takes for this event to happen is an exponentially distributed variable with a mean equal to the sum of the mean waiting times of the original events; that is, the mean time for a B cell to proliferate is given by the mean stimulation time 1/σ_{B} plus the mean proliferation time of stimulated B cells 1/r, thus the resulting proliferation rate \(\tilde{r}={\sigma }_{B}r/({\sigma }_{B}+r)\). In the same manner, the effective antibody production rate \(\tilde{k}\) is obtained from the B cell stimulation rate σ_{B} and the antibody production rate of plasma cells k_{Ab} as \(\tilde{k}={\sigma }_{B}{k}_{Ab}/({\sigma }_{B}+{k}_{Ab})\)—recall that the probability of differentiation into plasma cells is explicitly included in Eq. (2).
Furthermore, we choose to fix d_{min} = 0 for simplicity, which amounts to neglecting degeneracy in the number of B cell genotypes presenting full binding affinity to the glycoproteins^{27}—we expect that a d_{min} > 0 would have a negligible effect in the model output—while we set ε = 10^{10}. This latter choice is based on the fact that for a Euclidean shape space to match immunological data, the normalized stimulation radius can be chosen to lie somewhere between 0.15 and 0.22^{34}, and we adjust the value of ε in order to allow a 10–10^{3}fold increase in binding affinities from naive to fully mature B cells^{27}, with the exact number determined by the value of d_{max}. We also fix a small value for the mutational variance—in this case, ν = 10^{−4}, so that a mutation of the order of one standard deviation translates into a shift of the order of one percent in genotype space—and the width of H(x) so that the pool of naive B cells occupies one percent of the nonzero affinity zone on each side of the shape space. More explicitly, we have
where \(\tilde{H}\) is the height of the pool, which we have set to \(\tilde{H}=100\) cells. These choices are further commented on in the Discussion section.
All fixed parameters in the model are listed in Table 1. The protein immunogenicities, the affinity cutoff and the GC activity decay rate are free parameters.
Effects of adjuvants
The main features of the experimental data correspond to enhanced antibody response and enhanced response towards the NA protein for the adjuvanted vaccine formulations, as opposed to the nonadjuvanted case. We introduce these effects in the form of factors β_{i}—which are free parameters in the model—modifying the immunogenicities and antibody production parameters as follows:
Here, all β_{i} ≥ 1, with the equality holding in the nonadjuvanted case. The different values β_{i} regulate the strength of the effects and are adjuvantdependent. We expect that β_{NA} > β_{HA}, reflecting a selection pressure towards the production of NAspecific antibodies.
Parameter estimation
The free parameters in the model, as described above, correspond to the affinity cutoff d_{max}, the base immunogenicities γ_{NA} and γ_{HA}, the decay rate in GC activity μ, and the adjuvanticity factors β_{NA}, β_{HA} and β_{Ab}. The first four parameters in this list can be estimated from the data for the nonadjuvanted formulation of the vaccine, while the other three can be obtained for each adjuvant in the data set after fixing these.
Specifically, we fit the output of the model defined by Eqs (1) and (2), y^{(model)}, to the data from Schmidt et al.^{17} corresponding to the virusneutralizing antibody titers as a function of time following vaccination and immunization boost. This is compared to the total area under the curve of the distribution given by Ab(x, t); that is, \({y}^{({\rm{model}})}(t)=\int \,{\rm{d}}x\,Ab(x,t)\). We use the Differential Evolution (DE) algorithm^{35} to estimate the best parameter values by minimizing the root mean squared error (RMSE). The cost function takes the form
where i runs over all the N individual points in the data set, and Θ corresponds to a particular set of parameters. In other words, we aim to find the optimal set Θ* that minimizes RMSE_{Θ}. As mentioned above, this process is carried out in two steps, with
We note that we could have, alternatively, minimized the mean squared error (MSE) or residual sum of squares (RSS) yielding equivalent results, the advantage of the RMSE being that it has interpretable units: in this case, the error is given in units of Ab titer.
The equations themselves are solved using noflux boundary conditions, and with initial condition B(x, 0) = 0 and Ab(x, 0) = 1; the latter due to the fact that the first data point for all vaccine formulations satisfies \({y}_{0}^{({\rm{data}})}=1\). We use a grid of N_{x} = 1500 points, with locations given by \({\rm{d}}x/2,3\,{\rm{d}}x/2,\ldots ,1{\rm{d}}x/2\), where \({\rm{d}}x\equiv 1/{N}_{x}\).
The best fit parameters, along with the bounds used for them in the estimation procedure, are shown in Table 2 for model A, and Table S2 for the case of model B. We have given γ_{HA} more freedom of movement than γ_{NA}, due to the fact that HA is more immunogenic than NA^{13}; however, we have not made γ_{HA} > γ_{NA} a hard constraint. Instead, in order to reflect the imbalance between immunogenicities we have made the upper bound for γ_{HA} in the estimation three times larger than that of γ_{NA}—see Table 2.
Additionally, Table 2 also shows the results of carrying out the parameter estimation on 2500 bootstrapping samples from the data, constructed by selecting, at random, one datapoint per timepoint in the original dataset. As for the case discussed above, the parameters for the adjuvanted vaccine formulations are estimated after fixing the values of parameters from the nonadjuvanted case to those providing the best fit when considering all datapoints, i.e., the values from Table 2. There exists large variability on the estimates and, in particular, we see that the immunogenicities tend to explore their full range of values, which we understand as a consequence of the fact that these two parameters are generally expected to rescale one another—see the Discussion for more details on this.
Finally, in order to evaluate the models, we employ a baseline consisting in a nonspatial version of model A, i.e., without taking proteinspecificity into account, which we denote by model 0. In this case, we replace all xdependent terms by scalar ones, so that H(x) becomes H and represents a naive influx rate, while Q(x) becomes an effective measure of affinity Q—see Eqs (S1) and (S2) for details. Table 3 compares the models to this baseline in terms of their goodness of fit and complexity, showing both the RMSE and Akaike Information Criterion (AIC)^{36}. In general, while qualitative differences between the models are easily observable, as discussed below, a quantitative evaluation based on AIC clearly favors model A only for the first part of the sequential parameter estimation, corresponding to the nonadjuvanted vaccine formulation, for which its performance is equivalent to the baseline. For the adjuvanted cases, models A and B—as well as C, introduced below—behave similarly.
All code is implemented in Python and is freely available at: https://github.com/systemsmedicine/adjuvantedvaccine.
Results
Kinetics of antibody response
Figure 3 shows the output from model A as given by the best fit parameters for the nonadjuvanted and all three adjuvanted formulations of the vaccine. The antibody titers obtained from the model defined by Eqs (1) and (2) as the area under the curve of Ab(x, t) are compared to the raw data^{17}. We found that this variant of the model is able to account for the patterns appearing in the data—as can be seen from Fig. S1, this is also true for model 0, which does not treat NA and HAspecific antibodies separately. The vaccine formulations with squalenecontaining adjuvants result in antibody responses surpassing the detection limit within the first week post immunization, and that remain well above this level for an extended period of time. This is not true for the vaccine containing Diluvac and the nonadjuvanted vaccine, which require the immunization boost to achieve sustained responses above detection level. We also see that the response decays and approaches the detection limit towards 130 days post immunization, which does not happen in the cases of MF59 and AS03adjuvanted formulations.
Model B, on the other hand, fails to properly capture the decay in functional antibody titers, as can be seen from Fig. S2. At the same time, this has the effect of pulling the immunogenicities towards lower values, in order to compensate for the fast proliferation of B cells at longer times; the resulting dynamics for antibody titers fails to account for the rapid mounting of the response after the first dose, due to the effective decrease in antibody production rates driven by γ_{NA} and γ_{HA}—see Table S2. This suggests that our modulating function needs to play a role on all of the processes involved in the proliferation of the B cell population, rather than only have an effect over those representing recruitment of naive or memory cells.
An additional variant of the model was considered, corresponding to model A for the nonadjuvanted formulation of the vaccine, but letting the adjuvants affect the antigen immunogenicities only; that is, we chose β_{Ab} = 1, effectively removing a free parameter from the model. This simpler version of adjuvant influence, which we denote by model C, did not reproduce the behavior observed in the data—see Table S3 and Fig. S3. In order to compensate for the absence of boost in the proteinindependent rate of antibody production, larger immunogenicity boosts are required. The subsequent shift in weighted affinity results in a much larger selection effect in B cell proliferation, ultimately yielding an excessive rate of Ab production immediately after each of the immunizations.
B cell and antibody distributions in shape space
Focusing on model A, we next looked directly at the profiles of B(x) and Ab(x) at specific times for the four different vaccine formulations. This is shown in Figs 4 and 5 for B cells and antibodies, respectively. We show snapshots of the B cell and Ab distributions at the end of the first week, the end of the third week before boost, the beginning of the fourth week, and the end of the fifteenth week.
As expected from the model construction, the profile of B cells is initially characterized by the influx of naive cells represented by H(x). As time progresses, mutations widen the profile, and highaffinity cells proliferate faster and decay more slowly than lowaffinity ones, resulting in the distribution caving towards the center of the interval. When the GC is no longer active and the naive cell influx has stopped, we are left solely with highaffinity, slowlydecaying B cells.
Since the dynamics of B cells depends, for the most part, on their unweighted binding affinity—with weighted affinity having an evident influence only towards the edges of the interval, due to the recruiting of ‘memory’ cells—we do not clearly observe the effects of adjuvants on their overall distribution. This can, however, be seen from the antibody profiles in Fig. 5, where we find that the relative heights of the left and righthand side peaks of the distribution on the onedimensional shape space for the adjuvanted vaccine formulations tend to be more even that for the nonadjuvanted vaccine throughout the whole process. This is a direct consequence of the adjuvantdriven boost in glycoprotein immunogenicities, which is primarily directed towards γ_{NA}, while γ_{HA} remains essentially unchanged, as can be seen from Table 2—see also Fig. 6, showing the resulting weighted binding affinity Q(x) for all vaccine formulations.
The most striking difference between base and adjuvantboosted parameters corresponds to the rate of antibody production \(\tilde{k}\)—represented by β_{Ab}—with values tens of times larger for the squalenecontaining formulations—over 50 in the case of AS03—as shown in Table 2. This is also the parameter that clearly discriminates between squalenecontaining adjuvants and Diluvac, with the latter only resulting in a lower than twofold increase in antibody production rate.
Finally, from Fig. 5 we note that the effect of the boost immunization on antibody titers is strongest towards the very highaffinity edges of the interval, x = 0 and x = 1, and not so easily observable towards the peaks of the distribution. This is consistent with the higher rate of proliferation of fully mature ‘memory’ B cells, which also benefit more from the boost that their lowaffinity counterparts, as seen in Fig. 4.
Proteinspecific antibodies
Furthermore, we can calculate the total titer of antibodies with nonzero binding affinity towards NA (resp. HA), but zero binding affinity towards HA (resp. NA) at a given moment in time as
This allows us to clearly visualize the kinetics of the NA or HAspecific antibodies in the system, as illustrated in Fig. 7.
These dynamics may be compared, albeit only qualitatively, to the data for total antibody titers against different HA and NA proteins obtained from samples collected at day 76 post immunization^{17}. The proteins correspond to H1 (H1N1), H3 (H3N2), H5 (H5N1), H7 (H7N9), H9 (H9N2), N1 (H1N1), N1 (H5N1), and N2 (H3N2); in this case, we consider explicitly only the H1 and both N1 proteins, since these are the ones contained in the vaccine, and assume all other antibody titers to be below the detection level. The results for Ab_{NA}(t = 76) and Ab_{HA}(t = 76) are compared to the data in Fig. 8, and we see that in very broad terms the trends in the data are somewhat reproduced. However, as one may expect, the model in its current version requires a means to determine the proportion of functional antibodies in total antibody titers from the original output, in order to permit a quantitative comparison.
Discussion
We have presented a model of antibody production dynamics in response to vaccination with adjuvanted and nonadjuvanted influenza vaccines. Our focus was to construct a parsimonious description of antibody dynamics that is capable of reproducing experimentally observed behavior. With this in mind, we derived a deterministic onedimensional representation of the distribution of B cell and antibody lineages in shape space. Two different variants of the base, nonadjuvanted model were explored: model A, considering a modulation in B cell growth as a whole; and model B, consisting in a modulation of cell recruitment into the GC only. The results obtained favor a picture consistent with a broader regulation inside the GC itself, as the decay in B cell numbers is not fast enough to compensate for their proliferation and cannot, by itself, account for the decay in antibody titers observed at later time points^{17}.
For the adjuvanted formulations of the vaccine, the specific adjuvants studied were MF59, AS03 and Diluvac Forte. The latter is a veterinary approved adjuvant containing vitamin E in watery suspension^{37}. MF59 and AS03, on the other hand, are oilinwater emulsions based on squalene oil, which are approved for use in humans and count with extensive evidence for their properties as adjuvants. Squalene emulsions stimulate immune cell migration to the injection site and antigen uptake, resulting in adjuvanted vaccines that yield increased antibody titers with respect to their nonadjuvanted counterparts—see, e.g., refs^{38,39,40}. At the same time, they have shown effects on crossreactivity^{41,42} and reduction of virus replication and transmission^{43}, and are well tolerated, with a strong safety record^{38,44}. These features make squalenebased adjuvants excellent candidates for antigensparing vaccination strategies.
We have introduced the influence of adjuvants in the model as a boost in glycoprotein immunogenicities, and found that all adjuvanted formulations of the vaccine result in enhanced antibody response against NA. We also found that the boost in immunogenicities alone—illustrated by model C, shown in Table S3 and Fig. S3—is not sufficient to account for the observed titers, and an additional parameter representing an adjuvantdriven boost in the overall rate of antibody production in the GC is required to yield quantitative agreement with the data. The latter influences both the NA and HAspecific responses in the same way and is, in fact, the strongest effect of the adjuvants on the resulting antibody titers. In other words, there exist a qualitative change in the adjuvantdriven antibody response, brought about by the skewed boost in the reaction towards the NA protein. However, the most important effect induced by the adjuvants, and the key feature that separates Diluvac from squalenecontaining adjuvants, is the sheer scale of the resulting overall antibody response, irrespective of protein specificity. It is important to note that model A, with β_{Ab} being a free parameter, is the minimal model that reproduces the observed experimental behavior for the adjuvanted vaccine formulations in relation to model C given the assumptions and fixed parameters governing the nonadjuvanted case. In other words, we do not rule out the possibility that a different formulation for the base, nonadjuvanted model might work in conjunction with the assumptions underpinning model C, i.e., that the adjuvants have an effect on protein immunogenicities only.
The resulting dynamics from the model can be explored under different conditions from those in Schmidt et al.^{17}. It has been observed in ferrets that, while a second dose of the vaccine is required in order to obtain high and sustained immune responses, its timing does not have a significant impact on their overall behavior for the case of AS03^{45}. Similar results have been observed in clinical trials of MF59adjuvanted influenza vaccines, with betweendose intervals of up to six weeks for adults^{46}, and high antibody titers up to one year following vaccination with four weeks between doses in young children^{47}. Using the best fit parameters from Table 2, we can simulate Eqs (1) and (2) with different values of t_{boost} and assess the model output for all vaccine formulations, as shown in Fig. 9.
We observe that, in the context of model A, the longterm behavior of functional antibody titers is largely unaffected by the timing of the immunization boost for all four vaccine formulations; only the maximum titer achieved shows a small variation, being larger when the initial vaccination and the boost are more closely spaced. Furthermore, a strong response is not maintained for the Diluvac and nonadjuvanted vaccines. In contrast, the squalenebased adjuvants result in antibody titers well above the detection limit for extended periods after the boost. This is consistent with previous findings^{45,46,47}, and suggests that vaccines formulated with squalenecontaining adjuvants are generally robust to changes in the interval between the first and second doses. Further experiments involving AS03 and MF59 in different primeboost schedules would help elucidate this claim.
An interesting outcome resulting from the fact that model A is based on distributions in shape space—in contrast to model 0, which treats B cells and Abs as scalar variables—corresponds to the option of tracking the dynamics of the NA and HAspecific immune responses independently of one another, as we have shown in Fig. 7. This opens up the possibility of exploring the introduction of additional virus strains, and determining the antibody response against different glycoprotein variants. In particular, one may assess the effects of the qualitative shift on proteinspecific responses induced by the adjuvanted formulations on the antibody response towards strains carrying the same NA subtype, and their implications for crossreactive immunity. However, within the onedimensional representation we have adopted for shape space, it is not immediately clear how to include different variants of NA and HA. This is most evident from Fig. 8, where we have not made distinctions between different NA proteins belonging to the same subtype.
A possible way of extending the model to accommodate different HA and NA proteins may be to adopt a picture of an antibody landscape such as that from Fonville et al.^{30}, in which different virus strains are mapped to a onedimensional space, but doing so proteinwise. In other words, employing a twodimensional space and mapping HA and NA independently to positions along either dimension, such that HA and NAspecific immune responses give rise to separate landscapes. The model may be further extended by coupling it to additional equations describing the time evolution of withinhost influenza dynamics, such as the targetcell limited model described in Baccam et al.^{48}. This would allow us to assess the effects of the different vaccine formulations on postchallenge viral load dynamics; additional experiments could then be conducted in order to test these results.
As a final note, we must remark on an important limitation of our work. While we have attempted to derive a model which is as minimal as possible, we are nevertheless interested in being able to account for Ab distribution along the shapespace. As a consequence, we are left with a large number of parameters given the size of the dataset and, in order to focus on the relative Ab responses between adjuvanted and nonadjuvanted vaccine formulations, we have chosen to rely on assumptions and fix parameter values whenever possible. In particular, we have fixed most of the parameters that are part of the base, nonadjuvanted model, and that we do not expect to change for the adjuvanted formulations. We have taken these from previous estimations and assumptions^{27,32,33,34}, while others have been fixed either because we expect them to have a negligible effect—such as setting d_{min} = 0—or rescale one another in the nonadjuvanted case; the latter is the case of the height of the naive pool, \(\tilde{H}\), and the variance ν, which determines how fast the lowaffinity B cells diffuse into the regions of higher affinity. Nevertheless, we expect that choosing a working combination of values for \(\tilde{H}\) and ν does not represent a problem for the purpose of comparing different vaccine formulations, since these are kept constant throughout the comparison. Another obvious rescaling pair corresponds to the protein immunogenicities, which compete directly against one another in antibody production in order to match the model output to the experimental data. Here we have not fixed either of them, but have rather used the fact that HA is known to be more immunogenic than NA^{13} and that the neutralizing response is skewed towards HA^{7} in order to constrain the possible values of γ_{NA} while giving more freedom of movement to γ_{HA}. This biologically inspired choice constitutes an assumption of the model, and it is not a direct result of the parameter estimation.
A sensitivity analysis reveals that, for the base model, changes to the rate of decay in GC activity, μ, and to the size of the affinity region, d_{max}, have the largest quantitative effect on the outcome of the model—see Fig. S4(C,D). At the same time, the immunogenicity of NA is the parameter with the least impact—Fig. S4(E). Changes to the parameters controlling the effects of the adjuvants—β_{NA}, β_{HA}, and β_{Ab}—result in similar outcomes, with the Ab titer being to some extent more robust to changes in β_{NA}, as shown in Fig. S5. Additionally, we see that, as anticipated, the fixed parameters \(\tilde{H}\) and ν have a similar qualitative effect on the outcome of the model, so that larger values of one of them will very require smaller values of the other; this is illustrated in Fig. S6, showing the sensitivity function—as defined in Soetaert & Petzoldt^{49}—of the two parameters for the log_{2} of the Ab titer in the nonadjuvanted case. We note, however, that ν has a larger impact on the resulting titers after the boosting time. This can be explained by the fact that \(\tilde{H}\) is only relevant in the lowaffinity region of the shape space, while ν controls the rate of diffusion towards higher affinities in the entire interval, and the boost has a stronger effect on highaffinity B cells.
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Acknowledgements
This work was supported by the Alfons und Gertrud KasselStiftung.
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C.P.R. constructed the model and performed the simulations. V. von M. provided the data. E.H.V. envisaged and supervised the project. All authors discussed and wrote the paper.
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ParraRojas, C., Messling, V.v. & HernandezVargas, E.A. Adjuvanted influenza vaccine dynamics. Sci Rep 9, 73 (2019). https://doi.org/10.1038/s41598018364269
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