# How a well-adapting immune system remembers

^{a}Lewis–Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544^{b}Laboratoire de Physique de l’École Normale Supérieure, PSL University, Centre National de la Recherche Scientifique, Sorbonne University, University Paris-Diderot, 75005 Paris, France;^{c}David Rittenhouse Laboratories, University of Pennsylvania, Philadelphia, PA 19104;^{d}Initiative for the Theoretical Sciences, The Graduate Center, The City University of New York, New York, NY 10016

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Edited by Herbert Levine, Rice University, Houston, TX, and approved March 21, 2019 (received for review July 25, 2018)

## Significance

The adaptive immune system is able to protect us from a large variety of pathogens, even ones it has not seen yet. Can predicting the future pathogen distribution help in protection? We find that a combination of probabilistic forecasting and occasional sampling of the current environment reduces infection costs—a scheme easily implemented by the memory repertoire. The proposed theoretical framework offers a modular recipe for updating the memory repertoire, which quantitatively predicts the strength of the immune response in flu-vaccination experiments, unlike other update schemes. It also links the observed early life dynamics of the memory pool to the sparseness properties of the pathogen distribution and competitive receptor dynamics for pathogens.

## Abstract

An adaptive agent predicting the future state of an environment must weigh trust in new observations against prior experiences. In this light, we propose a view of the adaptive immune system as a dynamic Bayesian machinery that updates its memory repertoire by balancing evidence from new pathogen encounters against past experience of infection to predict and prepare for future threats. This framework links the observed initial rapid increase of the memory pool early in life followed by a midlife plateau to the ease of learning salient features of sparse environments. We also derive a modulated memory pool update rule in agreement with current vaccine-response experiments. Our results suggest that pathogenic environments are sparse and that memory repertoires significantly decrease infection costs, even with moderate sampling. The predicted optimal update scheme maps onto commonly considered competitive dynamics for antigen receptors.

All living systems sense the environment, learn from the past, and adapt predictively to prepare for the future. Their task is challenging because environments change constantly, and it is impossible to sample them completely. Thus, a key question is how much weight should be given to new observations vs. accumulated past experience. Because evidence from the world is generally uncertain, it is convenient to cast this problem in the language of probabilistic inference where past experience is encapsulated in a prior probability distribution which is updated according to sampled evidence. This framework has been successfully used to understand aspects of cellular (1⇓⇓–4) and neural (5⇓⇓–8) sensing. Here, we propose that the dynamics of the adaptive immune repertoires of vertebrates can be similarly understood as a system for probabilistic inference of pathogen statistics.

The adaptive immune system relies on a diverse repertoire of B- and T-cell receptors to protect the host organism from a wide range of pathogens. These receptors are expressed on clones of receptor-carrying cells present in varying copy numbers. A defining feature of the adaptive immune system is its ability to change its clone composition throughout the lifetime of an individual, in particular via the formation of memory repertoires of B and T cells following pathogen encounters (9⇓⇓⇓⇓–14). In detail, after a proliferation event that follows successful recognition of a foreign antigen, some cells of the newly expanded clone acquire a memory phenotype. These cells make up the memory repertoire compartment that is governed by its own homeostasis, separate from the inexperienced naive cells from which they came. Upon reinfection by a similar antigen, memory guarantees a fast immune response. With time, our immune repertoire thus becomes specific to the history of infections and adapted to the environments we live in. However, the commitment of part of the repertoire to maintaining memory must be balanced against the need to also provide broad protection from as-yet-unseen threats. What is more, memory will lose its usefulness over time as pathogens evolve to evade recognition.

How much benefit can immunological memory provide to an organism? How much memory should be kept to minimize harm from infections? How much should each pathogen encounter affect the distribution of receptor clones? To answer these questions, we extend a framework for predicting optimal repertoires given pathogen statistics (15) by explicitly considering the inference of pathogen frequencies as a Bayesian forecasting problem (16). We derive the optimal repertoire dynamics in a temporally varying environment. This approach can complement more mechanistic studies of the dynamics and regulation of immune responses (12, 17⇓⇓⇓⇓–22) by revealing adaptive rationales underlying particular features of the dynamics. In particular, we link the amount of memory production to the variability of the environment and show that there exists an optimal time scale for memory attrition. Additionally, we demonstrate how biologically realistic population dynamics can approximate the optimal inference process and analyze conditions under which memory provides a benefit. Comparing predictions of our theory to experiment, we argue for a view in which the adaptive immune system can be interpreted as a machinery for learning a highly sparse distribution of antigens.

## Theory of Optimal Immune Prediction

The pathogenic environment is enormous, and the immune system can only sample it sparsely, as pathogens enter into contact with it at some rate λ. We consider an antigenic space of K different pathogens with time-varying frequencies *A* and Materials and Methods). We reason that the immune system should efficiently use the information available through these encounters, along with prior knowledge of how pathogens evolve encoded in the system dynamics, to build an internal representation of the environment (Fig. 1*B*). Biologically, we can think about this representation as being encoded in the composition of the adaptive immune repertoire (the size and specificity of naive and memory lymphocyte clones), but generally further cellular memory mechanisms might also contribute. Based on this representation of the world, the immune system should organize its defenses to minimize harm from future infections (Fig. 1*C*).

How could the immune system leverage a representation of beliefs about pathogen frequencies to provide effective immunity? Each lymphocyte (B or T cell) of the adaptive immune system expresses on its surface a single receptor r out of L possible receptors. This receptor endows the lymphocyte with the ability to specifically recognize pathogens (labeled a) with probability ** Q** with certainty, but has an internal belief

The internal representation of the environment can be regarded as a system of beliefs, or guesses, about pathogen frequencies. Formally, these beliefs can be represented in the form of a probability distribution function **2**–**4** can be turned from abstract belief updates into dynamical equations for a well-adapting immune repertoire.

The Bayesian forecasting framework provides a broad account of the possible adaptive value of many features of the adaptive immune system without the need for additional assumptions. Immune memory formed after a pathogenic challenge is explained as an increase in optimal protection level resulting from an increase in estimated pathogen frequency, following Eq. **3**. Attrition of immune memory is also adaptive, because it allows the immune repertoire to forget about previously seen pathogens which it should do in a dynamically changing environment (Eq. **4**). Lastly, some of the biases in the recombination machinery and initial selection mechanisms (24) represent an evolutionary prior (Eq. **2**) which tilts the naive repertoire toward important regions of antigenic space.

We are proposing an interpretive framework for understanding adaptive immunity as a scheme of sequential inference. This view provides two key insights. First, it confirms the intuition that new experience should be balanced against previous memory and against unknown threats in order for adaptive immunity to work well. Second, it suggests a particular dynamics of implicit belief updates that can globally reorganize the immune repertoire to minimize harm from the pathogenic environment. Going beyond these broad ideas, in Results, we analyze in detail a model for optimal immune prediction in which all these statements can be made mathematically precise. We also show a plausible implementation that the immune system could follow to approximate optimal Bayesian inference, and we compare the resulting dynamics with specific features of the adaptive immune system.

## Results

### A Lymphocyte Dynamics for Approximating Optimal Sequential Inference.

For concreteness, we consider a drift-diffusion model of environmental change (Eq. **11**). The drift-diffusion model, while clearly a much-simplified model of real evolution, captures two key features of changing pathogenic environments: the coexistence of diverse pathogens and the temporal turnover of dominant pathogen strains. The aim of this model of pathogen evolution is not to provide a realistic description of short-term pathogen dynamics within a population such as during an epidemic, but rather to capture overall features of the long-term dynamics of many pathogens over a host’s lifetime. The drift-diffusion model is mathematically equivalent to a classical neutral stochastic evolution of pathogens (25) driven by genetic drift happening on a characteristic time scale τ and immigration from an external pool with immigration parameters **11**). Generally, pathogens are under selective pressure to evade host immunity, and strains are replaced faster than under the sole action of genetic drift. Matching the time scale of pathogen change to those observed experimentally, the model then provides a simple, effective description of the pathogen dynamics.

Can a plausible dynamics of lymphocyte receptor clones approximate the optimal repertoire dynamics? In particular, is there an approximate autonomous dynamics for the repertoire composition, which does not require access to the full latent high-dimensional belief distribution **1**) is the identity,

For the pathogen dynamics of Eq. **10**, we show, using a decomposition of *SI Appendix*, section 1B), that there exists an approximate but autonomous dynamics for the number of lymphocyte receptors N. It takes the following simple form (parallel to Eqs. **2**–**4**):*SI Appendix*, section 1C) and found that they approximate the optimal dynamics closely (Fig. 2).

These dynamics have a plausible biological implementation. Each pathogen encounter leads to a fixed increment χ of the number of specific lymphocytes (Eq. **6**), implying a regulation mechanism that controls the number of cell divisions upon clonal expansion as a function of precursor frequency. Through thymic output, the repertoire starts at and is renewed by the naive repertoire—encoded in **5** and **7**). By definition, this naive repertoire is optimally tuned to the baseline frequencies of pathogens **7**), which can be done through some global homeostatic mechanisms.

To gain intuition about what the dynamics of Eqs. **5**–**7** mean, let us define the rescaled variables **6** increases **7** discounts these old encounters, giving more relative weight to recent ones. If there are no new pathogen encounters,

The assumption that **1**) remains approximately invertible, leading to autonomous dynamics of the lymphocyte clone sizes **5**–**7** (*SI Appendix*, section 2C).

### Learnability of Pathogen Distribution Implies a Sparse Pathogenic Landscape.

The immune system must be prepared to protect us not just from one pathogen but a whole distribution of them. Even restricting recognition to short peptides and accounting for cross-reactivity (26), estimates based on precursor frequencies for common viruses give an effective antigen environment of size

This apparent paradox can be resolved by the fact that the pathogenic environment may be sparse, meaning that only a small fraction of the possible pathogens are present at any given time. In our model of the pathogen dynamics, this sparsity is controlled by the parameter θ. In the scenario that we are considering, typical pathogen landscapes Q are drawn from the steady-state distribution *SI Appendix*, section 1, Eq. **S1**). When *SI Appendix*, section 3C). Since the system only needs to learn about the pathogens that are present, the condition for efficient learning should naively be

Our theory can be used to quantify the benefit of memory as a function of the different immunological parameters. We compute the optimized cost function *A* shows that the benefit of memory increases with pathogen sparsity—when θ is small, even a few encounters suffice to seed enough memory to reduce the cost of future infections. The cost saturates with age to a value *B*) since learning and prediction are limited by decorrelation of the environment. The pathogen dynamics is sped up when there are strong selection pressures to evade immunity. Faster dynamics decrease learning efficiency and in turn reduce selective pressures. The effective time scale should in practice be set by a coevolutionary balance between both effects.

Analytical arguments show that in the limit of few samples the relative cost *SI Appendix*, section 3). In general, we find that the cost is a function of *SI Appendix*, section 1D for derivation from Eqs. **6** and **7**). Plotted in terms of this variable, the relative cost gap as function of time, *C*). Fig. 3*C* shows that the cost drops by a factor of

### Optimal Attrition Time Scale.

Our theory suggests that there is an optimal time scale for forgetting about old infections which is related to the time scale over which the environment varies. Eq. **7** shows that memory should optimally be discounted on an effective time scale *SI Appendix*, Eq. **37**), where

Interestingly, our theory predicts that memory should be discounted more quickly when the immune system has gathered more information (larger **8**,

### Memory Production in Sparse Environments Should Be Large and Decrease with Prior Exposure.

The theory can be used to make quantitative and testable predictions about the change in the level of protection that should follow a pathogen encounter. Consider an infection cost function that depends as a power law on the coverage, *SI Appendix*, section 4).

In the simplest model for repertoire updates, recognition of pathogens leads to proliferation proportionally to the number of specific precursor cells, followed by a homeostatic decrease of the memory pool (18, 33). Thus, the fold change *Materials and Methods*). Setting

To understand this prediction, first consider the effect of a primary infection on a naive repertoire, *SI Appendix*, section 2A) and Eq. **9** predicts a fold change of *B*). Combined with the estimate

To test Eq. **9** on immunological data, we fit the Bayesian update model to experiments reporting fold changes in antigen titers upon booster vaccinations against influenza from ref. 34 (Fig. 4*A*) using least squares. Titers correspond to the concentration of antibodies that are specific to the antigen a and can thus be viewed as an experimental estimate of

Interestingly, for T cells, Quiel et al. (38) have shown that fold expansion to peak cell numbers in an adoptive transfer experiment depends on the initial number of T cells as a power law with exponent **9**, which is for memory production. This implies a nonlinear relationship between peak T-cell level and memory production, which further suppresses memory production at high precursor numbers. This prediction might be checked in experiments measuring memory production after infection clearance, as well as the expansion peak.

### Long-Term Dynamics of a Well-Adapting Repertoire.

Our model makes predictions for the dynamics of growth and attrition of memory over time, with consequences for immunity and for the diversity of the immune repertoire. We quantify the dynamics in terms of a memory fraction defined as a sum of the coverage fractions *A*). At the same time, the memory fraction increases rapidly (Fig. 5*B*), but the growth of memory slows as subsequent infections lead to less memory production following the optimal fold-change rule in Eq. **9** and as attrition begins to play a role. The fraction of the repertoire devoted to memory in midlife is largely determined by how the cost of infections scales with coverage. The observed memory fraction of *B*). The diversity of the memory repertoire increases with time at a rate that slows with age (quantified in Fig. 5*C* by richness, which measures the number of unique specificities and the Shannon entropy of the repertoire frequency distribution).

To gain insight into these dynamics of our model, we average the stochastic equations over the statistics of pathogen encounters. We show in *SI Appendix*, section 2B that this mean-field approximation yields a differential equation for the population fraction of different clones with two opposing contributions which balance alignment of the immune repertoire with the current pathogenic environment (i.e., memory production) against alignment with the long-term mean environment (i.e., attrition). Interestingly, the mean-field equation broadly coincides with dynamics that were proposed in ref. 15 to self-organize an optimal immune repertoire. The essential difference here is that the time scale of learning slows down with increasing experience following the rules of optimal sequential update in Eq. **9**.

We then asked which features of the proposed repertoire dynamics are most relevant to ensure its effectiveness. How important is the negative correlation between fold expansion and prior immune levels, and how important is attrition? Furthermore, if the immune system follows Bayesian dynamics, it must have integrated on an evolutionary time scale a prior about composition and evolution of the pathogen environment through the parameters θ and τ—however, the prior may be inaccurate. How robust is the benefit of memory to imperfections of the host’s prior assumptions about pathogen evolution? To answer these questions, we compare the long-term immune-repertoire dynamics using the optimal Bayesian scheme to other simplified schemes. We find that a constant fold expansion dynamics quickly leads to very suboptimal repertoire compositions (Fig. 5*D*, pink line), since the exponential amplification of cells specific to recurrent threats quickly leads to a very large fraction of the repertoire consisting of memory of those pathogens (Fig. 5*E*, pink line). This suboptimality persists, even if we assume that some global regulation caps the constant fold expansion such that no individual receptor clone can take over all of the repertoire (Fig. 5 *D* and *E*, gray line). Thus, negative feedback in T-cell expansion to individual antigens is very important to maintain a properly balanced diverse repertoire. In contrast, within dynamics with a negative correlation, the precise levels of updating do not need to be finely tuned to the environmental statistics: Varying the assumed sparsity of the pathogen distribution, which controls fold expansion upon primary infection in the optimal dynamics, leads to a relatively modest deterioration of the convergence speed of the learning process (*SI Appendix*, Fig. S2*A*) and does not matter asymptotically (*SI Appendix*, Fig. S2*B*). Attrition does not matter at young age, but can play an important role for long-term adaptation to relatively rapidly changing pathogen distributions (Fig. 5*F*). However, the attrition time scale need not be finely tuned to get close to optimal dynamics (*SI Appendix*, Fig. S3).

### Adapting a Cross-Reactive Repertoire.

Above, we described adaptation of immune repertoires in terms of changes in the effective coverage

Following Perelson and Oster (41), we will represent the interaction of receptors and antigens by embedding both in a multidimensional metric recognition “shape space,” where receptors are points surrounded by recognition balls. Antigens that fall within a ball’s radius will be recognized by the corresponding receptor. In this presentation, a and r are the coordinates of antigens and receptors, respectively, and their recognition propensity depends on their distance,

Earlier sections have already discussed the optimal dynamics of the coverage *SI Appendix*, section 2C reveals a general qualitative phenomenon—competitive exclusion between clones expressed in the repertoire and their close neighbors within the cross-reactivity radius (*SI Appendix*, Fig. S4, blue line). This exclusion is not an assumption of the model, but rather stems from the optimal Bayesian theory. Given a receptor clone that covers one region of antigenic shape space, the global likelihood of detecting infections increases by placing other clones to cover other regions. This can be shown analytically when cross-reactivity is limited, memory updates are small in magnitude, and the pathogen distribution is assumed to be uncorrelated (*SI Appendix*, section 2C).

In general, the frequencies of pathogens might be correlated in antigenic space, for example, because mutations from a dominant strain give rise to new neighboring strains. An optimally adapting immune system should incorporate such correlations as a prior probability favoring smoothness of the pathogen distribution. Such priors work their way through the optimal belief update scheme that we have described and weaken the competitive exclusion between clones with overlapping cross-reactivity (*SI Appendix*, Fig. S4, orange line).

In general, when cross-reactivity is wide or the required clone fraction update is large, numerical analysis shows that achieving optimally predictive immunity after a pathogen encounter requires a global reorganization of the entire repertoire (*SI Appendix*, Fig. S5, blue line). There is no plausible mechanism for such a large-scale reorganization since it would involve up- and down-regulation, even of unspecific clones. However, in *SI Appendix*, section 2C we show that the optimal update can be well-approximated by changes just to the populations of specific clones with pathogen-binding propensities *SI Appendix*, Fig. S5, orange line). The local update rule provides protection that comes within 1% of the cost achievable by the best global update. Thus, reorganization of pathogen-specific receptor clone populations following an infection, as seen in vertebrates, can suffice to achieve near-optimal predictive adaptation of the immune repertoire.

## Discussion

The adaptive immune system has long been viewed as a system for learning the pathogenic environment (10). We developed a mathematical framework in which this notion can be made precise. In particular, we derived a procedure for inferring the frequencies of pathogens undergoing an immigration-drift dynamics and showed how such inference might approximately be performed by a plausible population dynamics of lymphocyte clones. Additionally, we analyzed how quickly the immune system can learn about its environment and find that the antigenic environment must be effectively sparse to be learnable with a realistic rate of pathogen encounters.

The optimal repertoire dynamics recapitulate a number of properties of real adaptive immune systems. Two repertoire compartments emerge from the theory: memory, which encodes lived experience, and naive, which encodes prior expectations. Memory is effective in reducing harm from infection despite the high dimensionality of pathogenic space; having encountered circulating pathogens only once on average can reduce the overall cost of infections by half. The first encounter of a naive individual with a pathogen leads to a large response, which increases protection levels by several orders of magnitude. Memory production depends on prior protection levels and eventually is predicted to saturate, as seen in vaccination experiments (34). This saturation is predicted to lead to a sublinear increase with age of the fraction of the total repertoire taken up by memory, consistent with observations in human cohorts (39, 40).

Our work makes a number of concrete predictions amenable to further testing. We make quantitative predictions about how much memory should be formed following an infection, as a function of number of pathogens effectively present in the environment and the number of previous infections. These dependencies might be tested by using a comparative approach, which relates the amount of formed memory in different species to how many pathogens to which they are susceptible. Our model also makes predictions about changes in the size and diversity of the memory compartment with aging, which might be tested in future repertoire-sequencing studies similar to those done by Britanova et al. (42), by sorting cells into naive and memory types before sequencing. Ultimately, we hope that our work might help motivate studies with longitudinal tracking of the long-term repertoire dynamics in model organisms living in controlled pathogenic environments.

Our framework can be extended to incorporate additional constraints on immune-system function or further aspects of pathogen evolution. An example of a constraint would be to introduce a maximal rate of change in the optimization to model maximal cellular division and death rates. A more complex pathogen dynamics might model explicitly their mutational dynamics in antigenic space. Such dynamics will lead to correlations in the pathogen distribution, which we showed will influence the structure of the optimal conjugate repertoire. In particular, the optimal response should spread around the currently dominant antigens to also provide protection against potential future mutations. Hypermutations in B cells may play a role in this diversification, in addition to their known function of generating receptors with increased affinity for antigens of current interest. It would also be interesting to extend our framework to other immune-defense mechanisms, including innate immunity, where the role of memory has received recent attention (43).

Although our study was motivated by the adaptive immune system, some of our main results extend to other statistical inference problems. We have extended earlier results on exactly computable solutions to the stochastic filtering problem for Wright–Fisher diffusion processes (44) to derive an efficient approximate inference procedure. This procedure might be of use in other contexts where changing distributions must be inferred from samples at different time points, e.g., in population genetics. Additionally, we have derived the convergence rate for Bayesian inference of categorical distributions in high dimensions in the undersampled regime, showing that effectively sparse distributions can be inferred much more quickly. These results add to the growing literature on high-dimensional inference from few samples (45, 46), which has arisen in the context of the big-data revolution.

We propose that the adaptive immune system balances integration of new evidence against prior knowledge, while discounting previous observations to account for environmental change. Similar frameworks have been developed for other biological systems. In neuroscience, leaky integration of cues has been proposed as an adaptive mechanism to discount old observations in change-point detection tasks (47, 48), and close-to-optimal accumulation and discounting of evidence has been reported in a behavioral study of rat decision-making in dynamic environments (49). Inference from temporally sparse sampling has been considered in the framework of infotaxis, which is relevant for olfactory navigation (50). In the context of immunity, related ideas about inference and prediction of pathogen dynamics have been used to predict flu-strain and cancer-neoantigen evolution in silico (51, 52). Finally, ideas similar to those developed here could be used in ecology or microbiome studies to reconstruct evolutionary or ecological trajectories of population dynamics from incomplete sampling of data at a finite number of time points, e.g., from animal sightings or metagenomics.

## Materials and Methods

### Modeling Pathogen Dynamics by an Immigration-Drift Process.

In our model, we describe the stochastic dynamics of the pathogenic environment (Fig. 1*A*) by a Fokker–Planck equation for the conditional probability distribution *SI Appendix*, section 1E. This dynamics retains key features of real pathogen environments. First, at a given point in time, the environment contains many different pathogens with different frequencies determined by genetic drift and immigration. Second, the dominant pathogens change over time, such as is the case for many viruses, e.g., the flu or HIV.

### Minimizing the Cost of Infection.

To solve the optimization problem Eq. **1** analytically, a set of necessary conditions for optimality, the so-called Karush–Kuhn–Tucker conditions, can be derived. When all receptors are present at a nonzero frequency in the optimal repertoire

### Change in Protection upon a Pathogen Encounter.

The inference dynamics induces via the mapping from **1**) a dynamics of an optimally adapting immune repertoire. To get intuition we derive how the coverage changes in a simple setting in which Eq. **13** holds (further cases are considered in *SI Appendix*, section 2).

Rewriting Eq. **6** in terms of the expected pathogen frequencies **13**, it follows that coverages are updated as**9**. To fit the dataset, we note that a proportional rescaling of

## Acknowledgments

The work was supported by European Research Council Starting Grant 306312, a Lewis–Sigler fellowship (to A.M.), Simons Foundation Mathematical Modeling of Living Systems Grant 400425, and NSF Grant PHY-1734030. Work on this project at the Aspen Center for Physics was supported by NSF Grant PHY-1607611.

## Footnotes

↵

^{1}A.M.W. and T.M. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: awalczak{at}lpt.ens.fr or tmora{at}lps.ens.fr.

Author contributions: A.M., V.B., A.M.W., and T.M. designed research; A.M., V.B., A.M.W., and T.M. performed research; A.M., V.B., A.M.W., and T.M. contributed new reagents/analytic tools; A.M., V.B., A.M.W., and T.M. analyzed data; and A.M., V.B., A.M.W., and T.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812810116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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