Optimal evolutionary decision-making to store immune memory

Essential revisions:

1. As promised in the introduction, the authors could comment on how their approach could be applied to memory T cells in future work, even if the dynamics of such memory is different. (Reviewer 1)

We have now included a paragraph in the Discussion section highlighting the parallels between T cell and B cell memory generation (lines 436-452). Of course, as the reviewers have pointed out the mechanisms are distinct but conceptually one may be able to give a similar evolutionary rationale for understanding memory differentiation in these B-cells and T-cells.

From the text:

“Although mechanistically distinct from B-cells, T-cells also differentiate into effector and memory in response to infections. […] Taken together, there are parallels between differentiation of T-cells and B-cells to memory, and it will be interesting to investigate the advantages of storing cross-reactive (and plausibly low-affinity) T-cell memory as a strategy to counter evolving pathogens.”

2. Improve Figure 1 so it defines the model more comprehensively without the reader having to consult the Methods section. (Reviewer 1)

We have extended Figure 1 to indicate more details about the model in the schematic.

3. Cross-reactive Abs are observed to take long to evolve. But the authors results seem to suggest that much cross-reactivity is evolved early in the affinity maturation. The authors should clarify. (Reviewer 2)

It is true that BnAbs, which are known for their cross-reactivity, take long to evolve within individuals. However, it should be noted that BnAb’s breadth is often achieved by targeting highly vulnerable regions of the virus, including the CD4 binding site in HIV or the influenza stem. In a sense, BnAbs achieve their breadth by specifically targeting conserved regions of a virus that are shared among a broad panel of strains. This is of course not true for all BnAbs and some are broad in a true sense. Still, we argue that BnAbs are more of an exception than a rule when thinking about cross-reactive interactions. As we are now clarifying in the text, we consider cross-reactivity as a feature that makes naive (or pre-class switched) antibodies flexible to interact with different pathogenic targets. With maturation, a receptor’s affinity increases and it becomes more rigid and specific to its cognate antigen (Fern´andez-Quintero et al. (2020); Li et al. (2003); Mishra and Mariuzza (2018); Wedemayer et al. (1997a)).

We are now clarifying these points in the main text (Lines 111-117) and in the Methods (lines 545560) when discussing our model of receptors’ affinity and cross-reactivity.

4. How would the assumed lifetime of memory B cells affect your results? It appears you have assumed that memory B cells persist through the lifespan of the organism. A discussion of how your results would change if memory lasted for less time would be useful. (Reviewer 2)

As the reviewer has pointed out, memory B-cells can persist for a very long time (e.g. over 50 years after smallpox vaccination). However, long-term memory does not necessarily imply that individual cells have a long life span, but rather it can be due to the persistence of (renewing) clones specific to a given antigen. Indeed, the mechanisms to maintain a long-lived immune memory are not well understood. Notably, experiments in mice have shown that a memory B-cell’s half-life can exceed the lifespan of the organism (Jones et al. (2015)) but we still lack such estimates for memory B-cells in humans.

While we do not model time limits to memory, we effectively model only one dominant memory at a time. This process is the consequence of modeling the memory as only being beneficial until a novel immune response is triggered resulting in the storage of an updated memory centered around a more recent antigen. After such update, the old memory is no longer relevant as antigens have drifted away.

In our current setup, the characteristic time for a novel response (and memory update) is set by the expected antigenic divergence. Accordingly, cross-reactivity of memory is optimized so that the organism can mount effective responses against evolved forms of antigens in this window of time. However, if the lifetime of memory were to be shorter than this characteristic time of memory update, we expect the organism to store more specific memory since this memory would be utilized to counter a more limited antigenic evolution before it is lost. In other words, the shorter of either the memory lifetime or the characteristic time for memory updates determines the optimal cross-reactivity for immune memory.

We are now discussing these effects in the manuscript (lines 717-729).

Reviewer #2:

1. The role of B cell precursor frequencies in the decision-making process of mounting a memory versus naïve B cell response. The authors define an effective deliberation time for the naïve B cell population to reach a level of activity that is similar to the memory B cell population, based on the argument that memory B cells "can respond quicker and in larger numbers" than naïve B cells. However, one could imagine a scenario where the memory B cell response is quickly outcompeted and overshadowed by the naïve B cell response due to the high precursor frequencies of the naïve B cells.

We thank the reviewer for raising this interesting question. A series of experiments have investigated different aspects of this question (Blanchard-Rohner et al. (2009); Tangye and Hodgkin (2004); Tangye et al. (2003)). In brief:

Previous experiments indicate that memory cells are fast responders and initiate cell division about τ₀ ≈ 1−2 days before na¨ıve cells (Blanchard-Rohner et al. (2009); Tangye and Hodgkin (2004); Tangye et al. (2003)). Moreover, the number of memory cells that are recruited to proliferate and differentiate to effector cells is b ≈ 2−3 times larger than the number of na¨ıve cells (Tangye and Hodgkin (2004); Tangye et al. (2003)). Once recruited, however, memory and na¨ıve cells have approximately a similar doubling time of about t_1/2 ≈ 0.5 − 2 days (Macallan et al. (2005); Tangye et al. (2003)). Putting these together, we can define an effective deliberation time τ for the na¨ıve population to reach an activity level (i.e., a population size) comparable to the memory. Assuming an exponential growth during the early stages of memory and na¨ıve proliferation, the deliberation time can be estimated in terms of the kinetic factors by τ = τ₀ + t_1/2 lnb/ln2 and it is within a range of τ ≈ 1.5 − 5 days. Therefore, given the experimental evidence, if memory is available and is activated, it is unlikely that it would be outcompeted and overshadowed by the naive B cell response.

These arguments were previously only presented in the Methods section (lines 521-537). We have now included these arguments in the main text (Lines 137-142) to better justify our assumptions.

2. The impact of the paper could potentially be heightened if some discussion of how the principles gleaned on optimal immune memory strategies could be translated to, e.g., vaccine design against fast-evolving pathogens.

Thanks for bringing up this question. We have added a discussion on this matter to the manuscript (lines 388-406).

General comments:

3. Lines 60-61, It is stated that "as in most molecular interactions, immune pathogen recognization is cross-reactive". I am confused by this statement, as many molecular interactions are indeed not cross-reactive (e.g., lock-and-key binding of enzymes and ligands, etc.). Immune-pathogen recognition would also not typically appear to be cross-reactive unless the pathogen is highly mutable or there have been multiple infections of an evolved pathogen, so this sentence further confuses me. Please note that if the sentence is kept as is, I believe the authors meant to use "recognition" instead of "recognization".

We have removed the misleading part of the sentence.

4. As it appears to be defined, antigenic divergence characterizes two distinct infections by a given pathogen. How relevant is this model and its results for a pathogen like HIV that mutates within its host, where a range of antigenic distances/divergence values quickly become relevant for a single infection?

We expect the model to still hold for the intra-host evolution of pathogens like HIV. Indeed, Immune response to chronic HIV involves turnover of B-cell lineages over time (Nourmohammad et al. (2019)), which one may argue that resembles elicitation of novel responses to diverged antigens, against which prior memory is ineffective. It would be interesting to see how the constant pressure from HIV on the host’s immune system and the gradual depletion of helper T-cells would impact the efficacy of memory over short, intermediate, and long time scales.

We have included these discussions to the manuscript (Lines 426-435).

5. Line 142: in what kinds of scenarios (or against what types of pathogens), might prior preferences be important to consider? To clarify, all of the analyses carried out here assume no prior preferences?

The reviewer is correct that we use no prior preferences in our analyses, and we are now explicitly stating this fact in the manuscript (Lines 156-158).

The implications of this question are very interesting. We expect (in principle) that extreme choices of priors impact the outcome of our model. One can however turn this question around and ask whether such priors can emerge as optimal strategies in light of immune-pathogen coevolution. One possibility to imagine is whether there is an evolutionary advantage to optimize immune machinery (over evolutionary time scales) so that it generates repertoires that can respond more efficiently to commonly observed pathogens. Among other factors, the outcome of this optimization depends on the space of pathogens, the diversity of receptor repertoires, and the constraints imposed by self/ non-self discrimination (see e.g. Mayer et al. (2015, 2019)). It would be very interesting to study how such priors could naturally emerge in the context of immune-pathogen coevolution, but this topic is beyond the scope of this manuscript.

6. In regard to Equation 2, it would seem that the same maximum net utility value could be obtained with either a particularly high expected utility or a particularly low Kdiss. Would the optimal memory protocol look different in these two cases, despite them having the same net utility value? Perhaps this is already addressed, but the answer is not immediately clear to me.

The reviewer is correct that dissipation and the expected utility could in principle compensate each other and result in degenerate states with the same optimum net utility. If such degenerate solutions were to be available, we expect to see multiple optima for the net utility function in the specificity / deliberation parameter space (α^∗,β^∗). However, as shown in Figure 2, for each value of antigenic divergence, we only obtain a single global optimum and do not see any degeneracy in the solutions.

7. Lines 215-218, it is stated that "This optimization [for the case of moderately evolving pathogens] results in a smaller deliberation factor β compared to the scenario with slowly evolving pathogens, yet a long enough deliberation to allow the energetically suboptimal memory to react to an infection". This appears to be true for a range of δ about 0.1-0.5. Yet, still within in the defined range of δ for moderately evolving pathogens, δ values between about 0.6 and 0.8 for the lower amplitude cases appear to result in β values above those for slowly evolving pathogens. Can the authors please clarify this?

We have adjusted the regime to 20% − 60% to avoid any confusion.

8. Line 104, it is not clear what is meant by the statement that "Physico-chemical constraints in protein structures can introduce a tradeoff between immune receptors' affinity and cross-reactivity". Is this tradeoff not determined by the antigenic divergence that an immune system encounters upon a new infection compared to a past infection?

This tradeoff is rooted in the biophysics of protein (and specifically antibody) structures. There has been accumulating evidence that as B-cell receptors maturate, hypermutations strengthen the binding affinity of the corresponding antibodies and at the same time make their structures more rigid and specific to their cognate antigens (Fern´andez-Quintero et al. (2020); Frank (2002); Li et al. (2003); Mishra and Mariuzza (2018); Wedemayer et al. (1997b); Wu et al. (2017)). In contrast, naive (or pre-class switched) antibodies are more flexible and they can interact with different pathogenic targets (i.e., they are more cross-reactive). Taken together, these structural studies suggest that the physico-chemical constraints in protein structures of antibodies cause a tradeoff between affinity and cross-reactivity, at least for a broad class of such molecules. Of course, there are always exceptions to this rule and further studies are necessary to better understand the biophysical constraints on antibody structures.

These points were previously discussed in the Methods (lines 545-560) and we have now included these statement earlier in the text when the model is introduced (Lines 108-111).

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