Understanding Memory B Cell Selection
Stephen Lindsly, Maya Gupta, Cooper Stansbury, Indika Rajapakse
UUnderstanding Memory B Cell Selection
Stephen Lindsly , Maya Gupta , Alnawaz Rehemtulla , Indika Rajapakse , , ∗ Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor Google Research, Mountain View, CA Department of Hematology/Oncology, University of Michigan, Ann Arbor Department of Mathematics, University of Michigan, Ann Arbor ∗ To whom correspondence should be addressed; E-mail: [email protected].
Abstract
The mammalian adaptive immune system has evolved over millions of years to become an incrediblyeffective defense against foreign antigens. The adaptive immune system’s humoral response creates plasmaB cells and memory B cells, each with their own goals. The affinity maturation process is widely viewedas a heuristic to solve the global optimization problem of finding B cells with high affinity to the antigen.However, memory B cells appear to be purposely selected earlier in the affinity maturation process andhave lower affinity. We propose that this memory B cell selection process may be a heuristic solution totwo expected risk optimization problems: optimizing for affinity to similar antigens in the future despitemutations or other minor differences, and optimizing to warm-start future plasma B cell training. We usesimulations to provide evidence for our hypotheses, taking into account data showing that certain B cellmutations are more likely than others.
The immune system is an effective evolved learning system that deploys a number of classifiers and learningalgorithms. While the innate immmune system is adept at identifying foreign invaders, or antigens, it mustengage the adaptive immune system to create a more massive and specific response. A core aspect of theadaptive immune system’s humoral response is training two types of classifiers through a process calledaffinity maturation (AM). These classifiers consist of plasma B cells which generate antibodies to identify thecurrent antigen, and memory B cells which are used in subsequent immune responses to similar antigens inthe future. The AM process is highly unusual, in that a specific region of DNA within B cells undergoingAM is mutated to generate offspring which may have higher affinity to the antigen. The preservation of DNAsequences is usually of utmost importance in most cells, but the region of the genome which defines theshape of the B cell receptor must be modified for the B cell receptor to have a chance of becoming better atrecognizing the antigen of interest [1]. These mutations are responsible for the B cells’ incredible ability torecognize practically any antigen that they are presented, making the mammalian adaptive immune systemone of the most effective classifiers in the natural world.In this paper, we consider whether the plasma B cell and memory B cell generation processes can beinterpreted as trying to satisfy specific goals, and if so, can we state their goals precisely? We borrow standardtools from machine learning, where it is common to first specify a precise mathematical objective to beoptimized (such as minimize the expected number of errors the system will make), then propose heuristicalgorithms that approximately optimize that mathematical objective. Similarly, we hypothesize that due toevolutionary pressures, the AM processes heuristically optimize for immunological objectives. We hypothesizewhat those goals might be, then compare how well differently-trained B cells are at those goals when facedwith adversarially-mutated antigens via simulations. These findings lead us to propose new hypotheses aboutthe implicit goals of the immune system’s training of memory B cells.First in Section 2 we review how plasma B cells are trained, and present the standard hypothesis for thegoal of plasma B cell training in mathematics. Then in Section 3, we consider the more enigmatic question ofwhat the memory B cell training process is trying to mathematically optimize. We test our hypotheses viasimulations in Section 4, and conclude with a discussion of open questions in Section 5.1 a r X i v : . [ q - b i o . CB ] D ec Plasma B cell training
We review the AM process that trains plasma B cells, then consider what mathematical criteria the plasma Bcell training may have evolved to optimize.
AM begins by recruiting naive B cells with some initial affinity to the antigen to secondary lymphoid organs.These naive B cells, along with T follicular helper (T FH ) cells and follicular dendritic cells, concentrate intotemporary structures known as germinal centers (Fig. 1A) [1, 2]. Germinal centers (GCs) ensure that thesecells are in close proximity, and lead to rapid mutations of B cells. During AM, B cells are evaluated by T FH cells for their affinity to the antigen. If the initial affinity of a B cell is high, it receives a chemical signal fromthe T FH cell to move to a separate area of the GC and proliferate. While the B cell is proliferating, a veryspecific section of the genome called the hypervariable region is exposed to an enzyme, activation-inducedcytidine deaminase (AID) [3, 4]. AID is able to deaminate cytosine creating uracil, a nucleotide that is notnormally found in DNA. The operation that repairs these changes is error-prone, leading to mutations in theDNA sequence [5].The process of deamination and mutations during repair is referred to as somatic hypermutation (SHM).After proliferating, the B cells return to the area of the GC containing T FH cells and are reevaluated for theiraffinity towards the antigen. This iterative process of proliferation, mutation, and affinity evaluation continuesuntil the B cells have a sufficiently high affinity to the antigen. At this point, the B cells differentiate intoplasma B cells and begin to produce antibodies which allow for the immune system to eradicate the antigen. We model AM in Algorithm 1, which we will use to simulate AM in our experiments. We simplify a fewknown or uncertain issues about AM, detailed in Subsection 2.3.Each naive
B cell is generated randomly by a combinatorial mix of its V, D, J, and C gene segments,as well as through junctional diversity between these segments [6]. Naive B cells span at least 100 millionpossibilities [6]. The naive B cells recruited to a germinal center are ones that already have some promisingaffinity s to the antigen A . SHM then mutates the initial B cells, mostly by changes to a part of the DNAsequence roughly 1,500 base pairs (bp) long. Mutations in the hypervariable region are on the scale of times more likely than mutations outside of this region [5]. Mutations can be swaps, insertions and deletionsin a categorical space modeled as { A, T, C, G, ∅} , where ∅ connotes a deletion. Many of these randommutations are not feasible though; they occur, but are immediately killed. AM optimization is parallelized anddistributed over G germinal centers, which algorithmically can be thought of as G different parallel processors.We model the different germinal centers as working independently, though there may be biochemical signalingbetween them. In practice, a body trains for multiple unrelated antigens simultaneously, but for simplicity,we consider one antigen A .The g th germinal center randomly samples an epitope e g from the given antigen that is a signatureroughly 5-15 amino acids long. The epitope selection problem is analogous to a feature selection, and is stillregarded as mysterious: somehow the cells know how to select epitopes that are particularly immunogenic in that they tend to be good identifiers for foreign bodies. The fact that the G different germinal centerswork on optimizing B cells for different epitopes { e g } for g = 1 , . . . , G is analogous to machine learningensembles like random forests. Unlike machine learning ensembles however, the set of plasma B cells { b g } do not vote. Each plasma B cell can generate antibodies to identify the antigen, so they act more like acondensed nearest neighbor classifier [7]. Here we model the algorithm as T discrete iterations. In practice,AM is partly time-limited because the antigen will decay away. We note that our Algorithm 1 simplifies a few known issues about AM, but we believe these issues to beminor and not affect the major conclusions of this work.AM is a continuous-time process, but we model this as discrete time steps t . Before SHM begins, theinitial random B cells may have already been proliferating without supervision, which means the initial2igure 1: High-level illustration of the adaptive immune system. First, an antigen enters the body, then theinnate immune system identifies pieces of the antigen as non-self. (Top:) The adaptive immune system’sresponds to the identified antigen. A diverse set of random naive B cells that have some initial affinity to theantigen flock together and form a germinal center [1, 2]. These B cells proliferate and mutate when selectedby T FH cells for their affinity to the antigen. B cells with moderate affinity are stored for later use as memoryB cells. High affinity B cells differentiate into plasma B cells, which are the solution to a particular antigen.(Bottom:) A mutated version of a previously encountered antigen, or an antigen from a related pathogen, ispresented to the adaptive immune system. It responds by forming germinal centers with both random naiveB cells with some initial affinity to the antigen, and memory B cells from the first encounter.random sample may be better modeled as random clusters of B cells. Algorithm 1 allows the germinal centersto grow without bounds, though the die-off rate d will tend to keep it to a limited size. In practice, the sizeof germinal centers are bounded by physical volume constraints and biochemical resource constraints. We usea constant die-off probability d , but there is some evidence die-off rate decreases as affinity increases [8].T FH cells measure nearby B cells for their affinity, so there is only some probability that a specific Bcell will have its affinity measured, and that probability a B cell’s affinity gets measured is thought to bea function of spatial organization (which is indirectly affected by affinity) and direct affinity. The affinityof a particular B cell influences its likelihood to go to the dark zone and proliferate. As affinity of a B cellincreases, so does the likelihood that the T FH cell will send a proliferation signal. In addition, the strength ofthe proliferation signal is generally proportional to the affinity of the B cell such that a high affinity B cell ismore likely to proliferate many times in the dark zone before returning to the light zone for another affinitycheck than a moderate or low affinity B cell [9, 10]. Algorithm 1 may oversimplify AM in other ways as wellthat we are not aware of, or that are not yet known. AM is a process that has long been framed as the immune system acting as a global optimization algorithmtrying to find a B cell that best identifies a given antigen through SHM [11]. That is, AM acts as if it were aheuristic to solve, arg max b ∈B s ( b, A ) (1)where A is an antigen, b is a B cell from the set B of all possible B cells, and s is an affinity function thatmodels the quality of the lock-and-key physical and biochemical interaction of b and A . Note that B is an3mmense categorical space defined by the B cell’s possible variable-length DNA sequence which encodes its Bcell receptor.However, the objective (1) does not recognize the fact that a plasma B cell does not need to be a perfectmatch to the antigen. In fact, there appears to be an underlying sufficient affinity τ , and once that affinity isreached, the process is successful and the B cell becomes a plasma B cell. Further, the immune system isunder time pressure to produce such sufficiently high-affinity plasma B cells as fast as possible.Therefore, we propose a more sophisticated model of what the plasma B cell generation process isoptimizing should also depend on the sufficient affinity τ ∈ R , and the given set B of initial naive B cellsin the germinal center. To capture the time pressure, given a naive B cell b ∈ B , let M ( b ) ∈ M be arandom mutation that randomly changes the inital B cell b into a new B cell M ( b ) , where the set M isall biochemically possible such mutations. Let M K ( b ) = M ( M ( . . . ( M ( b ) . . . ))) denote the random B cellgenerated after K random mutations M , so that the random B cell M K ( b ) is distributed as some P b,M,K that has support on all the B cells reachable by K mutations M of the initial B cell b .Then we hypothesize the plasma B cell selection process is a heuristic evolved to minimize the expectednumber K of random B cell mutations M it will take for some B cell in the germinal center to reach sufficientaffinity τ to the antigen: arg min K ∈ N K (2)such that max b ∈B E M (cid:20) s ( M K ( b ) , A ) (cid:21) ≥ τ. (3)This objective is consistent with Algorithm 1.Clearly, the immune system is not a sentient entity that could explicitly formulate the criteria (2) andthen devise a heuristic to optimize it. Rather, our hypothesis is that evolutionary pressures have selected forplasma B cell generation mechanisms that better optimize (2). Similar to plasma B cells, memory B cells are created within the germinal center, but there are key differencesin the generation of these two cell types to achieve their respective objectives [10, 12, 13]. We first reviewhow memory B cells are created, and then consider what criteria they are optimized for, analogous to ourcriteria (2) for plasma B cells.
Memory B cells do not undergo the entire AM process like plasma B cells do. In fact, memory B cells arecharacterized by their relatively low affinity compared to plasma B cells, and low SHM load (number ofmutations gathered from SHM) [13]. This implies that while memory B cells have initially high affinity to anantigen relative to the naive B cell repertoire, they do not undergo AM to the extent of plasma B cells, whichwe hypothesize is to ensure that their specificity is not “overfit" to the present antigen, to use a machinelearning concept.The gene
BACH2 plays an important role in the development of memory B cells within the germinalcenter, and in their eventual differentiation [13, 14].
BACH2 has been found to be inversely correlated withthe help a B cell receives from T FH cells, and the resulting weak interactions with T FH cells allow for BACH2 expression to remain high. Critically, the relationship between
BACH2 expression and T FH cells allowsfor some help from T FH cells in order for cell survival within the germinal center, but prevents the B cellprecursor from entering the dark zone where it would proliferate and mutate via SHM. This leads to threesubsets of B cells within the germinal center: (1) high affinity B cells which are selected for by T FH cells toproliferate and mutate via SHM, and eventually lead to plasma B cell differentiation, (2) moderate affinity Bcells (low compared to plasma B cell precursors, high compared to the average naive B cell) whose selectionby T FH cells is tempered by BACH2 , leading to memory B cells, and (3) low affinity B cells which receivelittle or no help from T FH cells leading to apoptosis [15, 16].Memory B cells are very similar to naive B cells in terms of their transcriptional profiles, enabling themto circulate freely within the body and survey for future instances of antigens. Despite these similarities, they4xhibit overexpression of anti-apoptotic genes which allows for the memory B cell to live for extraordinarilylong periods of time and therefore the ability to recognize antigens in the future [13]. If the immune system’s goal was to simply memorize the best B cell to identify the exact antigen A in thefuture, then we would expect the memory B cells to simply be copies of the plasma B cells, but they are not.One might alternatively expect AM to take advantage of the luxury of time it has before it needs the memoryB cells to train them with more mutations to have even higher affinity s ( b, A ) to the antigen than plasma Bcells. This also does not appear to happen. While both those options should be biologically feasible, insteadthe immune system does something radically different to train its memory B cells. As described in Alg. 1,the immune system selects memory B cells earlier in AM than plasma B cells, and thus the memory B cellson average have lower affinity s ( b, A ) than plasma B cells (Fig. 1A).To explain why the memory B cells are so underfit to the current antigen A , we propose two hypothesesfor what the objective is that memory B cell training is heuristically trying to optimize. Our two hypothesesfollow from the dual role of memory B cells [17, 12]: when future incarnations of the antigen A attack, thememory B cells are used as-is to differentiate into plasma B cells and eradicate the antigen; and memory Bcells are used to warm-start AM’s secondary training of plasma B cells. In fact, recent evidence shows that alarge portion of the plasma B cells in the secondary response are memory B cells from the first response, andsome memory B cells are also used to seed the new germinal centers to optimize secondary response plasmaB cells [15]. We propose that the key issue for memory B cells is that the future incarnation of A they are designed for islikely a mutation of A . Thus we hypothesize that memory B cells are trained to maximize their expectedaffinity with respect to randomness in the antigen. That is, let ˜ A be a random antigen drawn from somedistribution P A whose support is centered on the given antigen A , but which models the likelihood of differentrandom mutations of A over the lifespan of the memory B cell. Then if only a single memory B cell wastrained, its training might aim to optimize: arg max b ∈B E ˜ A (cid:104) s ( b, ˜ A ) (cid:105) . (4)In fact, AM produces a set of N memory B cells. We hypothesize that AM is evolved to try to train adiverse set of N memory B cells such that their average affinity to a mutated antigen is high. That is, wepropose that the implicit goal is to find the set of N B cells that solve: arg max { b n ∈B} ,n =1 ,...,N N (cid:88) n =1 E ˜ A (cid:104) s ( b n , ˜ A ) (cid:105) . (5)The goal (5) does not explicitly require diversity of the memory B cells however. A variant criteria thatwould reward diversity would require at least one memory B cell to have high affinity to the mutated antigenin expectation: arg max { b n ∈B} ,n =1 ,...,N E ˜ A max n =1 ,...N (cid:104) s ( b n , ˜ A ) (cid:105) . (6)We believe (5) is easier to implement biologically than (6), as it requires less coordination between the differentmemory B cells being trained, so we focus on it in our simulations. Both criteria reward a larger number N of memory B cells, and it is an open question how the number of memory B cells N for a particular antigen isset; there is downward pressure on N due to the physical resources needed to store and maintain those cells. The second role of memory B cells is that they can warm-start future rounds of AM, and thus our secondhypothesis is that memory B cells are trained so that fewer mutations are likely to be needed to optimize afuture plasma B cell for high affinity to the mutated antigen (Fig. 1B).5his would suggest that the memory B cell selection process should be trying to choose a good initial setof B cells for the future. That is, the set of N memory B cells { b n ∈ B} should be chosen to minimize thenumber of random mutations K needed to achieve sufficiently high-affinity B cells M K ( b n ) in the secondaryresponse to meet some affinity threshold τ with respect to a randomly mutated antigen ˜ A on average: b ∗ = arg min { b n ∈B} ,n =1 ,...,N (cid:32) K such that N (cid:88) n =1 E M, ˜ A (cid:20) s ( M K ( b n ) , ˜ A ) (cid:21) ≥ τ (cid:33) . (7)Goal (5) and goal (7) will probably have different optimal solutions depending on the distribution ofmutations of the antigen and the B cells, though a heuristic selection process might be a good approximationfor both goals. It is not well known how important memory B cells are to the secondary response plasmaB cell training in germinal centers, yet some evidence shows that secondary response germinal centers arecomprised of more naive B cells than one might expect [15]. Both (5) and (7) appear to require knowledge of the likelihood P ˜ A of different mutations the antigen mayundergo, and the likelihood P M | b of different mutations of a B cell. However, we note that for many simplechoices of P ˜ A and P M | b , the exact distributions would not matter: the immune system could cheaply achievea good approximate solution to (5) and (7) by making the memory B cells copies of the plasma B cells. Forexample, suppose P ˜ A is a distribution centered on the initial antigen a , with support on up to K swaps ofthe antigen’s amino acids, and that all such swaps were equally likely, and that affinity is simply a Hammingsimilarity on the amino acids. If one could only choose one memory B cell to protect against a random ˜ A from that distribution, choosing it to be a plasma B cell that had the highest affinity to a would likely be agood choice in terms of Hamming similarity. It would be easy for the immune system to make the memory Bcells copies of the plasma B cells. That is, there does not appear to be any biochemical restriction preventingthis. However, the memory B cells do in fact appear to trained with a different goal than the plasma B cells.We present two hypotheses as to why.Our first hypothesis is that because one is training a set of memory B cells in (5), it pays to have morediversity in the memory B cells than one gets by copying the plasma B cells. Plasma B cells tend to be lessdiverse because they are trained to optimize (1), which even with multiple local minima, will limit theirdiversity. Memory B cells are more diverse than plasma B cells because they start with random naive B cellsand stop earlier in the maturation process. We believe this diversity is important to optimize even the averageaffinity to the mutated antigen as per (7) because the true affinity function s is a highly nonlinear function ofthe amino acid sequences of a B cell b and antigen A that arise from complex biochemical properties andphysical lock-and-key structures [18, 19, 20].Similarly, if all possible mutations of a B cell were equally likely, and one could only select one memory Bcell, choosing it to be a copy of a high-affinity plasma B cell would likely be a very good bet. Our secondhypothesis is that the warm-start goal (7) is not well-optimized by a copy of the plasma B cell set becausethere is evidence that the mutations of the B cells in SHM P M | b are asymmetric . That is, certain mutationsof B cells are much more likely than others. That makes some B cells a much more flexible starting point forwarm-starting the new plasma B cells. This is analogous in machine learning training to local minima thatare difficult to get out of. Evidence for asymmetric P M | b is that many researchers have noted AID preferentialtargeting of specific motifs [5, 21, 22, 23]. As mutations would, by definition, change the specific sequencethat AID targets, it is reasonable to infer that the first mutation of this location is easier than future ones.Once the sequence is changed, AID is less likely to target this location. Overall, the preferential targeting ofAID would make it harder for this region to mutate further or reverse back to the original sequence.This asymmetry in the probability of moving around the space of all B cells via mutations during AMcreates a disconnect between the plasma B cell goal (1) and the goal of being a good warm-start solutionto future plasma B cell training as per (7). Specifically, a plasma B cell can make many difficult-to-reversemutations to optimize (1) for the current antigen a . In contrast, in machine learning parlance, the memory Bcell is regularized such that it underfits the current antigen a , but can more easily mutate in a secondaryresponse AM into a new plasma B cell with high-affinity to the random future antigen ˜ A . Overall, we notethat how well the goals (1), (5) and (7) align depends on the symmetry of P ˜ A , P M | b , and the nonlinearity of s .6 lgorithm 1 Affinity Maturation Algorithm for Training Plasma and Memory B Cells
Require: an antigen a ∈ A Require: an affinity score s ( b, a ) → R for B cell b ∈ B and antigen a ∈ A Require: a low affinity threshold (cid:15) to enter a germinal center
Require: a high affinity threshold τ >> (cid:15) to become a plasma B cell
Require: a survival rate d ∈ [0 , for candidate B cells Require: probability p that a B cell is measured by a T FH cell Require: probability function q ( t, s ) of producing a memory B cell on iteration t given affinity score s thatis monotonically increasing in s , and might be monotonically decreasing or unimodal in t Require: a random number generator rand ( p ) that outputs 1 with probability p and 0 otherwise initialize the set of plasma B cells B ∗ = ∅ and the set of memory B cells V ∗ = ∅ for g = 1 , . . . , G germinal centers do sample epitope e g from antigen a sample an initial set of J g naive B cells B g such that s ( b, e g ) > (cid:15) for all b ∈ B g for t = 1 , . . . , T iterations do for b ∈ B tg do if rand(p) == 1 then the B cell b is observed by some nearby T FH cell which measures s ( b, e g ) if s ( b, e g ) ≥ τ then insert b into the set of plasma B cells B ∗ such that b ∈ B ∗ break end if if rand( q ( t, s ( b, e g )) ) == 1 then insert b into the set of memory B cells V ∗ such that b ∈ V ∗ break end if if rand( r ( s ( b, e g )) ) == 1 then proliferate: B cell b sent to the dark zone to divide and mutate some number of times, and itsmutated copies are added to the set B t +1 g break end if if rand(d) then b dies break end if end if b is added to the set B t +1 g end for end for end for return the set of plasma cells B ∗ and the set of memory B cells V ∗ We use the AM algorithm (Alg 1) to model how plasma B cells and memory B cells are trained, and showthrough two simulations that the simulated memory B cells are better than the simulated plasma B cells atoptimizing our hypothesized goals (5) and (7), thus providing evidence for these goals.These simulations do not account for all of the real-world issues at play, such as that a viral mutation mustnot harm the virus’s functionality, and the issues described in 2.3. Despite these limitations, we argue thesesimulations capture many of the key issues needed to illustrate that our hypothesized goals are consistentwith the difference in plasma and memory B cell training. Complete code for our simulations will be madeavailable upon request. 7 .1 Simulation Set-up
Our simulations follow Alg. 1 for the AM process. We initialize a naive B cell repertoire (10,000 cells) with Bcell receptors that are a random sequence of 10-50 amino acids, where each amino acid is drawn uniformlyover the space of 61 non-stop codons (creating a non-uniform distribution over the amino acids). We simulatethe antigen as a sequence of 17 amino acids derived from a known antigenic sequence of chicken ovalbumin[24]. We simulate the affinity metric s between a B cell and the antigen using the standard localalign MATLAB function, which finds the optimal alignment between two sequences using the
BLOSUM50 matrix[25] and returns a score reflecting how similar two sequences are in this alignment. We use this score as ameasure of affinity between the B cell receptor and the antigen for simplicity, but note that it is only a roughapproximation of the more complex structural compatibility between the B cell receptor and antigen.Mutations of the B cell during AM are modeled in the codon space, where codons of the B cell receptorare replaced with one of the 61 codon possibilities. While SHM mutates B cells on a single nucleotide level,working in the codon space prevents the added complication of filtering out nonsensical B cell receptorsequences. In addition to the replacement of codons, codons in B cell receptor sequences can be inserted ordeleted. Insertions and deletions are given equal likelihood of occurring as any of the 61 non-stop codons, butare inherently less likely than the replacement of an individual amino acid as the translation from codons toamino acids is not an injective function. The codons defining each B cell receptor are chosen to mutate atrandom, but we simulate codons that contain C cytosines to be C + 1 times more likely to be mutated thancodons without cytosine, reflecting biological biases to nucleotide sequence motifs [5, 21, 22, 23].For our initial simulation of a primary adaptive immune response to an antigen (primary response), werandomly select 50 naive B cells from the B cell repertoire from the top 1,000 of the 10,000 naive B cellrepetoire in terms of affinity to said antigen. This reflects the recruitment of naive B cells with some partialaffinity by T FH cells to germinal centers [10, 2]. These 50 ‘founder’ B cells are then duplicated 20 times toform a germinal center population of 1,000 cells, reflecting the growth period of germinal center formation[26]. From here, after undergoing random mutations as defined above, 50 random B cells from the bottomhalf (in terms of affinity) are replaced by 50 mutated B cells from the top half. These mutations have thepossibility to increase, decrease, or have no effect on the affinity of the B cell receptors. This process isreflective of low affinity B cells undergoing apoptosis from lack of T FH cell help and the proliferation of Bcells with high affinity after being selected by T FH cells, and this process repeats for 100 iterations.As a basis for both Simulations 1 and 2, we extract 50 cells during the first half of our primary responsesimulation as the simulated memory B cells. The cells are randomly chosen from the top 50 th percentile ofgerminal center B cells. Another 50 cells are selected at the very end of the primary response to representplasma B cells, where these cells exhibit the highest 50 affinity scores to the antigen. As expected, thesimulated memory B cells have overall lower affinity to the antigen compared to the plasma B cells, buthigher than the initial set of naive B cells.Our simulated mutations of the antigen for the secondary response are derived from a uniform randomswap of any of the codons for any other (including possibly itself, i.e. a no-op), which creates a non-uniformdistribution over the amino acids as some amino acids are coded for by multiple codons. Mutations of theantigen can also include insertions or deletions of codons, similar to how B cell receptor mutations are defined. A secondary infection could involve an antigen that has been mutated or an antigen that is similar froma related pathogen. We simulate the changes in the antigen from the primary to secondary infection bycausing adversarial mutations to our antigenic sequence as follows. The antigenic mutations will be a swapof any amino acid to any other, an insertion of any amino acid, or a deletion of an amino acid. First, wegenerate 1,000 uniformly random mutations of the antigen a ’s codons as described above. Of those candidatemutations, we keep the one that has the least average affinity to the set of plasma B cells from the primaryresponse, to reflect that a potentially dangerous secondary infection would likely be from a challengingmutation. We repeat this random process a total of K times to produce an antigen with K mutations. Wetake that adversarial antigen ˜ a to be a realization of P ˜ A .We then test the different B cell populations by the hypothesized goal of (5): Fig. 2 shows the averageaffinity between a mutated antigen ˜ a with K = 1 , . . . , adversarial mutations and the set of 50 plasma B8igure 2: Simulation results averaged over 100 independent runs. Plot shows average affinity of naive, plasma,and memory B cells to adversarially mutated antigens with K = 1 , . . . , mutations. Ranges of mutationswhere plasma, memory, and naive B cells are optimal are shaded blue, orange, and yellow respectively.cells, the set of 50 memory B cells, and the naive B cell repertoire of 10,000 B cells. Results were averagedover 100 independent runs of the entire simulation.Fig. 2 shows that plasma B cells are the best choice to maximize (5) for a small number of mutations,memory B cells are the best choice between 3-6 mutations, and naive B cells are best after many mutations(7+). Our simulations are too simplified for the specific transition points to be meaningful, but we argue theydo provide strong evidence that the plasma B cells are likely not optimal for identifying substantially-mutatedantigens, and that there is some regime of likely mutations where memory B cells are most useful.We hypothesize that the fact that plasma B cells do not always have the highest affinity to mutatedantigens is driven by the greater diversity of the memory B cells and naive B cells. While the mutationsoccur in DNA space, the relevant diversity is in the resulting nonlinear physical and biochemical space thatdefines the affinity to the antigen. Roughly modeling this using the pairwise BLOSUM similarities in each setshows substantial diversity differences, with the plasma B cells having average within-set BLOSUM similarityof 46.1, the memory B cells having much lower average within-set BLOSUM similarity of 26.9, and the naiveB cells having even lower within-set BLOSUM similarity of 9.4. To understand why diversity is important,note if the goal was for a set of B cells to have at least one B cell close to the mutated antigen, then onewould seek B cells that cover the space of where the mutated antigen might show up. Our actual hypothesisin (5) is that one hopes for high mean affinity over the set, which will not show as strong a preference fordiversity as the max affinity over the set as in (6), but still appears to reward greater diversity than theoverfit simulated plasma B cells produce. In this simulation, we mimic a secondary response training of a new set of plasma B cell’s optimized forhigh affinity to the mutated antigen ˜ a (described in Simulation 1). For this secondary response, we initializethe secondary response plasma B cell optimization with one of three choices: (i) the N = 50 plasma B cellsgenerated in the primary response for the original antigen a , or (ii) the N = 50 memory B cells generated inthe primary response for the original antigen a , or (iii) N = 50 naive B cells. We populate the germinalcenter in an identical way to the primary response, using the three sets of 50 B cells as our new foundercells. As in the first SHM round (primary response), the naive B cells are random, but chosen to have someinitial affinity to the now-mutated antigen to simulate recruitment to the germinal center. Each of thesethree secondary response germinal centers undergo AM in an identical fashion to the primary response.Results reported in Fig. 3 show that for just one or two antigen mutations, the plasma B cells have the9igure 3: Average affinity of the secondary response plasma B cells to the mutated antigen as a function ofSHM iterations when warm-started by a set of plasma, memory, or naive B cells (as marked), for K=1...10mutations, averaged over 100 independent runs.highest initial affinity to the mutated antigen and reach the primary response’s average plasma B cell affinityto the original antigen ( τ ) through the fewest number of SHM iterations. For 4 or more adversarial mutations,naive and memory B cells are better choices to warm-start the secondary response than the primary responseplasma B cells, reaching an affinity of τ with fewer iterations of B cell maturation.Surprisingly, our simulations did not show a regime where memory B cells were the best warm-starts. Wehypothesize this is due to the selection of naive B cells for the secondary germinal center being biased tohave some initial affinity to the mutated antigen (as described in the primary response simulation). That is,the naive B cells in Fig. 3 are selected for having some initial affinity to the mutated antigen, and so showgreater initial affinity to ˜ a than the primary response naive B cells in Fig. 2, which were selected for havingsome initial affinity to the original antigen a . In real secondary germinal centers, both naive and memory Bcells are recruited simultaneously in order to have the best chance of creating new plasma B cells, and ourresults using independent populations of naive and memory cells supports this biological phenomena. In fact,recent experimental evidence suggests that secondary response germinal centers are comprised of more naiveB cells than previously thought, consistent with the diminished value of memory B cells for this purposethat we see in this simulation. [15]. It is also possible that there is an antigen mutation regime for which thememory B cells are indeed more effective than naive B cells for warm-starting, but that our simulations werenot realistic enough to capture it.We also note that after a few mutations, the secondary response simulation takes more than 100 iterationson average to reach the affinity threshold τ as shown in Fig. 4. We hypothesize that this is due to theadversarial mutations of ˜ a creating a more difficult problem for the germinal center to solve. We hypothesized that the dual role of memory B cells can be captured by two objectives (5) and (7) thatminimize expected risks, and that these goals would not be as well-optimized by plasma B cells that aretrained for (1), due to their overfitting the original antigen. Our simulations, while limited, provide strongevidence that plasma B cells would not optimize (5) or (7) once the antigen underwent sufficient adversarialmutatations. We believe this suboptimality of plasma B cells against mutated antigens provides a role for thedifferent selection mechanism used for memory B cells.10igure 4: Time of convergence to τ for plasma, memory, and naive B cells during simulated secondarygerminal centers’ warm starts for K = 1 , . . . , mutations averaged over 100 independent runs. Here, lowervalues indicate a faster time to convergence, showing that the plasma B cells are the best warm-starts for 1, 2or 3 of the simulated adversarial mutations, and naive B cells are the best warm-starts for more mutations.There are missing bars for 7 and 8 mutations because the average simulated warm start of plasma B cells forthe 7 and 8 adversarial mutations and memory cells for the 7 adversarial mutations scenarios did not reach τ after 250 iterations.Our simulations show a limited range of antigen mutations over which our simulated memory B cells areoptimal; for substantial mutations, we show random naive B cells can work even better. These findings areconsistent with our knowledge of the adaptive immune system: Plasma B cells are a one-time solution andare highly overfit to the current antigen of interest. Memory B cells provide a more approximate solution tothe current antigen, which is kept within the body to recognize future antigens with similar characteristics. Ifa future antigen is so different from what has been previously encountered by the immune system that nomemory B cells are able to identify it, a new solution is formed from scratch using naive B cells.Memory B cells play two roles, both differentiating into plasma B cells and re-initiating germinal centers,but these roles may be played by distinct subpopulations [17, 12]. These distinct subpopulations of memoryB cells might have resulted from distinct AM processes, or changes in the AM process over the AM timespantime that we have not explicitly modeled in our Alg. 1 [27]. Thus our two memory B cell objectives in (5)and (7) may apply to independent memory B cell populations. Here we investigated a simplified model,where memory B cells were considered a unified group. However, we were not able to show via simulations aregime in which the memory B cells were clearly better than secondary naive B cells for re-initiating thesecondary germinal response. Our results suggest this re-initialization task might be a weaker or rarer roleof the memory B cells; recent experimental evidence similarly noted memory B cells were less successfulat becoming secondary plasma B cells than previously assumed [15]. However, even if memory B cells arenot always needed for the secondary response, it might be that in some cases they are very important forwarm-starting, which might still exert evolutionary pressure on their selection process.Our two memory B cell objectives in (5) and (7) are both stated in terms of average performance overthe set of memory B cells. We chose this representation for its robustness, but other set functions, such asthe median affinity over the set of memory B cells, could be better models. The true objective functions ofmemory B cells in nature are not known, but may be elucidated through the integration of computationalsimulations and biological experiments. Actively monitoring the affinity of B cells during affinity maturation,as well as detecting when and why GC B cells become memory B cells, may assist with the development of11ore accurate and complex models in the future.A key lesson that we believe applies more broadly to the practice of machine learning is the difficulty ofwarm-starting model training when the target distribution has changed. In a machine learning setting, assumewe have trained a model f for distribution P X,Y , but now wish to update f for a “mutated" distribution ˜ P X,Y with further training. Two analogies from this work stand out. First and foremost, the original classifiermay be overfit to P X,Y . If stochastic gradients are used to train, then much like the non-uniform randommutations of SHM, it may be difficult to find the stochastic gradients needed to escape from the local minimain the new loss landscape with P X,Y . Similar to our simulations showing more naive re-starts will convergemore quickly than pre-trained plasma B cells, completely random re-starts or underfit solutions may in factproduce better solutions faster when updating machine learned models for shifting data distributions, thoughlikely at the cost of greater churn [28].
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