Predicting states of internal tree nodes

The goal of the discrete trait analysis framework presented here is to characterize the distribution of predicted trait values along the tips of B cell lineage trees. Given an alignment of sequences inferred to descend from the same naive ancestor (i.e. the same clonal lineage), lineage tree topologies and branch lengths were estimated using maximum parsimony using dnapars v3.967 [26]. While other methods are available for inferring B cell lineage trees, maximum parsimony has been shown to be effective in some cases [27] and is faster than more complex maximum likelihood models designed for B cell lineages [4,28]. Importantly, the statistics presented here are not limited to tree topologies inferred through maximum parsimony; maximum parsimony is used here primarily for computational expediency.

Maximum parsimony is also used to infer the discrete character states (e.g. cell subtype, isotype, tissue) of internal nodes, given a tree in which each tip is associated with a given character state. Nodes with different states from their immediate ancestors are counted as state changes. More specifically, internal node states were reconstructed using the Sankoff dynamic programming maximum parsimony algorithm [29], which, given a weight matrix for each type of state change, determines the minimum number of state changes that must be made along the tree given the states at the tips. The backtrace step of this algorithm can be used to determine a set of most parsimonious internal node states. Often there are multiple such maximum parsimony sets. To represent state changes across ambiguous internal node sets, trajectories with equal parsimony were randomly chosen in the backtrace step of the Sankoff algorithm, beginning at the root of the tree and moving towards the tips. This process was performed 100 times for each tree, and the mean of each type of state change was reported.

Strictly bifurcating B cell lineage trees frequently have clusters of nodes separated by zero-length branches (soft polytomies), which represent a high degree of uncertainty in tree topology. This uncertainty in the order of bifurcating nodes can result in a potentially large number of uninformative state changes along the polytomy. Multiple steps were taken to minimize the effects of random polytomy resolution (S1 Appendix). Briefly, nodes within each polytomy were first re-ordered to minimize the number of state changes along the tree. To represent the uncertainty in the order of state changes, nodes within each polytomy were grouped together into separate subtrees according to their predicted state. These state-specific subtrees were then joined together in a balanced manner, ensuring that state changes could occur in any direction among the states contained within the polytomy (S1 Appendix).

Testing trait-phylogeny association

Analysis begins with a B-cell lineage tree topology with discrete character states (trait values) associated with each tip, and internal node states reconstructed through maximum parsimony. The goal of our discrete trait analysis framework is to determine how the distribution of discrete character states along the internal nodes of a tree differs from its expectation if traits are randomly distributed among the tips. The statistics introduced herein are shown graphically in Fig 1. More formally, if there are m possible discrete character states, and o_ij is the number of state changes from type i to type j, the three statistics investigated are defined as: (1) (2) (3)

The PS (parsimony score) statistic is the total number of state changes along a tree. The SC (switch count) statistic from i to j is the number of state changes from state i to j. The SP (switch proportion) statistic from i to j is the proportion of state changes from state i to j.

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Fig 1. Hypothetical phylogenetic trees used to illustrate tree/trait association statistics.

Trait values at internal nodes of the tree are inferred using maximum parsimony given the trait values at the tips, which are shown using different colors and shapes. Two example state changes are highlighted. (left) No association between tip-trait values and tree: Distribution of traits across this tree is indistinguishable from randomly distributed traits by any statistic used. (middle) Tip-trait values clustered in tree: Cells with the same trait value are more closely related to each other in the tree, which will yield significantly fewer switches than in the same tree with permuted tips, and therefore a significantly low PS statistic. (right) Asymmetric ancestor/descendant relationships among trait values: All switches in this tree are from A to B (SP = 1), significantly higher than expected from the same tree with permuted tips. This indicates an asymmetric relationship between these states along the tree.

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We calculate the significance of these three statistics using a permutation test. This is done by randomizing traits at the tips of the lineage tree, re-calculating each statistic on the permuted tree, and repeating for a specified number of replicates. For each replicate we calculate the difference between the statistic calculated on the observed tree and the same statistic calculated on the permuted tree. We refer to this difference as δ. If mean δ > 0 (hereafter mean δ is indicated by δ), this indicates the statistic is on average higher in observed trees than in permuted trees. For a one-tailed test, we calculate the p value that δ > 0 as the proportion of replicates in which δ ≤ 0. Similarly, we calculate the p value that δ < 0 as the proportion of replicates in which δ ≥ 0. For a two-tailed test, we calculate the p value that δ > 0 as the proportion of replicates in which δ < 0, plus half of the replicates in which δ = 0. We refer to the calculation of these p values for the statistics in Eqs 1–3 as the PS test, SC test, and SP test, respectively. The process of performing the SP test is shown graphically in S2 Fig. The overall process is the same for all three tests–they only differ in the statistic calculated.

These three statistical tests capture different aspects of how the distribution of characters observed along a lineage tree differs from random association between tree topology and trait values. The PS test determines the extent to which trait values are clustered together within the tree. A significantly low PS statistic (i.e. δ < 0, p < 0.05) indicates identical trait values are more closely clustered together within the tree than expected from random association between tree topology and trait values. By contrast, a significantly high PS statistic indicates identical trait values are less clustered together than expected by chance. Variations of this test were previously developed in [23] and applied in [30] to study spread of influenza.

While the PS test only determines a general association between trait values and tree topology, the SC and SP tests are both aimed at determining whether a particular trait value is more ancestral to another in the tree. A significantly high SC statistic (Eq 2) from state i to state j indicates a greater number of switches from state i to state j than expected from random association between tree topology and trait values. The SC test was used by [24] in the context of virus phylogeography, and by [6] to decompose the phylogenetic relationships among B cell subtypes within HIV infection. The SC test, however, is expected to have a high false negative rate because trees with randomized tip states often have more state changes events in general. Because of this, SC test δ values tend to be negative even when there is no polarity in the ancestor/descendant relationship among type i and j (Fig 1).

To normalize for differences in the total number of state changes between observed and permuted trees, we introduce the SP test. A significantly high SP statistic (Eq 3) from state i to state j indicates a greater proportion of switches from state i to state j than expected from a random distribution of trait values at the tips. In contrast to the SC test, a significant association between the tree and trait may exist, but only associations in which one trait is more often ancestral to the other than expected will give rise to a significantly high SP statistic (Fig 1C). Further, the denominator of the SP statistic can be altered to test other hypotheses. For instance, to test whether a greater proportion of state changes to state j come immediately from state i than expected by chance, one can restrict the analysis to consider only state changes towards j.

Accounting for uncertainty in tree topology

To account for uncertainty in tree topology, we bootstrap the multiple sequence alignment of each clone [31]. This is performed by randomly sampling multiple sequence alignment columns with replacement. Lineage tree topology and branch length estimation then proceeds as before. Test statistics are calculated for each bootstrap replicate tree, and then for a single permutation of the traits at that tree’s tips. Calculations for δ values proceed as before, but with each o_ij value indexed for each replicate. This procedure is very similar to that proposed in [32] and can in principle be extended to other sets of tree topologies, such the posterior distribution of tree topologies generated by MCMC sampling under Bayesian phylogenetic inference.

From trees to repertoires

B cell repertoire datasets often consist of hundreds or thousands of B cell lineage trees. Often hypotheses do not concern individual lineages, but instead the behavior of the collection of B cell lineages as a group. To characterize multiple B cell lineages, the observed and permuted summary statistics are summed across all lineages of a repertoire for each bootstrap replicate. Additionally, traits may be permuted among trees, which may increase statistical power and detect nonrandom association among trait types within trees.

Simulations

We tested the performance of the three proposed statistics using simulations based on B cell lineage trees estimated from an empirical dataset. Sequences were obtained from peripheral blood samples taken from one subject at ten time points, from eight days before to 28 days after influenza vaccination [33] (subject 420IV). Sequence preprocessing and clonal clustering are described in [4]. Sequences were down-sampled by 50%, and only clones with >10 unique sequences were retained. A total of 399 clones containing 11 to 370 (mean = 26.2) unique sequences remained. Tree topologies and branch lengths were estimated for each clone using dnapars v3.967 [26] via the R package alakazam v0.3.0 [34] and dowser v0.0.1. We simulated state changing down each tree using a Markov model parametrized by initial frequencies π for each state, relative rate parameters r_ij for each pair of possible states i and j, and r, the average rate of state changes per mutation per site. The mean value of this rate matrix was calculated as the sum of the diagonal elements weighted by their initial state frequencies. All values of the matrix were then divided by this mean and multiplied by r. This calibration was performed so that r*l state change events are expected to occur across a branch of length l mutations/site under the initial state frequencies. Because the frequency of each state can change along the tree (i.e. the Markov chain is not necessarily at equilibrium), r may not necessarily equal the average rate of state change events further down the tree. For each tree, the state at the germline node was randomly drawn based on each state’s π value. For each node after the germline, the rate matrix is multiplied by the node’s ancestral branch length and exponentiated to give the probability of each state at that node, given the state at its immediate ancestor node. The state was then randomly chosen based on these probabilities. This process begins at the germline node and continues down the tree until each tip node has a state. Because each tip corresponds to a sequence, this forms a dataset of sequences paired with simulated discrete characters. Internal node states are not included in the final simulated dataset used for analysis. Simulations were performed with two state models (A and B) that explored a large parameter space (π_a = 0.5, 1; r_ab = 1, 10; r = 10, 25, 50, 100, 1000), and four state models (A, B, C, D) that explored more complex patterns of state change at low overall rate (r = 10). Twenty simulation repetitions were performed for each parameter combination. Statistical tests were performed as described in Methods; however, to improve computational efficiency simulation analyses did not use bootstrapped multiple sequence alignments, and instead performed 100 permutations on a fixed maximum parsimony tree for each clone. Only clones with more than one state type were analyzed.

To benchmark the performance of the SP test under more extreme conditions, we performed similar simulations along large phylogenies with asymmetrical topologies. In these simulations, perfect ladder topologies were used, with each internal node bifurcating to at least one tip and at most one internal node. All branch lengths were set to a length of 0.001 mutations/site. Simulations using these tree topologies used two states and otherwise continued as detailed above.

Controlling the false positive rate using down-sampling

Initial analyses using simulations along large ladder phylogenies demonstrated that the SP test had a high false positive rate when the overall rate of switching was extremely low (S6 Fig). These problematic cases are identifiable as they result in trees with a high ratio of tips to predicted state changes (S7 Fig). We adjust for these cases by using a simple down-sampling algorithm. The goal of the down-sampling algorithm is to randomly draw a subset of the original data whose resulting phylogenetic tree cannot have a tip-to-state change ratio greater than a pre-specified value k. A tree containing m state types at its tips must have at least m -1 state changes along its edges. If a tree contains m state types and k*(m-1) tips, then the ratio of tips to state changes (which must be ≥ m-1) along its branches can be no larger than k. For the desired tip-to-state change ratio of k, the algorithm proceeds by randomly sampling one tip of each state type (m) from a tree, and then randomly sampling k*(m-1)–m remaining sequences without replacement. This produces a tree with m states, k*(m-1) tips, and a maximum tip-to-state change ratio of k. Tree building and summary statistic calculation then proceed as above. This down-sampling is repeated each repetition. For computational expediency, this process can be repeated on tree topologies that have been already estimated. Tips are simply sampled from the tree with the remaining topology and branch lengths intact. This down-sampling algorithm was implemented in dowser v0.1.0, and all analyses presented that use this algorithm use dowser v0.1.0, R v4.0.3 [35], and ape v5.5–3 (www.github.com/emmanuelparadis/ape, commit 4fd8482). All other analyses used dowser v0.0.1, R v3.6.1, and ape v0.5.3.

Empirical datasets

We demonstrated the utility of the proposed discrete trait framework by analyzing two empirical datasets. The first was aimed at understanding B cell differentiation during HIV infection, and consisted of BCR mRNA sequences taken from sorted populations of unswitched memory B cell (MBC), CD19^hi MBC, CD19^lo MBC, and germinal center B cells (GCBC) from three HIV viremic subjects (subject 1–3; [6]). Each dataset was subsampled to a maximum of 50,000 total sequences, and only clones with more than 10 sequences were retained. Unique sequences associated with more than one cell type were kept distinct. This resulted in 138, 203, and 166 multi-cell type clones with a mean of 49.9, 39.4, and 30.3 unique sequences per clone, for subjects 1–3 respectively. State changes across all lineages for each subject were calculated over 100 bootstrap replicates. To control the false positive rate of the SP test, all trees were down-sampled to a maximum tip-to-state change ratio of 20.

The second dataset was aimed at understanding isotype switching patterns in human children, and consists of BCR mRNA sequences obtained from peripheral blood samples taken from a human subject each year from age 1 to 3 years old [36]. Preprocessing, including grouping of sequences into clonal clusters, is detailed in S2 Appendix. Only clones with at least 4 unique sequences and more than one isotype were retained. Identical sequences associated with more than one isotype were kept distinct so each sequence was associated with one isotype. This resulted in 792 multi-istoype clones with a mean of 9.1 unique sequences each. State changes across all lineages were calculated over 100 bootstrap replicates. To control the false positive rate of the SP test, all trees were down-sampled to a maximum tip-to-state change ratio of 20.

Differentiating state change patterns with two states

We used simulations to assess the performance of our proposed tests. We model B cell migration and differentiation using a Markov model with two states, A and B, and empirically-derived lineage tree topologies (Methods). Briefly, the pattern of state changes along a tree was determined by the probability that the state at the root was A (π_a = 0.5, 1; π_b = 1 –π_a), the average rate of state change (r = 10, 25, 50, 100, 1000 changes/mutation/site), and the relative rate of change from A to B (r_ab = 1, 10; r_ba = 1/r_ab). These parameters represent a range of slow, fast, biased, and unbiased state change patterns along a B cell lineage. Each simulation resulted in a dataset of BCR sequences, each associated with a single trait value (A or B) resulting from the simulation process. The goal of our simulation analysis is to determine if the summary statistics provide useful information about the mode and tempo of trait evolution.

We ran 20 simulation repetitions for each parameter combination, and tested the significance of each of the proposed statistics to assess their statistical power. Our simulations are designed to generate trees whose tip-states are more clustered together than if the tips states are randomly distributed across the tree tips. Consistent with this expectation, 320/320 simulation repetitions in which r < 1000 (i.e. overall rate of state change < 1000 changes/mutation/site) showed a significantly low PS statistic regardless of other parameters (δ < 0; p < 0.05; S3 Fig). This confirms the PS test’s usability for detecting nonrandom association between tree topology and trait values. However, at r = 1000, only 3/80 repetitions showed a significantly low PS statistic (S3 Fig), indicating this relationship is difficult to detect at high rates of state change.

We used the same simulations to test whether the SC statistic was capable of detecting the direction of state changes in B cell repertoires. A total of 300 simulation repetitions were performed using parameters expected to give biased (directed) state changes; namely, with lineages always beginning in A (π_a = 1) and/or highly biased rates of state change from A to B (r_ab = 10). Surprisingly, only 3/300 of these simulations showed a significantly high SC from A to B (δ > 0; p < 0.05; S4 Fig). By contrast, 186/300 showed a significantly low SC from A to B (δ < 0; p < 0.05). This indicates that significantly high SC statistics are highly conservative, while significantly low SC statistics are primarily driven by overall phylogenetic association with a trait. This issue is likely exacerbated as dataset size grows, hence the SC test is likely still useful for single lineages [24,25] or for detecting very strong trends in large datasets [6]. However, given these results the SC test does not appear appropriate as a general solution for detecting biased migration and differentiation in B cell repertoire datasets.

We next tested whether biased state change patterns were detected by the SP test. To test this method’s false positive error rate, we first investigated simulations with totally unbiased state changes; namely, in which lineage trees were equally likely to begin at state A or B (π_a = 0.5) and had equal rates of state changes between A and B (r_ab = r_ba = 1). SP tests from A to B on these datasets resulted in a roughly uniform distribution of p values across all tested migration rates (δ > 0, p < 0.05 in 5/100; Fig 2A). This indicates that completely unbiased state changing is consistent with the null hypothesis of this test. Simulations in which lineages always had state A at the root (π_a = 1) and/or the relative rate of state change was higher from A to B (r_ab = 10) were expected to give high SP statistics. At low overall rates of state change (r = 10), 55/60 of these simulations had significantly high SP statistics from A to B (δ > 0; p < 0.05; Fig 2B–2D). As the rate of state change increased (r = 25, 50, 100, or 1000), this relationship diminished as the distribution of trait values became less distinguishable from random association (Fig 2B–2D). These results indicate that, under this two state Markov model framework, a significantly high SP statistic is associated with biased origination, biased rate of state change, or both depending on the overall rate.

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Fig 2. Distribution of SP test p values from A to B from two state simulation analyses in which state change between state A and B was determined by the probability of starting in A (π_a), relative rate of migrating from A to B (r_ab), and the average rate of state change (r).

To the left of each plot, possible starting states are circled, relative rates are shown by arrowhead size. (a) π_a = 0.5, r_ab = 1, fully unbiased state change, shows roughly uniform distribution of p values at all tested rates. (b) π_a = 0.5, r_ab = 10 shows low p values at rates < 50. (c) π_a = 1, r_ab = 1 shows low p values at low rates (10) but not at higher rates. (d) π_a = 1, and r_ab = 10 shows low p values at rates < 100. (e) π_A = 0.5, r_ab = 1 shows low p values at rate < 50 if 50% of A sequences are discarded. Compared to (a), this shows that p values are sensitive to biased sampling of sequences. Red lines show the cutoff of p value = 0.05.

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Finally, we used these simulated datasets to test whether the SP test is affected by biased data sampling, as this potential bias is important for some other phylogeographic methods of trait evolution e.g. [37]. We tested this by randomly discarding half of the sequences associated with A in simulations with totally unbiased state change (π_a = 0.5, r_ab = 1). Though SP tests from A to B on these datasets gave a uniform p value distribution when all sequences were included (Fig 2A), SP statistics from A to B became significantly high when half of A sequences were discarded (Fig 2E). This indicates that severely biased sampling may give a similar signature as biased origination or state change for the SP test (Fig 2B–2D). Biased sampling may be caused by a variety of experimental factors, and applications of these statistics to empirical datasets will need to carefully consider possible effects of biased data collection for each trait type.

Differentiating complex relationships among trait values

All the tests detailed above are extendable to data with more than two states; however, due to its superior performance in two state simulations, we will focus in the rest of this study on the SP test. The permutation step of the SP test usually permutes trait values within each tree separately (Methods). However, when more than two states are present it may be advantageous to randomize trait value assignments among trees rather than just within each tree. This changes the null hypothesis, which is now that the proportion of state changes observed is the same as that expected if trait values are randomly distributed among all trees. Deviations from this null hypothesis may be due not only to biased ancestor/descendant relationships within individual trees, but also co-occurrence of trait values within different trees. To demonstrate the difference between these two mechanisms, we performed simulations with four trait values: A, B, C, and D. To test the difference between simple association and biased ancestry, these simulations used unbiased state change between A and B, and unidirectional state change from C to D. Trees began with states A, B, or C in equal probability; state changes were allowed in both directions from A to B and unidirectionally from C to D. For each repetition, the rate of state change r = 10, and relative rates were equal among allowed state changes. Performing the SP test on these simulations while permuting among trees showed significantly higher SP statistics in both directions between A and B, and between C and D than expected (20/20 for each; Fig 3A). This indicates the SP test when permuting among trees detected the association between these trait values but not the directionality of C to D state changes. In contrast, the SP test when permuting only within trees correctly yielded a significantly high SP statistic from C to D in 20/20 simulations; further, no simulation yielded a significantly high SP statistic from D to C, indicating a low false positive rate (Fig 3B). No simulation using either permutation method showed a significantly high SP statistic between unassociated trait values (e.g. A and C), indicating a low false positive rate. These results indicate that permuting trait values within trees is a more effective means of detecting biased ancestor/descendant relationships, while permuting between trees is more appropriate for detecting associations among traits.

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Fig 3. Distribution of SP test p values from four state simulation analyses under multiple modes of evolution diagrammed to the left of each plot.

Twenty repetitions were performed in each scenario. In simulations, possible starting states are circled and possible state changes are shown with arrows. All allowed state changes occurred at the same relative rate and the total rate of state change (r) was 10 changes/mutation/site (see Fig 2). (a) Permuting trait values among trees reveals low p values for all state changes between A and B, and between C and D. (b) Permuting within each tree reveals low p values from C to D, but not between A and B. Both a and b imposed no constraints on the types of state changes allowed in the maximum parsimony algorithm. (c) Direct switching simulations result in low p values from A to D, but not from other states to D. (d) Sequential switching simulations result in low p values from B to D but not from other states to D. (e) Irreversible switching simulations result in low p values from B and C to D, but not from A. (f) Unconstrained switching simulations also result in low p values from B and C to D, but not from A. The strange results of e and f are likely artefacts of the constrained parsimony algorithm, which forbids reverse alphabetical state changes (e.g. D to C), used to count state changes in simulations c-d.

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Differentiating constrained modes of state change

In some instances, there are known constraints to the direction that state changes can occur, such as in isotype switching. Isotype-determining constant regions in humans are ordered as IgM/IgD, IgG3, IgG1, IgA1, IgG2, IgG4, IgE, IgA2. Human B cells begin with IgM/IgD, and because the mechanism of class switching is irreversible, these events can only occur sequentially in the order specified. For instance, IgA1 can switch to IgG4, but not to IgM or IgG1. This constraint may be naturally incorporated into the Sankoff parsimony algorithm [29] by making impossible isotype switches have an arbitrarily high weight. A frequent focus of isotype switching analysis is whether a particular isotype (e.g. IgE) arises from direct switching from IgM or from sequential switching from an intermediate isotype [9,10]. These types of hypotheses could be investigated using the SP test.

To determine if the SP test can usefully distinguish between types of constrained relationships among trait values, we simulated datasets to represent possible isotype switching patterns. As above, datasets contained four trait values: A, B, C, and D under different modes of evolution. Because questions often focus on the origin of a particular isotype [10] we only counted state changes leading to D when calculating SP statistics. Further, because state changes can only occur in a particular direction, we permute trait values among trees in these tests to increase power. While we previously showed that permuting among trees confuses biased association with biased ancestry (Fig 3A), switching between these states can only occur in one direction. Because of this, association between two states implies a direction of switching and among tree permutation is justifiable. We first simulate direct switching in which trees always had state A at the root and only state changes from A to the other states were allowed (Fig 3C). We expected these simulations to show a significantly high SP statistic only from A to D. Confirming this expectation, all 20 of these simulations had a high SP statistic from A to D (δ > 0, p < 0.05; Fig 3C). We next simulated sequential switching, where arriving at state D requires transitioning through B. All trees began in A and state changes were allowed from A to B and C, but D arose only from B. We expected these simulations to show a significantly high SP statistic only from B to D. All 20 of these simulations showed a significantly high SP statistic from B to D (δ > 0, p < 0.05; Fig 3D). These results demonstrate that the SP test using constrained parsimony can discriminate between simple hypotheses of isotype switch patterns, such as direct versus sequential switching.

We next investigated whether the SP test can distinguish between more complex types of constrained switching. We simulated irreversible isotype switching in which trees begin with state A, and only state changes moving alphabetically (A to D) were allowed. Naively, we may expect these simulations should show similar SP test results from A, B, and C to D. However, all 20 of these simulation repetitions showed a significantly high SP statistic to D from B and C, but not from A (Fig 3E). As a control, we simulated unconstrained switching in which trees begin randomly at any state and may change between all states. Using a constrained parsimony model, these simulations showed the same significantly high SP to D from B and C, but not from A (Fig 3F), indicating that this pattern is possibly an artifact of the constrained parsimony model. These results demonstrate that, while the SP test outlined here can distinguish between simple types of constrained state change, its relationship to more complex modes of constrained state change such as irreversible evolution are difficult to predict, and should be interpreted cautiously.

Controlling false positives in the case of slow rates of state change

Simulations based on empirical data have shown that the SP test can reliably detect biased state change along phylogenetic trees in many cases (Figs 2 and 3). However, because the null hypothesis of the SP test is that trait values are randomly distributed along tree tips, the SP test may have a high false positive rate in situations where the total number of predicted state changes along the tree is significantly different than expected from the null hypothesis. One such case is that of slow state change along a large tree. We confirmed this using 50 repetitions of two-state simulations using unbiased (π_a = 0.5, r_ab = 1) switching along a single ladder phylogeny of 1000 tips and a slow rate of state change r = 1 state change/mutation/site (Methods, S6 Fig). Though the switching process in these trees was unbiased, 42% of repetitions had a significantly high SP statistic from A to B, and 32% had a significantly high SP statistic from B to A, for a total false positive rate of 74%. This indicates that, without correction, the SP test is unreliable when the rate of state change is slow and the tree is large.

We investigated whether the issue of slow state change rates over large trees was a significant problem in empirical datasets by characterizing the ratio of tips-to-state transitions for each tree. Slow rates of state transition are expected to produce fewer state transitions along a tree. Simulations using ladder phylogenies confirm this prediction: simulations using a slow rate of state change (r = 1) had a mean tip-to-state change ratio of 326.8; however, when r = 10 this ratio dropped to a mean of 37.6, and when r = 100 this ratio was only 5.2 (S7 Fig). We next calculated the tip-to-state change ratio in three empirical datasets in lineages with at least 20 sequences. The first of these characterized cell type transitions in HIV infection, and had a mean tip-to-state change ratio of 33.4. The second empirical dataset focused on isotype switching in a food allergic patient, and had a mean tip-to-state change ratio of 11.1 (S7 Fig). Finally, we included a recently published dataset of B cells obtained from blood and thymus samples from myasthenia gravis patients [8], which had a mean tip-to-state change ratio of 35.4 (S7 Fig). These results show that rates of state change in empirical BCR datasets are characteristic of simulations in which r is between 10 and 100 state changes/mutation/site. The SP test performs well under these rates (Fig 2). Collectively, these results show that trees with rates of state change that are problematically slow for the SP test are identifiable in practice by calculating the ratio of tips to predicted state changes, and uncharacteristic of the BCR datasets analyzed in this study.

To further address the issue of poor performance during slow rates of state change, we developed a simple down-sampling algorithm to control the false positive rate when the rate of state change (r) is low (Methods). This algorithm randomly down-samples each clonal lineage such that its ratio of tips to state changes is less than or equal to a pre-specified threshold. To assess whether this controlled the false positive rate, we first performed simulations in which the state change process was completely unbiased. In these simulations, the proportion of repetitions with a significantly high SP test statistic from A to B indicated the rate of false positives from A to B. As expected, the false positive rate was high when the ratio of tips to state changes was high. At a tip-to-state change ratio of 500, 32% of repetitions simulated with r = 1 had a significantly high SP test statistic from A to B in repertoires with 100 trees (Fig 4A). When these same lineages were down-sampled to a maximum ratio of 10, however, the false positive rate from A to B was only 2%, less than expected given the alpha value of 0.05. To test if down-sampling to a low tip-to-state change ratio led to a significant reduction in power, we repeated these analyses with simulations in which the relative rate of switching from A to B (r_ab) was 10. In these simulations, repetitions with significantly high SP statistics from A to B represent true positives. We found that at all levels of down-sampling, simulations with biased (r_ab = 10) and slow rates of state change (r = 1 and 10) had true positive rates ranging from 82% to 100% in repetitions containing at least 20 lineages (Fig 4B). This demonstrated that down-sampling did not significantly lower power in these cases. Down-sampling to a ratio of 10–20 did have low power when performing analyses in faster rates of state change r = 100 (mean true positive = 6.0%) or when there was only one tree within a simulated repertoire (mean true positive = 14.7%). Down-sampling also did not increase the false positive rate of SP test from B to A in the simulations tested (S8 Fig). Repeating previous simulations (Fig 2) with down-sampling to a tip-to-state change ratio of 20 showed consistent results, with a reduction in power at higher rates of state change (S9 Fig). Collectively, these results suggest that ensuring all trees analyzed have a maximum tip-to-state change ratio of 10–20 adequately controls the false positive rate in the case of slowly changing discrete states without severely limiting power. This could be achieved by down-sampling lineages with a higher ratio of tips to state changes.

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Fig 4. Controlling false positive rates using down-sampling.

Y axis shows the proportion of 50 simulation replicates in which the SP test from A to B was significant. Simulations were performed on large ladder phylogenies (Methods) at varying rates, biases, and trees per repetition (representing lineages within a simulated repertoire). In each case, simulated lineages were down-sampled to the specified maximum tip-to-state change ratio. (a) Unbiased simulations, significant SP test values indicate false positives. These show that the SP test has a high rate of false positives when the rate of state change is low (i.e. 1 change/mutation/site) and the maximum tip-to-state change ratio is high (> 20). Down-sampling lineages to a maximum tip-to-state change ratio of 10–20, however, controls this false positive rate. (b) Simulations with rates biased from A to B, in which significant SP test values indicate true positives. These results show that down-sampling does not simply reduce power. However, power is lowered when the switching rate is high (>100) or the number of trees/repertoire is low (< 20).

https://doi.org/10.1371/journal.pcbi.1009885.g004

Differentiation of B cell subtypes during HIV infection

Over the course of the immune response, B cells differentiate into multiple cellular subsets with distinct properties. Recent studies have focused on the role of T-bet, a transcription factor usually associated with differentiation of T cells, in shaping B cell responses during infection. For example, [6] used data from three HIV+ patients to demonstrate that CD19^hi memory B cells (CD19^hi MBCs, a surrogate for T-bet+ B cells) represented earlier states in the affinity maturation process than germinal center B cells (GCBCs), and to define the relationships among other B cell subtypes including CD19^lo MBCs and un-switched MBCs. More specifically, [6] used the SC test with trait values permuted among trees. However, the simulation analyses performed here demonstrated the SC test is highly conservative, and that permuting among trees may only detect unstructured association among trait values (Fig 3). It is therefore not clear whether the relationship from CD19^hi MBCs to GCBCs observed in [6] was driven by biased ancestor/descendant relationships among these cell types within trees. Our results above suggests that the SP test using within tree permutation would be a more appropriate test of this relationship.

We characterized the relationships among B cell subsets with a two-tailed SP test using within tree permutation for each of the three subjects. These analyses showed a significantly high SP statistic from CD19^hi MBCs to GCBCs in all three subjects, and to CD19^lo MBCs in one subject (δ > 0, p < 0.025; Figs 5B–5D and S10). These analyses confirm the conclusions in [6] that CD19^hi MBCs are significantly closer, cladistically, to the predicted germline sequence than GCBC sequences than expected by chance. Naively, one may interpret this as evidence that GCBCs derive from CD19^hi MBCs. However, because GCBCs are expected to have far higher mutation rates than MBCs, the observed patterns are also consistent with early production of CD19^hi MBC from GCBCs, followed by a near cessation of mutations in CD19^hi MBCs. This is consistent with the conclusions of [6] that CD19^hi MBCs represent earlier stages in the GC reaction, rather than the direction of differentiation. Overall, these SP tests confirm that the previously observed relationships from CD19^hi MBCs and CD19^lo MBCs to GCBCs are driven by biased ancestor/descendant relationships within trees rather than simply association in the same trees, as may have been the case from the previously used SC tests with among tree permutations [6].

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Fig 5. Analysis of B cell subtypes in three HIV+ subjects.

(a) Example tree visualized using ggtree [38,39] showing observed relationship between CD19^hi MBCs and GCBCs. (b-d) Direction of significant SP test δ values for subjects 1 (b), 2 (c), and 3 (d). Arrows within each diagram show the direction of significantly high (blue) or significantly low (red) SP statistics between CD19^hi MBCs, CD19^lo MBCs, unswitched MBCs (Unsw), and GCBCs in each subject.

https://doi.org/10.1371/journal.pcbi.1009885.g005

Sequential isotype switching to IgE and IgG4

Antibody isotypes are a major determinant of function. Of particular interest is characterizing whether IgE antibodies, the primary antibody isotype associated with allergic response, arise directly from IgM switching, or through sequential switching from another downstream isotype [9,10]. Previous studies have shown evidence that IgE in mice and human adults arises from sequential switching primarily from IgG [9,10], though a recent study in 27 humans in the first three years of life found evidence of a greater association between IgA1 and IgE in children with food allergy and eczema [36]. Specifically, [36] showed a higher number of shared clones between IgE and IgA1 than between IgE and other isotypes in these subjects. A phylogenetic test of this relationship would confirm that IgE and IgA1 sequences show a direct ancestor-descendant relationship within these B cell trees rather than just being part of the same clone.

We applied our discrete trait framework to determine the origins of IgE in a single subject (id = 2442) from [36]. This subject was selected due to reported history eczema, food allergy, and B cell clones containing IgE and other isotypes [36]. Using an SP test in which only state changes leading to IgE were considered and trait values were permuted among trees, we found a significantly high SP statistic from IgA1 to IgE (Fig 6B). No other isotype showed a significantly high SP statistic to IgE. These results favor IgE arising from sequential switching through IgA1 over direct switching from IgM in this subject. Performing a similar test using only state changes leading to IgG4 revealed a significantly high SP statistic from IgG1, IgG2, and IgG3 to IgG4 (Fig 6B). This pattern is similar to irreversible switching within the IgG family (Fig 3E). As shown in simulation analyses, this test is not suited to infer relative rates of switching from different isotypes if all kinds of switches are considered. However, these results are most consistent with origin of IgG4 through sequential switching with other IgG isotypes rather than direct switching from IgM or sequential switching from IgA1. Overall, these results are consistent with the conclusions of [36] that IgE arose preferentially through sequential switching from IgA1 in this subject. Our results further suggest IgG4 arose preferentially through sequential switching from other IgG subtypes in this subject.

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Fig 6. Analysis of antibody isotypes from a single subject.

(a) Example trees visualized using ggtree [38,39] showing observed relationships between cells expressing BCRs with IgA1 and IgE, as well as IgG1 and IgG4 isotypes. IgE and IgG4 are indicated on each tree using larger tip circles. (b) Distribution of SP test δ values to IgE from each of the other isotypes (different colors), and distribution of SP test δ values to IgG4 from each of the other isotypes (different colors). State changes in b were calculated using constrained parsimony which forbids state changes that violate the geometry of the Ig heavy chain locus, and SP tests were performed using permutation among trees.

https://doi.org/10.1371/journal.pcbi.1009885.g006