We worked with TCRβ-immunosequencing data representing 666 individuals (Emerson et al., 2017). V(D)J recombination scenarios were assigned to each sequence from each individual using the IGoR software which is designed to learn unbiased V(D)J recombination statistics from immune sequence reads (Marcou et al., 2018). Using these V(D)J recombination statistics, IGoR output a list of potential recombination scenarios with their corresponding likelihoods for each TCRβ-chain sequence in the training data set. We annotated each sequence with a single V(D)J recombination scenario by sampling from these potential scenarios according to the posterior probability of each scenario (see Materials and methods for further details).

Annotated TCR sequences can be separated into two categories: ‘productive’ rearrangements which code for a complete, full-length protein and ‘non-productive’ rearrangements which do not. Non-productive sequences are generated when the V(D)J recombination process produces a sequence that is either out-of-frame or contains a stop codon. Each T cell contains two loci which can undergo the V(D)J recombination process. When the first recombination fails to generate a functional receptor (creating a non-productive sequence), followed by a successful rearrangement on the T cell’s second chromosome (a productive sequence), the non-productive rearrangement can be sequenced as part of the repertoire. Non-productive sequences do not generate proteins that undergo functional selection in the thymus, and their recombination statistics should reflect only the V(D)J recombination generation process (Robins et al., 2010; Murugan et al., 2012; Sethna et al., 2019). In contrast, the recombination statistics of productive sequences should reflect both V(D)J recombination generation and functional selection. Because we are interested in nucleotide trimming during the V(D)J recombination generation process, prior to selection, we only include non-productive sequences in our training data set. Further, because V-gene sequences within the TRB locus contain more sequence variation than D- and/or J-genes, we only include V-gene sequences in our training data set.

The extent of nucleotide trimming varies substantially from gene to gene (Nadel and Feeney, 1995; Nadel and Feeney, 1997; Jackson et al., 2004; Murugan et al., 2012). Previous work has identified an interesting impact of sequence features, such as sequence nucleotide context, on trimming probabilities using a PWM model (Murugan et al., 2012). To our knowledge, this is the only model that takes nucleotide sequence identity into account when predicting trimming probabilities. Specifically, this model leverages a ‘trimming motif’ containing 2 nucleotides 5’ of the trimming site and 4 nucleotides 3’ of the trimming site to predict the probability of trimming at a given site. It was designed and trained using sequencing data from just nine individuals (Murugan et al., 2012), and has surprisingly good model fit and predictive accuracy across many V-genes despite its simplicity. Using a different, and much larger, repertoire sequencing data set, we have trained this PWM model and replicated previous work (Figure 2—figure supplement 1). We will refer to this model as the 2×4 motif model. It is important to note that this PWM model is not the primary model described in Murugan et al., 2012, but again is the only one that relates nucleotide identity to trimming.

While the 2×4 motif model has good predictive accuracy and model fit (Murugan et al., 2012), it is limited by its assumption that the trimming mechanism relies solely on a sequence motif. Here, we have generalized this PWM model to a model that allows for arbitrary sequence features, and trained each new model using conditional logistic regression (see Materials and methods). With this set-up, we were able to evaluate the relative importance of new mechanistically interpretable features for predicting trimming probabilities. Specifically, we designed features to measure the effects of DNA-shape, length, and GC nucleotide content in both directions of the wider sequence on the probability of trimming at a given position in a gene sequence.

We parameterize each of these features as follows. An example of how an arbitrary V-gene sequence is transformed into features for modeling is shown in Figure 1. To parameterize DNA-shape, we used previously developed methods (Zhou et al., 2013; Chiu et al., 2016) to estimate various DNA-shape values (i.e. roll, twist, electrostatic potential, minor groove width, etc.) for each single-nucleotide position within a sequence window surrounding the trimming site. To parameterize length, we measure the sequence-independent distance from the end of the gene (i.e. the number of nucleotides from the 3’-end of the sequence) as an integer-valued variable. We parameterize GC nucleotide content using the raw counts of AT and GC nucleotides on both sides of the trimming site (the two-side base-count). By using raw nucleotide counts, this measure also serves to parameterize length. Because AT nucleotides have a greater potential for sequence-breathing compared to GC nucleotides within a sequence (Jose et al., 2009), these two-side base-count terms may be serving as a proxy for the capacity of a sequence to breathe. As such, because sequence-breathing potential is only relevant for nucleotides that are paired, we do not include the nucleotides within the 3’ single-stranded-overhang when counting 3’ AT and GC nucleotides (see Appendix 2).

With these features, we designed models containing various feature combinations (Figure 1B). Collectively, these models allow us to explore other possible sequence-level determinants of nucleotide trimming, in addition to the previously proposed (Murugan et al., 2012) “trimming motif” hypothesis. We trained each model using the V-gene training data set described above (see Materials and methods for further model training details), and evaluated performance using a suite of different held-out data groups (Figure 2). Specifically, to evaluate model fit, we computed the expected per-sequence conditional log loss of each model using the full V-gene training data set.

To evaluate model generalizability, we computed the expected per-sequence conditional log loss using the following held-out groups:

For each of these held-out group analyses, each model was re-trained using the full V-gene training data set with the held-out group-of-interest removed (see Materials and methods and Appendix 3 for further details) prior to computing the loss. A lower expected per-sequence conditional log loss indicated better model fit and/or model generalizability. Following this model evaluation, we validated a subset of the models by using the model coefficients from the previous TCRβ V-gene training run and computing the expected per-sequence conditional log loss of the model using several independent testing data sets (Figure 2).

In an effort to capture the complex underlying biochemistry of the deletion process, we trained models containing various combinations of sequence-level feature types (Figure 1B) and evaluated their ability to accurately predict V-gene trimming probabilities. With this approach, we found that a model containing parameterizations of local sequence context, length, and GC nucleotide content in both directions of the wider sequence (the 1×2 motif + two-side base-count beyond model) had the best model fit and generalizability across different data sets (Figure 3 and Figure 3—figure supplement 1). This model contains a 1×2 motif, including 1-nucleotide position 5’ of the trimming site and 2-nucleotide positions 3’ of the trimming site within the trimming window, and includes only bases beyond this trimming window in the AT and GC two-side base-count terms (Figure 1). Despite containing fewer total parameters than the original 2×4 motif model (Murugan et al., 2012) (12 parameters compared to 18 parameters), the 1×2 motif + two-side base-count beyond model had better predictive accuracy (Figure 4 and Figure 4—figure supplement 1).

We considered the significance of the inferred model coefficients using a Bonferroni-corrected significance threshold of 0.0033 (corrected for the total number of model coefficients). With this threshold, we found that many of the inferred model coefficients were significant and quantified mechanistic patterns. Each coefficient represents the change in log10 odds of trimming at a given site resulting from an increase in the feature value, given that all other features are held constant. Within the nucleotides immediately surrounding the trimming site, bases 5’ of the trimming site have a slightly stronger effect on the trimming probability than bases 3’ of the trimming site (Figure 4B). Specifically, 5’ of the trimming site, C nucleotides have a strong positive effect on the trimming probability (log10⁡coefficient=0.2388) whereas A and T nucleotides have a negative effect (log10⁡coefficientA=−0.108 and log10⁡coefficientT=−0.137). In contrast, immediately 3’ of the trimming site, G and T nucleotides have a positive effect on the trimming probability (log10⁡coefficientG=0.093 and log10⁡coefficientT=0.125) whereas C nucleotides have a negative effect (log10⁡coefficient=-0.174). These results suggest a different possible mechanistic pattern than previous motif-only models (Murugan et al., 2012; Figure 2—figure supplement 1B). Further, beyond the 1×2 motif sequence window, the count of GC nucleotides 5’ of the motif (within a 10-nucleotide window) has a strong positive effect on the trimming probability (log10⁡coefficient=0.164) (Figure 4C). The counts of both AT and GC nucleotides 3’ of the motif have a strong negative effect on the trimming probability (log10⁡coefficientAT=−0.123 and log10⁡coefficientGC=−0.126). Interestingly, the magnitude of these negative effects are very similar between AT and GC counts. This suggests that the raw number of nucleotides 3’ of the motif (e.g. the length) is more important for predicting the trimming probability at a given site compared to the identity of the nucleotides. p-values for each of these coefficients were reported to be smaller than machine tolerance (2.23×10−308). We noted minimal variation in the magnitude of each inferred coefficient even when changing the number of sequences included in the training data set (Figure 4—figure supplement 7).

Because we were interested in parameterizing sequence-breathing effects using the two-side base-count terms, we only included nucleotides that are considered to be double-stranded after hairpin-opening within each count. In our modeling, we assume that the DNA hairpin is opened at the +2 position, leading to a 4-nucleotide-long 3’-single-stranded-overhang (the 2 nucleotides furthest 3’ are considered P-nucleotides) (Ma et al., 2002; Lu et al., 2007). As such, the first 2 nucleotides of the gene sequence can be considered single-stranded, and we do not include them in the two-side base-count terms. When we train a model that ignores this distinction, and include all gene sequence nucleotides in the two-side base-count terms, we note very similar inferred coefficients and model fit (Figure 4—figure supplement 2). We acknowledge that other hairpin-opening positions may be possible. To explore whether the +2-hairpin-opening-position assumption could be affecting our inferences, we trained the 1×2 motif + two-side base-count beyond model with other possible hairpin-opening-position assumptions and noted minimal variation in model fit (Figure 4—figure supplement 3).

We also evaluated the predictive accuracy of motif + two-side base-count beyond models containing different ‘trimming motif’ sizes. We find that models containing a small motif (e.g. a 1×2 motif) achieve similar predictive accuracy and are more generalizable compared to models containing a larger motif (Figure 4—figure supplement 4).

Because the trimming mechanism is thought to be consistent across V-, D-, and J-genes from both productive and non-productive sequences, we were also interested in whether the inferred coefficients for the 1×2 motif + two-side base-count beyond model would be consistent between the model trained using the non-productive V-gene training data set, a model trained using a non-productive J-gene data set, and a model trained using a productive V-gene data set. As such, we trained a new 1×2 motif + two-side base-count beyond model using only non-productive J-gene sequences and a separate, new 1×2 motif + two-side base-count beyond model using only productive V-gene sequences (both sequence sets were from the same cohort of individuals as the V-gene training data set). We found that the inferred coefficients were highly similar between the three models (Figure 4—figure supplement 5 and Figure 4—figure supplement 6).

When evaluating models containing only a single feature type, we find that the two-side base-count model which parameterizes GC nucleotide content on both sides of the trimming site (and, indirectly, length) has the best model fit and generalizability across all held-out groups tested (Figure 3). As such, these GC-content features, which are likely parameterizing the capacity for the sequence to breathe, are more predictive of V-gene trimming probabilities than local sequence context or DNA-shape alone. This finding supports previous observations that Artemis may act as a structure-specific nuclease as opposed to a nuclease that binds specific DNA sequences (Ma et al., 2005; Chang et al., 2015; Chang and Lieber, 2016; Yosaatmadja et al., 2021).

A persistent concern with the 1×2 motif + two-side base-count beyond model was that the motif coefficients could be driven by certain genes, instead of representing an actual gene-segment-wide signal. When comparing the inferred trimming profiles from the two-side base-count model to those from the 1×2 motif + two-side base-count beyond model, we identified a group of V-genes which had drastically lower prediction error when the 1×2 motif terms were included. These V-genes had a difference in per-gene root mean squared error between the two models that was greater than –0.13 (Figure 5A). The genes included in this group were TRBV5-3, TRBV7-3*01, TRBV7-3*04, TRBV7-4, TRBV9, TRBV11, and TRBV13. To evaluate whether these genes could be driving the observed motif signal, we explored whether the prediction error for these genes changed when they were removed from the model training data set.

In fact, we found that the inferred trimming profiles for these genes still had very low prediction error despite the genes not being included in the model training data set (Figure 5B and C), showing the generalizability of these features. The inferred model coefficients from this 1×2 motif + two-side base-count beyond model fit using the subsetted training data set were highly similar to those from the original model fit using the full training data set. Because genes which are highly similar sequence-wise to the group of held-out genes could still be present in the training data set and be driving these similarities, we defined a new data set that excluded this larger group of genes. When we repeated the same experiment with this new, more-restricted training data set, we observed similar results (Figure 5B and C). As such, both of these experiments provided evidence that the motif signal may actually represent a gene-segment-wide sequence motif that appears to get preferentially trimmed, independent of GC-content-related effects.

Using a subset of the V-gene training data set used here, we previously identified a set of single nucleotide polymorphisms (SNPs) within the gene encoding the Artemis protein that are associated with increasing the extent of V- and J-gene trimming (Russell et al., 2022b). This result suggested that trimming profiles may subtly vary in the context of these SNPs. As such, we were interested in whether these SNPs could be mediating (or serving as a proxy for) a change in the trimming mechanism. To explore this, we worked with paired SNP array and TCRβ-immunosequencing data representing 611 of the original 666 individuals in the V-gene training data set used here. Our previous work Russell et al., 2022b used data from only 398 of these individuals, however, the conclusions of that paper held when using this expanded group of 611 individuals in the analysis. With these data, we asked whether the inferred coefficients from the V-gene-specific 1×2 motif + two-side base-count beyond model varied significantly in the context of the non-coding Artemis-locus SNP (rs41298872) that was found to be most strongly associated with increasing the extent of V-gene trimming in our previous work (Russell et al., 2022b). As such, we re-defined the model to include an interaction coefficient between the SNP genotype and each model parameter (see Materials and methods). We then used a Bonferroni-corrected significance threshold of 0.0033 (corrected for the total number of interaction coefficients) to evaluate the significance of each interaction coefficient. For each significant interaction coefficient, we concluded that the corresponding model coefficient varied significantly in the context of the SNP genotype.

Using these methods, we found that several of the 1×2 motif + two-side base-count beyond model coefficients varied significantly in the context of the Artemis-locus SNP rs41298872 (Figure 6). Specifically, we found that 3’ of the trimming site, the negative effect of A nucleotides on the trimming odds varied in the context of the SNP for the position immediately 3’ of the trimming site (log10⁡interaction coefficient=0.006, p=0.0006) and one position away (log10⁡interaction coefficient=0.007, p=0.0006). Further, we found that the negative effect of the count of AT nucleotides 3’ of the motif varied strongly in the context of the SNP (log10⁡interaction coefficient=0.010, p=1.47×10−12). No other motif or two-side base-count coefficients were found to significantly vary.

Because the 3’-side base-count-beyond terms parameterize both GC nucleotide content and length in their definition, we were interested in whether the significance of the 3’-AT-nucleotide count SNP variation effect was related to GC nucleotide content, length, or both. To do this, we re-defined the 3’-side base-count-beyond parameters to be a proportion instead of raw AT/GC nucleotide counts and included an additional length term in the model to remove length-related effects from the inferred 3’-side base-count-beyond coefficients. Using this new model, we repeated the analysis and found that the length coefficient varied significantly in the context of the SNP (log10⁡interaction coefficient=0.005, p=6.24×10−23), but the 3’-AT-nucleotide-proportion term did not (Figure 6—figure supplement 1). This result is fully consistent with the fact that the Artemis-locus SNP is known to be associated with increasing the extent of trimming (a proxy for length).

To validate our previously trained models, we worked with TCRα- and TCRβ-immunosequencing data representing 150 individuals, TCRγ-immunosequencing data representing 23 individuals, and IGH-immunosequencing data representing 9 individuals from three independent validation cohorts. Before analyzing these data, we ‘froze’ our trained model coefficients in git commit 093610a on our repository. In contrast to the training data cohort, these validation cohorts contain different demographics and were each processed using different sequence annotation methods (see Materials and methods). To explore the potential effects of using a different sequence annotation method, we re-annotated the TCRβ training data set using the same annotation method as the TCRα-β testing data and found that it had little to no effect on the model fit or performance (Figure 7—figure supplement 1).

To evaluate the performance of the 1×2 motif + two-side base-count beyond model using these testing data, we used the model coefficients from the previous TCRβ V-gene training run and computed the expected per-sequence conditional log loss of the model using each testing data set (TCRβ V-gene sequences, TCRα V-gene sequences, TCRγ V-gene sequences, IGH V-gene sequences, TCRβ J-gene sequences, etc.). We found that the model has high predictive accuracy (i.e. low expected per-sequence conditional log loss) for both non-productive V- and J-gene sequences from the TCRβ testing data set (Figure 7). The model also extends well to non-productive V- and J-gene sequences from the TCRα and TCRγ testing data sets and to non-productive V-gene sequences from the IGH testing data set. The model has relatively poor predictive accuracy for non-productive IGH J-gene sequences, however. We noted very similar results when validating model performance using productive V- and J-gene sequences from each testing data set (Figure 7—figure supplement 2).

We hypothesized that the weight of the 1×2 motif and two-side base-count beyond model terms may vary across each testing data set. To explore this for each data set, we again used the model coefficients from the previous TCRβ V-gene training run and trained a new two-parameter model containing one coefficient scaling the 1×2 motif terms and a second coefficient scaling the two-side base-count beyond terms (see Materials and methods). With this approach, we found that the two-side base-count beyond terms were dominant compared to the 1×2 motif terms for every data set (Figure 7—figure supplement 3A). The scale coefficient for the 1×2 motif terms was very small for several of the data sets, especially the IGH data set, indicating only a weak motif-related signal. The sequence motifs that lead to a large increase in trimming probabilities in the model appear at relatively low frequencies within the germline IGH genes (Figure 7—figure supplement 4), perhaps explaining the weakness of the motif-related signal. When evaluating the expected per-sequence conditional log loss of these partially re-trained models, we note a small improvement in model fit for each re-trained model compared to the original model (Figure 7—figure supplement 3B).

TCRβ repertoire sequence data for 666 healthy bone marrow donor subjects was downloaded from the Adaptive Biotechnologies immuneACCESS database using the link provided in the original publication (Emerson et al., 2017). V(D)J recombination scenarios were assigned to each sequence for each individual using the IGoR software (version 1.4.0) (Marcou et al., 2018) as follows. The IGoR software can learn unbiased V(D)J recombination statistics from immune sequence reads. Using these statistics, IGoR can output a list of potential recombination scenarios with their corresponding likelihoods for each sequence. As such, using the default IGoR V(D)J recombination statistics, the 10 highest probability V(D)J recombination scenarios were inferred for each TCRβ-chain sequence in the training data set (Marcou et al., 2018). We then annotated each TCRβ-chain sequence with a single V(D)J recombination scenario by sampling from these 10 scenarios according to the posterior probability of each scenario. We filtered these sequences for rearrangements which contained more than 1 trimmed nucleotide and less than 15 trimmed nucleotides (see the ‘Notation’ section for further details). We further subset the data to include only non-productive sequences, and used these data for all subsequent model training. After these processing and filtering steps, we used V-gene trimming length distributions from 21,193,153 non-productive sequences for all model training. To test each trained model, we used V-gene trimming length distributions from the remaining 107,121,841 productive sequences (as described in Appendix 3). From this same data set, we also used J-gene trimming length distributions from 107,255,406 productive sequences and 20,204,801 non-productive sequences to test each model.

Genome-wide SNP array data corresponding to 611 of the training data set individuals was downloaded from The database of Genotypes and Phenotypes (accession number: phs001918). Details of the SNP array data set, genotype imputation, and quality control have been described previously (Martin et al., 2020). We only used SNP data corresponding to the Artemis locus (rs41298872) which we previously found to be strongly associated with increasing the extent of V-gene trimming (Russell et al., 2022b).

Let I be a set of individuals. For each subject i∈I, assume we have a TCR repertoire consisting of sequences indexed by k such that k=1,…,Ki. We assume that each sequence can be unambiguously annotated with being from a specific V-gene and J-gene sequence, and having a number of deleted nucleotides from each gene. For modeling purposes, we combine TRB V-gene or J-gene alleles that have identical terminal nucleotide sequences (last 24 nucleotides of each sequence) into TRB V-gene allele groups and TRB J-gene allele groups. As such, each TCR sequence is annotated with being from a V-gene allele group and J-gene allele group. Because we are requiring that each gene allele group originates from the same TRB V-gene or J-gene, there may still be overlap in terms of sequence identity between allele groups. For simplicity, we orient all sequences in the 5’-to-3’ direction, and use the top strand for V-gene sequences and the bottom strand for J-gene sequences. We will be introducing modeling methods as they relate to V-genes and V-gene trimming, however, with this sequence orientation, the same methods can be applied to J-genes and J-gene trimming. We will use σ to represent a gene sequence oriented in the 5’-to-3’ direction and n to represent the number of nucleotides deleted from the 3’ end of this sequence as we describe our modeling.

We are interested in modeling the probability of trimming n nucleotides from a given gene sequence σ, P(n∣σ). We can define an empirical conditional probability density function to estimate this probability. To start, we can uniformly sample from any given individual’s repertoire. Let S be a random variable that represents the gene-allele-group sequence from such a sample. Let N be a random variable that represents the number of deleted nucleotides, which for notational convenience we assume take on a non-negative integer value (nonsensical values will have probability zero). Let 0≤C(i)⁢(σ)≤Ki represent the number of TCRs that use gene allele group σ. Let 0≤C(i)⁢(n,σ)≤Ki represent the number of TCRs that have gene allele group σ and n gene nucleotides deleted. With these data, we can form the empirical conditional probability density function:(1)Pemp(N=n∣S=σ,i)=C(i)(n,σ)C(i)(σ).

Using these TCRβ repertoire data, we want to model the influence of various sequence-level parameters on P(n∣σ) . With this assumption, let L and U be lower and upper bounds, respectively, on n such that N′={L,…,U} is the set of all reasonable nucleotide deletion amounts. The precise location of hairpin opening and its relationship to deletion is unclear. Hence, we have chosen to define L=2 since smaller trimming amounts may result from an alternative, hairpin-opening-position-related (or other) trimming mechanism. Likewise, we have chosen to define U=14 since trimming amounts greater than 14 nucleotides are uncommon and could also result from an alternative trimming mechanism. We will subset the training data set, after IGoR annotation (see details in a previous section), such that we will only consider TCRs that have 2≤n≤14. Similarly, the one existing analysis (Murugan et al., 2012) exploring the relationship between sequence context and nucleotide trimming only considered TCRs that had 2≤n≤12 for their modeling. We summarize all of the notation discussed in this section, as well as in the following sections, in Appendix 1—table 1.

For our model, we make the following assumptions about V(D)J recombination biology:

With these assumptions, we can resolve the nucleotide sequence on both sides of the trimming site and define mechanistically interpretable model features using these two sequences. Specifically, we define a ‘trimming motif’ consisting of several nucleotides on either side of the trimming site, the predicted ‘DNA-shape’ of the nucleotides and bonds in close proximity to the trimming site, the counts of GC or AT nucleotides on either side of the trimming site beyond the ‘trimming motif’ region (e.g. the ‘two-side base-count beyond’), and the sequence-independent ‘length’ from the end of the gene to the trimming site (see Appendix 2 for further details). An example of how an arbitrary V-gene sequence is transformed into features for modeling is shown in Figure 1. We will assume that observations can be drawn from a model in which these features vary across trimming lengths n for a given gene allele group σ. We can then explore the influence of these features on the probability of trimming at a certain site given a gene sequence.

With the features summarized above, we can define a model covariate function f than contains any unique combination of parameter-specific covariate functions (Table 1). This function f will be the sum of each of the desired parameter-specific covariate functions. This framework allows us to generalize the existing PWM model (Murugan et al., 2012) to a model that allows for arbitrary sequence features. For example, we replicate this PWM model using the model covariate function, f1(n,σ;βmtif,a=2,b=4), where n represents the number of trimmed nucleotides, σ represents the gene-allele-group sequence, βmtif represents motif-specific parameter coefficients, and a and b are non-negative integer values that represent the number of nucleotides 5’ and 3’ of the trimming site, respectively, that are included in the ‘trimming motif’. This function is described further in (Equation 14). To extend this model to a model containing motif parameters and base-count-beyond parameters, the model covariate function will be(2)f(n,σ;βmotif,βAT,βGC,a,b,c):=f1(n,σ;βmotif,a,b)+f2(n,σ;βAT,βGC,a,b,c)

where f2 represents the base-count-beyond model covariate function (Equation 17), βAT and βGC represent base-count-beyond-specific parameter coefficients, and c represents the number of nucleotides 5’ of the trimming site to be included in the base-count. We will use this motif and base-count-beyond model example to discuss the model formulation in the following sections, however, many other parameter combinations are possible. We will not define a model covariate function that contains two parameters that model the same feature. For example, length and base-count-beyond coefficients will never be included in a model covariate function together (since they both parameterize length). Likewise, motif and DNA-shape coefficients will never both be included in a model covariate function.

We will be using the motif and base-count-beyond parameters given by (Equation 2) as examples for the remainder of this section, however, we could also formulate a model with any other parameter of interest, as described in the previous section (Table 1). As such, we can fit a conditional logit model which posits that(3)P(n∣σ;βmotif,βAT,βGC,a,b,c):=exp⁡(f(n,σ;βmotif,βAT,βGC,a,b,c))∑n′∈N′exp⁡(f(n′,σ;βmotif,βAT,βGC,a,b,c))

where N′ is the set of all reasonable trimming lengths, a and b represent the number of nucleotides 5’ and 3’ of the trimming site to be included in the ‘trimming motif,’ respectively, c represents the number of nucleotides 5’ of the trimming site to be included in the base-count parameters, and f(n,σ;βmotif,βAT,βGC,a,b,c) is the model covariate function for the motif and base-count-beyond model given by (Equation 2). We will let P(n∣σ;βmotif,βAT,βGC,a,b,c) denote the conditional probability that a given gene will be trimmed by n nucleotides.

Let yi⁢k⁢σ⁢n equal 1 if a gene allele group σ is trimmed by n nucleotides for TCR k from subject i, and equal 0 otherwise. With this, we can define a likelihood function, L⁢(βmtif,βAT,βGC,a,b,c), such that for a random sample of subjects, L⁢(βmtif,βAT,βGC,a,b,c), is the likelihood of the model parameters, βmtif, βAT, and βGC, given that we observed a set of trimming amounts for a set of given genes. As such, the log-likelihood function can be written aslog⁡L(βmotif,βAT,βGC,a,b,c)=∑i,k,σ,nyikσn⋅log⁡P(n∣σ;βmotif,βAT,βGC,a,b,c)

where P(n∣σ;βmotif,β AT,βGC,a,b,c) is given by (Equation 3). Instead of maximizing this log-likelihood directly, we may wish to aggregate the data to reduce the number of observations and simplify model fitting. Recall that for subject i, C(i)⁢(σ) represents the number of TCRs which use gene allele group σ and C(i)⁢(n,σ) represents the number of TCRs which have gene allele group σ and n gene nucleotides deleted. As such, C(i)⁢(n,σ) is the count of observations which will have the same trimming probabilities P(n∣σ;βmotif,β AT,βGC,a,b,c) and will have been trimmed by n for subject i and gene allele group σ. Thus, using this aggregated data from all subjects i∈I, we can re-write the log-likelihood function equivalently as(4)log⁡L(βmotif,βAT,βGC,a,b,c)=∑i,σ,nC(i)(n,σ)⋅log⁡P(n∣σ;βmotif,βAT,βGC,a,b,c).

As above, for a random sample of subjects, L⁢(βmtif,βAT,βGC,a,b,c) is the likelihood of the model parameters, βmtif, βAT, and βGC, given that we observed a set of trimming amounts for a set of given genes.

With this likelihood formulation, all observations in the sample get uniform treatment in the construction of the likelihood. However, subjects may differ in their repertoire size and composition for reasons other than trimming. For example, it is known that gene usage differs across subjects. Thus, to avoid having these differences pollute our βmotif^, βAT^, and βGC^ inference, we propose a subject and gene weighting scheme.

As such, we can define the expected likelihood of a process where we first draw a subject i uniformly at random, then we sample TCR sequences from their repertoire according to a given distribution, as follows. For a single TCR sequence from such a sample, let S be a random variable representing the gene of the sequence, and let N be a random variable representing the number of deleted nucleotides. We can sample each TCR sequence with probability Psamp(N=n,S=σ) which we will specify later. Also, given random S and N, the log-likelihood of the model parameters, βmtif, βAT, and βGC, is given bylog⁡L(βmotif,βAT,βGC,a,b,c;N,S)=log⁡P(N∣S;βmotif,βAT,βGC,a,b,c).

With this, the expected log-likelihood of the model parameters, βmotif, βAT, and βGC given this random sample is given byE[log⁡L(βmotif,βAT,βGC,a,b,c)∣I=i]=∑n,σPsamp(N=n,S=σ)⋅log⁡P(N=n,S=σ;βmotif,βAT,βGC,a,b,c)=∑n,σPsamp(N=n,S=σ)⋅log⁡P(N=n∣S=σ;βmotif,βAT,βGC,a,b,c).

We can define a new, weighted log-likelihood function, log⁡Lexpected⁢(βmtif,βAT,βGC,a,b,c), equivalent to this expected log-likelihood:(5)log⁡Lexpected(βmotif,βAT,βGC,a,b,c):=∑iE[log⁡L(βmotif,βAT,βGC,a,b,c)∣I=i].

For a random sample of subjects, the weighted likelihood, Lexpected⁢(βmtif,βAT,βGC,a,b,c), represents the likelihood of the model parameters, βmtif, βAT, and βGC, given that we observed a set of trimming amounts for a given set of gene allele groups after weighting observations according to the sampling procedure Psamp(N=n,S=σ). We can use whichever sampling procedure, Psamp(N=n,S=σ), we want. For example, recall that we originally formed the empirical conditional PDFs in (Equation 1) for each subject i by uniformly sampling from each TCR repertoire to get a total repertoire size of Ki:Pemp(N=n∣S=σ,i)=C(i)(n,σ)C(i)(σ),Pemp(S=σ∣i)=C(i)(σ)Ki,

andPemp(i)=1I.

With this, we can define a sampling procedure equivalent to this empirical joint PDF as follows:(6)Psamp(N=n,S=σ):=Pemp(N=n,S=σ)=Pemp(n∣σ,i)⋅Pemp(σ∣i)⋅Pemp(i)

With this sampling procedure,(7)log⁡Lexpected(βmotif,βAT,βGC,a,b,c)=∑i,σ,nPemp(n∣σ,i)⋅Pemp(σ∣i)⋅Pemp(i)⋅log⁡P(n∣σ;βmotif,βAT,βGC,a,b,c).

As such, each subject, instead of each observation, gets uniform treatment in the construction of the weighted likelihood.

While this procedure would correct for individual subjects having different repertoire sizes, it does not account for gene usage differences. To avoid having these differences pollute our βmotif^, βAT^, and βGC^ inference, we propose a subject-independent gene-allele-group sampling scheme. While we could use any distribution on σ, including a uniform weight by gene allele groups, we have chosen to define:Pmarg(σ)=1I∑iPemp(σ∣i).

We can reformulate the sampling procedure which is an empirical average per-gene-allele-group frequency such that:(8)Psamp(N=n,S=σ):=Pemp(n∣σ,i)⋅Pmarg(σ)⋅Pemp(i).

With this subject-independent gene sampling procedure, we can define a weighted likelihood LW⁢(βmtif,βAT,βGC,a,b,c) such that(9)log⁡LW(βmotif,βAT,βGC,a,b,c):=∑i,σ,nPemp(n∣σ,i)⋅Pmarg(σ)⋅Pemp(i)⋅log⁡P(n∣σ;βmotif,βAT,βGC,a,b,c)

As such, each gene and each subject get uniform treatment in the construction of the weighted likelihood.

From here, we can maximize this weighted log-likelihood, log⁡LW⁢(βmtif,βAT,βGC,a,b,c), to estimate the log-probabilities βmtif, βAT, and βGC, where βmtif is equivalent to a (log) position-weight-matrix. To estimate each coefficient, we can solve the weighted maximum likelihood estimation problem:(10)(βmotif^,βAT^,βGC^)=argmaxβmotif,βAT,βGC log⁡LW(βmotif,βAT,βGC,a,b,c)

using the mclogit package in R. We can formulate a weighted maximum likelihood problem in a similar way for any model covariate function f containing a unique combination of parameter-specific covariate functions (Table 1).

We compare our inferred coefficients to the existing PWM model which was designed and trained using least squares (Murugan et al., 2012). When replicating this model using our methods described above (i.e. the 2×4 motif model), we note highly similar results (Figure 2—figure supplement 1).

In order to evaluate the model fit and generalizability of each model, we use a variety of training and testing data sets to train each model and calculate the log loss. We will describe our general model evaluation procedure here. We describe variations of this general model evaluation procedure in Appendix 3. Let T represent a training data set and H represent a held-out testing data set. With the training set T, we can train each model of interest as described above in (Equation 10). After this model fitting, we can calculate the expected per-sequence conditional log loss of the model with given coefficients, ℳ, for a given held-out testing set, H, such that(11)ℓ(M∣H):=−∑i,σ,nPempH(n,σ,i)⋅log⁡P(n∣σ;M)=−∑i,σ,nPempH(n∣σ,i)⋅PempH(σ∣i)⋅PempH(i)⋅log⁡P(n∣σ;M)

where i represents a subject, n represents a trimming length, and σ represents a gene allele group. Because we are incorporating the empirically observed frequency of each subject, trimming length, and gene allele group within each ‘held-out testing set,’ PempH(n,σ,i), in this formulation, the expected per-sequence conditional log loss values are guaranteed to be directly comparable between held-out testing sets with varying compositions. Models that have lower expected per-sequence conditional log loss will indicate that the model has a better fit.

During model fitting, we estimated the model coefficients βmotif^, βAT^, and βGC^ by maximizing the weighted likelihood function given by (Equation 9). To measure the significance of each of these model coefficients β^∈{βmotif^,βAT^,βGC^} we want to test whether each coefficient β^=0. To do this, we can first estimate the standard error of each inferred coefficient using a clustered bootstrap (with subject-gene pairs as the sampling unit). As such, for each bootstrap iterate, we sampled subject-gene pairs from the full V-gene training data set with replacement. Using this re-sampled data, we maximized the weighted likelihood function given by (Equation 9) to re-estimate each coefficient. We repeated this bootstrap process 1000 times and used the resulting 1000 coefficient estimates to estimate a standard error for each model coefficient. With this estimated standard error of each inferred model coefficient β^∈{βmotif^,βAT^,βGC^}, we test whether β^=0 by calculating the test statistic(12)T(β^)=β^se⁡(β^)

and comparing T(β^) to a N(0,1) distribution to obtain each p-value. We consider the significance of each model coefficient using a Bonferroni-corrected threshold. To establish the threshold, we corrected for the total number of model coefficients being evaluated in the given model.

With the motif and base-count-beyond model, we are interested in quantifying variation in model coefficients in the context of genetic variations within the gene encoding the Artemis protein that were previously identified as being associated with increasing the extent of trimming (Russell et al., 2022b). Recall that we trained this model using the model covariate function given by (Equation 2). During model fitting, we estimated the model coefficients βmotif^, βAT^, and βGC^ by maximizing the weighted likelihood function given by (Equation 9).

We have previously identified a set X of SNPs within the gene encoding the Artemis protein that are significantly associated with increasing the extent of trimming (Russell et al., 2022b). For each SNP x∈X and individual i∈{1,…,I}, we measure the number of minor alleles in the genotype, gix∈{0,1,2}. We are interested in whether each of the inferred model coefficients β^∈{βmotif^,βAT^,βGC^} vary in the context of genotype for each genetic variant x∈X. As such, for each SNP of interest, we can adapt the 1×2 motif + two-side base-count beyond model covariate function to allow for genotype-specific variation of each model coefficient by incorporating additional interaction coefficients βx∈{βxmotif,βxAT,βxGC} to model the relationship between each model parameter and the SNP x genotype. We can then estimate the coefficients of this new model, βmotif^, βAT^, βGC^, βxmotif^, βxAT^, and βxGC^, as before by maximizing the weighted likelihood given by (Equation 9) using the adapted model covariate function. We can measure the significance of each of the model coefficients using the methods described in the previous section. Ultimately if a SNP-coefficient interaction term β^x∈{βxmotif^,βxAT^,βxGC^} is significant, we can conclude that the corresponding model coefficient β^ varies significantly in the context of the genotype of SNP x∈X. We use this same procedure to evaluate whether each model coefficient varies in the context of each SNP of interest.