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The ability to learn complex patterns in data has tremendous implications in immunogenomics. T-cell receptor TCR sequencing assesses the diversity of the adaptive immune system and allows for modeling its sequence determinants of antigenicity. Our results highlight the flexibility and capacity for deep neural networks to extract meaningful information from complex immunogenomic data for both descriptive and predictive purposes.
Next-generation sequencing NGS has allowed a comprehensive description and understanding of the complexity encoded at the genomic level in a wide variety of organisms. The applications of NGS have grown rapidly as this technology has become a molecular microscope for understanding the genomic basis for the fundamental functions of the cell 1. In parallel to this explosion of NGS applications, in the machine learning world, deep learning has seen a similar expansion of applications as computational resources have grown; there exist many opportunities to apply deep learning in genomics as the data generated from NGS are very large and highly complex 2 , 3 , 4 , 5 , 6 , 7.
T cell receptor sequencing TCR-Seq is an application of NGS that has allowed scientists across many disciplines to characterize the diversity of the adaptive immune response 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , With this new sequencing technology, there has arisen a need to develop analytical tools to parse and draw meaningful concepts from the data such as those pertaining to shared sequence concepts or motifs , since antigen-specific T cells exist within a sea of T cells with specificities irrelevant to the microbe or tumor cell being assessed.
In recent work, investigators have applied conventional sequence analytics, where either targeted motif searches or sequence alignment algorithms have been used to begin parsing the data within TCR-Seq 20 , 21 , However, identifying signal over noise is particularly challenging in studying in vivo T cell responses such as tumor-specific T cell responses, which are mediated by a small proportion of the overall pool of tumor-infiltrating lymphocytes and peripheral blood lymphocytes 23 , 24 , In light of this need to better featurize TCR sequences, we turned to deep learning primarily through the use of convolutional neural networks CNNs as a powerful means to extract important features from sequencing data for both descriptive and predictive purposes.
As has been demonstrated in previous genomic applications of deep learning, the main advantage of CNNs in this application is the ability to learn these sequence motifs referred to as kernels in this context through some objective function given to the network 4. These learned motifs can then be used as part of a complex deep learning model to either describe the data in a new latent space or be used for a classification task. Furthermore, since the initial conception and presentation of this work, multiple groups have begun to recognize the value of deep learning and broader machine learning techniques in this endeavor to learn these sequence concepts of immune receptors 26 , 27 , 28 , 29 , We present DeepTCR, a platform of both unsupervised and supervised deep learning that is able to be applied at the level of individual T cell receptor sequences as well as at the level of whole T cell repertoires, which can learn patterns in the data that may be used for both descriptive and predictive purposes.
Across these various datasets, the level of non-specific signal varies given the technical difficulties associated with extracting true non-specific signatures of TCR responses, and we seek to demonstrate the value of applying deep learning in these scenarios to leverage knowledge about sequence homology to extract the true antigen-specific signals.
Subsequently, a three-layer CNN is used to extract sequence-based features from both chains. The CDR3 sequences serve as input to the encoder side of the network where convolutional layers are then applied to learn sequence motifs from these regions.
Finally, these inputs are concatenated together passed through fully connected layers to result in the latent representation of the TCR. Finally, the weights of the neural network are trained via gradient descent to jointly minimize both the reconstruction and variational loss. The trained network is then used to take a given TCR and represent it in a continuous numerical domain for downstream analysis such as clustering.
Featurization methods that encourage high-quality clusters that capture a high degree of information of the label i. We first implemented this method of TCR featurization within the unsupervised learning setting in order to learn the underlying distribution of the sequence data in high-dimensional space for the purpose of clustering TCR sequences that likely recognize the same antigen as this is a commonly performed analysis in TCR-Seq.
Our implementation of a VAE Fig. First, to benchmark these various methods of featurization in clustering antigen-specific TCRs, we ran an agglomerative clustering algorithm varying the number of clusters from 5 to and then assessed the variance ratio criterion of the clustering solutions and the adjusted mutual information from the clustering solutions to the ground truth antigen labels scikit-learn 34 , We noted that the VAE methods maintained the highest variance ratio criterion while also maintaining a high adjusted mutual information to the ground truth labels for both murine and human datasets Fig.
To further query the value of these learned features in correctly clustering sequences of the same specificity, we applied a K -Nearest Neighbors Algorithm across a wide range of K values using a fivefold cross-validation strategy and assessed performance metrics of the classifier including AUC, Recall, Precision, and F 1 Score for all featurization methods Fig.
We noted that across all performance metrics, the VAE-based methods at least one outperformed current state-of-the-art approaches for TCR featurization. To do so, we developed a fully supervised model that learns sequence-specific motifs to correctly classify sequences by their antigen-specific labels Fig. In addition, being able to extract knowledge from the network can inform relevant motifs for antigen-specific recognition.
Therefore, we established a method by which we could identify the most predictive i. Motifs that were highly associated with the predicted probability of binding a given antigen were displayed with Logomaker The binding of a TCR to a peptide-major histocompatibility complex pMHC is not usually considered a binary phenomenon but rather one that is characterized by a binding affinity.
We specifically removed any TCR sequences from this independent validation cohort that were in the data used to train the models. Predictive performance of Residue Sensitivity analysis to identify known contact residues shown in Supplementary Fig.
One of the advantages of training a predictive model is the ability to perturb its inputs and measure the change in the output of the model; in other words, being able to conduct a sensitivity analysis. Assuming the model has correctly learned the rules of antigen specificity, one can identify residues in a TCR sequence that are highly sensitive to change in a causal fashion and thus describe the relative importance of any given residue to antigen-specific binding.
We first took the sequence data from the CDR3 regions of these TCRs and permuted each position of each sequence with all other 19 amino acid and obtained predicted affinities for each single amino acid mutation in order to see which residues were sensitive to change Fig. We first noted that certain positions were highly sensitive to any change in amino acid i. To represent this information about each residue in a more compact visualization, we created Residue Sensitivity Logos RSLs which allow for rapid comparison of many sequences in a logo type format where the size of the residue corresponds to sensitivity at that particular position and the color scheme represents the average direction of changes at that position to the binding affinity Fig.
Finally, we correlated information from these RSLs to the crystal structure Fig. These results suggest that by combining the high-throughput nature of single-cell technology with deep learning as illustrated in these examples, one can obtain a robust understanding of the sequence determinants of TCR antigenicity as well as provide guidance for TCR engineering.
Building on the supervised sequence classifier, we then wanted to design an architecture that could learn from a label applied to a whole repertoire of TCR sequences, most of which are irrelevant to the antigen of interest. Our supervised repertoire classifier was formulated as a supervised multi-instance learning algorithm that is able to extract meaningful concepts that may lie within large repertoires of many sequences Fig.
This scenario is akin to many use-cases of TCR-Seq where ground truth labels i. To test the utility of this approach, we collected data from published TCR-Seq data of an assay where T cells from an elite suppressor ES8 of HIV were cultured with autologous HIV-1 Gag and Nef epitope variants and sequenced after culture to determine the immune repertoire against each epitope 31 , 45 , 46 , 47 , 48 , In the original work, TCR sequences were deemed to be antigen specific if they met certain statistical requirements based on the read count of a given sequence, a proxy for clonal expansion.
However, given that T cell expansion in culture in the presence of stimulatory cytokines can occur independent of antigen recognition, we wanted to take advantage of deep learning to leverage the TCR sequence and not just the read count in determining whether an epitope elicited an antigen-specific immune response.
We hypothesized that if a well had an antigen-specific response, its T cell repertoire should be distinguishable via its sequence concepts from those not specific for the stimulating peptide s CEF, AY9, No Peptide. In contrast to previously described models herein, our model would make a prediction about the entire T cell repertoire in a well and not any individual sequence, as we would not expect the majority of T cells within a well expanding to a given epitope to be antigen specific.
Therefore, we trained a repertoire classifier to predict if the well had been treated by the cognate epitope, or non-cognate conditions CEF, AY9, No Peptide given its T cell repertoire Fig. A representative positive cognate epitope is shown where the AUC for the cognate epitope in this classification problem is 1. Furthermore, we can measure the magnitude of the difference between the cognate epitope repertoire and the controls by measuring the difference between the average predictions for epitope-specific wells vs controls, termed Delta Prediction Fig.
While we utilize the AUC as a non-parametric rank based statistical test, the difference in average prediction values between the antigen-specific well and controls is a measure of the magnitude of this difference or the effect size. Once the classifier has been trained, single sequence predictions can be obtained by running each sequence separately through the trained model. This allows us to identify the most predictive sequences against a given epitope.
As can be seen for the example epitope, the highly predictive sequences represent only a minority of unique TCRs in the antigen-specific wells and are often not the sequences with the highest read counts Fig. Following previously described the TCR Featurization Block, we implement a multi-head attention mechanism to make sequence assignments to concepts within the sample.
The number of concepts in the model is a hyperparameter, which can be varied by the user depending on the heterogeneity expected in the repertoires. Of note, this assignment of a sequence to a concept is done through an adaptive activation function that outputs a value between 0 and 1, allowing the network to put attention on the sequences that are relevant to the learning task.
When taking the average of these assignments over all the cells in a repertoire, this results in a value within the neural network that directly corresponds to the proportion of the repertoire that is described by that learned concept.
These proportions of concepts in the repertoire are then sent into a final traditional classification layer. The model is trained to learn the distinguishing TCR sequence features of the cognate epitope from the controls through Monte-Carlo simulations where the model is trained on two out of the three triplicates and performance is assessed on the left-out well for each set of conditions, ensuring that any predictions used for downstream interpretation have been obtained from data not used in training.
When we ran this pipeline across the 25 tested epitopes, we noted that our model predicted that 19 of these epitopes covering 5 out of the 6 epitope families elicited highly distinguishable sequence features of the T cell repertoire within this elite suppressor when selecting for epitopes that passed a statistical threshold for AUC above 0.
Consensus epitopes are denoted in red. UMAP dimensionality reduction was applied to the per-sequence prediction values to generate visualizations for the antigen-specific TCRs. Intensity of coloring corresponds to density as computed by Gaussian kernel density estimation. Given the breadth of the epitopes this particular elite suppressor responded to, we wanted to characterize the immune repertoire that responded to these epitopes and whether the number and extent of escape variants affected the sequence diversity of the immune repertoire.
In order to ask this question, we took epitopes from the epitope families that had at least two autologous variants with detectable immune responses via previously described method and conducted all pairwise comparisons of these escape variants within a given epitope family.
By training a model for each pair of epitopes within an epitope family, we could measure how distinguishable the repertoire was between any two given variants. A model that could not distinguish between two variants would suggest that the immune repertoire was homologous and thus cross-reactive to both of these variants. On the contrary, if a model could distinguish the immune repertoire between variants, then it would suggest that divergent immune responses were elicited by these variants.
For ES8, the original investigators queried 1 consensus and 11 escape variants of the GAG TW10 epitope family, 10 of which this particular elite suppressor had acquired and as detected via sensitive RT-PCR in either plasma or pro-viral samples, representing the epitope family of highest acquired escape variants.
Following training models on all pairwise variants as described previously, a clustered heatmap was used to visualize the pairwise 1-Delta Predictions to compare the immune repertoires of these 10 autologous epitope variants Fig. Following training this in this multi-class fashion, we extracted the per-sequence predictions across all classes and applied a UMAP dimensionality reduction to visualize the sequences for each variant Fig. Interestingly, when comparing the immune repertoires of the escape variants to the consensus epitope, we noted that while the consensus epitope elicited a relatively focused repertoire, many of the escape variants elicited rather heterogeneous responses based on TCR diversity.
These findings lead us to believe that the GAG TW10 epitope is under considerable immune pressure where escape variants often create TCR repertoires that are not only distinguishable from the repertoire against the consensus epitope but also are far more heterogeneous, suggesting less specific immune responses are generated against these escape variants.
To write your own document using our LaTeX style, math notation, or to copy our notation page, download our template files. Errata in published editions. No, our contract with MIT Press forbids distribution of too easily copied electronic formats of the book. In other words, this is a rule which can be used to learn in a neural network. There are a number of challenges in applying the gradient descent rule. We'll look into those in depth in later chapters.
But for now I just want to mention one problem. Unfortunately, when the number of training inputs is very large this can take a long time, and learning thus occurs slowly. An idea called stochastic gradient descent can be used to speed up learning. Then we pick out another randomly chosen mini-batch and train with those.
And so on, until we've exhausted the training inputs, which is said to complete an epoch of training. At that point we start over with a new training epoch. Incidentally, it's worth noting that conventions vary about scaling of the cost function and of mini-batch updates to the weights and biases. This is particularly useful when the total number of training examples isn't known in advance.
This can occur if more training data is being generated in real time, for instance. But when doing detailed comparisons of different work it's worth watching out for. We can think of stochastic gradient descent as being like political polling: it's much easier to sample a small mini-batch than it is to apply gradient descent to the full batch, just as carrying out a poll is easier than running a full election.
In practice, stochastic gradient descent is a commonly used and powerful technique for learning in neural networks, and it's the basis for most of the learning techniques we'll develop in this book. Exercise An extreme version of gradient descent is to use a mini-batch size of just 1. Then we choose another training input, and update the weights and biases again. And so on, repeatedly. This procedure is known as online , on-line , or incremental learning. In online learning, a neural network learns from just one training input at a time just as human beings do.
Let me conclude this section by discussing a point that sometimes bugs people new to gradient descent. Some people get hung up thinking: "Hey, I have to be able to visualize all these extra dimensions". And they may start to worry: "I can't think in four dimensions, let alone five or five million ".
Is there some special ability they're missing, some ability that "real" supermathematicians have? Of course, the answer is no. Even most professional mathematicians can't visualize four dimensions especially well, if at all. The trick they use, instead, is to develop other ways of representing what's going on. People who are good at thinking in high dimensions have a mental library containing many different techniques along these lines; our algebraic trick is just one example.
Those techniques may not have the simplicity we're accustomed to when visualizing three dimensions, but once you build up a library of such techniques, you can get pretty good at thinking in high dimensions. I won't go into more detail here, but if you're interested then you may enjoy reading this discussion of some of the techniques professional mathematicians use to think in high dimensions. While some of the techniques discussed are quite complex, much of the best content is intuitive and accessible, and could be mastered by anyone.
Implementing our network to classify digits. Alright, let's write a program that learns how to recognize handwritten digits, using stochastic gradient descent and the MNIST training data. We'll do this with a short Python 2. If you're a git user then you can obtain the data by cloning the code repository for this book,. If you don't use git then you can download the data and code here.
Actually, we're going to split the data a little differently. We'll leave the test images as is, but split the 60,image MNIST training set into two parts: a set of 50, images, which we'll use to train our neural network, and a separate 10, image validation set.
We won't use the validation data in this chapter, but later in the book we'll find it useful in figuring out how to set certain hyper-parameters of the neural network - things like the learning rate, and so on, which aren't directly selected by our learning algorithm.
See this link for more details. I obtained this particular form of the data from the LISA machine learning laboratory at the University of Montreal link. If you don't already have Numpy installed, you can get it here.
Let me explain the core features of the neural networks code, before giving a full listing, below. The centerpiece is a Network class, which we use to represent a neural network. Here's the code we use to initialize a Network object:.
In this code, the list sizes contains the number of neurons in the respective layers. This random initialization gives our stochastic gradient descent algorithm a place to start from. In later chapters we'll find better ways of initializing the weights and biases, but this will do for now. Note that the Network initialization code assumes that the first layer of neurons is an input layer, and omits to set any biases for those neurons, since biases are only ever used in computing the outputs from later layers.
Note also that the biases and weights are stored as lists of Numpy matrices. So, for example net. It's not the first and second layers, since Python's list indexing starts at 0. Since net. With all this in mind, it's easy to write code computing the output from a Network instance. We begin by defining the sigmoid function: def sigmoid z : return 1. Here, n is the number of inputs to the network. If you try to use an n, vector as input you'll get strange results. Although using an n, vector appears the more natural choice, using an n, 1 ndarray makes it particularly easy to modify the code to feedforward multiple inputs at once, and that is sometimes convenient.
Of course, the main thing we want our Network objects to do is to learn. To that end we'll give them an SGD method which implements stochastic gradient descent. Here's the code. It's a little mysterious in a few places, but I'll break it down below, after the listing.
The other non-optional parameters are self-explanatory. This is useful for tracking progress, but slows things down substantially. The code works as follows. In each epoch, it starts by randomly shuffling the training data, and then partitions it into mini-batches of the appropriate size. This is an easy way of sampling randomly from the training data. This is done by the code self. I'm not going to show the code for self. We'll study how backpropagation works in the next chapter, including the code for self.
For now, just assume that it behaves as claimed, returning the appropriate gradient for the cost associated to the training example x. Let's look at the full program, including the documentation strings, which I omitted above. Apart from self. SGD and self. The self. You can get the gist of these and perhaps the details just by looking at the code and documentation strings.
We'll look at them in detail in the next chapter. Note that while the program appears lengthy, much of the code is documentation strings intended to make the code easy to understand. In fact, the program contains just 74 lines of non-whitespace, non-comment code. All the code may be found on GitHub here. Gradients are calculated using backpropagation.
Note that I have focused on making the code simple, easily readable, and easily modifiable. It is not optimized, and omits many desirable features. For example, if the list was [2, 3, 1] then it would be a three-layer network, with the first layer containing 2 neurons, the second layer 3 neurons, and the third layer 1 neuron.
The biases and weights for the network are initialized randomly, using a Gaussian distribution with mean 0, and variance 1. Note that the first layer is assumed to be an input layer, and by convention we won't set any biases for those neurons, since biases are only ever used in computing the outputs from later layers.
It's a renumbering of the scheme in the book, used here to take advantage of the fact that Python can use negative indices in lists. Note that the neural network's output is assumed to be the index of whichever neuron in the final layer has the highest activation. How well does the program recognize handwritten digits? We execute the following commands in a Python shell,.
Of course, this could also be done in a separate Python program, but if you're following along it's probably easiest to do in a Python shell. We do this after importing the Python program listed above, which is named network ,. Network [ , 30 , 10 ]. Note that if you're running the code as you read along, it will take some time to execute - for a typical machine as of it will likely take a few minutes to run. I suggest you set things running, continue to read, and periodically check the output from the code.
If you're in a rush you can speed things up by decreasing the number of epochs, by decreasing the number of hidden neurons, or by using only part of the training data. Note that production code would be much, much faster: these Python scripts are intended to help you understand how neural nets work, not to be high-performance code! And, of course, once we've trained a network it can be run very quickly indeed, on almost any computing platform. For example, once we've learned a good set of weights and biases for a network, it can easily be ported to run in Javascript in a web browser, or as a native app on a mobile device.
In any case, here is a partial transcript of the output of one training run of the neural network. The transcript shows the number of test images correctly recognized by the neural network after each epoch of training.
As you can see, after just a single epoch this has reached 9, out of 10,, and the number continues to grow,. That's quite encouraging as a first attempt. I should warn you, however, that if you run the code then your results are not necessarily going to be quite the same as mine, since we'll be initializing our network using different random weights and biases. To generate results in this chapter I've taken best-of-three runs.
As was the case earlier, if you're running the code as you read along, you should be warned that it takes quite a while to execute on my machine this experiment takes tens of seconds for each training epoch , so it's wise to continue reading in parallel while the code executes. Using the techniques introduced in chapter 3 will greatly reduce the variation in performance across different training runs for our networks. As I mentioned above, these are known as hyper-parameters for our neural network, in order to distinguish them from the parameters weights and biases learnt by our learning algorithm.
If we choose our hyper-parameters poorly, we can get bad results. If we do that, we get better results, which suggests increasing the learning rate again. If making a change improves things, try doing more! So even though we initially made a poor choice of hyper-parameters, we at least got enough information to help us improve our choice of hyper-parameters. In general, debugging a neural network can be challenging.
This is especially true when the initial choice of hyper-parameters produces results no better than random noise. Of course, we know from our earlier experiments that the right thing to do is to decrease the learning rate.
But if we were coming to this problem for the first time then there wouldn't be much in the output to guide us on what to do. We might worry not only about the learning rate, but about every other aspect of our neural network. We might wonder if we've initialized the weights and biases in a way that makes it hard for the network to learn? Or maybe we don't have enough training data to get meaningful learning?
Perhaps we haven't run for enough epochs? Or maybe it's impossible for a neural network with this architecture to learn to recognize handwritten digits? Maybe the learning rate is too low? Or, maybe, the learning rate is too high? When you're coming to a problem for the first time, you're not always sure. The lesson to take away from this is that debugging a neural network is not trivial, and, just as for ordinary programming, there is an art to it.
You need to learn that art of debugging in order to get good results from neural networks. More generally, we need to develop heuristics for choosing good hyper-parameters and a good architecture.
We'll discuss all these at length through the book, including how I chose the hyper-parameters above. Try creating a network with just two layers - an input and an output layer, no hidden layer - with and 10 neurons, respectively. Train the network using stochastic gradient descent. What classification accuracy can you achieve? It's pretty straightforward. For completeness, here's the code.
The data structures used to store the MNIST data are described in the documentation strings - it's straightforward stuff, tuples and lists of Numpy ndarray objects think of them as vectors if you're not familiar with ndarray s :. The first entry contains the actual training images. This is a numpy ndarray with 50, entries. Those entries are just the digit values
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