DARSI: A convolutional neural network for gene expression prediction from MPRA data

To develop a model capable of predicting gene expression from regulatory sequence variation, we began by evaluating the landscape of existing machine learning architectures in genomics. In particular, large-scale pretrained transformer architectures such as DNA BERT and the Nucleotide Transformer [35–38] have demonstrated the ability to learn complex long-range interactions in biological sequences. Likewise, deep convolutional neural networks, including BPNet and Enformer [39,40], have achieved high-resolution predictions of regulatory activity directly from raw DNA input. As described in detail in S2 Text, we reviewed these and related frameworks to assess their potential utility in the context of MPRA-derived data, weighing considerations such as predictive accuracy, interpretability, scalability, and suitability for modestly sized experimental libraries.

Guided by our evaluation, we concluded that a convolutional neural network provided the best compromise between prediction accuracy and interpretability when trained on small datasets. As a result, we implemented a convolutional neural network specifically designed to learn from MPRA-derived regulatory sequences and predict gene expression levels. To reach predictive power over regulatory regions and capture correlations between nucleotides, the model leverages convolutional filters capable of detecting both local and long-range interactions between base pairs [41], potentially enabling the identification of regulatory features distributed across the input sequence. The network takes as input the sequence variants of a given operon and their corresponding expression levels. After iterative optimization, we settled on a 12-layer architecture (see S3 Text) that shares design elements with previously established models in the field [39–41].

As a case study, we utilized data from a recent MPRA study conducted by [8] in E. coli. This work dissected the regulatory information of 114 bacterial operons by randomly mutating a 160 bp region upstream of the transcription start site of each operon at a mutation rate of 10%. This process generated a dataset akin to that featured in the schematic shown in Fig 2A that correlates sequence and gene expression. To ensure sufficient coverage of mutations, we selected operons with at least 1,000 sequence variants. This number guaranteed that, for every base pair along the sequence, our dataset contained at least 100 sequence variants in which that base pair is mutated. We further cross-referenced all sequences with the annotated E. coli genome available on EcoCyc [42] to verify that the sequences encompassed regions upstream of the genes. The lower bound used for number of variants and the cross-validation of the data with annotated databases reduced the dataset to 95 mutagenized operons, each originating from E. coli colonies cultivated in LB medium. Across the 95 operons, the mean number of unique sequences per operon is , with unique barcodes per operon. This results in an overall mean of sequence variants across all operons. The sequence data for each operon served as input to the network, while discretized normalized mRNA counts (described below) were used as the output. A separate convolutional neural network was trained for each operon, resulting in a total of 95 independently trained networks.

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Fig 2. The DARSI pipeline.

(A) MPRA dataset that makes it possible to correlate regulatory sequence with gene expression. (B) One-hot encoding scheme used to convert each DNA sequence into a binary image. (C) Distribution of the together with the bin of gene expression assigned to each value for the illustrative case of the yqhC operon. Note that the overlap observed between the low and high expression bins is an artifact of the histogram binning and does not reflect an actual overlap between the classes of expression. (D) The images and the corresponding gene expression bins constitute the inputs and outputs of our convolutional neural network, respectively.

https://doi.org/10.1371/journal.pcbi.1014092.g002

Convolutional neural networks are designed to take images or matrices as inputs. Thus, to prepare the DNA sequence data for use as input to our networks, we transformed the sequences into a two-dimensional matrix representation. Specifically, each 160 bp regulatory sequence from the MPRA dataset was encoded as a binary image using a so-called one-hot encoding scheme, as illustrated in Fig 2B and detailed in the “One-hot encoding” section of the Materials and methods. Consequently, the data for each operon is represented as a stack of images, with each image corresponding to a specific sequence variant for that operon.

In designing the network outputs, we considered two possible approaches: predicting continuous gene expression levels through regression, or discretizing the outputs and framing the problem as a classification task. To ensure comparability across sequences, we first normalized mRNA counts by dividing the number of sequenced mRNAs by the corresponding DNA copy number for each variant (Fig 2A). To assess the regression approach, we trained regression models on the ten operons with the largest number of sequence variants (leuABCD, rumB, zupT, yncD, uvrD, mscK, ftsK, yqhC, groSL, and xylA). As detailed in the S4 Text, regression resulted in substantially lower predictive performance, with average R² value of 0.06 and normalized RMSE value of 1.02, indicating poor generalization.

Given the poor performance of our networks in the context of predicting continuous gene expression levels through regression, we decided to explore the discretization approach. Specifically, we discretized normalized gene expression values into expression bins, referred to as “classes”, and trained the networks to predict the bin associated with each regulatory sequence. This binning allowed us to cast the task as a multi-class classification problem.

Mutations within the sequences can lead to various outcomes, such as a complete absence of detectable gene expression, a measurable reduction, or an increase in gene expression compared to typical levels observed across the sequence variants for each operon. To capture this variation in expression level, we examined the distribution of the logarithm of the normalized expression counts. Based on this distribution, we defined three distinct expression classes: (1) sequences resulting in no detectable gene expression (zero expression bin), (2) sequences yielding low but measurable levels of gene expression (low expression bin), and (3) sequences associated with high levels of gene expression (high expression bin). While the decision to use three bins was informed by the natural clustering of data in the logarithmic space, this choice is ultimately a simplification that, as we will show in the next sections, can still lead to predictive power and the ability to identify transcription factor binding sites.

For each operon we determined the thresholds of for each bin to partition the gene expression counts into the three classes. The zero gene expression bin corresponds to sequences that yielded no detectable mRNA. The threshold between the low and high gene expression bins were chosen so as to lead to statistically significant differences in mean gene expression levels between these two classes, as described in detail in the “RNA count labeling” section of the Materials and methods. Fig 2C shows the distribution of and the associated bins color-coded for the illustrative yqhC operon from the MPRA dataset by [8]. This operon is used throughout the text to illustrate our pipeline and its results, as it represents the average performance of the pipeline. Similar plots to Fig 2C, showing the distribution of expression counts for the rest of the operons in the dataset can be accessed through our GitHub repository.

The number of observations in each expression bin vary significantly. Indeed, as shown in Fig 2C, the bin corresponding to zero gene expression was typically overrepresented with respect to the low and high gene expression bins. To account for this over-representation, the zero gene expression bin was under sampled when training the networks, while the low and high bins were over sampled to create an evenly split processed dataset [43,44].

Training for each network utilized 70% of the processed data, following adjustments for data imbalance. Training was conducted in MATLAB using standard optimization toolboxes, with parameters optimized via stochastic gradient descent [45–47]. An additional 15% of the data (3,000–5,000 variants across the 95 operons) was reserved for validation during training, serving to optimize network architecture as discussed below. The remaining 15% of the data was allocated for final evaluation of the predictive power of each network.

Before engaging in the training of all 95 networks, we optimized the overall network architecture for accuracy in predicting gene expression in our dataset. While adding more convolutional layers should allow the network to extract longer-range interactions between nucleotides along the sequence, increasing the depth of the network leads to a substantial rise in the number of trainable parameters, potentially resulting in overfitting [24]. As a result, we systematically and iteratively modulated the network architecture to assess its impact on prediction accuracy.

To optimize the network architecture, we focused on data from the 10 operons with the largest number of sequence variants. As expected, our optimization revealed that increasing model complexity (e.g., by adding layers and channels) generally improves training accuracy but can lead to overfitting, where the model performs poorly when validated using unseen data. We converged onto an optimal architecture, detailed in Table A in S3 Text, that strikes a balance between model complexity and performance. This chosen architecture is consistent with similar networks implemented in prior studies [27,39,40]. Using this optimized architecture, we independently trained 95 convolutional neural networks, one for each operon in our dataset. Further details on the DARSI architecture, its optimization, and training specifications can be found in the S3 Text.

Finally, we also explored whether training a single model across all operons could enhance predictive power by leveraging a pooled dataset across variants. However, as shown in S5 Text, this approach performed no better than random classification, further motivating our operon-specific modeling strategy.

DARSI can predict gene expression from raw sequence

As outlined above, each trained network, corresponding to an individual operon in the dataset, was evaluated using the reserved 15% of the processed data designated as the test set. For each operon, raw sequences from the test partition were input into the trained network, which then predicted the corresponding output gene expression bin. The predicted bins were compared against the experimentally measured gene expression values to calculate an accuracy score for each network. As illustrated in Fig 3A, the networks achieved an average predictive accuracy of 79.8% across all 95 operons.

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Fig 3. Predictive Performance of DARSI Models on Bacterial MPRA Data.

(A) The accuracy of the DARSI model in predicting gene expression levels from previously unseen DNA sequences for each E. coli operon in the dataset. The average prediction accuracy of 79.8% is shown as a horizontal line. (B) Confusion matrix illustrates the comparison between measured and predicted expression levels for the yqhC operon in E. coli, highlighting model performance. Each entry in the matrix represents the number of sequences classified as true positives, true negatives, false positives and false negatives for each gene expression bin. The row projections of the confusion matrix in blue and red are true positive and false positive rates, respectively, while the column projections in blue and red are the true negative and false negative rates, respectively. (C) Average F1 score values for the three expression bins are shown with the ideal classifier represented by the solid horizontal line. The values of the F1 score are close to 1, corresponding to an ideal classifier.

https://doi.org/10.1371/journal.pcbi.1014092.g003

To more rigorously evaluate the effectiveness of our DARSI model in predicting gene expression from raw sequence input, we generated confusion matrices. In these matrices, each column represents predicted expression bin (i.e., zero expression, low expression or high expression), while each row indicates the actual bin to which the sequences belong as reported by measurements. Each entry within the matrix indicates the number of sequences belonging to each combination of predicted and measured gene expression bins. Consequently, these matrices provide a summary of false positives, false negatives, true positives, and true negatives for each of the three discrete expression bins.

The confusion matrix for the yqhC operon is displayed in Fig 3B. This matrix indicates that the trained model classifies the majority of unseen data for each operon with high specificity (low false positive rate) and high sensitivity (low false negative rate), as evidenced by the diagonal dominance and the row and column projections shown in Fig 3B. To access a full list of confusion matrices for all the models trained, the reader is referred to the GitHub repository.

Further to benchmark the efficacy of DARSI in predicting gene expression level (bin) from raw sequence, we investigated how its predictive power compares to pre-trained foundational models that have been published recently—see [38,48] for examples. We utilized the 50 million-parameter-all-species model from Nucleotide Transformer (NT) foundation model [38] to generate embeddings from raw sequence data. Said embeddings were then used to train a logistic regression to classify the bin of expression for each gene. The mean classification accuracy across all operons for this model was 48.5% as opposed to 79.8% for the same task from DARSI. This shows that a much simpler DARSI model not only outperforms the large foundational model in this predictive task. The detail of this one-shot foundational model-based classification method is discussed in the S6 Text.

To evaluate the overall performance of DARSI across all 95 operons, we computed the average F1 score for each expression bin. The F1 score is a metric that assesses both specificity and sensitivity of classifiers, and is commonly employed to gauge classifier performance [24,49–51]. The F1 score for a given bin of expression is defined as

(1)

where, for example, “true positive” indicates the number of true positives resulting from our model for a specific bin. According to this definition, an ideal classifier with 100% true positive and 0% false positive and false negative rates will have an F1 score of one. True positives, false negatives, and false positives have been highlighted for the zero gene expression bin of the representative yqhC operon in Fig 3B, leading to an average F1 score of 0.64 across the three bins for this operon.

By averaging the F1 score of all DARSI networks, we can compare the average network performance to that of an ideal classifier. Fig 3C presents the F1 score values for the zero expression bin (), low expression bin () and high expression bin () averaged over all 95 trained convolutional neural networks, where the error bars indicate the standard deviation. The F1 scores for all three expression bins exceed the threshold of 0.7 that is commonplace in most fields [52,53], indicating that the model effectively distinguishes sequences within these bins with high specificity and sensitivity. Thus, we deemed the gene expression predictions made by the trained DARSI models to be reliable for them to be leveraged in our exploration of regulatory architectures in E. coli.

Uncovering binding sites using saliency maps

We next leveraged the predictive power of our networks to identify transcription factor binding sites within unmapped regulatory regions. To achieve this task using MPRA datasets, existing methods typically analyze the mutual information or the shift in gene expression resulting from mutations at individual base pairs (Fig 1F; [8,22,54,33]). These analyses aim to isolate the impact of mutations at each nucleotide on the overall gene expression levels. To make this possible, the contributions of mutations across all other nucleotides within the sequence are averaged. Consequently, these approaches assume that the contributions of individual base pairs to gene expression are independent from one another. In contrast, the convolutional neural network architecture employed in this study makes it possible to account for spatial correlations between nucleotides throughout the sequence.

To examine the capacity of our networks to identify clusters of nucleotides corresponding to binding sites, we created so-called saliency maps. These saliency maps are better understood in the context of convolutional neural networks for image classification. For example, a neural network can be trained to classify images between those featuring a dog and those not featuring a dog [55,56,57]. A saliency map is a heat map that reports on how important each pixel within an image was in making the decision of how to classify that image. In the specific case of the classification of images featuring a dog, we would expect pixels that fall within the dog to carry more information than those pixels that are in the background of the image.

Similarly, because regulatory sequences (Fig 4A) are represented as images using the one-hot encoding approach (Fig 4B), the saliency map of each sequence describes how important each pixel within the image—a binary pattern unique to each sequence—is for determining the gene expression bin corresponding to that sequence (Fig 4C). As a result, these saliency maps can be loosely thought of as heatmaps reporting on the sensitivity of the predicted gene expression level to mutating each nucleotide at every position along the regulatory sequence. As described in detail in the “Saliency maps” section of the Materials and methods, the generation of these maps involves calculating the derivative of the network loss function—a measure of how well the network does at predicting output gene expression—with respect to each pixel of the images encoding for the DNA sequence in the binary input layer of the network.

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Fig 4. Saliency map generation and binding site identification.

(A) Sequence variants from the test subset of a given operon, each containing random point mutations, are processed through the pre-processing pipeline to generate one-hot encodings, as illustrated in (B). (C) These one-hot encoded sequences are input into the trained DARSI model, where the gradient of the network loss is computed with respect to each pixel in the input image for each variant. These gradients measure the sensitivity of the network output to each nucleotide. Gradients are averaged across all variants in the test subset to generate saliency maps, which are represented as heatmaps. These heatmaps indicate the pixels containing the most information used by the network to classify the gene expression bin of the input sequence. (D) An example saliency map for the illustrative yqhC operon highlights regions of high sensitivity. Notably, the saliency values at the same sequence position are relatively insensitive to base pair identity, suggesting that DARSI primarily relies on positional information rather than specific nucleotide identity to predict gene expression. (E) Maximum saliency values at each position are normalized and exponentiated to produce unitless plots of gene expression sensitivity to mutations, as shown for the yqhC operon. (F) These sensitivity plots are further refined by filtering peaks that exceed one standard deviation above the mean (dashed line in (E)), span at least 10 bp, and show contiguous effects as either activators or repressors. (G) The refined plots enable the identification of potential binding sites and operon regulatory architectures. For the yqhC operon, two previously annotated regions were identified: a promoter associated with the operon and an activator binding site, which, although annotated, had not been previously associated with the regulation of yqhC.

https://doi.org/10.1371/journal.pcbi.1014092.g004

By applying this process to all sequence variants of a given operon in the test dataset, we generated an ensemble of saliency maps, one for each sequence variants. These maps are then averaged to produce a final saliency map for the operon. As shown in Fig 4D, the saliency map for the illustrative yqhC operon used throughout this study reveals several segments along the sequence that exhibit higher information content for predictions made by the trained DARSI model. Notably, minimal variation is observed along individual columns, suggesting that the network primarily considers the positional context of the base pair rather than its specific nucleotide identity when classifying expression levels. These clusters of highly sensitive positions form the initial hypotheses for the locations of binding sites within the sequence.

To interpret the information encoded within the saliency maps, the maximum saliency value among the four nucleotides at each position along the regulatory sequence—that is, along each column of the saliency map—is calculated. The result is a saliency vector that reports on the sensitivity of output gene expression to mutation along the regulatory sequence. Note that the absolute values of saliency maps generated by the network are not inherently interpretable; only relative changes in these values are meaningful. As a result, we normalize the saliency vector by subtracting its mean and dividing by its standard deviation. Subsequently, the normalized saliency vectors are exponentiated to represent likelihoods or probabilities (see the “Binding site identification” section in the Materials and methods). These processed values are visualized as “gene expression sensitivity to mutation” plots. Fig 4E shows an example of this plot for the yqhC operon. Because we are after binding site-sized features within these plots, we smoothed the curve by averaging the data using a sliding window of size 5 bp as in previous studies [8]. Further examples are provided in the S1 Fig.

To identify transcription factor binding sites, we examined the smoothed values of the gene expression sensitivity to mutation shown in Fig 4E. Here, red bars correspond to positions along the sequence where mutations led to an increase in expression, suggesting potential repressor binding sites. Conversely, blue bars represent nucleotides where mutations resulted in decreased expression, indicating potential activator binding sites.

To predict binding sites along regulatory sequences, we identify clusters of base pairs with high sensitivity. Specifically, following the approach of [58], we detect positions where the maximum sensitivity exceeds one standard deviation above the mean sensitivity across the entire sequence (horizontal line in Fig 4E). In accordance with the minimum length of DNA binding sites in E. coli reported by [59–61], potential binding sites are defined as regions exceeding this threshold and spanning at least 10 base pairs. Fig 4F presents a filtered expression sensitivity-to-mutation plot, highlighting two prominent peaks (blue) corresponding to regulatory regions annotated in RegulonDB [7]. The first peak aligns with a promoter previously mapped to yqhC, while the second corresponds to activator binding sites that, although annotated in RegulonDB, had not been associated with the regulation of yqhC. This finding highlights DARSI’s ability to identify functional connections between regulatory elements and their target genes. Notably, these activator binding sites are not obvious when examining the mutual information (S2 Fig), which constitutes the basis of previous approaches for identifying binding sites using MPRA data [8]. This particular example demonstrates DARSI’s capacity to reveal regulatory features overlooked by traditional methods. A detailed description of the filtering steps employed to generate these expression shift plots is provided in the “Binding site identification” section of the Materials and methods. Further, a comparison of the sensitivity of all operons predicted by DARSI to the same analysis based on mutual information can be found in the GitHub repository.

Using this pipeline, we identified a total of 172 binding sites across all 95 operons, successfully capturing 88.4% of the previously documented sites in published and curated databases [7]. In addition to these annotated binding sites, DARSI predicted 73 hypothetical novel binding sites, spanning more than one-third of the operons in the MPRA dataset (Fig 5A). We classify sites as promoters only if they have been previously mapped as such; otherwise, we label them as activator sites. Additionally, binding sites located within 5–6 bp of each other are reported as a single site for a more conservative assessment.

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Fig 5. Benchmarking binding sites sites identified by DARSI against curated RegulonDB dataset.

(A) Venn diagram showing the number of binding sites—both transcription factor binding sites and promoters—identified by DARSI and how that number is compared to the previously known sites. DARSI identified a total of 172 binding sites across all 95 operons, capturing 88.4% of previously known sites documented in published and curated databases [7], and predicted the existence of 73 new binding sites. (B) Bar plots illustrating the number of sites uncovered by DARSI, the number of sites previously annotated in RegulonDB [7], and the newly identified sites and the sites missed by DARSI across all 95 operons analyzed in this study.

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Fig 5B provides a detailed summary of the binding sites identified by DARSI, the missed binding sites, and the newly predicted hypothetical sites for each operon, alongside annotations from the RegulonDB database [7]. Notably, DARSI failed to identify 13 previously annotated sites in RegulonDB for the ykgE, yicJ, rapA, yeiQ, ybeZ, yjjJ, tff-rpsB-tsf, poxB, rspA, ompR, yjiY, znuA, and leuABCD operons. These missed sites were primarily located within regulatory architectures containing multiple binding sites, such as the ykgE and ompR operons. Examples of regulatory architectures inferred through DARSI are presented in S2 Fig, with comprehensive visualizations for all operons accessible via the GitHub repository. Further, detailed information, including the sequences of each binding site, their genomic coordinates, strand orientation, and prior annotations, is provided in the supplementary table, available for download on the GitHub repository.

MPRA dataset from E. coli

The dataset used throughout this article was generated through the work by [8]. Here, as shown diagrammatically in Fig 1A, a 160-bp long region around the transcription start site of 114 operons in E. coli were randomly mutated at a 10% rate (i.e., each base pair along the sequence had a 10% chance of being mutated from its wild-type base to any of its three alternative bases). This library was then cloned into plasmids driving the expression of a reporter gene (Fig 1B and 1C). Plasmid libraries were transformed into cells and grown in various growth conditions (though in this work we only focus on bacteria grown in LB). The expression from each operon variant was measured by sequencing (Fig 1C).

To normalize for the variation in copy number for each reporter construct, DNA counts of the barcode were also included in the table and were used to normalize the expression counts as discussed in the main text. Processed dataset, therefore, provides both the sequence of the regulatory region and the normalized expression count of the gene regulated by that sequence (Fig 1E). In this study, we only considered operons with enough sequence variants to ensure that, on average, each base pair was mutated in at least 100 variants (i.e., 100x coverage). Given a 10% mutation rate, this corresponds to a minimum of variants. Examples of some of these sequence variants within this dataset for the yqhC operon are provided in S1 Text.

RNA-seq raw data processing

The sequencing datasets used in this work are deposited in the SRA database as PRJNA599253 and PRJNA603368. Code for sequence processing is provided in the Github repository together with example datasets and Jupyter Notebooks that display how to use the data to generate, for example, the processed data outlined in S1 Text. Here, we give a brief description of the process.

Random barcodes were cloned between the promoter and the reporter gene in order to identify the promoter variant through the RNA reads, as well as provide multiple distinct data points that reduce possible bias introduced by barcodes. In an initial sequencing run the promoter sequence and barcodes were sequenced simultaneously to obtain a map that links a regulatory region variant with the corresponding barcode. Pair end reads were merged, quality filtered, and filtered for read length using “fastp” [69]. Promoter sequence and barcode were extracted from each read and the number of occurrences of each barcode and promoter combination counted. A promoter variant can have multiple barcodes associated with it, however, a barcode has to be unique. If a barcode was observed for multiple promoter variants, the barcode was then removed. Additionally, combinations with less than 3 reads were removed due to the possibility of sequencing errors.

In Reg-Seq, the promoter library is grown in various growth conditions to assess a variety of regulatory conditions. For the purpose of this paper, we take one of these growth conditions: growth in LB. From each culture both RNA and DNA (plasmids) are extracted. Using specific primers, the reporter gene mRNA, including the barcode, is reverse transcribed and amplified to generate cDNA and measure gene expression. Barcodes are also amplified from plasmids using PCR in order to count the number of plasmids present with a specific regulatory sequence. Sequencing adapters are added by another PCR and both barcodes obtained from cDNA and plasmid DNA are sequenced. Reads are trimmed and quality filtered using ‘fastp’ [69]. The occurrence of each barcode is counted in the RNA-Seq and DNA-Seq datasets. Finally, using the results from the initial sequencing run, each corresponding promoter variant is identified through its barcode.

One-hot encoding

Every 160 bp long regulatory sequence from the MPRA dataset is converted to a two-dimensional binary image , where each entry of the matrix A_i,j takes the form

(2)

Here, n_j is the j^th nucleotide in each of the sequences. Fig 2D shows an example of these binary images generated for the regulatory sequences for the yqhC operon.

RNA count labeling

The RNA count corresponding to each sequence variant reports on the gene expression level driven by that mutated regulatory region. These values are normalized by dividing the RNA count by the DNA copy number count to ensure that variability in the normalized RNA count is not due to variability in the plasmid copy number.

To bin the normalized expression counts, we developed a binning algorithm to categorize sequence variants into three discrete groups based on their normalized RNA counts: (1) sequences that resulted in zero gene expression, (2) sequences that resulted in low gene expression levels, and (3) sequences that resulted in high gene expression levels. The pipeline first bins all the zero expression counts to the zero expression bin. It then automatically determines the best threshold for separating the remaining gene expression data into the low and high gene expression bins based on their . To make this possible, a t-test is conducted on the difference of the mean gene expression of the low and high expression bin, leading to a separation of data that minimizes the p-value between the bins.

The vector encodes for the expression bins associated with each of the N_i sequence variants for the operon i in the dataset. Each sequence variant for a given operon i is given a label from a set K_i = {1,2,3} where 1, 2, and 3 denote the zero, low, and high expression bins respectively. The algorithm we implemented attempts to partition the vector into three bins such that the p-value associated with a t-test conducted between each pair of bins is minimized [71,70]. The iterative algorithm used to bin the normalized mRNA counts is given in Algorithm 1.

Algorithm 1 RNA count binning algorithm

 1: Take vector as input, where

 2: Values that are infinite in the input vector y_i are associated with a label of 1 for the zero expression bin

 3: Initialize best_bins as an empty cell array of size 2 to store the label associated with variant that have finite log(normalized expression count)

 4: Set best_p_values as an array of size 2 to store lowest p-values obtained throughout the algorithm with all values initialized to ∞

 5: for iter = 1 to max_iteration (set to 10,000 in this paper) do

 6:  Generate random thresholds for 1 bin edge separating the low and high expression classes in the range [min(RNA count), max(RNA count)]

 7:  bins ← Label observations with labels {2,3} that fall within the thresholds for each bin

 8:  p-values ← the p-value associated with a t-test performed on the mean of the low and high expression bins log(normalized expression count)

 9:  if then

 10:   Update best_p_values ← p_values

 11:   Update best_bins ← bins

 12:  end if

 13: end for

Saliency maps

Trained DARSI models demonstrate the capability to predict operon expression levels directly from their nucleotide sequences. Beyond prediction, these models can be utilized to identify potential binding sites within regulatory sequences. This is achieved by analyzing the derivative of the network’s loss function (described in detail below) with respect to the input sequence. By computing these gradients, the nucleotides most critical to the model’s predictive understanding of gene expression can be identified.

The performance of the network is quantified using a cross-entropy loss function defined as

(3)

where, denotes the ground truth label vector for each variant of a given operon. For instance, a variant assigned to the zero expression bin is represented as . Similarly, represents the probability vector predicted by the network for the same sequence. This three-dimensional vector specifies the predicted probabilities of the sequence belonging to each expression bin, as determined by the model.

This loss function measures the discrepancy between the network’s predictions and the ground truth, serving as an indicator of model accuracy. To assess the sensitivity of classification outputs to perturbations in nucleotide sequences, we leverage the gradient-weighted class activation mapping (Grad-CAM) approach [55,57,72]. Grad-CAM computes the gradient of a selected, differentiable output—such as the cross-entropy loss—with respect to neurons or nodes in a specified layer of the network, typically a convolutional layer. For a toy example demonstrating the computation of saliency maps through backpropagation, please refer to S8 Text.

This method allows for the visualization of features critical to the model’s predictions by backpropagating the gradients through the network and overlaying them on the input sequence. The resulting gradient map highlights the pixels within the input image (corresponding to nucleotides within the sequence) that significantly contribute to the network’s decision-making process. By identifying these key features, Grad-CAM enhances the interpretability of deep learning models and provides insights into the regulatory architecture underlying gene expression [55,57,72].

For a two-dimensional image classifier such as DARSI, the saliency score for any given channel in a convolutional layer with k channels is computed by

(4)

where y^c is the predicted posterior for the bin c, is the pixel located at (i,j) position of the k^th channel of the chosen convolutional layer, and N is the total number of pixels [55].

Equation 4 generates a weighting score for every channel k within a convolutional layer. In order to plot this score as a heatmap for any given sequence, similar to the ones shown in Fig 4C and 4D, a weighted-average mask U is computed such that

(5)

where f represents a non-linear activation function such as the rectified linear unit function ReLU [55]. Algorithm 2 summarizes the steps that are taken to generate these saliency maps for each of the genes within the expression shift dataset.

Algorithm 2 Saliency map generation.

 1: for each operon i do

 2:  Train and find the optimized DARSI model

 3:  Load the test subset of the data for the operon i

 4:  Run the Grad-CAM script to generate the masks (U^c) for each variant in the test subset

 5:  Overlay the mask for each image to the input binary image (one-hot encoding representation of the sequence (Fig 4B) and plot this as a heatmap that shows the relative importance of each base pair (Fig 4C)

 6:  Compute a final saliency map by taking an average over all the maps generated across the test data (Fig 4D)

 7:  Compute a 1-dimensional saliency vector for each operon i from the heatmap by taking the maximum value at each nucleotide position, namely for 𝑗 ∈{1, 2,…, 160}

 8:  Plot each of the as a function of base pair position (Fig 4E and 4F).

 9: end for

Binding site identification

As discussed in Algorithm 2 above, the saliency vector computed for a given operon i captures the saliency of each regulatory sequence. The vector is normalized about its mean and standard deviation, namely

(6)

where and are the mean and standard deviation of the vector , respectively.

The vector represents a difference from the mean in sensitivity of expression level to mutation at any given position j. Therefore we assume that this vector is proportional to the derivative of the dissociation constant with respect to that nucleotide, or more formally

(7)

where K_D and n_j are the dissociation constant and nucleotide at position j, respectively.

Finally, we used this proportionality to estimate the probability of occupancy P(j)_i as

(8)

where denotes the absolute value. The use of absolute values is necessary due to prior normalization of the vector , which ensures that both strongly negative and strongly positive normalized values contribute to the probability estimate. This adjustment is crucial to account for regions associated with activators (negative values) and repressors or other functional elements (positive values), ensuring a strong signal is captured in both cases.

The probability was computed for every position j for every operon i and was plotted as bar charts to show the expression shift (Fig 4E). The peaks in probability that were more than one standard deviation from the mean were selected (Fig 4F). These filtered peaks were then passed through a secondary filter to select only regions where the length of a continuous region of repression or activation (i.e., the predicted binding site) is more than 10 bp long—the minimum length of the binding sites in E. coli [59–61]. The process for generating filtered expression shift plots is shown in Algorithm 3.

Algorithm 3 Binding sites identification.

 1: for Each operon i do

 2:  Generate saliency map .

 3:  Normalize vector using equation 6 to generate vector .

 4:  Generate an exponential vector .

 5:  Find elements within that are one standard deviation above the mean of

 6:  Initiate an empty vector to store final expression shift data.

 7:  for Each peak k found in previous step do

 8:    Check that the region around the peak is continuous in either repression or activation using the sign of the normalized saliency vector and that the region is larger than 10 nucleotide

 9:    If a region is found to be above 10 bp long, store the region into the final expression shift vector

 10:  end for

 11:  Plot for the operon i

 12: end for

The DARSI pipeline repository

To enhance the accessibility and usability of DARSI for gene expression prediction and related applications, we have designed the implementation with a modular architecture. Each MATLAB script operates independently, accompanied by comprehensive documentation for ease of understanding. A master script is also provided to sequentially execute the necessary functions, offering detailed guidance on processing raw RNA-Seq data and training a DARSI model.

The complete set of scripts is available in a dedicated GitHub repository. The repository includes modules for processing raw RNA-Seq data, generating expression shift datasets, training DARSI models, performing cross-validation, evaluating model performance, and identifying binding sites. Additionally, all data used to train the DARSI model, along with outputs such as saliency maps, confusion matrices, and expression shift plots, are made available in the repository.