About MCKcat Model

1 Extracting Multi-Scale Convolutional Features

Because the dimensions of the pre-trained representations of enzymes and reactions are different, we extract hierarchical convolutional features from the pre-trained representations of enzymes to align the enzyme embeddings and reaction representations, which will then be used in the subsequent bidirectional cross-attention block.

Specifically, first of all, we use three convolutional layers in parallel. The sizes of the convolutional kernels are (2, 1024), (3, 1024) and (4, 1024) respectively. The number of input channels is 1, and the number of output channels is num_filters = 256. Then, we concatenate these three convolutional results along the sequence dimension to obtain the feature matrix \( M \in \mathbb{R}^{(3L-6) \times 256} \). In this way, we capture the feature information of the enzyme sequence at different local scales.

2 Cross-Attention Fusion Block

The appropriate integration of the enzyme residue embeddings and the reaction representations is crucial to our prediction. Our goal is to predict the \( k_{cat} \) value of an enzyme-catalyzed reaction, which can be interpreted as the influence of the active site of an enzyme on the corresponding catalytic reaction. Therefore, the integration module is responsible for cross-talking between the reaction and the protein sequence context.

Here, a cross-attention fusion module is used to integrate these two representations. The concept of cross-attention is derived from Perceiver, which is a variant of the Transformer architecture. Perceiver performs excellently in handling multi-modal inputs, being able to compress dimensions and improve computational speed. Previous studies have applied Perceiver to the prediction of compound-protein interactions as well as the prediction of functional phosphorylation sites and their regulatory types.

Here, we input the feature matrix \( M \) of the enzymes into the cross-attention fusion block along with the reaction representation to obtain fully integrated embeddings for enzyme-reaction combinations. Specifically, we set the query \( Q \), the key \( K \), and the value \( V \) with the reaction representation \( h_r \), the enzyme feature matrix \( M \), and \( M \), respectively, and obtain the updated reaction embeddings \( \hat{h}_r \) as follows:

where \( C=256 \) is the embedding dimension and \( layer\_norm \) is the layer normalization operation. Similarly, since the reaction is a fixed-dimensional vector with sequence length 1, the attention weight from the enzyme to the reaction, \( attention_2 \), degenerates to an all-one vector, i.e., \( attention_2 = \mathbf{1} \in \mathbb{R}^{(3L-6) \times 1} \). We can obtain an updated enzyme feature matrix \( \hat{M} \) with the following equation:

Then, we perform the max pooling in \( \hat{M} \) along the sequence dimension to obtain the vector representation of the enzyme, which is then concatenated with the updated reaction representation \( \hat{h}_r \) to serve as the embedding for the enzyme-reaction combination.

3 Prediction Network & Ensemble Strategy

Finally, we input the embeddings of enzyme-reaction combinations into a three-layer fully connected network. The number of neurons in each layer is 256, 64, and 1 respectively. The output denotes the predicted \( k_{cat} \) value. Here, we choose the mean squared error (MSE) loss function and the Adam optimizer to train the model. Five-fold cross-validation on the training set is used to determine the optimal hyperparameters. The learning rate is set to \( 5\times10^{-5} \), and the weight decay is set to \( 1\times10^{-5} \).

To further enhance prediction performance and robustness, we adopt an ensemble learning strategy. Specifically, we use 10 different seeds of random number to generate different initial weight parameters for the models, train 10 prediction models, and then average their prediction results as the final prediction outcome.