Stage I: Disentangled representations of biological variations and measuring processes

We modeled the measured expression vector as the coupled effects of true biological variations and inevitable measurement noises. Although the measuring process may have distinctive effects on different cell types, it is reasonable to assume the true biological variations are independent of measuring noises. Considering that distilling the underlying biological contents from transcriptome measures is the critical step to remove the batch effects, we first designed a novel autoencoder structure to build representations of biological variations, which are expected to be disentangled from measuring noises.

Three forward neural networks are deployed in this stage, including one content encoder E, two generators (decoders) G1 and G2 (Fig. 1b). The inputs to the model include the expression vector of one cell denoted as x, and its batch indicator vector b. One-hot encoding strategy is used to indicate the batch of the cell. For instance, in the case of three batches, cells from the first batch have their batch indicator vector b = [1, 0, 0]T. The output of the encoder E is denoted as c = E(x), which is expected to exclusively represent the biological contents of cells, and be ignorant of the measuring process. The neural network G1 is deployed to generate the representation of measurement noise G1(b), since the measurement noise cannot be fully captured by a simple one-hot vector. Another generator G2 is further used to finish the reconstruction of the original expression vector. The inputs to the generator G2 include both E(x) and b, because intuitively, it is possible for the generator to reconstruct the original measured expression vector only if both the biological content and measurement noise are simultaneously provided. The final reconstructed expression vector is G(E(x), b) = f(G1(b) + G2(E(x), b)), where f is a non-linear transformation, and is used to match the range of reconstructed vector with the original expression vector. The ReLU function f(x) = max(0, x) can be the default candidate for non-negative expression vectors. The reconstruction loss ( represents expectation) can be formalized as:

The key to successfully extract biological contents of one cell is to disentangle the biological representation c from the corresponding cell batch indicator b. We achieve this by deliberately generating a random batch indicator vector b~ for each cell, where randomly selected one element is set to 1 while others to 0. Well-trained generators G1 and G2, with E(x) and b~ as inputs, should fabricate one cell with the same content as x. This inspired our content loss as:

In summary, the overall loss function of the first stage is:

where λc and λr are tunable hyperparameters to make tradeoffs between the content and reconstruction loss. In our experiments, this loss function can be optimized at low operating cost, to obtain sufficiently good representations, especially for the identification of the batch-specific cells. However, the overwhelming researches in the field of deep learning have confirmed that it is hard to generate images indistinguishable from true ones by only optimizing the reconstruction loss of autoencoders [40], which inspired us to add the adversarial structures in the stage II, further removing the batch effects from the original expression profiles.

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