Inverse Problem Antidote (IPA): Modeling of Systems Biology Model with Invertible Neural Networks

In computational biology, accurately modeling biological systems is essential for understanding the underlying mechanisms of biological processes. One of the primary challenges in modeling lies in the inability to measure certain biological parameters directly. The task of identifying the parameters is known as the inverse problem and it entails estimating these unobservable parameters from available data. Existing modeling methods primarily focus on single-direction prediction. These methods include traditional Partial Differential Equation (PDE) modeling and machine learning approaches where given parameters are provided to predict the output and further compare the output with experimental evidence. However, these unidirectional methods struggle to effectively apply the experimental evidence directly to address the inverse problem. We contend that a single biological process should be modeled bidirectionally to simultaneously address the inverse problem. To this end, we propose leveraging the capabilities of the invertible neural network (INN) to establish a connection between the input parameter space and the model output space. We meticulously designed a bidirectional training technique that effectively applies real-world experimental data in guiding the INN in modeling the biological process. We tested our approach on a PDE-based Bone Morphogenic Protein (BMP) signaling network system in the zebrafish embryo and found the bidirectional modeling approach significantly enhances the alignment between simulation and experimental data. This method achieves a 94.65% reduction in Root Mean Square Error (RMSE) compared to the single-direction model when reconstructing experimental data using simulation with INN-identified parameters. Moreover, our method is rapid to implement, facilitating the precise identification of parameter ranges from experimental data. It makes parameter optimization feasible and provides a guideline for determining simulation parameter ranges, bypassing the traditional and laborious trialand-error method.