Mathematical Modeling and Inference of Epidermal Growth Factor-Induced Mitogen-Activated Protein Kinase Cell Signaling Pathways

Abstract

The mitogen-activated protein kinase (MAPK) pathway is an important intracellular signaling cascade that plays a key role in various cellular processes. Understanding the regulatory mechanisms of this pathway is essential for developing effective interventions and targeted therapies for related diseases. Recent advances in single-cell proteomic technologies have provided unprecedented opportunities to investigate the heterogeneity and noise within complex, multi-signaling networks across diverse cells and cell types. Mathematical modeling has become a powerful interdisciplinary tool that bridges mathematics and experimental biology, providing valuable insights into these intricate cellular processes. In addition, statistical methods have been developed to infer pathway topologies and estimate unknown parameters within dynamic models. This review presents a comprehensive analysis of how mathematical modeling of the MAPK pathway deepens our understanding of its regulatory mechanisms, enhances the prediction of system behavior, and informs experimental research, with a particular focus on recent advances in modeling and inference using single-cell proteomic data.

Keywords: cell signaling pathway; epidermal growth factor; inference; mathematical model; mitogen-activated protein kinase; single cell.

Figure 1

Schematic overview of the Ras–Raf–MEK–ERK module: In the cytosolic subsystem, the Ras–Raf–MEK–ERK pathway begins when the input signal Ras–GTP activates Raf, which subsequently activates MEK through a single-step processive module. MEK then activates ERK kinase in a two-step distributive manner. Both active and inactive forms of MEK and ERK are capable of freely diffusing between the cytosol and the nucleus. In the nuclear subsystem, activated MEK can further activate ERK. Specific phosphatases, such as Raf phosphatase, MKP, and STP, deactivate the active forms of Raf*, MEKpp, and ERKpp at various subcellular locations [71].

Figure 2

Schematic overview of the MAP kinase pathway and the PI3K/AKT pathway activated by EGF receptors. The box with the green dashed line includes the Ras–Raf–MEK–ERK module, as shown in Figure 1, while the box with the blue solid line encompasses the EGF-induced MAPK pathway.

Figure 3

Mechanistic and data dual-driven approaches for modeling cell signaling pathways. (A) Mechanistic modeling approaches rely on experimentally discovered regulatory mechanisms, kinase activity data, and dynamic models to simulate cell signaling pathways. (B) Data-driven modeling approaches utilize static correlation network models, omics datasets, and statistical methods or machine learning algorithms to analyze signaling pathways. Two main combination techniques are employed for dual-driven approaches. The parallel structure approach uses weighting techniques to merge results from different models into a single output, whereas the serial structure approach uses the prediction of one model as the input for another model. (C) Inferred network model: The final network model is constructed by integrating predictions from the dual-driven approaches.

Figure 4

Modeling and simulation of cell signaling pathways using single-cell data. (A) Data types: Single-cell data include time-lapse and snapshot proteomic data, with pseudo-time trajectories generated from snapshot data using bioinformatics methods. (B) Model types: Stochastic models may involve chemical reaction systems (CRS) or stochastic differential equations (SDEs). Multi-scale models combine multiple model types, such as ODEs, CRS, and SDEs. (C) Simulation types: Stochastic simulations can be generated from either stochastic models or multi-scale models. (Red-line in deterministic: the average simulation of all simulations; lines in stochastic and NLMEM with different color: different simulations of the stochastic model and NLMEM model, respectively).