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Mathematical models of biomolecular networks are commonly used to study mechanisms of cellular processes, but their usefulness is often questioned due to parameter uncertainty. Here, we employ Bayesian parameter inference and dynamic network analysis to study dominant reaction fluxes in models of extrinsic apoptosis. Although a simplified model yields thousands of parameter vectors with equally good fits to data, execution modes based on reaction fluxes clusters to three dominant execution modes. A larger model with increased parameter uncertainty shows that signal flow is constrained to eleven execution modes that use 53 out of 2067 possible signal subnetworks. Each execution mode exhibits different behaviors to in silico perturbations, due to different signal execution mechanisms. Machine learning identifies informative parameters to guide experimental validation. Our work introduces a probability-based paradigm of signaling mechanisms, highlights systems-level interactions that modulate signal flow, and provides a methodology to understand mechanistic model predictions with uncertain parameters.