Fractal Principles of Cellular Systems for Unified Biology

The complexity of cellular systems, encompassing metabolic and signaling networks, poses significant challenges for classical systems biology approaches that rely on linear, reductionist, or mean-field frameworks. Here, we introduce a novel Pellis-based systems biology framework that leverages the mathematical properties of the Pellis Function, a φ-scaled fractal operator, to capture the multi-scale, non-linear, and adaptive dynamics of cellular networks. This framework formalizes cellular processes as fractal energy-information flows, unifying metabolic flux distributions, signal transduction cascades, and cross-scale regulatory interactions within a single, scale-invariant formalism. We demonstrate that metabolic networks exhibit φ-scaled hierarchical modularity, with fractal flux distributions optimizing both energy efficiency and robustness. Similarly, signaling networks are shown to obey temporal and spatial fractality, enabling noise filtering, signal amplification, and oscillatory coordination across cellular compartments. By integrating metabolic and signaling layers into a multilayer Pellis network, emergent behaviors-including adaptation, resilience, and antifragility-are quantitatively captured and predicted. The framework extends to multifractal and temporal analyses, revealing critical transitions, long-range correlations, and system-level memory effects that classical models fail to describe. Furthermore, the Pellis Function is operationalized for computational modeling, parameter inference from omics and imaging data, and informed predictive simulations, opening avenues for personalized and precision systems biology. Applications range from disease modeling-including cancer and metabolic disorders-to synthetic network design, highlighting universal scaling laws that bridge evolutionary, developmental, and pathological contexts. Overall, this work establishes a foundational, scale-invariant, and mathematically rigorous framework for systems biology, positioning the Pellis Function as a unifying operator for the quantitative understanding, prediction, and control of cellular network dynamics.