A Mathematical Model of the Cell Cycle: Exploring the Impact of Zingerone on Cancer Cell Proliferation

TL;DR

The study addresses how CDK1–APC interactions shape cancer cell-cycle oscillations and whether the natural compound $Zingerone$ can slow proliferation. It builds a two-variable dynamical system with Hill-type nonlinearities, analyzes invariant domains, equilibria, and a unique stable limit cycle, and shows the oscillation period scales with the cyclin-synthesis rate $\alpha_1$. A theoretical cancer-population model links cell-cycle timing to population growth via $\tau$, $r$, and the viability $CV(t,z)$, with experimental melanoma data used to fit $CV(t,z)$ and extract IC$_{50}(t)$ curves. The results suggest that Zingerone reduces cell viability by modulating $\alpha_1$-driven period shortening, providing a quantitative framework for evaluating natural compounds in cancer therapy.

Abstract

This paper presents a mathematical model that explores the interactions between Cyclin-Dependent Kinase 1 (CDK1) and the Anaphase-Promoting Complex (APC) in cancer cells. Through the analysis of a dynamical system simulating the CDK1-APC network, we investigate the system's behavior and its implications for cancer progression and potential therapeutic interventions. Our findings highlight the critical role of CDK1-APC interactions in regulating the cell cycle and examine the impact of Zingerone, a compound derived from ginger, on modulating the period of the oscillatory dynamics. These results provide new insights into the potential of Zingerone to influence cell proliferation and offer avenues for less harmful cancer treatments. The quantitative analysis is conducted by first theoretically deriving the cell viability as a function of time and Zingerone concentration, and then validating this function by using experimental data.

Figures (27)

• Figure 1: Image Source: cell_cycle_image - Progression through the cell cycle. The cell cycle consists of two main phases: Interphase, in which the cell undergoes initial growth (G1), DNA synthesis (S), and secondary growth (G2), and Mitotic phase (M), in which the cell undergoes mitosis and cytokinesis.
• Figure 2: A simplified diagram illustrating CDK1 activation through its binding to Cyclin B.
• Figure 3: A simplified diagram depicting APC targeting Cyclin B for degradation, resulting in CDK1 inactivation.
• Figure 4: Image Source: oscillating_model - This Figure illustrates a simplified depiction of the cell cycle. This figure highlights the main interactions between CDK1 and APC. For further details, the reader is referred to oscillating_model.
• Figure 5: Solution curves displaying the level of activations of CDK1 and APC over time.

Theorems & Definitions (11)

• Definition 3.1: Dynamical System
• Definition 3.2: Flow-Invariance
• Theorem 3.1: Tangent Condition
• Proposition 3.1: Locally Lipschitz Condition
• Theorem 3.2: Picard-Lindelöf Theorem
• Theorem 3.3: Invariance Theorem
• Theorem 3.4: Local Existence
• Theorem 3.5: Existence and Uniqueness for Locally Lipschitz Problems
• Definition 3.3: Positive Semi-orbit
• Definition 3.4: Limit Set