A singular limit for an age structured mutation problem.

Area of Science:

• Mathematical Biology
• Cell Population Dynamics
• Systems Biology

Background:

• Cell population dynamics are often modeled using ordinary differential equations (ODEs) to track subpopulation sizes based on genome.
• Cellular processes like vital dynamics and mutations, particularly during cell division, introduce complexity not always captured by simple ODEs.
• McKendrick-type equations, akin to network transport problems, offer an alternative framework for these complex dynamics.

Purpose of the Study:

• To reconcile two distinct modeling approaches for cell population dynamics: ODEs and McKendrick-type equations.
• To demonstrate the asymptotic equivalence between these two modeling frameworks under specific conditions.
• To provide a unified theoretical basis for understanding trait spread and cell evolution.

Main Methods:

• Development and analysis of a system of ordinary differential equations for trait spread.
• Formulation of a McKendrick-type equation system incorporating cell vital dynamics and mutations.
• Application of appropriate scaling transformations to the McKendrick-type equations.
• Asymptotic analysis to compare the long-term behavior of both model types.

Main Results:

• The study establishes a formal link between ODE models and McKendrick-type models.
• It is shown that under specific scaling, the McKendrick-type equations asymptotically converge to the ODE description.
• This equivalence holds for modeling the spread of traits within cell populations with inherent cellular dynamics.

Conclusions:

• The findings bridge the gap between simplified ODE models and more complex McKendrick models for cell populations.
• This unification offers a more comprehensive understanding of trait evolution, incorporating cell division and mutation dynamics.
• The research provides a valuable theoretical tool for analyzing complex biological systems and predicting population behavior over time.