Time Inhomogeneous Mutation Models with Birth Date Dependence

Area of Science:

• Mathematical Biology
• Evolutionary Biology
• Genetics

Background:

• The Luria-Delbrück model is foundational for understanding spontaneous mutations in microbial populations.
• Classic models often assume independent and identically distributed (i.i.d.) cell division times and do not account for cell death.

Purpose of the Study:

• To develop an extended Luria-Delbrück model incorporating non-i.i.d. cell division times and cell deaths.
• To derive new probability distributions for mutant cell counts under more complex biological conditions.
• To provide a robust mathematical framework for analyzing mutation dynamics.

Main Methods:

• Extension of the Bellman-Harris integral equation to account for birth-date-dependent cell lifetimes.
• Inclusion of cell death probabilities for both normal and mutant cells.
• Development of an analytic method to prove convergence theorems for final mutant counts.

Main Results:

• A generalized family of probability distributions for mutant cell counts is derived.
• The model successfully recovers the classic Luria-Delbrück distribution when cell deaths are excluded.
• Specific cases, such as the Haldane model and proportional hazard functions, are analyzed.

Conclusions:

• The extended model offers a more comprehensive approach to fluctuation analysis, applicable to non-exponential growth and cell death scenarios.
• The derived probability distributions and convergence theorems enhance the predictive power of mutation analysis.
• A computational algorithm is provided for practical application of the model.