Accumulation of Mutations in Stochastically Growing Colonies: Theory and Applications

The work described herein extends results for cellular growth. Previous endeavors have used linear rates with birth and death processes to model mutation and obtain expected numbers of mutant cells relative to a stopping criterion. Much of this work has followed from the seminal results of Luria and Delbruck in 1943.;This paper gives an overview of previous results based on linear rates and develops models that use nonlinear rates of growth. The primary focus is on logistic rates that exhibit an asymptotic behavior with growth of population. An asymptotic model is more realistic than a linear rate that promotes growth without limit since most actual systems exhibit a slowing of growth as competition for resources increases. A fundamental overview is presented to relate results to the Kolmogorov equations and boundary layer methods that permit approximate solution of the resulting equations. After contrasting logistic and linear mutations, subsequent discussion focuses on a general deterministic model with attention to properties that regulate the number of mutants in a colony.;Results for nonlinear models indicate that the population of mutant cells increases with death rate for fairly general models. Moreover, the studies reveal that logistic models give a higher mutant population than linear models. This is significant since the natural limit of population raises the question of the effect on the ultimate mutant population. Results from the general modeling indicate that colonies with primary regulation being the death rate show a maximum number of mutants if no back-mutation is allowed. Correspondingly, the number of mutants is a minimum if the mechanism for regulation is division-control.;Additional work examines models for the resistance of mutations to radiation after adaptation to small doses. A novel model for communication based on mass action is presented that describes development of resistance by exposure to a small initial dosage. A second model for adaptation to radiation is a memory model which seems more widely accepted. Monte Carlo methods are used to implement spatial constraints for cellular interaction, and the memory model and communication model are compared to determine conditions under which the models can be distinguished.