Optimality of the spontaneous prophage induction rate

Lysogens are bacterial cells that have survived after being infected by bacterial viruses called bacteriophages. Instead of being killed by the virus, the infected cell survives by integrating the viral DNA into its own genome. This is only possible with "temperate" bacteriophages which do not always lyse their host to reproduce, but sometimes replicate passively using the lysogenic pathway. After an infection resulting in lysogeny, the lysogen continues to grow and divide normally, seemingly unaffected by the integrated viral genome which is now referred to as a prophage. However, the prophage can have an impact on the hosts phenotype and overall fitness in certain environments. This makes competition between the lysogen and its nonlysogen counterpart possible because both cells have different genomes and potentially different growth rates. Additionally, the prophages within the lysogens are capable of spontaneously reverting back to the lytic pathway via spontaneous prophage induction (SPI), causing death of the lysogen and the release of new progeny phages. These new phages can then lyse or lysogenize other susceptible nonlysogens, thereby impacting the competition between lysogens and nonlysogens. In a scenario with differing growth rates, it is not clear whether SPI would be beneficial or detrimental to the lysogens since it directly causes cell death but also attacks nonlysogenic competitiors, either lysing or lysogenizing them. In this work we study the evolutionary dynamics of a mixture of lysogens and nonlysogens and derive general conditions on the rate of SPI resulting in lysogens displacing nonlysogens. We show that there exists an optimal SPI rate, and apply the model to bacteriophage{lambda} . We find that the model can explain why the experimentally measured SPI rate for phage{lambda} is so low. We also investigate the impact of stochasticity and conclude that even at low copy numbers the SPI rate can still be fairly low while still providing an advantage to the lysogens.