From Bench to Keyboard and Back Again: A Brief History of Lambda Phage Modeling.

Cellular decision making is the process whereby cells choose one developmental pathway from multiple possible ones, either spontaneously or due to environmental stimuli. Examples in various cell types suggest an almost inexhaustible plethora of underlying molecular mechanisms. In general, cellular decisions rely on the gene regulatory network, which integrates external signals to drive cell fate choice. The search for general principles of such a process benefits from appropriate biological model systems that reveal how and why certain gene regulatory mechanisms drive specific cellular decisions according to ecological context and evolutionary outcomes. In this article, we review the historical and ongoing development of the phage lambda lysis-lysogeny decision as a model system to investigate all aspects of cellular decision making. The unique generality, simplicity, and richness of phage lambda decision making render it a constant source ofmathematical modeling-aided inspiration across all of biology. We discuss the origins and progress of quantitative phage lambda modeling from the 1950s until today, as well as its possible future directions. We provide examples of how modeling enabled methods and theory development, leading to new biological insights by revealing gaps in the theory and pinpointing areas requiring further experimental investigation. Overall, we highlight the utility of theoretical approaches both as predictive tools, to forecast the outcome of novel experiments, and as explanatory tools, to elucidate the natural processes underlying experimental data.

Optimality of the spontaneous prophage induction rate.

Lysogens are bacterial cells that have survived after genomically incorporating the DNA of temperate bacteriophages infecting them. If an infection results in lysogeny, the lysogen continues to grow and divide normally, seemingly unaffected by the integrated viral genome known as a prophage. However, the prophage can still have an impact on the host's phenotype and overall fitness in certain environments. Additionally, the prophage within the lysogen can activate the lytic pathway via spontaneous prophage induction (SPI), killing the lysogen and releasing 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 kills the host cell but also attacks nonlysogenic competitors, either lysing or lysogenizing them. Here we study the evolutionary dynamics of a mixture of lysogens and nonlysogens and derive general conditions on SPI rates for lysogens to displace nonlysogens. We show that there exists an optimal SPI rate for bacteriophage λ and explain why it is so low. We also investigate the impact of stochasticity and conclude that even at low cell numbers SPI can still provide an advantage to the lysogens. These results corroborate recent experimental studies showing that lower SPI rates are advantageous for phage-phage competition, and establish theoretical bounds on the SPI rate in terms of ecological and environmental variables associated with lysogens having a competitive advantage over their nonlysogenic counterparts.

Coupling of DNA Replication and Negative Feedback Controls Gene Expression for Cell-Fate Decisions.

Cellular decision-making arises from the expression of genes along a regulatory cascade, which leads to a choice between distinct phenotypic states. DNA dosage variations, often introduced by replication, can significantly affect gene expression to ultimately bias decision outcomes. The bacteriophage lambda system has long served as a paradigm for cell-fate determination, yet the effect of DNA replication remains largely unknown. Here, through single-cell studies and mathematical modeling we show that DNA replication drastically boosts cI expression to allow lysogenic commitment by providing more templates. Conversely, expression of CII, the upstream regulator of cI, is surprisingly robust to DNA replication due to the negative autoregulation of the Cro repressor. Our study exemplifies how living organisms can not only utilize DNA replication for gene expression control but also implement mechanisms such as negative feedback to allow the expression of certain genes to be robust to dosage changes resulting from DNA replication.

Late-Arriving Signals Contribute Less to Cell-Fate Decisions.

Gene regulatory networks are largely responsible for cellular decision-making. These networks sense diverse external signals and respond by adjusting gene expression, enabling cells to reach environment-dependent decisions crucial for their survival or reproduction. However, information-carrying signals may arrive at variable times. Besides the intrinsic strength of these signals, their arrival time (timing) may also carry information about the environment and can influence cellular decision-making in ways that are poorly understood. For example, it is unclear how the timing of individual phage infections affects the lysis-lysogeny decision of bacteriophage λ despite variable infection times being likely in the wild and even in laboratory conditions. In this work, we combine mathematical modeling with experimentation to address this question. We develop an experimentally testable theory, which reveals that late-infecting phages contribute less to cellular decision-making. This implies that infection delays lower the probability of lysogeny compared to simultaneous infections. Furthermore, we show that infection delays reduce lysogenization by providing insufficient CII for threshold crossing during the critical decision-making period. We find evidence for a cutoff time after which subsequent infections cannot influence the cellular decision. We derive an intuitive formula that approximates the probability of lysogeny for variable infection times by a time-weighted average of probabilities for simultaneous infections. We validate these theoretical predictions experimentally. Similar concepts and simplifying modeling approaches may help elucidate the mechanisms underlying other cellular decisions.