Modeling Co-evolutionary Dynamics of CRISPR Immunity and Phage Mutations Using Stochastic Simulations

Viruses evolve by accumulating mutations that help them evade recognition by host immune systems, driving an ongoing co-evolutionary arms race with adaptive immune responses. In this study, we develop a computational model that simulates the interaction between bacteriophages and bacteria, focusing on the CRISPR-Cas immune system, which enables bacteria to defend against phage infections. Using a stochastic Gillespie algorithm, we model key biological processes such as bacterial growth, phage reproduction, spacer acquisition (immune memory), and phage mutation. When bacteria successfully defend against phages, they incorporate viral DNA fragments (spacers) into their CRISPR arrays, allowing them to recognize and neutralize the same phages in future encounters. This dynamic system, where bacteria continually adapt to phage mutations by acquiring new spacers, reflects the co-evolutionary tension between bacterial immune memory and viral escape strategies. Our results show that the structure and efficiency of CRISPR arrays, particularly the addition of new spacers at the leader-distal end, play a critical role in bacterial population stability. We compare systems with symmetric and non-symmetric spacer efficiency distributions and explore the implications of these different efficiency patterns, as observed in experimental systems. Additionally, we find that high phage mutation rates can overwhelm bacterial defenses, leading to continuous cycles of infection and adaptation. This study provides insights into the co-evolutionary dynamics between bacteria and phages, emphasizing the importance of CRISPR-based immune memory and spacer retention for bacterial survival. Our computational model offers a framework to explore how CRISPR immunity shapes microbial ecosystems and influences long-term population stability.