I will present two studies of systems under tight regulation. The first example concerns epidemics spread in a population under strong control measures. We developed a realistic, analytically solvable framework (SPEIRD) that explicitly incorporates the influence of social distancing measures on COVID-19 dynamics [1]. Starting from SPEIRD, we derived closed-form expressions for epidemiological observables, such as detected cases and fatalities, revealing simple quantitative relationships with model variables [2]. These analytical solutions offer valuable insights into the cause-effect connections underlying outbreak dynamics, often obscured in numerical approaches that were up to now exclusively used to study the effects of social distancing. As the second example, we consider the dynamics of bacterial restriction-modification systems, which modulate horizontal gene transfer, including the transmission of pathogenic genes, such as determinants of antibiotic resistance or virulence. These systems are often tightly regulated by a specialized control (C) protein through a complex regulation that involves cooperative (nonlinear) positive and negative feedback loops. We develop a nonlinear-dynamics model of the system regulation [3] and analytically derive the system stability diagram, where a single free parameter determines if the system is bistable. We infer this parameter from experimental measurements for three systems with different architectures. We show that our model can explain the experimental data and that modulation of the barrier to horizontal gene transfer may be achieved through qualitatively different mechanisms (with or without bistability). Overall, these two examples illustrate that while systems under tight and strongly nonlinear control are commonly analyzed only numerically, analytical results can provide a simple and intuitive understanding of the system dynamics.
[1] Djordjevic M, Djordjevic M, Ilic B, Stojku S, Salom I, Understanding infection progression under strong control measures through universal COVIDâ€19 growth signatures, Global challenges, 5(5), 2000101, 2021.
[2] Ilic B, Salom I, Djordjevic M, Djordjevic M, An analytical framework for understanding infection progression under social mitigation measures, Nonlinear Dynamics, in press, 2023.
[3] Djordjevic M, Zivkovic L, Djordjevic M, Bistability in the regulatory dynamics of bacterial restriction-modification systems, in preparation, 2023.
Acknowledgments: This work is supported by the Science Fund of the Republic of Serbia (grant no. 7750294, q-bioBDS) and by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia.