110 Stability of Genetic Oscillators with Distributed Delayed Feedback

ABSTRACT

Genetic oscillators govern periodic phenomena in biology including circadian rhythms and are also the basis of biological clocks used in the design of synthetic genetic circuits. Models of genetic oscillators tend to neglect biological detail, however, because biological systems tend to be too complicated to model efficiently. One way to incorporate additional biological detail into models of genetic oscillators is to use distributed delay differential equations. To investigate the utility of distributed delay differential equations for modeling genetic oscillators, we constructed delayed differential equation models of genetic oscilla- tory motifs. We found that these models are equivalent to higher-dimension models, which are reflective of more granular biological detail. We also characterized the stability of these models. Our findings may inform future modeling efforts in the domains of synthetic and systems biology, where delayed differential equations could pose advantages over ordinary differential equation models.

INTRODUCTION

Genetic oscillators, which are intrinsic time-keeping devices in living organisms, have been the subject of research in the field of mathematical biology for some time [1, 2]. The study of these oscillators has led to a deeper understanding of the molecular mechanisms that underlie a wide range of biological processes, including circadian rhythms [3], cell cycle regulation [4], and developmental processes [5]. Circadian rhythms, for instance, are endogenous rhythms that persist in the absence of external cues and are responsible for co- ordinating a wide range of physiological and behavioral processes [6]. Biological clocks, based on the principles of genetic oscillators, have been engineered to regulate the expres- sion of genes in synthetic circuits [7, 8], with applications ranging from biosensors [9] to drug delivery systems [10].

While genetic oscillators have become increasingly well understood, modeling their behavior presents a significant challenge [11]. Mathematical models of genetic oscilla- tors must balance the need for accuracy with computational efficiency, as the large number of molecular interactions involved in these systems can be computationally expensive to model in detail [12]. As a result, models of genetic oscillators tend to simplify the un- derlying biology, often neglecting important biological details in favor of simplicity and efficiency [13, 14]. These simplified models may fail to reflect dynamics observed in vivo by, for example, altering system stability [11].

Distributed delay differential equations are a promising approach to modeling genetic oscillators with greater biological detail [15]. Unlike ordinary differential equations which ignore time delays entirely and discrete delay differential equations which assume constant delay in feedback steps [16, 17], distributed delay differential equations allow for a distri- bution of delays that more closely approximates the complex biological interactions that occur in living organisms. These models have been shown to accurately capture the be- havior of some simple genetic oscillators [18], while also remaining intuitive and simple to simulate. To investigate the utility of distributed delay differential equations for model- ing genetic oscillators, we constructed and analyzed distributed delay differential equation models of repressilators, one important class of synthetic genetic oscillator [2, 7].

ACKNOWLEDGEMENTS

We would like to thank Professor Elena Cherkaev at the University of Utah who pro- vided mentorship and expertise throughout this project.

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