Modeling and minimization of dCas9-induced competition in CRISPRi-based genetic circuits

Implementing logic functions in living cells is a fundamental area of interest among synthetic biologists. The goal of designing biochemical circuits in synthetic biology is to make modular and tractable systems that perform well with predictable behaviors. Developing formalisms towards the design of such systems has proven to be difficult with the diverse retroactive effects that appear with respect to the context of the cell. Repressor-based circuits have various applications in biosynthesis, therapeutics, and bioremediation. Particularly using CRISPRi, competition for components of the system (unbound dCas9) can affect the achievable dynamic range of repression. Moreover, the toxicity of dCas9 via non-specific binding inhibits high levels of expression and limits the performance of genetic circuits. In this work, we study the computation of Boolean functions through CRISPRi based circuits built out of NOT and NOR gates. We provide algebraic expressions that allow us to evaluate the steady-state behaviors of any realized circuit. Our mathematical analysis reveals that the effective non-cooperativity of any given gate is a major bottleneck for increasing the dynamic range of the outputs. Further, we find that under the condition of competition between promoters for dCas9, certain circuit architectures perform better than others depending on factors such as circuit depth, fan-in, and fan-out. We pose optimization problems to evaluate the effects engineerable parameter values to find regimes in which a given circuit performs best. This framework provides a mathematical template and computational library for evaluating the performance of repressor-based circuits with a focus on effective cooperativity.