Implementing Arithmetic and Other Analytic Operations By Transcriptional Regulation

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Abstract
The transcriptional regulatory machinery of a gene can be viewed as a computational device, with transcription factor concentrations as inputs and expression level as the output. This view begs the question: what kinds of computations are possible? We show that different parameterizations of a simple chemical kinetic model of transcriptional regulation are able to approximate all four standard arithmetic operations: addition, subtraction, multiplication, and division, as well as various equality and inequality operations. This contrasts with other studies that emphasize logical or digital notions of computation in biological networks. We analyze the accuracy and precision of these approximations, showing that they depend on different sets of parameters, and are thus independently tunable. We demonstrate that networks of these “arithmetic” genes can be combined to accomplish yet more complicated computations by designing and simulating a network that detects statistically significant elevations in a time-varying signal. We also consider the much more general problem of approximating analytic functions, showing that this can be achieved by allowing multiple transcription factor binding sites on the promoter. These observations are important for the interpretation of naturally occurring networks and imply new possibilities for the design of synthetic networks. The biochemistry of the cell is daunting in its complexity. In order to understand this complexity, we are often forced to use metaphors or construct analogies to systems that we understand better. One long-standing analogy is to digital computers, with their large networks of interacting components that manage to act in coherent and useful ways. Indeed, we know from both theoretical models and empirical observations that biological entities such as genes can sometimes be described accurately by digital, or logical, expressions—turning on or off in response to regulatory signals. However, far more sophisticated computations can also take place, as has been documented in the responses of genes such as the lac operon or Endo-16. We analyze chemical kinetic models of transcriptional gene regulation and show that even simple models are capable of nearly arbitrary analog computations, ranging from elementary arithmetic operations to general analytic functions. Understanding the computational capacities of genes, and of biochemical systems more generally, tells us what to look for when studying natural systems and tells us what we can hope to build by biological engineering.