In natural biological systems, digital behaviour is appropriate in settings where decision making is necessary, such as in developmental circuits. The digital approach is an abstraction of graded analog functions, where values above a threshold are classified as ‘1’ and values below this threshold are classified as ‘0’ (Fig. 1a). Digital computation using synthetic gene circuits has included switches, counters, logic gates, classifiers and edge detectors (see references 28–40 in Supplementary Information). However, given that there is often unwanted crosstalk amongst synthetic devices and cellular resource limitations, it may be challenging to scale digital logic functions to the level needed for complex computations in living cells. Analog functions can be found in natural biological systems, where they enable graded responses to environmental signals. For example, neurons can implement both digital and analog computation. Furthermore, electronic circuits that perform analog computation on logarithmically transformed signals have been used in commercially valuable electronic chips for several decades.

A central goal of synthetic biology is to achieve multi-signal integration and processing in living cells for diagnostic, therapeutic and biotechnology applications. Digital logic has been used to build small-scale circuits, but other frameworks may be needed for efficient computation in the resource-limited environments of cells. Here we demonstrate that synthetic analog gene circuits can be engineered to execute sophisticated computational functions in living cells using just three transcription factors. Such synthetic analog gene circuits exploit feedback to implement logarithmically linear sensing, addition, ratiometric and power-law computations. [...] Our circuits can be composed to implement higher-order functions that are well described by both intricate biochemical models and simple mathematical functions. By exploiting analog building-block functions that are already naturally present in cells, this approach efficiently implements arithmetic operations and complex functions in the logarithmic domain