Signal-to-Noise Ratio Measures Efficacy of Biological Computing Devices and Circuits

Abstract

Engineering biological cells to perform computations has a broad range of important potential applications, including precision medical therapies, biosynthesis process control, and environmental sensing. Implementing predictable and effective computation, however, has been extremely difficult to date, due to a combination of poor composability of available parts and of insufficient characterization of parts and their interactions with the complex environment in which they operate. In this paper, the author argues that this situation can be improved by quantitative signal-to-noise analysis of the relationship between computational abstractions and the variation and uncertainty endemic in biological organisms. This analysis takes the form of a ΔSNRdB function for each computational device, which can be computed from measurements of a device's input/output curve and expression noise. These functions can then be combined to predict how well a circuit will implement an intended computation, as well as evaluating the general suitability of biological devices for engineering computational circuits. Applying signal-to-noise analysis to current repressor libraries shows that no library is currently sufficient for general circuit engineering, but also indicates key targets to remedy this situation and vastly improve the range of computations that can be used effectively in the implementation of biological applications.

Keywords: Boolean logic; analysis; controls; digital circuits; signals; synthetic biology.

Figure 1

Example of a Boolean signal with values that cannot be perfectly distinguished: (A,B) show histograms for 50,000 cells sampled from typical distributions for Boolean true and false states, respectively. These distributions overlap, however, and in the overlapping range it is difficult or impossible to distinguish between true and false values.

Figure 2

The input/output curve of a device can be used to analyze its ΔSNR_dB. (A) Three example input/output curves, Device A (blue) is stronger than Device B (red) and also has more similar input and output ranges, while Device C (black) has a similar range to Device A but flatter slope. (B) For any given input levels, the observed ΔSNR_dB depends on the amount of noise σ_g, converging to a maximum at low noise and falling as the noise increases.

Figure 3

Variation of maximum ΔSNR_dB and low-noise output SNR_dB (σ_g = 1.02) with respect to input levels for three example devices. (A–C) show ΔSNR_dB for device A, B, and C, respectively, while (D–F) show low-noise output SNR_dB for the same devices. Note that the color scales are truncated at the lower end to provide better resolution in the upper range.

Figure 4

With significant expression noise, ΔSNR_dB may be significantly worse than under ideal conditions. For example, the charts above present the same analysis of Device A as in Figure 3, (A) shows ΔSNR_dB, (B) shows output SNR_dB, but with a more typical σ_g = 3 level of expression noise. Note that the color scales are truncated at the lower end to provide better resolution in the upper range.

Figure 5

Low noise ΔSNR_dB in a chain of inverters: (A–D) show chains of one to four Device A elements, (E–H) show chains of Device B elements, and (I–L) show chains of Device C elements. Notice that for Device A there is a range of widely separated inputs (approximately μ_g,false < 10^5.5, μ_g,true > 10⁶), where it is possible for SNR to remain strong; Device B is weaker and less well matched between input and output, and thus any computation with more than a single element has a badly degraded signal strength for all possible inputs. Device C, on the other hand, degrades incrementally in SNR across a broad range. Note that the color scales are truncated at the lower end to provide better resolution in the upper range.

Figure 6

The efficacy of a circuit with noisy distributions can be estimated from the (SNR_dB for individual devices under the same noise conditions. For example, estimates of chains of Device A repressors with σ_g = 3 (A–D) are a good conservative bound on the behavior observed in simulation (E–H). Note that the color scales are truncated at the lower end to provide better resolution in the upper range.

Figure 7

Parameter scan of ΔSNR_dB for the TetR homolog library from Stanton et al. (2014) with σ_g = 2.0, sorted by maximum ΔSNR_dB. (A–T) show ΔSNR_dB for each device in the library, sorted in descending order of maximum ΔSNR_dB. Colors use the same range as previously, from −20 to 5 dB. Note that the color scales are truncated at the lower end to provide better resolution in the upper range.

Figure 8

TetR homolog library from Stanton et al. (2014): (A) maximum gate ΔSNR_dB with σ_g = 2.0, sorted by maximum ΔSNR_dB. (B) All input/output curves, computed from models, provided in Stanton et al. (2014).