Control theory meets synthetic biology

Abstract

The past several years have witnessed an increased presence of control theoretic concepts in synthetic biology. This review presents an organized summary of how these control design concepts have been applied to tackle a variety of problems faced when building synthetic biomolecular circuits in living cells. In particular, we describe success stories that demonstrate how simple or more elaborate control design methods can be used to make the behaviour of synthetic genetic circuits within a single cell or across a cell population more reliable, predictable and robust to perturbations. The description especially highlights technical challenges that uniquely arise from the need to implement control designs within a new hardware setting, along with implemented or proposed solutions. Some engineering solutions employing complex feedback control schemes are also described, which, however, still require a deeper theoretical analysis of stability, performance and robustness properties. Overall, this paper should help synthetic biologists become familiar with feedback control concepts as they can be used in their application area. At the same time, it should provide some domain knowledge to control theorists who wish to enter the rising and exciting field of synthetic biology.

Keywords: control theory; feedback; gene regulation; genetic circuits; robustness; synthetic biology.

Figure 1.

Feedback control set-ups in synthetic biology. (a) General feedback control architecture where a controller measures an output y of interest of a process, compares it with a desired value u, and applies it as an input to the process. (b) In-cell feedback control implementation: the process and the controller are both ‘running’ in the cell and, as such, are implemented by biomolecular reactions. (c) In silico feedback control implementation: the process is the cell itself with all its molecular circuitry while the controller is implemented in a computer. (Microscopy image courtesy of Cell Image Library [2].)

Figure 2.

Condensed timeline of synthetic biology. (a) The development of synthetic biology is grounded on molecular biology, genetic engineering and genomics. (b) The early phases of synthetic biology were focusing mostly on forward engineering simple modules, such as switches and oscillators. (c) After the ‘era’ of modules, synthetic biology is heading towards the era of systems, in which modules will serve as functional units to create more complex and sophisticated systems with potential applications to energy, environment and medicine.

Figure 3.

The essence of negative feedback. (a) Negative feedback extends the linear regime of an amplifier and provides robustness to uncertainty. The top diagram shows the amplifier within a negative feedback loop. The bottom diagram shows the equivalent input/output mapping corresponding to the closed loop feedback system. The graph in the box shows the mapping (i.e. the dose–response curve) between the input (signal on incoming arrow) and the output (signal on outgoing arrow). (b) High-gain negative feedback attenuates disturbances and speeds up the temporal response. The purple block(s) represent the ordinary differential equation (ODE) that links the input (incoming arrow) to the output (outgoing arrow). For a desired constant value u, the open loop system's response is obtained by setting z = u and simulating the open loop system. The closed loop system response is obtained by simulating the closed loop system with K = 1 and G large. In this case, the steady-state error between y and the desired value u can be decreased by increasing G, that is, (c) High-gain negative feedback can lead to oscillations and amplifies high-frequency disturbances. The open loop system is simulated as before by setting z = u. The closed loop system is simulated with G large and K = 1. The left-hand plot shows the time response of the system. The right-hand plot shows the frequency response of y to disturbance d. The horizontal axis represents the frequency ω of a periodic disturbance d(t) = sin(ωt) and the vertical axis shows the amplitude of the resulting y(t) signal. (d) Negative integral feedback completely rejects disturbances. The open loop system is as in panel (b) and simulated similarly. The closed loop system is simulated for two different values of G (as shown) and for K = 1. In all diagrams, the circle represents a summing junction: the outgoing arrow is a signal given by the weighted sum with the indicated signs of the signals on the incoming arrows. Also, we have used the shortened notation . The simulation codes used to generate this figure are available in the electronic supplementary material.

Figure 4.

Enhancing robustness through feedback and feed-forward control. (a) Decreased variability of gene expression through negative autoregulation. (b) Negative autoregulation shifts noise to higher frequency. (c) Feed-forward circuits decrease the sensitivity of the output to input disturbances.

Figure 5.

Negative feedback implementation. (a) Transcriptional negative feedback by inhibition of protein transcription [46]. (b) Translational negative feedback by inhibition of protein translation [68]. (c) Translational negative feedback by increased mRNA degradation enabled by a microRNA (z) [69]. (d) Transcriptional negative feedback implemented through competitive binding with a scaffold protein s [70]. (e) Transcriptional negative feedback implemented by deactivation of transcriptional activator K* [71].

Figure 6.

Improving modularity through feedback control. (a) Failure of modularity in genetic circuits. A synthetic genetic clock in isolation displays sustained oscillation (black, solid line), but, once it is connected to a downstream system, oscillations disappear (red, dashed line). Loading on the upstream system's transcription factor is formally modelled as a signal s called retroactivity, which affects as a disturbance the dynamics of the upstream system. (b) Insulation devices buffer from retroactivity. Insulation devices attenuate retroactivity to the output s and have small retroactivity to the input r; they can be placed as buffering elements between an upstream and a downstream system. (c) High-gain negative feedback to design insulation devices. High-gain negative feedback can be used to attenuate the effect of retroactivity s on the system's dynamics. A phosphorylation cycle where the output y results from u-mediated activation of inactive y_in and is converted back to y_in by a phosphatase P can implement the high-gain negative feedback design to attenuate s. Gain G can be increased by increased concentrations y_in and P. (d) Two-stage insulation device. This allows decoupling of the requirements of attenuating s from those of having small r. While the second stage is a high-gain feedback device as in (c), the first stage has low protein amounts (‘low-gain’) to have low retroactivity to the input r. It attenuates any load-induced slow down owing to large y_in by cycling at a fast rate compared with the speed of gene expression (time scale of input u). The simulation codes used to generate this figure are available in the electronic supplementary material.

Figure 7.

Coordinated population control system [113]. LuxI and LuxR are produced constitutively in each cell. LuxI catalyses the synthesis of small molecule AHL, which can diffuse freely across the membrane. As cell number grows, AHL concentration increases, binding with LuxR to activate a ‘killer gene’ to reduce cell count. Blue arrows indicate biochemical reactions, black arrows show the diffusion of AHL across the membrane, and dashed red arrows show the population control feedback loop.