Realizing 'integral control' in living cells: how to overcome leaky integration due to dilution?

Abstract

A major problem in the design of synthetic genetic circuits is robustness to perturbations and uncertainty. Because of this, there have been significant efforts in recent years in finding approaches to implement integral control in genetic circuits. Integral controllers have the unique ability to make the output of a process adapt perfectly to disturbances. However, implementing an integral controller is challenging in living cells. This is because a key aspect of any integral controller is a 'memory' element that stores the accumulation (integral) of the error between the output and its desired set-point. The ability to realize such a memory element in living cells is fundamentally challenged by the fact that all biomolecules dilute as cells grow, resulting in a 'leaky' memory that gradually fades away. As a consequence, the adaptation property is lost. Here, we propose a general principle for designing integral controllers such that the performance is practically unaffected by dilution. In particular, we mathematically prove that if the reactions implementing the integral controller are all much faster than dilution, then the adaptation error due to integration leakiness becomes negligible. We exemplify this design principle with two synthetic genetic circuits aimed at reaching adaptation of gene expression to fluctuations in cellular resources. Our results provide concrete guidance on the biomolecular processes that are most appropriate for implementing integral controllers in living cells.

Keywords: adaptation; integral control; robustness; synthetic biology; time-scale separation.

Figure 1.

Quasi-integral control mitigates the effect of leaky integration due to dilution. (a) Two types of ideal ICMs. The controller reactions are boxed in pink, and the rest of the circuit belongs to the process to be controlled. Dilution of the controller species is neglected in ideal ICMs. (b) When dilution of the controller species is considered, ideal ICMs become leaky ICMs that cannot carry out integration. The adaptation error, defined as the extent to which the output is affected by the disturbance, can be arbitrarily large. (c) All controller reactions in a quasi-ICM are 1/ε times faster than those in the corresponding leaky ICM, with ε ≪ 1. (d) Simulations of the type I and type II ideal, leaky and quasi-ICMs. Simulation parameters for both motifs: α = γ = δ = k = 1 h⁻¹, θ = 1 nM⁻¹ h⁻¹, ε = 0.02, u = 10 nM and d = 5 nM h⁻¹. Disturbance input d is applied at 15 h. (e) A general ε-quasi-integral control system. Output of the process y becomes an input to the quasi-integral controller. Variable z₂ is the leaky memory variable, and z₁ represents the remaining controller states, if any. All controller reactions are much faster than dilution, as characterized by parameter ε. Output of the controller v drives the process to set-point u and adapts to d. (Online version in colour.)

Figure 2.

Two physical realizations of quasi-ICMs. (a) Genetic circuit diagram of the phosphorylation-based quasi-integral controller. Chemical reactions realizing the controller are boxed in pink. (b) Simulation of the circuit's response according to (3.1). A set-point input u = 20 nM is applied at time 0 and a disturbance input d = 0.5 is applied at 20 h. The vertical axis represents the ratio between output y, defined to be proportional to p (y = σp), and set-point u. The dashed black line is the response of the phosphorylation-based control system assuming no dilution of the active substrate b^*. The dotted blue line, the thin green line with square markers and the solid red line represent circuit's response in the presence of nonzero substrate dilution (γ = 1 h⁻¹) and decreasing ε, which corresponds to increasing catalytic rates (k_i, i = 1, 2). (c) Genetic circuit diagram of the sRNA-based quasi-integral controller. (d) Simulation of the circuit's response according to (3.4). A set-point input u = 1 is applied at time 0 and a disturbance input d = 0.5 is applied at 30 h. The vertical axis represents the ratio between output y, defined in (3.6), and set-point input u. The dashed black line represents response of an ideal integral control system, where RNA decay rate δ is set to 0. The dotted blue line, the thin green line with square markers and the solid red line represent circuit's responses in the presence of nonzero RNA decay rate (δ = 3 h⁻¹, corresponding to half-life of approx. 13 min) and decreasing ε. Parameter ε is decreased by increasing the mRNA–sRNA removal rate (θ/β). The DNA copy numbers of the regulated gene and the sRNA are increased simultaneously by a factor of 1/ε as ε decreases. Simulation parameters are listed in electronic supplementary material, section S5, table S2. (Online version in colour.)