Emergent bistability by a growth-modulating positive feedback circuit

Abstract

Synthetic gene circuits are often engineered by considering the host cell as an invariable 'chassis'. Circuit activation, however, may modulate host physiology, which in turn can substantially impact circuit behavior. We illustrate this point by a simple circuit consisting of mutant T7 RNA polymerase (T7 RNAP*) that activates its own expression in the bacterium Escherichia coli. Although activation by the T7 RNAP* is noncooperative, the circuit caused bistable gene expression. This counterintuitive observation can be explained by growth retardation caused by circuit activation, which resulted in nonlinear dilution of T7 RNAP* in individual bacteria. Predictions made by models accounting for such effects were verified by further experimental measurements. Our results reveal a new mechanism of generating bistability and underscore the need to account for host physiology modulation when engineering gene circuits.

Figure 1. Bistability induced by a T7 RNAP* positive-feedback circuit

(a) The circuit consists of T7 RNAP* regulated by its own promoter (P_T7lac). IPTG activates the circuit. CFP serves as the circuit readout. (b) A microcolony of MC4100Z1 cells carrying the circuit, induced with 1000μM IPTG. The colony exhibited bimodal CFP expression. The corresponding movie is included as Supplementary Movie 1. (c) The bimodal expression represented steady state behavior. The cell culture gave rise to ~31% ON cells both before dilution and 8 hours after dilution. (d) The OFF subpopulation (solid line) could be induced to generate a similar distribution as the ON subpopulation (dotted line). The results in (c) and (d) are representative data from three replicates (Figs. S3 and S4). Cell counts were normalized by total number of cells in each sample. The percentages of ON cells were different in (c) and (d) due to different experiment protocols (see Materials and Methods). (e) Time series of sample OFF, INT, and ON colonies (see Fig. S5 for additional data).

Figure 1. Bistability induced by a T7 RNAP* positive-feedback circuit

(a) The circuit consists of T7 RNAP* regulated by its own promoter (P_T7lac). IPTG activates the circuit. CFP serves as the circuit readout. (b) A microcolony of MC4100Z1 cells carrying the circuit, induced with 1000μM IPTG. The colony exhibited bimodal CFP expression. The corresponding movie is included as Supplementary Movie 1. (c) The bimodal expression represented steady state behavior. The cell culture gave rise to ~31% ON cells both before dilution and 8 hours after dilution. (d) The OFF subpopulation (solid line) could be induced to generate a similar distribution as the ON subpopulation (dotted line). The results in (c) and (d) are representative data from three replicates (Figs. S3 and S4). Cell counts were normalized by total number of cells in each sample. The percentages of ON cells were different in (c) and (d) due to different experiment protocols (see Materials and Methods). (e) Time series of sample OFF, INT, and ON colonies (see Fig. S5 for additional data).

Figure 2. Circuit activation reduces bacterial growth rates

(a) In microcolonies, increasing CFP expression reduced the growth rate. Error bars indicate the standard deviation of 65 OFF colonies, 12 INT colonies, and 16 ON colonies respectively. One-way analysis of variance shows that the growth rates of OFF, INT, and ON colonies are significantly different (p-value~1×10⁻⁹). A multiple comparison test using 95% confidence interval shows that INT colonies grew slower than OFF colonies, ON colonies grew slower than INT colonies, and ON colonies grew slower than OFF colonies. (b) At the population level, increasing CFP expression also reduced the growth rate. Under each condition, the growth rate of ON cells (open squares, grey line) was estimated by assuming 30% ON cells in each population (Supplementary Methods).

Figure 3. Interplay between growth modulation and the positive feedback loop can lead to bistability

(a) T7 RNAP* (X) expression reduces the growth rate, which reduces dilution rate of T7 RNAP*. The two steps of negative regulation (red lines) form a positive feedback loop. Furthermore, T7 RNAP* activates its own expression (blue line). T7 RNAP* is thus regulated by two positive feedback loops. Phase planes of the basic circuit model (Eq. 2) (b) without (γ=0) and (c) with growth retardation (γ=10). Filled circles indicate stable steady states and open circles indicate unstable steady states. The nullclines were calculated by using δ=0.01, α=10, and ϕ=20. See Supplementary Methods for detailed model derivation (Eqs. S3–S11).

Figure 4. Modulation of circuit dynamics by initial culture density

(a) Stochastic simulations show that the percentage of ON cells would increase with increasing N₀ (the initial number of cells per population). See Supplementary Methods for modeling details (Table S1 & Eq. S15). (b) Experimental validation of (a): the percentage of ON cells increased with OD and [IPTG] (see Fig. S9 for the corresponding CFP distributions). The lines are drawn as a guide for the eyes. Each error bar indicates the standard deviation of four replicates collected from two independent experiments.

Figure 5. Modulation of circuit dynamics

(a) A bifurcation diagram obtained by solving Eq. 2 (δ=0.01; γ=10) analytically with standard methods . (b) Modeling stochastic switching between the OFF and ON cells and their differential growth. See Supplementary Methods for detailed model description (Eqs. S12–S14). (c) The simulated time series of ON cells percentage in an OFF → ON culture (grey line) and an ON → OFF culture (black line) (α=19 and ϕ=100). The difference between the two (black dotted line) was fitted to a₀exp(−τ/τ_memory) to obtain τ_memory. (d) The CFP level of ON → OFF cell cultures (black lines, filled squares) eventually converged to that of the OFF → ON cultures (grey lines, open squares) at the twelfth hour. (e) Simulations predict that decreasing initial percentage of ON cells reduced memory. (f) ON → OFF cultures (black lines, filled squares) were prepared by mixing varying ratios of ON cultures and OFF cultures (as indicated). Consistent with the model predictions (e), the CFP levels of ON → OFF cultures converged faster to that of the OFF → ON culture (grey line, open squares) with decreasing initial ON culture percentage. Each error bar indicates the standard deviation of three replicates. (g) The maximal τ_memory is predicted to occur at low α for a low ϕ (=60) and at high α for a high ϕ ( = 90). (h) Experimental validation of (g). The τ_memory was calculated by using average time series of at least four replicates (Fig. S11).

Figure 5. Modulation of circuit dynamics

(a) A bifurcation diagram obtained by solving Eq. 2 (δ=0.01; γ=10) analytically with standard methods . (b) Modeling stochastic switching between the OFF and ON cells and their differential growth. See Supplementary Methods for detailed model description (Eqs. S12–S14). (c) The simulated time series of ON cells percentage in an OFF → ON culture (grey line) and an ON → OFF culture (black line) (α=19 and ϕ=100). The difference between the two (black dotted line) was fitted to a₀exp(−τ/τ_memory) to obtain τ_memory. (d) The CFP level of ON → OFF cell cultures (black lines, filled squares) eventually converged to that of the OFF → ON cultures (grey lines, open squares) at the twelfth hour. (e) Simulations predict that decreasing initial percentage of ON cells reduced memory. (f) ON → OFF cultures (black lines, filled squares) were prepared by mixing varying ratios of ON cultures and OFF cultures (as indicated). Consistent with the model predictions (e), the CFP levels of ON → OFF cultures converged faster to that of the OFF → ON culture (grey line, open squares) with decreasing initial ON culture percentage. Each error bar indicates the standard deviation of three replicates. (g) The maximal τ_memory is predicted to occur at low α for a low ϕ (=60) and at high α for a high ϕ ( = 90). (h) Experimental validation of (g). The τ_memory was calculated by using average time series of at least four replicates (Fig. S11).