Circuit-Host Coupling Induces Multifaceted Behavioral Modulations of a Gene Switch  Andrew E. Blanchard, Chen Liao, Ting Lu  Biophysical Journal  Volume 114, Issue 3, Pages 737-746 (February 2018) DOI: 10.1016/j.bpj.2017.12.010 Copyright © 2017 Biophysical Society Terms and Conditions

Figure 1 An integrated circuit-host system. (A) Schematic of circuit-host interactions. The circuit regulates the host by placing a metabolic load through the consumption of the central resources; the host modulates the circuit by specifying the availability of resources needed for exogenous gene expression. (B) A minimal circuit-host model for a self-activating gene switch. The circuit enables self-activation through its internal regulation. In addition, it negatively regulates the host’s growth by placing a metabolic load. The host modulates the circuit by growth-dependent alteration of protein production and dilution. The two key parameters characterizing the interactions are loading factor α and production factor β. (C) Growth rate as a function of protein production for different values of loading factor (α) and fixed value of β = 0.0. The fold activation (ϕ) was fixed to a value of 25 and the maximal production (μ) was fixed at 2.25κ (κ = 250 nM). Each plot is normalized by its maximal growth rate. (D) The production modulation function W as a function of growth rate for different values of production factor (β). To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 2 Single-cell dynamics of the switch modulated by the loading factor α with a fixed production factor (β = 0.0). (A) Time derivative of protein concentration as a function of protein concentration. With increasing loading factor, the time derivative decreases but the steady states remain the same. Notably, reduction of the time derivative increases with the loading factor. (B) Magnitudes of the eigenvalues as a function of the loading factor. The magnitudes decrease monotonically with the load but those of the high state decrease more dramatically. (C) Deterministic time trajectories of the switch starting from the vicinity of the unstable steady state for different loading factors. A generation is the doubling time for a maximally growing cell. (D) Probability density function (PDF) of the switch from stochastic simulations (histograms) and Fokker-Planck formalism (black lines). The probability of the high state increases with the load. The dashed lines show the minimum for the Fokker-Planck distribution, and the numbers represent the percent of the simulated distribution below (low state) and above (high state) the line. For both stochastic and deterministic simulations the fold activation (ϕ) was fixed to a value of 25 and the maximal production (μ) was fixed at 2.25κ (κ = 250 nM). To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 3 Single-cell steady states modulated by the circuit-host coupling. (A) Deterministic phase diagram of the switch with or without coupling. Without coupling corresponds to α = 0.0 and its boundary is shown with yellow solid lines, whereas with coupling corresponds to α = 0.0026 and β = −0.5 and its boundary is shown with blue dashed lines. Modifying resource allocation as a function of growth leads to a shift in the phase diagram from yellow (without coupling) to blue (with coupling). The values μ and ϕ are the maximal production and fold activation constants, respectively. (B) Deterministic steady-state values of the switch as a function of the fold activation. μ = 2.25κ (κ = 250 nM) is used here. Blue dashed lines, with coupling; yellow solid lines, without coupling. (C) Time trajectories of the switch for three parameter sets that correspond to the dots P1, P2, and P3. (D) Stochastic phase diagram of the switch with (blue dashed lines) or without (yellow solid lines) coupling. Different from the deterministic phase diagram where monostability and bistability are defined, the phase space in the stochastic case is partitioned into unimodal and bimodal regimes. (E) Maxima and minima in the probability distributions from the Fokker-Planck formalism as a function of fold activation. μ = 2.25κ (κ = 250 nM) is used here. (F) Steady-state probability distributions from stochastic simulations (purple histograms) and Fokker-Planck formalism (black lines). The three cases from left to right correspond to the parameter sets P1–P3 in (D). Notice that the inset for P2 in the bottom row shows that the distribution is indeed bimodal. (G) Difference between the steady-state probability distributions of the switch with and without coupling. Parameters used correspond to the dot P2 of (D). (H) Heat map for the absolute difference of the probability distributions with and without coupling. To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 4 Cellular population distribution altered by circuit-host coupling. Here, the loading factor (α) is varied whereas the production factor (β) is fixed at zero. (A) Time evolution of population distributions with different loading factors. In each simulation, the population stays at 10,000 cells and their initial protein distribution follows the single-cell steady-state distribution. Increasing the load favors the population of cells with low protein production. (B) Difference of the initial and final probability distributions of the population shown in (A). All columns have a fixed fold activation (ϕ) of 13 and a maximal production (μ) of 2.25κ (κ = 250 nM). To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 5 Absolute difference between single-cell and population level distributions of the protein. (A) Absolute difference of the distributions for a given maximal production of μ = 2.25κ (κ = 250 nM) and β = 0.0. (Circles) Stochastic simulations; (lines) Fokker-Planck equation. (B) Heat map for the absolute difference of the single-cell and population level probability distributions. To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 6 Quantification of the total impacts of circuit-host coupling on switch behaviors. (A) Alteration of the switch’s phase diagram from the NC (with α = 0.0) to both SP (with α = 0.0026 and β = −0.5) coupling. It is the summation of the difference between NC and SC and the difference between SC and SP. (B) Mean and standard deviation of the protein concentration for the cases of NC, SC, and SP. Here maximal production (μ) is 2.25κ (κ = 250 nM). Dots and error bars are the results from stochastic simulations; solid lines and shading are the results from Fokker-Planck analysis. (C) Probability distributions of the protein for four cases that correspond to the dots P1–P4 in (A). (D) Leading component of the coupling effects that contribute the most to the alteration of circuit behaviors. In region I, single-cell effect dominates; in region II, population effect dominates; in region III, both effects are negligible. To see this figure in color, go online. Biophysical Journal 2018 114, 737-746DOI: (10.1016/j.bpj.2017.12.010) Copyright © 2017 Biophysical Society Terms and Conditions