Introduction
Robust control of population-level behaviors
Motivation
In these cases, single-cell-level design is not sufficient for robust population-level behavior. In this chapter, we first present the design of single-cell-level and population-level feedback control circuits for robust population-level behavior in Sections 2.2 and 2.3.
Single cell-level control circuit design
The dashed lines in the “non-productive” cell represent that the circuit is not working properly due to the altered metabolic state of the cell under stress. On the other hand, the bottom panel in Figure 2.5B is the circuit in the 'non-productive' 𝑗 cell.
Population-level control circuit design
It shows that population-level production always converges to the constant value𝜇 despite the disturbance𝑤. In contrast to single-cell-level designs, population-level production at steady state does not depend on fractions of cells entering the 'non-producing' phenotype.
Population control of density-dependent functionalities
The total production of AHL in a population is proportional to the population density with the degree𝛽𝑆. 1, the population density 𝑁1 decreases to a lower level in the open-loop circuit, leading to a large error in the population ratio, as shown in Figure 2.14B (right).
Principle I: Signal integral feedback control
𝑆 is synthesized from 𝑋 and degraded by a constitutive enzyme 𝐸, and there is no external dilution in the environment, as shown in Figure 2.16A. Enzymatic degradation is modeled using Michaelis-Menten kinetics with dissociation constant𝐾. The dynamics of 𝑆 can be described as. In the closed-loop circuit, GFP is sensed by an AHL quorum-sensing signal, and AHL molecules activate a transcription factor TetR that represses GFP and the AHL synthase LuxI. The enzyme Aiia degrades AHL and its concentration can be tuned. The overall population level expression indicates adaptation to disturbance. dissociation constant
In simulation, we find similar dynamics of GFP expression under the control of the intracellular integral feedback loop, as shown in Figure 2.19B. The induction expression of ClpXP can set the desired concentration of GFP in the population, as shown in Figure 2.19C. Then we consider the same environmental disturbance that leads to 'produce' and. low-producing' phenotypes in the population, as illustrated in Figure 2.20A. Perturbation causes a greater proportion of 'non-producing' cells B. Figure 2.21: Robustness performance of signaling and intracellular integrated circuits.
Principle II: Bistable state switching feedback control
Still, it is possible to stabilize total production at the population level in the ultrasensitive regime. Based on a stochastic simulation of 𝑁 = 100 cells, we plot the steady-state input-output mapping of multiple bistable switches in parallel at the population level (𝑦versus 𝑢) in Figure 2.24D. The orange colored mapping shows a graded response of the controller at the population level with an ultrasensitive regime in the target gene expression 𝑦.
To find the population-level expression in the closed loop, we need to find the intersection of input-output mappings of both the controller and the process. Meanwhile, the population-level mapping exhibits ultrasensitivity in the controller response, leading to a robust closed-loop steady state of population-level expression𝑦.
Layered control circuit design
The population-level controller enables robust adaptation to environmental perturbations that disrupt target chemical production. However, the population-level controller only ensures robust overall dosage by inducing overexpression in other cells. Thus, a single cell level controller can set a reasonably high expression level for cells in the ON state, when coupled with the population level control circuit.
Here, we add an integrated single-cell-level feedback controller to a switch-based population-level bistable control circuit. Therefore, the population-level controller provides robust total dosage, while the cell-level controller provides robust expression v.
Discussion
The general Monod model for the growth of a 𝑛cell population on a single nutrient 𝑀 in a batch culture is given by. More examples are shown in Figures A.1 and A.2 in Appendix A. An example of a cooperative interaction includes various mechanisms of metabolite cross-feeding, mutual activation of growth by quorum-sensing signaling molecules, and toxin removal. Quorum sensing in bacteria: the LuxR-LuxI family of cell density-responsive transcriptional regulators. In: Journal of bacteriology176.2, p.
The dynamics of bacterial populations are maintained in the chemostat.” In: The dynamics of bacterial populations maintained in the chemostat. Achieving 'integral control' in living cells: how can we overcome leaky integration due to dilution?” In: Journal of the Royal Society Interface15.139, p.
Design of population interactions in well-mixed environments
Motivation
Microbes form communities involving multiple cell types with complex interactions in natural environments such as soil, food, and the human gut (Fierer and Jackson, 2006; Gill et al., 2006). Compared to monocultures, multiple cell populations contain more diverse functionalities with promoted stability and robustness to environmental fluctuations (LaPara et al., 2002; Burmølle et al., 2006). Microbial consortia of cooperative cell populations have been shown to exhibit promoted biomass (Schink, 2002; Burmølle et al., 2006) and resilience to cheaters and invaders (Pande et al., 2016).
It also plays an important role in antibiotic warfare among several bacterial species (Clemente et al., 2015). For example, rapidly growing cell populations have been shown to exhibit strong dominance and outcompete other cell populations to extinction in continuous cultures with limiting nutrients (Tilman, 1977; Sommer, 1983; Passarge et al., 2006).
Pairwise versus mechanistic models
Since the Monod model describes the cell growth and consumption dynamics of the nutrient, it can. 𝑀 is proportional to the population density with rate 𝛽𝑀 and𝛿, we can write the model for the population and 𝑀chemical dynamics:. 𝑆 is produced by cells at a constant rate 𝛽𝑆, we can write the model of the population with the accumulated chemical-mediated interaction:.
Cell population dynamics show markedly different outcomes for these two chemically mediated interactions, as shown in the simulations in Figure 3.2B. With the chemical consumed or degraded, population density depends not only on its growth and mortality rates, but also on its production and consumption/degradation rates. When there is strong production and poor consumption/degradation of 𝑀, the chemical concentration builds up and kills all cells (top right).
Batch versus chemostat cultures
We consider two cell populations of type I and type II grown together in media, as shown in Figure 3.3A. For chemostat cultures, there is a constant dilution of cells and media in the co-culture of two cell populations, as shown in Figure 3.3D. For example, as shown in Figure 3.4A,D, we consider cell populations involving mutual competition by bactericidal and bacteriostatic antibiotics in a batch culture.
A)(D) Schematic diagrams of a co-culture of two cell populations with competitive interactions via bactericidal or bacteriostatic antibiotics in a batch culture. A)(D) Schematic diagrams of a co-culture of two cell populations with competitive interactions via bactericidal or bacteriostatic antibiotics in a chemostat culture.
Stable coexistence of two cell populations
Furthermore, as we increase the chemical production rates 𝛽𝑖 of 𝑀𝑖, the stable region increases, as shown in Figure 3.7C. The population growth and intracellular dynamics model of the 𝑖th microcolony of cell type I is written as. Engineering for Biofuels: Harnessing Innate Microbial Capacity or Importing Biosynthetic Potential?” In: Nature Reviews Microbiology7.10, pp.
Competition for nutrients and light: stable coexistence, alternative stable states or competitive exclusion? In: Ecological Monographs 76.1, p. The second cooperation mechanism is through growth activation via quorum sensing molecules, as shown in Figure A.1B. The quorum sensing molecules can activate growth without being consumed.
Design of population interactions in spatially structured consortia 98
Robust population behaviors in defined spatial environments
We adopted the population density ratio control circuit design presented in Section 2.4 as shown in Figure 4.1A. To investigate which spatial distributions can maintain stable and robust control of the density ratio, we first perform simulations of all potential distributions using the full reaction–diffusion model in equation (B.1). We first present three different spatial distributions showing the desired density ratio 𝑦∗ = 1 at steady state in Figure 4.2A.
We then test for asymmetric distributions and find that the density ratio deviates from the value 1, as shown in Figure 4.2B. Specifically, to maintain a relatively high density ratio between two populations, a more mixed colony distribution of two cell types may promote robustness.
Stable coexistence of two cell populations with spatial structures
Therefore, interactions are non-local behaviors that depend on the spatial distribution of neighboring cells and signaling molecules. For a chemostat culture, assume there is a constant dilution rate𝐷 in the media and nutrient input rate 𝑀0. At 20 h/100 h/50 h, we perturb the density to show the convergence of the density ratio.
We show simulations of population growth dynamics and density ratio of the batch culture in Figure A.1, and of the chemostat culture in Figure A.2.
Conclusion
Competitive interactions in batch versus chemostat cultures
The population density of type I and II cells is denoted by 𝑁1 and 𝑁2, the concentrations of antibiotics I and II by 𝑇1 and 𝑇2, and the concentration of nutrients by 𝑀. We hypothesize that antibiotic production depends on population density and nutrient input to avoid infinite production even when the nutrient is used up. Parameters 𝛼1, 𝛼2 are cell growth rates, 𝛾 is cell death rate, 𝛾𝑀 is nutrient consumption dissociation rate, 𝐾𝑇 is antibiotic killing dissociation rate, 𝛽1, 𝛽2 are antibiotic production rates and 𝛿 is nutrient consumption rate.
For convenience, some parameters are set to be the same rate for two cell populations. Parameters are the same as in equation (A.1), except that 𝐾𝑇 is the dissociation rate of antibiotics that inhibit cell growth.
Cooperative interactions in batch versus chemostat cultures
At 15 hours into the simulation we dilute the cell populations to a low initial density and show their growth dynamics in fresh media. The third mechanism of cooperation occurs via the removal of toxins through the secretion of enzymes that break down toxins, as shown in Figure A.1C. Similarly, we can derive ODE models for chemostat culture populations by adding dilution and nutrient influx into the media.
At 20 h/100 h/50 h in the simulation, we dilute the cell populations to a low initial density and show their growth dynamics in fresh media. In Figure A.1, simulations of cell growth dynamics show the coexistence of two populations with cooperative interactions, yet the steady state densities at stationary phase depend on initial cell densities as well as nutrients.
Competition via accumulated versus consumed/degraded chemicals . 135
