Bistability and oscillations in co-repressive synthetic microbial consortia

Mehdi Sadeghpour^{1},Alan Veliz-Cuba^{2},Gábor Orosz^{1},Krešimir Josić^{3,}^{4,}^{5},Matthew R. Bennett^{5,}^{6}()

^{1}. Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA ^{2}. Department of Mathematics, University of Dayton, Dayton, OH 45469, USA ^{3}. Department of Mathematics, University of Houston, Houston, TX 77204, USA ^{4}. Department of Biology and Biochemistry, University of Houston, Houston, TX 77204, USA ^{5}. Department of Biosciences, Rice University, Houston, TX 77251-1892, USA ^{6}. Department of Bioengineering, Rice University, Houston, TX 77251-1892, USA

Background: Synthetic microbial consortia are conglomerations of genetically engineered microbes programmed to cooperatively bring about population-level phenotypes. By coordinating their activity, the constituent strains can display emergent behaviors that are difficult to engineer into isogenic populations. To do so, strains are engineered to communicate with one another through intercellular signaling pathways that depend on cell density.

Methods: Here, we used computational modeling to examine how the behavior of synthetic microbial consortia results from the interplay between population dynamics governed by cell growth and internal transcriptional dynamics governed by cell-cell signaling. Specifically, we examined a synthetic microbial consortium in which two strains each produce signals that down-regulate transcription in the other. Within a single strain this regulatory topology is called a “co-repressive toggle switch” and can lead to bistability.

Results: We found that in co-repressive synthetic microbial consortia the existence and stability of different states depend on population-level dynamics. As the two strains passively compete for space within the colony, their relative fractions fluctuate and thus alter the strengths of intercellular signals. These fluctuations drive the consortium to alternative equilibria. Additionally, if the growth rates of the strains depend on their transcriptional states, an additional feedback loop is created that can generate oscillations.

Conclusions: Our findings demonstrate that the dynamics of microbial consortia cannot be predicted from their regulatory topologies alone, but are also determined by interactions between the strains. Therefore, when designing synthetic microbial consortia that use intercellular signaling, one must account for growth variations caused by the production of protein.

Author Summary Recently it has been shown that synthetic microbial consortia can use intercellular signaling pathways to create transcriptional regulatory topologies that mimic genetic circuits. However, if the strains within the consortium compete for resources, an added layer of complexity emerges. Here, we use computational modeling to explore the behavior of a two strain, transcriptionally co-repressive microbial consortium. We find that, unlike its genetic counterpart the bistable toggle switch, the co-repressive consortium can exhibit oscillatory behavior if the strains’ growth rates depend on protein production.

Corresponding Authors:
Krešimir Josić,Matthew R. Bennett

Issue Date: 22 March 2017

Cite this article:

Mehdi Sadeghpour,Alan Veliz-Cuba,Gá, et al. Bistability and oscillations in co-repressive synthetic microbial consortia[J]. Quant. Biol.,
2017, 5(1): 55-66.

(A) Gene circuit diagram of a single cell co-repressive toggle switch [15]. (B) Proposed synthetic microbial consortium with a co-repressive network. Each strain contains a transcriptional inverter (mediated by LacI) and an enzyme that creates a quorum sensing molecule. Repression occurs when the quorum sensing molecule from one strain diffuses into the other strain, up-regulating the target transcriptional inverter (green dashed arrows). That inverter down-regulates production of the second, orthogonal quorum sensing molecule.

Fig.2 Two-strain population toggle with equal growth rates.

(A) The equilibrium ${x}_{\mathrm{1}}^{*}$ as a function of population ratio $r$. The dashed and solid lines correspond to unstable and stable equilibria, respectively. (B) Two trajectories of Equation (1) approaching one of the two stable equilibria marked by blue ♦ and ■ based on the initial conditions. The third, unstable equilibrium is denoted by red ○. The gray dashed line shows the separatrix between the two basins of attraction of the stable equilibria. The parameters are chosen as ${\beta}_{1}={\beta}_{2}=0.023{\mathrm{min}?}^{-1}$ corresponding to E. coli’s cell cycle of approximately 30 minutes, $\alpha =10{\mathrm{min}?}^{-1}$, $\theta =\mathrm{500}$, $N=\mathrm{200}$ , and $n=\mathrm{2}$. The simulations are carried out for constant population ratio $r=\mathrm{0.4}$ and initial conditions $({x}_{\mathrm{1}}(\mathrm{0}),{x}_{\mathrm{2}}(\mathrm{0}))=(\mathrm{100},\mathrm{200})$ proteins per cell and $({x}_{\mathrm{1}}(\mathrm{0}),{x}_{\mathrm{2}}(\mathrm{0}))=(\mathrm{300},\mathrm{100})$ proteins per cell.

Fig.3 Two-strain population toggle with different growth rates.

Simulations of Equations (1), (4), and (6) with ${\beta}_{0}=0.023{\mathrm{min}?}^{-1}$ and different $\epsilon $ values as indicated. Other parameters are the same as in Figure 1. Initial protein concentrations are $({x}_{\mathrm{1}}(\mathrm{0}),{x}_{\mathrm{2}}(\mathrm{0}))=(\mathrm{100},\mathrm{200})$ and initial population ratio is $r(\mathrm{0})=\mathrm{0.1}$.

Fig.4 Metabolic loading leading to relaxation oscillations.

Simulations of Equations (1), (4), and (7) for different values of $\epsilon $ and $\rho $ as indicated. Parameters are ${\beta}_{0}=0.023{\mathrm{min}?}^{-1}$, $\alpha =10{\mathrm{min}?}^{-1}$, $\theta =\mathrm{500}$, $N=\mathrm{200}$, and $n=\mathrm{2}$. Initial conditions are $({x}_{\mathrm{1}}(\mathrm{0}),{x}_{\mathrm{2}}(\mathrm{0}))=(\mathrm{100},\mathrm{200})$ proteins per cell and $r(\mathrm{0})=\mathrm{0.4}$.

Fig.5 Bifurcation diagrams for the two-strain toggle under metabolic load.

In panels (A, B, D, E), solid and dashed lines denote stable and unstable equilibria, respectively. In panels (D, E, F), the markers $\times $ and $*$ indicate transcritical and Hopf bifurcations, respectively. The solid magenta line in panels (C, F) shows the amplitude of the periodic solution. In panel (F) the periodic solution emerges from a Hopf bifurcation. Panels (A, B, C) correspond to the case $\epsilon =\mathrm{0}$ while panels (D, E, F) correspond to $\epsilon =\mathrm{0.25}$. Other parameters are the same as in Figure 3.

Fig.6 Stochasticity in the dynamics of the two-strain toggle consortium. (A, B) Simulations of Equations (8)?(11) with no metabolic loading ($\rho =\mathrm{0}$) and equal growth rates ($\epsilon =\mathrm{0}$) for different population sizes as indicated.

The blue and green curves show the mean number of proteins in strains 1 and 2, respectively. The red curve shows the population ratio. Simulations of the deterministic system from Equations (1) and (4) are also shown using gray curves. Parameters are the same as in Figure 3. Initial protein counts are ${x}_{\mathrm{1},i}(\mathrm{0})=\mathrm{100}$, $i=\mathrm{1},\dots ,{n}_{\mathrm{1}}$, ${x}_{\mathrm{2},j}(\mathrm{0})=\mathrm{200}$, $j=\mathrm{1},\dots ,{n}_{\mathrm{2}}$, and initial ratio of strain 1 is $r(\mathrm{0})=\mathrm{0.4}$. (C, D) Different stochastic simulations of the population ratio corresponding to panels (A, B), respectively, as well as the mean (${\mu}_{r}$) and the standard deviation (${\sigma}_{r}$).

Fig.7 Simulations of Equations (8) ?(11) with unequal growth rates of the strains and no metabolic loading ($\rho =\mathrm{0}$) with different $\epsilon $ and $N$ values as indicated.

The gray curves show the corresponding simulations of the deterministic model. Parameters are the same as in Figure 3. Initial conditions are ${x}_{\mathrm{1},i}(\mathrm{0})=\mathrm{100}$ proteins, $i=\mathrm{1},\dots ,{n}_{\mathrm{1}}$, ${x}_{\mathrm{2},j}(\mathrm{0})=\mathrm{200}$ proteins, $j=\mathrm{1},\dots ,{n}_{\mathrm{2}}$, and initial ratio of strain 1 is $r(\mathrm{0})=\mathrm{0.1}$.

Fig.8 Effects of the metabolic load on the stochastic dynamics of the two-strain toggle.

Simulations of Equations (8) ?(11) with metabolic loading $\rho =\mathrm{0.5}$ and $\epsilon =\mathrm{0}$ for different populations sizes as indicated. The gray curves show the simulations of the deterministic model. Parameters are the same as in Figure 3. Initial conditions are ${x}_{\mathrm{1},i}(\mathrm{0})=\mathrm{100}$ proteins, $i=\mathrm{1},\dots ,{n}_{\mathrm{1}}$, ${x}_{\mathrm{2},j}(\mathrm{0})=\mathrm{200}$ proteins, $j=\mathrm{1},\dots ,{n}_{\mathrm{2}}$, and initial ratio of strain 1 is $r(\mathrm{0})=\mathrm{0.4}$.

Fig.9 Effects of the metabolic load on the extinction times of the two-strain consortium.

Normalized histograms of the extinction times for different values of $\rho $ obtained from 500 simulations of the stochastic model from Equations (8) ?(11) with $N=\mathrm{40}$ and $\epsilon =\mathrm{0}$. Parameters are the same as in Figure 3. Initial conditions are ${x}_{\mathrm{1},i}(\mathrm{0})=\mathrm{100}$ proteins, $i=\mathrm{1},\dots ,{n}_{\mathrm{1}}$, ${x}_{\mathrm{2},j}(\mathrm{0})=\mathrm{200}$ proteins, $j=\mathrm{1},\dots ,{n}_{\mathrm{2}}$, and initial population ratio is $r(\mathrm{0})=\mathrm{0.5}$.

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