Cooperative interactions in batch versus chemostat cultures

Chapter V: Conclusion

A.2 Cooperative interactions in batch versus chemostat cultures

We show more examples of different chemical-mediation mechanisms for cooper- ative interactions, and show population density dynamics and coexistence in batch and chemostat cultures.

The first cooperation mechanism is via crossfeeding of metabolites, as shown in Figure A.1A. The metabolites produced by one population can be considered as nutrients to the other population and promote their growth. Meanwhile, both popu- lations grow on another shared nutrient. We denote the population densities of cell type I and II by 𝑁₁ and𝑁₂, concentrations of nutrient I and II by 𝑀₁and 𝑀₂, and the shared nutrient concentration by 𝑀. We assume that the production of 𝑀₁and 𝑀₂ depend on population densities as well as the shared nutrient uptake, to avoid infinite production even when the nutrient is used up. In a batch culture, we can write down an ODE model:

cell I population density: d𝑁₁ d𝑡

=𝛼₁ 𝑀₂ 𝐾_𝑀 +𝑀₂

𝑁₁, cell II population density: d𝑁₂

d𝑡

=𝛼₁ 𝑀₁ 𝐾_𝑀 +𝑀₁

𝑁₂, nutrient I concentration: d𝑀₁

d𝑡

=𝛽₁ 𝑀 𝐾_𝑀 +𝑀

𝑁₁−𝛿 𝑀₁ 𝐾_𝑀+𝑀₁

𝑁₂, nutrient II concentration: d𝑀₂

d𝑡

=𝛽₂ 𝑀 𝐾_𝑀 +𝑀

𝑁₂−𝛿 𝑀₂ 𝐾_𝑀+𝑀₂

𝑁₁, nutrient concentration: d𝑀

d𝑡

=−𝛿 𝑀 𝐾_𝑀 +𝑀

(𝑁₁+𝑁₂).

(A.5)

Parameters𝛼₁, 𝛼₂are cell growth rates, 𝐾_𝑀 is the dissociation rate in nutrient con- sumption,𝛽₁, 𝛽₂are production rates of nutrients, and𝛿is the nutrient consumption

cell II cell I

enzyme II enzyme I

cell II cell I

nutrient II nutrient I

B Population growth dynamics

time (h) population density _{cell I cell II}

C Coexistence w/ perturbation Crossfeeding in

batch culture A

time (h)

density ratio

perturb density

cell II cell I

signal II signal I

E Population growth dynamics

time (h) population density cell I cell II

F Coexistence w/ perturbation Quorum sensing

in batch culture D

time (h)

density ratio

perturb density

H Population growth dynamics

time (h) population density ^{cell I cell II}

I Coexistence w/ perturbation Toxin removal

in batch culture G

time (h)

density ratio

perturb density

Figure A.1: Cooperation via cross-feeding, quorum sensing and toxin removal in the batch culture. (A)(D)(G) Schematic diagrams of a co-culture of two cell populations with cooperative interactions in a batch culture. There is a constant inlow/outflow of media. (B)(E)(H) Simulations of population densities in the batch culture. At 15hr in the simulation, we dilute the cell populations to a low initial densities and show their growth dynamics in fresh media. (C)(F)(I) Simulations of the density ratio of two populations with various initial densities. At 15hr, we perturb the densities to show the convergence of the density ratio.

rate. For simplicity, some parameters are set to be the same rate for two cell populations.

The second cooperation mechanism is via growth activation via quorum sensing molecules, as shown in Figure A.1B. The quorum sensing molecules can activate growth without being consumed. We denote concentrations of signal I and II by𝑆₁ and𝑆₂, and assume that the production of𝑆₁and𝑆₂depend on population densities as well as the shared nutrient uptake. In a batch culture, we can write down an ODE

model:

cell I population density: d𝑁₁ d𝑡

=𝛼₁ 𝑀 𝐾_𝑀+𝑀

𝑆₂ 𝐾_𝑆+𝑆₂

𝑁₁, cell II population density: d𝑁₂

d𝑡

=𝛼₂ 𝑀 𝐾_𝑀+𝑀

𝑆₁ 𝐾_𝑆+𝑆₁

𝑁₂, signal I concentration: d𝑆₁

d𝑡

= 𝛽₁ 𝑀 𝐾_𝑀 +𝑀

𝑁₁, antibiotics II concentration: d𝑆₂

d𝑡

= 𝛽₂ 𝑀 𝐾_𝑀 +𝑀

𝑁₂, nutrient concentration: d𝑀

d𝑡

=−𝛿 𝑀 𝐾_𝑀 +𝑀

(𝑁₁+𝑁₂).

(A.6)

The parameter𝐾_𝑆 is the dissociation rate in signal-induced growth activation.

The third cooperation mechanism is via toxin removal by secretion of enzymes that degrade toxins, as shown in Figure A.1C. We assume there are two toxins in the media, denoted by 𝑇₁ and 𝑇₂. The enzyme I secreted by cell II population can degrade toxin I and relieve the death of cell I population, and vice versa. We denote concentrations of enzyme I and II by𝑅₁and𝑅₂, and assume that the production of 𝑅₁and𝑅₂ depend on population densities as well as the shared nutrient uptake. In a batch culture, we can write down an ODE model, assuming Hill-type functions of toxin removal kinetics by the enzyme:

cell I population density: d𝑁₁ d𝑡

=

𝛼₁ 𝑀 𝐾_𝑀+𝑀

−𝛾 𝑇₁ 𝐾_𝑇 +𝑇₁

𝑁₁, cell II population density: d𝑁₂

d𝑡

=

𝛼₂ 𝑀 𝐾_𝑀+𝑀

−𝛾 𝑇₂ 𝐾_𝑇 +𝑇₂

𝑁₂, toxin I concentration: d𝑇₁

d𝑡

=−𝜂 𝑅₁ 𝐾_𝑅+𝑅₁

𝑇₁, toxin II concentration: d𝑇₂

d𝑡

=−𝜂 𝑅₂ 𝐾_𝑅+𝑅₂

𝑇₂, enzyme I concentration: d𝑅₁

d𝑡

=𝛽₁ 𝑀 𝐾_𝑀 +𝑀

𝑁₂, enzyme II concentration: d𝑅₂

d𝑡

=𝛽₂ 𝑀 𝐾_𝑀 +𝑀

𝑁₁, nutrient concentration: d𝑀

d𝑡

=−𝛿 𝑀 𝐾_𝑀+𝑀

(𝑁₁+𝑁₂).

(A.7)

The parameter 𝛾 is cell death rate, 𝐾_𝑇 is the dissociation rate in toxin killing, 𝜂 is toxin degradation rate by enzymes, 𝐾_𝑅 is the dissociation rate in enzymatic degradation.

Similarly, we can derive ODE models for populations in the chemostat culture by adding dilution and inflow of nutrient in the media.

cell II cell I

enzyme II enzyme I

cell II cell I

nutrient II nutrient I

B Population growth dynamics

time (h)

population density

cell I cell II

C Coexistence w/ perturbation Crossfeeding

in chemostat A

cell II cell I

signal II signal I

E Population growth dynamics

cell I cell II

time (h)

population density

F Coexistence w/ perturbation Quorum sensing

in chemostat D

time (h) density ratio perturb

density

H Population growth dynamics cell I cell II

time (h)

population density

I Coexistence w/ perturbation Toxin removal

in chemostat G

time (h) density ratio ^perturb_density

time (h)

density ratio

perturb density

Figure A.2: Cooperation via cross-feeding, quorum sensing and toxin removal in the chemostat culture. (A)(D)(G) Schematic diagrams of a co-culture of two cell populations with cooperative interactions in a chemostat culture. There is a constant inlow/outflow of media. (B)(E)(H) Simulations of population densities in the chemostat culture. At 20hr/100hr/50hr in the simulation, we dilute the cell populations to a low initial densities and show their growth dynamics in fresh media.

(C)(F)(I) Simulations of the density ratio of two populations with various initial densities. At 20hr/100hr/50hr, we perturb the densities 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. In Figure A.1, simulations of cell growth dynamics show coexistence of two populations with cooperative interactions, yet the steady state densities at stationary phase depend on initial cell densities as well as nutrients. When diluting and regrowing cells in fresh media, the new steady state at stationary phase is steered to different levels. The same two population system in chemostat culture can maintain a stable coexistence

with cooperative interactions despite initial condition and perturbations in densities, shown in Figure A.2.

A.3 Competition via accumulated versus consumed/degraded chemicals