Numerical investigation of microbial quorum sensing under various flow conditions

View article
Microbiology

Main article text

 

Introduction

Materials and Methods

where particles fi(r, t) travel in the direction i with the lattice velocity ci (c0 = (0, 0), c1 = (1, 0), c2 = (0, 1), c3 = (−1, 0), c4 = (0, −1), c5 = (1, 1), c6 = (−1, 1), c7 = (−1, −1), c8 = (1, −1)) to a new position r + ciΔt after a time step Δt. The relaxation time (τ) was described by the commonly used Bhatnagar–Gross–Krook collision operator (Bhatnagar, Gross & Krook, 1954) and the D2Q9 lattice with the corresponding equilibrium distribution function: feqi(r,t)=ωiρ(1+ucic2s+(uc2i)2c4suu2c2s)
where ωi are lattice weights (ω0 = 4/9, ω1–4 = 1/9, ω5–8 = 1/36), cs is a lattice dependent constant (here, c2s = 1/3), and u is the macroscopic flow velocity. The moments of the discretized mesoscopic particles retrieve the macroscopic density ρ=fi and momentum ρu=cifi. The Chapman-Enskog expansion showed that this LB approach recovers the incompressible NSE with the viscosity ν=c2s(τΔt2) (Krüger et al., 2017). Once the flow field was obtained, we simulated solute transport with a particle distribution function g, using the regularized LB algorithm (RLB) for numerical accuracy (Latt & Chopard, 2006; Latt, 2007) and the D2Q5 lattice for numerical efficiency (Li, Mei & Klausner, 2017): gi(r+ciΔt,t+Δt)=geqi(r,t)+(1Δtτ)ωi2c4sQi:Πneqi+ΩRXNi(r,t)
where ci are the lattice velocities (c0 = (0, 0), c1 = (1, 0), c2 = (0, 1), c3 = (−1, 0), c4 = (0, −1), c5 = (1,1)) corresponding to the lattice weights ωi0 = 1/3, ω1–4 = 1/6), and Qi : Πneqi is the tensor contraction of the two tensors Qi=cicTic2sI and Πneqi=jcicTi(gj(r,t)geqj(r,t)). The reaction term in the Eq. (3) describes the production of signaling molecules: ΩRXNi(r,t)=Δtωi(1+FH[ˆAˆθ])ˆkˆB
where F represents a multiplication factor which was set to either 0 or 10 to reflect the magnitude of autoinduced signal production (Fekete et al., 2010), ˆA is a concentration of signaling molecules, ˆθ is the QS induction threshold, ˆk is the basal production rate constant of signaling molecules, and ˆB is the microbial density. QS induction often displays a switch-like behavior (Fujimoto & Sawai, 2013; Heilmann, Krishna & Kerr, 2015; Hense & Schuster, 2015), which is represented in the model by a step function with a higher signal production rate above the threshold concentration of signaling molecule: H[ˆAˆθ]={1,(ˆAˆθ)0,(ˆA<ˆθ)

with the molecular diffusivity ˆD=c2s(τΔt2). Note that we are ignoring the breakdown of signaling molecules (Lee et al., 2002), limiting us to settings where production and transport are the dominant processes.

Results and Discussion

QS processes of a single microbial aggregate

Empirical approximation of concentration profiles

Adn(x)|x>1=exp(adnln(x)bdn+cdn)
where Aup and Adn are 0 in the down- and up-stream directions, respectively, and aup=0.376exp(2.5975Pe)+2.7165exp(0.0244Pe)

cup=7.1289exp(0.0348Pe)+5.9469exp(0.4272Pe)
𝑎dn=8.6156𝑃𝑒0.066813.3056
bdn=0.1051Pe0.2522+0.1082
cdn=7.5322Pe0.0464+8.7195

The effect of QS induced signal production on transport distances

QS induction between spatially distributed multiple microbial aggregates

where n is the number of aggregates, di0 is the distance between xi and x0 (di0 = xix0), xi is the location of ith aggregate, x0 is the reference location (x0 = 1), Dai is the Da calculated only with the density of ith microbial aggregate (i.e., microscopic Da), and Aup and Adn are Eqs. (9) and (10), respectively. Here, an example system with macroscopic Da(DaT=Dai) = 3.2Daθ consist of four aggregates (A1–4) located at x1 = 0.4, x2 = 1, x3 = 1.096 and x4 = 1.7 with the evenly distributed microscopic Dai (i.e. Da1 = Da2 = Da3 = Da4 = 0.8Daθ) was tested. In using Eq. (18), the profile was first constructed for Dai = Da* that does not consider autoinduction (F = 0). Then, if there is an aggregate with A(xi) ≥ 1, the profile was recalculated with updated Dai = (1 + F) × Da* until all Dai with A ≥ 1 were updated.

Conclusions and Perspectives

Additional Information and Declarations

Competing Interests

Christof Meile is an Academic Editor for PeerJ.

Author Contributions

Heewon Jung conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, authored or reviewed drafts of the paper, and approved the final draft.

Christof D. Meile conceived and designed the experiments, authored or reviewed drafts of the paper, and approved the final draft.

Data Availability

The following information was supplied regarding data availability:

The LB code is available at BitBucket: https://bitbucket.org/MeileLab/jung_qsTpDistn.

Funding

This work was supported by the Genomic Sciences Program in the DOE Office of Science, Biological and Environmental Research DE-SC0016469 and DE-SC0020374. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

4 Citations 1,035 Views 258 Downloads

Your institution may have Open Access funds available for qualifying authors. See if you qualify

Publish for free

Comment on Articles or Preprints and we'll waive your author fee
Learn more

Five new journals in Chemistry

Free to publish • Peer-reviewed • From PeerJ
Find out more