# Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Framework of the MRC-CLB-ADT Model

#### 2.1. Module 1. Modified RapidCell (MRC) Model for Particle Chemosensing in Two Chemoattractant Fields

#### 2.1.1. Static (Time-Invariant) Concentration Fields

#### 2.1.2. Dynamic (Time-Variant) Concentration Fields

#### 2.2. Module 2. Colloidal Lattice Boltzmann (CLB) Model for Particle-Fluid Interactions

#### 2.2.1. Fluid Flow Submodule (FFS)

#### 2.2.2. Particle Flow Submodule

#### 2.3. Module 3. Advective-Diffusive Transport (ADT) Model for Chemoattractant Distributions

#### 2.4. Coupling of the Modules, MRC-CLB-ADT Model

#### 2.5. Simulation Parameters

## 3. Results

#### 3.1. Simulations with Imposed Temporally-Invariant, Spatially-Variant Chemoattractant Concentrations

#### 3.2. Simulations with Spatiotemporal Variations in Chemoattractant Concentrations Computed via ADT Model

- Case 1. “ADT Point: Initial”: At $t=0$, MeAsp and Ser were released into the fluid from point sources at $\left(x=50,y=101\right)$ and $\left(x=150,y=101\right)$, respectively. No additional chemoattractant releases occurred for $t>0$. Snapshots from this simulation are shown in Figure 8.
- Case 2. “ADT Point: Continuous”: After the initial condition was set up as in Case 1, $\left(\u25b5C\right)$ of each chemoattractant was released into the fluid in each time-step for $t>\u25b5t$ from their respective point source locations, at which their maximum concentrations were maintained throughout the simulation. Snapshots from this simulation are shown in Figure 9.
- Case 4. “ADT Imposed: Continuous”: After the initial concentration fields of chemoattractants were established as in Case 3, $\left(\u25b5C\right)$ of each chemoattractant was released into the fluid in each time-step for $t>\u25b5t$ from their respective point source locations. Snapshots from this simulation are shown in Figure 11.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Numerical Validation of the ADT Model

**Figure A1.**Benchmark problem used to validate the ADT model based on the LBM. Temporal and spatial distributions of a substrate were computed following its instantaneous injection into a Couette flow as a point source at $x=0$. The flow domain was sheared along the top and bottom boundaries with the same magnitude of velocity, u, but in the opposite directions at a vertical distance of ${x}_{2}$ from the injection point. The lateral boundaries are periodic. The velocity profile is shown on the right panel.

**Figure A2.**Comparison of analytically and numerically (by the ADT model) computed concentration gradient dynamics for solute following instantaneous release into a Coutte flow at center of domain. The ADT model simulation results at (

**a**) $t={\tau}_{s}$ and (

**b**) $t=2{\tau}_{s}$; analytic results at (

**c**) $t={\tau}_{s}$ and (

**d**) $t=2{\tau}_{s}$.

**Figure A3.**Comparison of analytically and numerically computed (by the ADT model) substrate concentration profiles along the horizontal and vertical cross-sections with respect to the center of the concentration field at (

**a**) $t={\tau}_{s}$ and (

**b**) $t=2{\tau}_{s}$. ${\mathbf{x}}_{c}=\left({x}_{c},{y}_{c}\right)=(51,51)$ is the center point, where the instantaneous point source was located.

#### Appendix A.2. ADT Model Simulations of Advective-Diffusive Substrate Transport in a Flow Channel

**Figure A4.**Advective and diffusive transport of a chemoattractant, characterized by $Pe=500$, in a horizontal flow channel (

**a**) without internal obstacles or (

**b**) with a circular obstacle shown in white. Snapshots are taken at at $\Omega =$ 0.0, 0.11, 0.18, 0.25, 0.35 and 0.49.

#### Appendix A.3. With Simulated ADT-Concentrations Moving around Obstacles

**Figure A5.**Snapshots of the chemoattractant concentration field (whose contour lines are shown by solid lines) and trajectory of a chemosensory particle (shown by a light green dashed line) at time-steps of $1000$; $\mathrm{28,000}$; $\mathrm{68,000}$; $\mathrm{80,000}$; $\mathrm{174,000}$; and $\mathrm{204,000}$ in (

**a**) through (

**f**), respectively. The total chemoattractant mass remained constant throughout this simulation.

**Figure A6.**Snapshots of the chemoattractant concentration field (whose contour lines are shown by solid lines) and trajectory of a chemosensory particle (shown by a light blue dashed line) at time-steps of $1000$; $\mathrm{14,000}$; $\mathrm{32,000}$; $\mathrm{42,000}$; $\mathrm{52,000}$; and $\mathrm{62,000}$ in (

**a**) through (

**f**), respectively. Because the chemoattractant was continuously added to the center of the domain, the total chemoattractant mass in the fluidic domain increased in time. As a result, the domain was saturated with the chemoattractant at later times, which led to the random walk in particle trajectory towards the end of the simulation.

#### Appendix A.4. Tables of Parameters and Variables for All the Modules

#### Appendix A.4.1. Prescribed Parameters for the MRC Module

**Table A1.**Descriptions and values of predetermined parameters in Equations (1)–(7).

Parameter | Description [Source] | Value |
---|---|---|

N | Number of chemoreceptors in receptor cluster [28] | 18 |

${v}_{a}:{v}_{s}$ | Ratio of Tar to Tsr receptors [28] | 1:1.4 |

${K}_{a}^{on}$ | Dissociation constant in the on state of Tar receptors [17] | 0.012 $\mathsf{\mu}$M |

${K}_{a}^{off}$ | Dissociation constant in the off state of Tar receptors [17] | 0.0017 $\mathsf{\mu}$M |

${K}_{s}^{on}$ | Dissociation constant in the on state of Tsr receptors [17] | 10${}^{6}$ $\mathsf{\mu}$M |

${K}_{s}^{off}$ | Dissociation constant in the off state of Tsr receptors [17] | 100 $\mathsf{\mu}$M |

${\left[CheR\right]}_{tot}$ | Total $CheR$ concentration [17] | 0.16 $\mathsf{\mu}$M |

${\left[CheB\right]}_{tot}$ | Total $CheB$ concentration [17] | 0.28 $\mathsf{\mu}$M |

${\left[CheZ\right]}_{tot}$ | Total $CheZ$ concentration [17] | * |

$m{b}_{0}$ | Basal motor bias [17] | $0.65$ |

H | Motor Hill coefficient [17] | $10.3$ |

a | Scaling factor for methylation [17] | $0.0625$ |

b | Scaling factor for demethylation [17] | $0.0714$ |

${k}_{Z}$ | Rate constant [17] | $30/{\left[CheZ\right]}_{tot}$$\mathsf{\mu}$M${}^{-1}$${s}^{-1}$ |

${k}_{Y}$ | Rate constant [17] | 100 $\mathsf{\mu}$M${}^{-1}$ ${s}^{-1}$ |

${k}_{s}$ | Scaling coefficient [17] | $0.45$$\mathsf{\mu}$M |

${\gamma}_{Y}$ | Rate constant [17] | 0.1 s${}^{-1}$ |

${C}_{a0}$ | Minimum chemoattractant concentration for MeAsp | 0.1 $\mathsf{\mu}$M |

${C}_{s0}$ | Minimum chemoattractant concentration for Ser | 0.1 $\mathsf{\mu}$M |

$\omega $ | Scaling parameter for MeAsp gradient | 1 |

$\nu $ | Scaling parameter for Ser gradient | 0.1 or 0.001 |

$({x}_{a},{y}_{a})$ | Location of the maximum MeAsp concentration initially | (14.3 cm, 28.9 cm) |

$({x}_{s},{y}_{s})$ | Location of the maximum Ser concentration initially | (42.9 cm, 28.9 cm) |

${L}^{*}$ | Domain length | 57 cm |

r | Scaling parameter for domain size | 14.3 cm |

_{tot}concentration does not need to be specified explicitly since it is canceled with itself when multiplying with k

_{Z}in Equation (3).

**Table A2.**Descriptions of functions and variables in Equations (1)–(7).

Variable | Description [Source] |
---|---|

F | Total free energy differences between ‘on’ or ‘off’ state [28] |

$h\left(m\right)$ | Offset energy given by $1-m/2$ [28] |

$\left[MeAsp\right]$ | Chemoattractant MeAsp concentration [28] |

$\left[Ser\right]$ | Chemoattractant Ser concentration [28] |

${A}_{c}$ | Probability of the cluster activity [17] |

[CheY-P] | Concentration of phosphorylated CheY [17] |

m | Receptor methylation [17] |

$mb$ | Motor bias [17] |

$x,y$ | Horizontal and vertical coordinates |

#### Appendix A.4.2. System Parameters and Variables for the CLB ad ADT Modules

**Table A3.**Notations used in CLB and ADT Modules. FFS and PFS correspond to the Fluid Flow Submodule (Section 2.2.1) and the Particle Flow Submodule (Section 2.2.2) of the CLB Module.

Notation | Type | Description | Used by |
---|---|---|---|

${c}_{s}$ | parameter | speed of sound | FFS, ADT |

$\mathbf{e}$ | parameter | unit velocity vectors | FFS, PFS , ADT |

${f}_{i}$ | variable | population densities associated with fluid flow | FFS, PFS |

${f}_{i}^{eq}$ | variable | equilibrium distribution associated fluid flow | FFS |

${f}_{m}$ | parameter | particle force strength | PFS |

$\mathbf{g}$ | parameter | acceleration due to external forces | FFS, PFS |

${g}_{i}$ | variable | population densities associated with substrate transport | ADT |

${g}_{i}^{eq}$ | variable | equilibrium distribution associated substrate transport | ADT |

i | variable | index | FFS, PFS, ADT |

${m}_{p}$ | parameter | particle mass | PFS |

$\mathbf{r}$ | variable | position vector | FFS, PFS, ADT |

${\mathbf{r}}_{b}$ | variable | position of boundary nodes of ECP | PFS |

${\mathbf{r}}_{b}^{c}$ | variable | covered lattice nodes by ECP motion | PFS |

${\mathbf{r}}_{b}^{u}$ | variable | uncovered lattice nodes by ECP motion | PFS |

${\mathbf{r}}_{c}$ | variable | position of the ECP’s centroid | PFS |

${\mathbf{r}}_{cl}$ | variable | location of the cluster receptor | PFS |

${\mathbf{r}}_{i}$ | variable | distance vector | PFS |

$|{\mathbf{r}}_{it}|$ | parameter | repulsive threshold distance | PFS |

${\mathbf{r}}_{p}$ | parameter | particle radius | PFS |

${\mathbf{r}}_{pw}$ | variable | surface to surface distance between the wall and ECP | PFS |

${\mathbf{r}}_{v}$ | variable | position of intra-particle lattice node | PFS |

t | variable | time | FFS, PFS, ADT |

${t}^{*}$ | variable | post-collision time | PFS |

$\mathbf{u}$ | variable | fluid velocity | FFS, PFS, ADT |

C | variable | chemoattractant concentration | ADT |

D | parameter | diffusion coefficient of chemoatractant | ADT |

${\mathbf{F}}_{run}$ | variable | forces associated with running motion of ECP | PFS |

${\mathbf{F}}_{{\mathbf{r}}_{b}}$ | variable | hydrodynamic forces | PFS |

${\mathbf{F}}_{{\mathbf{r}}_{b}}^{c,u}$ | variable | forces associated with (un)covered lattice nodes | PFS |

${\mathbf{F}}_{{\mathbf{r}}_{p}w}$ | variable | steric interaction forces between ECP and wall | PFS |

${\mathbf{F}}_{T}$ | variable | total force imposed on ECP | PFS |

${I}_{p}$ | parameter | moment of inertia of ECP | PFS |

${L}^{*}$ | parameter | domain length | FFS |

M | parameter | Mach number | FFS |

${\mathbf{T}}_{tumble}$ | variable | torque associated with tumble motion of ECP | PFS |

${\mathbf{U}}_{p}$ | variable | translation velocity of of ECP | PFS |

${\theta}_{cl}$ | variable | rotation angle of the receptor cluster | PFS |

$\tilde{\nu}$ | parameter | fluid kinematic viscosity | FFS |

$\rho $ | variable | fluid density | FFS, PFS |

$\tau $ | parameter | relaxation parameter associated fluid flow | FFS |

${\tau}_{c}$ | parameter | relaxation parameter associated substrate transport | PFS |

$\psi $ | parameter | stiffness parameter associated with steric interaction forces | PFS |

${\omega}_{i}$ | parameter | weights associated with the D2Q9 lattice geometry | FFS, PFS, ADT |

$\tilde{\phi}$ | variable | uniform deviate | PFS |

$\u25b5P$ | parameter | pressure differential | FFS |

$\u25b5t$ | parameter | temporal increment | FFS, PFS, ADT |

$\u25b5x$ | parameter | lattice spacing | FFS, PFS, ADT |

$\u25b5\theta $ | variable | angular rotation of ECP | FFS |

${\Omega}_{tumble}$ | variable | angular velocity of ECP due to its tumbling motion only | PFS |

${\Omega}_{p}$ | variable | angular velocity of of ECP | PFS |

$\mathsf{{\rm Y}}$ | parameter | time-scale factor associated with ECP’s angular rotation | PFS |

## References

- Stanton, M.M.; Sanchez, S. Pushing bacterial biohybrids to in vivo applications. Trends Biotechnol.
**2017**, 35, 910–913. [Google Scholar] [CrossRef] [PubMed] - Chien, T.; Doshi, A.; Danino, T. Advances in bacterial cancer therapies using synthetic biology. Curr. Opin. Syst. Biol.
**2017**, 5, 1–8. [Google Scholar] [CrossRef] [PubMed] - Felicetti, L.; Femminella, M.; Reali, G.; Lio, P. Applications of molecular communications to medicine: A survey. Nano Commun. Netw.
**2016**, 7, 27–45. [Google Scholar] [CrossRef] - Sylvain, M. Bacterial microsystems and microrobots. Biomed. Microdevices
**2012**, 14, 1033–1045. [Google Scholar] - Ceylan, H.; Giltinan, J.; Kozielski, K.; Sitti, M. Mobile microrobots for bioengineering applications. Lab Chip
**2017**, 17, 1705–1724. [Google Scholar] [CrossRef] - Carlsen, R.W.; Sitti, M. Bio-hybrid cell-based actuators for microsystems. Small
**2014**, 10, 3831–3851. [Google Scholar] - Purcell, E.M. Life at low Reynolds number. Am. J. Phys.
**1977**, 45, 3–11. [Google Scholar] [CrossRef] - Zhang, L.; Abbott, J.J.; Dong, L.; Kratochvil, B.E.; Bell, D.; Nelson, B.J. Artificial bacterial flagella: Fabrication and magnetic control. Appl. Phys. Lett.
**2009**, 94, 064107. [Google Scholar] [CrossRef] - Patino, T.; Mestre, R.; Mestre, S. Miniaturized soft bio-hybrid robotics: A step forward into healthcare applications. Lab Chip
**2016**, 16, 3626–3630. [Google Scholar] [CrossRef] [PubMed] - Park, S.J.; Park, S.H.; Cho, S.; Kim, D.M.; Lee, Y.; Ko, S.Y.; Hong, Y.; Choy, H.E.; Min, J.J.; Park, J.O.; et al. New paradigm for tumor theranostic methodology using bacteria-based microrobot. Sci. Rep.
**2013**, 3, 1–8. [Google Scholar] [CrossRef] - Shao, J.; Xuan, M.; Zhang, H.; Lin, X.; Wu, Z.; He, Q. Chemotaxis-Guided Hybrid Neutrophil Micromotor for Actively Targeted Drug Transport. Angew. Chem. Int. Ed.
**2017**, 56, 12935–12939. [Google Scholar] [CrossRef] [PubMed] - Nelson, B.J.; Kaliakatsos, I.K.; Abbott, J.J. Microrobots for minimally invasive medicine. Annu. Rev. Biomed. Eng.
**2010**, 12, 55–85. [Google Scholar] [CrossRef] - Anderson, J.C.; Clarke, E.J.; Arkin, A.P.; Voigt, C.A. Environmentally controlled invasion of cancer cells by engineered bacteria. J. Mol. Biol.
**2006**, 355, 619–627. [Google Scholar] [CrossRef] - Zoaby, N.; Shainsky-Roitman, J.P.; Badarneh, S.; Abumanhal, H.; Leshansky, A.; Yaron, S.; Schroeder, A. Autonomous bacterial nanoswimmers target cancer. J. Control. Release
**2017**, 257, 68–75. [Google Scholar] [CrossRef] - Felfoul, O.; Mohammadi, M.; Gaboury, L.; Martel, S. Tumor targeting by computer controlled guidance of magnetotactic bacteria acting like autonomous microrobots. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, CA, USA, 25–30 September 2011; pp. 1304–1308. [Google Scholar]
- Colin, R.; Sourjik, V. Emergent properties of bacterial chemotaxis pathway. Curr. Opin. Microbiol.
**2017**, 39, 24–33. [Google Scholar] [CrossRef] [PubMed] - Vladimirov, N.; Løvdok, L.; Lebiedz, D.; Sourjik, V. Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate. PLOS Comput. Biol.
**2008**, 4, e1000242. [Google Scholar] [CrossRef] - Cortez, R. The method of regularized Stokeslets. SIAM J Sci. Comput.
**2001**, 23, 1204–1225. [Google Scholar] [CrossRef] - Başağaoğlu, H.; Allwein, S.; Succi, S.; Dixon, H.; Carrola, J.T., Jr.; Stothoff, S. Two- and three-dimensional lattice-Boltzmann simulations of particle migration in microchannels. Microfluid Nanofluid
**2013**, 15, 785–796. [Google Scholar] [CrossRef] - Başağaoğlu, H.; Carrola, J.T., Jr.; Freitas, C.J.; Başağaoğlu, B.; Succi, S. Lattice Boltzmann simulations of vortex entrapment of particles in a microchannel with curved and flat edges. Microfluid Nanofluid
**2015**, 18, 1165–1175. [Google Scholar] [CrossRef] - Nguyen, H.; Başağaoğlu, H.; McKay, C.; Carpenter, A.; Succi, S.; Healy, F. Coupled RapidCell and lattice-Boltzmann models to simulate hydrodynamics of bacterial transport in response to chemoattractant gradients in confined domains. Microfluid Nanofluid
**2016**, 20, 1–14. [Google Scholar] [CrossRef] - Xu, F.; Bierman, R.; Healy, F.; Nguyen, H. A multiscale model of Escherichia coli chemotaxis from intracellular signaling pathway to motility and nutrient uptake in nutrient gradient and isotropic fluid environments. Comput. Math. Appl.
**2016**, 71, 2466–2478. [Google Scholar] [CrossRef] - Lai, R.Z.; Gosink, K.K.; Parkinson, J.S. Signaling Consequences of Structural Lesions that Alter the Stability of Chemoreceptor Trimers of Dimers. J. Mol. Biol.
**2017**, 429, 2823–2835. [Google Scholar] [CrossRef] - Pan, W.; Dahlquist, F.W.; Hazelbauer, G.L. Signaling complexes control the chemotaxis kinase by altering its apparent rate constant of autophosphorylation. Protein Sci.
**2017**, 26, 1535–1546. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ud-Din, A.I.M.S.; Roujeinikova, A. Flagellin glycosylation with pseudaminic acid in Campylobacter and Helicobacter: Prospects for development of novel therapeutics. Cell. Mol. Life Sci.
**2017**, 2018, 1163–1178. [Google Scholar] - Ma, Q.; Sowa, Y.; Baker, M.A.; Bai, F. Bacterial Flagellar Motor Switch in Response to CheY-P Regulation and Motor Structural Alterations. Biophys. J.
**2016**, 110, 1411–1420. [Google Scholar] [CrossRef] - Krembel, A.; Colin, R.; Sourjikr, V. Importance of multiple methylation sites in Escherichia coli chemotaxis. PloS ONE
**2015**, 10, e0145582. [Google Scholar] [CrossRef] - Edgington, M.; Tindall, M. Understanding the link between single cell and population scale responses of Escherichia coli in differing ligand gradients. Comput. Struct. Biotechnol. J.
**2015**, 13, 528–538. [Google Scholar] [CrossRef][Green Version] - Succi, S. The lattice-Boltzmann Equation for Fluid Dynamics and Beyond; Oxford University Press: New York, NY, USA, 2001. [Google Scholar]
- Bhatnagar, P.L.; Gross, E.P.; Krook, M. A model for collision process in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev.
**1954**, 94, 511–525. [Google Scholar] [CrossRef] - Qian, Y.H.; D’Humieres, D.; Lallemand, P. Lattice BGK models for Navier-Stokes equation. Europhys. Lett.
**1992**, 17, 479–484. [Google Scholar] [CrossRef] - Buick, J.M.; Greated, C.A. Gravity in a lattice Boltzmann model. Phys. Rev. E.
**2000**, 61, 5307–5320. [Google Scholar] [CrossRef][Green Version] - Hanasoge, S.M.; Succi, S.; Orszag, S. Lattice Boltzmann method for electromagnetic wave propagation. Europhys. Lett.
**2011**, 96, 14002. [Google Scholar] [CrossRef][Green Version] - Ladd, A.J.C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech.
**1994**, 271, 285–309. [Google Scholar] [CrossRef] - Ding, E.J.; Aidun, C. Extension of the Lattice-Boltzmann method for direct simulation of suspended particles near contact. J. Stat. Phys.
**2003**, 112, 685–708. [Google Scholar] [CrossRef] - Başağaoğlu, H.; Succi, S.; Wyrick, D.; Blount, J. Particle shape influences settling and sorting behavior in microfluidic domains. Sci. Rep.
**2018**, 8, 8583. [Google Scholar] [CrossRef] - Aidun, C.K.; Lu, Y.; Ding, E.J. Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech.
**1998**, 373, 287–311. [Google Scholar] [CrossRef] - Başağaoğlu, H.; Succi, S. Lattice-Boltzmann simulations of repulsive particle-particle and particle-wall interactions: Coughing and choking. J. Chem. Phys.
**2010**, 132, 134111. [Google Scholar] [CrossRef] [PubMed] - Başağaoğlu, H.; Meakin, P.; Succi, S.; Redden, G.R.; Ginn, T.R. Two-dimensional lattice-Boltzmann simulation of colloid migration in rough-walled narrow flow channels. Phys. Rev. E
**2008**, 77, 031405. [Google Scholar] [CrossRef] - Feng, J.; Hu, H.H.; Joseph, D.D. Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid Part 1. Sedimentation. J. Fluid. Mech.
**1994**, 261, 95–134. [Google Scholar] [CrossRef] - Gibbs, R.J.; Matthews, M.D.; Link, D.A. The relationship between sphere size and settling velocity. J. Sedimentary Petrol.
**1971**, 41, 7–18. [Google Scholar] - Hilpert, M. Lattice-Boltzmann model for bacterial chemotaxis. J. Math. Biol.
**2005**, 51, 302–332. [Google Scholar] [CrossRef] [PubMed] - Kang, Q.; Zhang, D.; Chen, S.; He, X. Lattice Boltzmann simulation of chemical dissolution in porous media. Phys. Rev. E
**2002**, 85, 036318. [Google Scholar] [CrossRef] - Landsberg, P.T. Grad v or grad(Dv)? J. Appl. Phys.
**1984**, 56, 1119. [Google Scholar] [CrossRef] - Schnitzer, M.J. Theory of continuum random walks and application to chemotaxis. Phys. Rev. E
**1993**, 48, 2553–2568. [Google Scholar] [CrossRef] - Boon, J.P.; Lutsko, J.F. Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations. J. Stat. Phys.
**2017**, 166, 1441–1454. [Google Scholar] [CrossRef] - Andreucci, D.; Cirillo, E.N.; Colangeli, M.; Gabrielli, D. Fick and Fokker-Planck diffusion law in inhomogeneous media. J. Stat. Phys.
**2019**, 174, 469–493. [Google Scholar] [CrossRef] - Wu, F.; Shi, W.; Liu, F. A lattice Boltzmann model for the Fokker–Planck equation. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 2776–2790. [Google Scholar] [CrossRef] - Ma, Y.; Zhu, C.; Ma, P.; Yu, K. Studies on the Diffusion Coefficients of Amino Acids in Aqueous Solutions. J. Chem. Eng. Data
**2005**, 50, 1192–1196. [Google Scholar] [CrossRef] - Frankel, N.W.; Pontius, W.; Dufour, Y.S.; Long, J.; Hernandez-Nunez, L.; Emonet, T. Adaptability of non-genetic diversity in bacterial chemotaxis. eLife
**2014**, 3, e03526. [Google Scholar] [CrossRef] - Jasuja, R.; Lin, Y.; Trentham, D.R.; Khan, S. Response tuning in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA
**1999**, 96, 11346–11351. [Google Scholar] [CrossRef][Green Version] - Fetter, G.W. Contaminant Hydrogeology; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Matlab R2017a. Available online: https://www.mathworks.com/ (accessed on 4 May 2019).
- Mello, B.A.; Tu, Y. Effects of adaptation in maintaining high sensitivity over a wide range of backgrounds for Escherichia coli chemotaxis. Biophys. J.
**2007**, 92, 2329–2337. [Google Scholar] [CrossRef] [PubMed] - Frank, V.; Piñas, G.E.; Cohen, H.; Parkinson, J.S.; Vaknin, A. Networked Chemoreceptors Benefit Bacterial Chemotaxis Performance. mBIO
**2016**, 6, e01824-16. [Google Scholar] [CrossRef] [PubMed] - Bolster, D.; Dentz, M.; Le Borgn, T. Hypermixing in linear shear flow. Water Resour. Res.
**2011**, 47, W09602. [Google Scholar] [CrossRef]

**Figure 1.**Coupling of the modules of the multiscale MRC-CLB-ADT model and information exchanges among the modules.

**Figure 2.**Chemotactic signaling by Tar and Tsr MCPs in E. coli. Chemoattractants such as N-methyl-L-aspartate and L-serine (represented by triangle and diamond shapes) are sensed by Tar and Tsr MCPs, respectively, and binding results in signal transduction across the cell membrane to a phosphorelay response circuit. Phosphoryl group (P) transfer to Che proteins controls direction of rotation of flagellar motor and MCP methylation-dependent adaptive response. Default flagellar rotation is counterclockwise, causing cell to run; switching to clockwise rotation results in reorientation of cell through tumbling motion due to flagellar unbundling. Circled letters represent Che proteins described in text, e.g., A represents CheA, Y represents CheY, etc.

**Figure 3.**D2Q9 (two-dimensional nine velocity vector) lattice geometry. The vector basis set for the D2Q9 model consists of a null vector (rest population), which improves the stability of the algorithm, four off-diagonal vectors of length unity directed towards the nearest neighbor nodes, and four diagonal vectors of length $\sqrt{2}$ directed toward the next-nearest neighboring nodes.

**Figure 4.**LB model representation of an ECP (left) and the momentum exchanges between the ECP and the fluid (right) [34,38,39]. Filled circles are the intra-particle virtual fluid nodes of ECP closest to its surface, filled triangles outside the ECP surface represent its extra-particle bulk fluid nodes, and the filled square represents the boundary node located half-way between the intra-particle node (${\mathbf{r}}_{v}$) and extra-particle node (${\mathbf{r}}_{v}+{\mathbf{e}}_{i}\u25b5t$). Hydrodynamic links along which the ECP exchanges momentum with the fluid are shown by red line segments.

**Figure 5.**Data exchanges between submodels (MRC, CLB and ADT) in each time step. $\left[MeAsp\right]$ and $\left[Ser\right]$ are MeAsp and Ser dynamic concentrations; $\mathbf{u}$ is the fluid velocity; ${\mathbf{F}}_{run}$ and ${\mathbf{T}}_{tumble}$ are the force and torque associated with direct run and tumble motion of an ECP.

**Figure 6.**Spatially-variant, temporally-invariant MeAsp and Ser concentration fields in a 2D fluidic domain. Axes represent distances across the domain and colorbars represent amino acid chemoattractant concentrations in $\u25b5C$, (red/yellow = MeAsp, leftward side of gradient and blue/magenta = Ser, rightward side of gradient). In MRC-CLB simulations, two different ratios of chemoattractant gradient were chosen, with MeAsp gradient parameter $\omega $ set at $\omega $ = 1, and Ser gradient parameter $\nu $ set at either (

**a**) $\nu $ = 0.1 or (

**b**) $\nu $ = 0.001.

**Figure 7.**Average number of time steps the ECP resided in the “MeAsp half” or in the “On Ser Half”, concluded from ten replicates of MRC-CLB simulations with “Imposed” chemoatrractant concentration fields at the end of 50,000 time steps. Simulations were performed for two different values of $\nu $ (MeAsp parameter $\omega $ is fixed at $\omega =1$, leading to $\omega /\nu $ ratios of $1/0.1$ and $1/0.001$).

**Figure 8.**Trajectories of the ECP computed by the MRC-CLB-ADT model for Case 1 “ADT Point: Initial” and $\nu =0.1$ at the dimensionless times (in LB units) of $\mathrm{10,000}$; $\mathrm{18,000}$; $\mathrm{30,000}$; and $\mathrm{50,000}$ are shown in (

**a**–

**d**). Simulation times can be expressed in seconds by multiplying the dimensionless times by a factor of $0.938$. Each snapshot shows the contour plots of the concentration fields of MeAsp (left color bar) and Ser (right color bar). The center of MeAsp was initially on the left half and the center of Ser was on the right half of the domain. The total mass of MeAsp and Ser remained unchanged.

**Figure 9.**Trajectories of the ECP computed by the MRC-CLB-ADT model for Case 2 “ADT Point: Continuous” and $\nu =0.1$ at the dimensionless times (in LB units) of $\mathrm{10,000}$; $\mathrm{18,000}$; $\mathrm{30,000}$; and $\mathrm{50,000}$ are shown in (

**a**–

**d**). Simulation times can be expressed in seconds by multiplying the dimensionless times by a factor of $0.938$. Each snapshot shows the contour plots of the concentration fields of MeAsp (left color bar) and Ser (right color bar). The center of MeAsp was initially on the left half and the center of Ser was on the right half of the domain.

**Figure 10.**Trajectories of the ECP computed by the MRC-CLB-ADT model for Case 3 “ADT Imposed: Initial” and $\nu =0.1$ at the dimensionless times (in LB units) of $\mathrm{10,000}$; $\mathrm{18,000}$; $\mathrm{30,000}$; and $\mathrm{50,000}$ are shown in (

**a**–

**d**). Simulation times can be expressed in seconds by multiplying the dimensionless times by a factor of $0.938$. Each snapshot shows the contour plots of the concentration fields of MeAsp (left color bar) and Ser (right color bar). The center of MeAsp was initially on the left half and the center of Ser was on the right half of the domain.

**Figure 11.**Trajectories of the ECP computed by the MRC-CLB-ADT model for Case 4 “ADT Imposed: Continuous” and $\nu =0.1$ at the dimensionless times (in LB units) of $\mathrm{10,000}$; $\mathrm{18,000}$; $\mathrm{30,000}$; and $\mathrm{50,000}$ are shown in (

**a**–

**d**). Simulation times can be expressed in seconds by multiplying the dimensionless times by a factor of $0.938$. Each snapshot shows the contour plots of the concentration fields of MeAsp (left color bar) and Ser (right color bar). The center of MeAsp was initially on the left half and the center of Ser was on the right half of the domain.

**Figure 12.**The source location of MeAsp and Ser is initially at ${P}_{1}=(50,101)$ and ${P}_{2}=(150,101)$, respectively. For ten replicates per simulation type, the time history of the average spatial distance between the position of the ECP and ${P}_{1}$ or ${P}_{2}$ is d${}_{(50,101)}$ or d${}_{(150,101)}$ in two different fluidic environments, characterized by $\nu =$ 0.1 or $\nu =$ 0.001. If d${}_{\left(\mathrm{50,101}\right)}-$d${}_{\left(\mathrm{150,101}\right)}<0$, the ECP retains its chemotactic activities mostly in the left half; otherwise, it would largely reside in the right half of the fluidic domain. The simulation types include (

**a**) imposed concentrations, (

**b**) Case 1, (

**c**) Case 2, (

**d**) Case 3, and (

**e**) Case 4.

**Figure 13.**Ten simulations of 50,000 time steps each were performed for the“Imposed” case, and Case 1 through Case 4, in which trajectories of an ECP in a fluidic environment with an $\omega $/$\nu $ ratio of (

**a**) 1/0.1 and (

**b**) 1/0.001 were traced. Heights of the bars correspond to the total residence time of an ECP either in the right half, in which the Ser concentration was initialized, or in the left half, in which the MeAsp concentration was initialized, of the fluidic domain. The first set of bars in both (

**a**) and (

**b**) are repeated from Figure 7 while the remaining sets are from ADT model simulations.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

King, D.; Başağaoğlu, H.; Nguyen, H.; Healy, F.; Whitman, M.; Succi, S. Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments. *Entropy* **2019**, *21*, 465.
https://doi.org/10.3390/e21050465

**AMA Style**

King D, Başağaoğlu H, Nguyen H, Healy F, Whitman M, Succi S. Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments. *Entropy*. 2019; 21(5):465.
https://doi.org/10.3390/e21050465

**Chicago/Turabian Style**

King, Danielle, Hakan Başağaoğlu, Hoa Nguyen, Frank Healy, Melissa Whitman, and Sauro Succi. 2019. "Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments" *Entropy* 21, no. 5: 465.
https://doi.org/10.3390/e21050465