A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration, Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands

Abstract

Bacterial chemotaxis can enhance the bioremediation of contaminants in aqueous and subsurface environments if the contaminant is a chemoattractant that the bacteria degrade. The process can be promoted by traveling bands of chemotactic bacteria that form due to metabolism-generated gradients in chemoattractant concentration. We developed a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to model chemotaxis, because LBMs are well suited to model reactive transport in the complex geometries that are typical for subsurface porous media. This MRT-LBM can attain a better numerical stability than its corresponding single-relaxation-time LBM. We performed simulations to investigate the effects of substrate diffusion, initial bacterial concentration, and hydrodynamic dispersion on the formation, shape, and propagation of bacterial bands. Band formation requires a sufficiently high initial number of bacteria and a small substrate diffusion coefficient. Uniform flow does not affect the bands while shear flow does. Bacterial bands can move both upstream and downstream when the flow velocity is small. However, the bands disappear once the velocity becomes too large due to hydrodynamic dispersion. Generally bands can only be observed if the dimensionless ratio between the chemotactic sensitivity coefficient and the effective diffusion coefficient of the bacteria exceeds a critical value, that is, when the biased movement due to chemotaxis overcomes the diffusion-like movement due to the random motility and hydrodynamic dispersion.

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Acknowledgments

This work was supported by NSF #0911425. We would like to thank our colleagues Dr. Yaqi You and Dr. Shao-Yiu Hsu for their helpful suggestions and comments.

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Correspondence to Zhifeng Yan.

Appendix: Chapman–Enskog Expansion

Appendix: Chapman–Enskog Expansion

Here, we present the elaborate derivation that shows the LB equations (8) and (9) approximate solutions to the continuum-scale equations (4) and (5) via a Chapman–Enskog expansion (McCracken and Abraham 2005). To simplify the notation, we first consider a generic reactive and chemotactic species that could represent either bacteria or substrate. For the special case of a substrate, \(\hat{\mathbf{v}}_\mathrm{c}= 0\) is used. We also set the reaction terms to zero, \(R_{\mathrm{b,s}}=0\). The case of \(R_{\mathrm{b,s}} \ne 0\) can be dealt with as described by Hilpert (2005).

The generic MRT-LB equation can be written as

$$\begin{aligned} | f (\mathbf{x_j} + \delta \mathbf{e}_i,t_n+\delta ) \rangle = | f(\mathbf{x_j},t_n) \rangle - \mathsf{\Omega }\left[ | f (\mathbf{x_j},t_n) \rangle - |f^{(0)} (\mathbf{x_j},t_n) \rangle \right] , \end{aligned}$$
(17)

where \(\mathsf{\Omega }\) is the collision matrix in particle space. In order to recover the MRT-LB equations (8) and (9), one just needs to insert the identity matrix \(\mathsf{M}^{-1}\mathsf{M}\) before and behind \(\mathsf{\Omega }\), and then obtains

$$\begin{aligned} | f (\mathbf{x_j} + \delta \mathbf{e}_i,t_n+\delta ) \rangle = | f(\mathbf{x_j},t_n) \rangle - \mathsf{M}^{-1}\mathsf{S}\left[ | m (\mathbf{x_j},t_n) \rangle - |m^{(0)} (\mathbf{x_j},t_n) \rangle \right] , \end{aligned}$$
(18)

where \(\mathsf{S}=\mathsf{M}\mathsf{\Omega }\mathsf{M}^{-1}\) is the collision matrix in moment space. This matrix is constructed to be diagonal,

$$\begin{aligned} \mathsf{S}= diag(s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9,s_{10},s_{11},s_{12},s_{13},s_{14},s_{15},s_{16},s_{17},s_{18},s_{19}) . \end{aligned}$$

In order to relate Eq. (17) to a macroscopic equation, we apply a Taylor-series expansion to the particle distribution function \(f\),

$$\begin{aligned} f_i(\mathbf{x} + \mathbf{e}_i \delta , t+\delta ) - f_i (\mathbf{x},t) = \sum _{n=1}^{\infty } \frac{\delta ^n}{n!} \left[ \mathbf{e}_i \cdot \varvec{\nabla } + \frac{\partial _{}{}}{\partial {t}} \right] ^n f_i (\mathbf{x},t) , \end{aligned}$$
(19)

and further expand the particle distribution function

$$\begin{aligned} f_i = \sum _{n=0}^{\infty } \delta ^n f_i^{(n)} \end{aligned}$$
(20)

and the time derivative

$$\begin{aligned} \frac{\partial {}}{\partial {t}} = \sum _{n=0}^{\infty } \delta ^n \frac{\partial _{n}{}}{\partial {t}} . \end{aligned}$$
(21)

We ensure that the expansion conserves mass locally by requiring that

$$\begin{aligned} \sum _{i=1}^{N} f_i^{(n)} = C^{(n)} = \left\{ \begin{array}{lll} C &{} \text{ for } &{} n = 0 \\ 0 &{} \text{ for } &{} n > 0 \\ \end{array} \right. , \end{aligned}$$
(22)

where \(C\) is the dimensionless concentration of the bacteria or substrate, and \(N\) is the number of lattice vectors. By substituting Eqs. (1921) into the LB equation (17), we obtain

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\delta ^n}{n!} \left[ \mathbf{e}_i \cdot \varvec{\nabla } + \sum _{m=0}^{\infty } \delta ^m \frac{\partial _{m}{}}{\partial {t}} \right] ^n \sum _{p=0}^{\infty } \delta ^p f_i^{(p)} = - \sum _{n=1}^{\infty } \delta ^n \sum _{j=1}^N \mathsf{\Omega }_{ij} f_j^{(n)} . \end{aligned}$$
(23)

Let us order this equation after powers of \(\delta \) and assume that the coefficients of the \(\delta ^n\) vanish. To the first order of \(\delta \), we obtain

$$\begin{aligned} \left[ \frac{\partial _{0}{}}{\partial {t}} + \mathbf{e}_i \cdot {\varvec{\nabla }} \right] f_i^{(0)} = - \sum _{j=1}^N \mathsf{\Omega }_{ij} f_j^{(1)} . \end{aligned}$$
(24)

To the second order of \(\delta \), we obtain

$$\begin{aligned} \frac{\partial _{1}{}}{\partial {t}} f_i^{(0)} + \left( \frac{\partial _{0}{}}{\partial {t}} + \mathbf{e}_i \cdot \varvec{\nabla } \right) \left[ f_i^{(1)} - \frac{1}{2} \sum _{j=1}^N \mathsf{\Omega }_{ij} f_j^{(1)} \right] = - \sum _{j=1}^N \mathsf{\Omega }_{ij} f_j^{(2)} . \end{aligned}$$
(25)

Multiplying (24) and (25) by the transformation matrix \(\mathsf{M}\) converts them into moment space:

$$\begin{aligned}&\frac{\partial _{0}{}}{\partial {t}} | m^{(0)}\rangle + \sum _{\alpha } \mathsf{A}_{}^{(\alpha )} {\varvec{\nabla }}_{\,\alpha } | m^{(0)} \rangle = - \mathsf{S}| m^{(1)} \rangle \end{aligned}$$
(26)
$$\begin{aligned}&\frac{\partial _{1}{}}{\partial {t}} | m^{(0)} \rangle + \frac{\partial _{0}{}}{\partial {t}} \left( \mathsf{I}{-}\frac{1}{2} \mathsf{S}\right) | m^{(1)} \rangle {+} \sum _{\alpha } \mathsf{A}_{}^{(\alpha )} {\varvec{\nabla }}_{\,\alpha } \left( \mathsf{I}{-} \frac{1}{2} \mathsf{S}\right) | m^{(1)} \rangle = {-} \mathsf{S}| m^{(2)} \rangle .\qquad \end{aligned}$$
(27)

Here \(\text{ A }^{(\alpha )}\) is a constant \(N\times N\) matrix, which is defined for all Cartesian directions \(\alpha \) and the components of which are given by

$$\begin{aligned} \mathsf{A}_{pq}^{(\alpha )} = \sum _i \mathsf{M}_{pi} \mathbf{e}_{i,\alpha } \mathsf{M}_{iq}^{-1}, \end{aligned}$$
(28)

where \(\mathbf{e}_{i,\alpha }\) is the \(\alpha \)th Cartesian component of the \(i\)th lattice vector.

For the D3Q19 model, the lattice vectors are

$$\begin{aligned} ( {\mathbf{e}}_1,{\mathbf{e}}_2,\ldots , {\mathbf{e}}_{19})\!=\!\left( \begin{array}{rrrrrrrrrrrrrrrrrrr} 0&{} 1 &{}-1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{}-1 &{} 1 &{}-1 &{} 1 &{}-1 &{} 1 &{}-1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0 &{} 0 &{} 1 &{}-1 &{} 0 &{} 0 &{} 1 &{}-1 &{}-1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0&{} 1&{}-1&{} 1&{}-1 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{}-1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{}-1 &{}-1 &{} 1&{} 1&{}-1&{}-1&{} 1 \end{array} \right) \!. \end{aligned}$$

We used methods from d’Humières et al. (2002)’s paper to construct the transformation matrix \(\mathsf{M}\), although we selected a different order of the lattice vectors in order to simplify numerical implementation.

$$\begin{aligned} \mathsf{M}= \left( \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ -30&{} -11&{} -11&{} -11&{} -11&{} -11&{} -11&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8&{} 8 \\ 12&{} -4&{} -4&{} -4&{} -4&{} -4&{} -4&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ 0&{} 1&{} -1&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1&{} 1&{} -1&{} 1&{} -1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} -4&{} 4&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1&{} 1&{} -1&{} 1&{} -1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 1&{} -1&{} 0&{} 0&{} 1&{} -1&{} -1&{} 1&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1 \\ 0&{} 0&{} 0&{} -4&{} 4&{} 0&{} 0&{} 1&{} -1&{} -1&{} 1&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} -1&{} 1&{} 1&{} -1&{} -1&{} 1 \\ 0&{} 0&{} 0&{} 0&{} 0&{} -4&{} 4&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} -1&{} 1&{} 1&{} -1&{} -1&{} 1 \\ 0&{} 2&{} 2&{} -1&{} -1&{} -1&{} -1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} -2&{} -2&{} -2&{} -2 \\ 0&{} -4&{} -4&{} 2&{} 2&{} 2&{} 2&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} -2&{} -2&{} -2&{} -2 \\ 0&{} 0&{} 0&{} 1&{} 1&{} -1&{} -1&{} 1&{} 1&{} 1&{} 1&{} -1&{} -1&{} -1&{} -1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} -2&{} -2&{} 2&{} 2&{} 1&{} 1&{} 1&{} 1&{} -1&{} -1&{} -1&{} -1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} 1&{} -1&{} -1&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} 1&{} -1&{} -1 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} 1&{} -1&{} -1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1&{} -1&{} 1&{} -1&{} 1&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} -1&{} 1&{} 1&{} -1&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} 1&{} -1 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} -1&{} -1&{} 1&{} -1&{} 1&{} 1&{} -1 \end{array} \right) \end{aligned}$$

This matrix transfers the particle distribution function \(f\) to the hydrodynamic moment vector \(|m\rangle \),

$$\begin{aligned}&|m\rangle = \mathsf{M}|f\rangle = (C,e,\psi ,j_x,q_x,j_y,q_y,j_z,q_z,3p_{xx},3\pi _{xx}, p_{ww},\pi _{ww},\\&\quad p_{xy},p_{yz},p_{xz},m_x,m_y,m_z)^T . \end{aligned}$$

We only consider the first, fourth, sixth, and eighth moments, which represent the concentration and the mass flux density in the continuum-scale equations (4) and (5). Since \(\mathbf{e}_i\) and \(\mathsf{M}\) are now known, we can calculate the three \(\mathsf{A}_{}^{(\alpha )}\) matrices according to Eq. (28),

$$\begin{aligned} \mathsf{A}_{}^{(x)} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 21/5 &{} 19/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 10/19 &{} 1/57 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 4/63 &{} 10/63 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2/9 &{} 5/9 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 6/5 &{} -1/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2/3 &{} 1/3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \end{aligned}$$
$$\begin{aligned} \mathsf{A}_{}^{(y)} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 21/5 &{} 19/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 10/19 &{} 1/57 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/6 &{} 0 &{} 1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 4/63 &{} 10/63 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/9 &{} -5/18 &{} -1/3 &{} 5/6 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3/5 &{} -1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/3 &{} -1/6 &{} -1/3 &{} -1/6 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \end{aligned}$$
$$\begin{aligned} \mathsf{A}_{}^{(z)} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 21/5 &{} 19/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 10/19 &{} 1/57 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/6 &{} 0 &{} -1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 4/63 &{} 10/63 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/9 &{} -5/18 &{} 1/3 &{} -5/6 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2/5 &{} 1/10 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1/2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/3 &{} 1/6 &{} -1/3 &{} -1/6 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \end{aligned}$$

Now we evaluate the first component of Eq. (26) and get

$$\begin{aligned} \frac{\partial _{0}{C^{(0)}}}{\partial {t}} + \frac{\partial {j_x^{(0)}{}}}{\partial {x}} + \frac{\partial {j_y^{(0)}}}{\partial {y}} + \frac{\partial {j_z^{(0)}}}{\partial {z}} = 0 , \end{aligned}$$
(29)

where \(C^{(0)} = C\). To ensure that this first-order mass balance equation describes the advective transport of the concentration \(C\), the equilibrium mass flux vector must be given by

$$\begin{aligned} \mathbf{j}^{(0)} = C (\hat{\mathbf{v}}+ \hat{\mathbf{v}}_\mathrm{c}) , \end{aligned}$$
(30)

where \(\hat{\mathbf{v}}\) is the known dimensionless fluid velocity, and \(\hat{\mathbf{v}}_\mathrm{c}\) is the dimensionless bacterial chemotactic velocity. For the substrate, \(\hat{\mathbf{v}}_\mathrm{c}= 0\). The concentration change due to diffusion is quantified by the first component of Eq. (27),

$$\begin{aligned} \textstyle \frac{\partial _{1}{C}}{\partial {t}} + \frac{\partial {}}{\partial {x}} \left[ \left( 1-\frac{1}{2} s_4 \right) j_x^{(1)} \right] + \frac{\partial {}}{\partial {y}} \left[ \left( 1-\frac{1}{2} s_6 \right) j_y^{(1)} \right] + \frac{\partial {}}{\partial {z}} \left[ \left( 1-\frac{1}{2} s_8 \right) j_z^{(1)} \right] = 0 . \end{aligned}$$
(31)

In order to determine \(j_x^{(1)}\), \(j_y^{(1)}\) and \(j_z^{(1)}\), we evaluate the fourth, sixth, and eighth components of Eq. (26),

$$\begin{aligned}&\textstyle \frac{\partial _{0}{j_x^{(0)}{}}}{\partial {t}} + \frac{\partial {}}{\partial {x}} \left( \frac{10}{19} C + \frac{1}{57} e^{(0)} + p_{xx}^{(0)} \right) + \frac{\partial {p_{xy}^{(0)}}}{\partial {y}} + \frac{\partial {p_{zx}^{(0)}}}{\partial {z}} = -s_4 j_x^{(1)}, \end{aligned}$$
(32)
$$\begin{aligned}&\textstyle \frac{\partial _{0}{j_y^{(0)}}}{\partial {t}} + \frac{\partial {p_{xy}^{(0)}}}{\partial {x}} + \frac{\partial {}}{\partial {y}} \left( \frac{10}{19} C + \frac{1}{57} e^{(0)} - \frac{1}{2} p_{xx}^{(0)} + \frac{1}{2} p_{ww}^{(0)} \right) + \frac{\partial {p_{yz}^{(0)}}}{\partial {z}} = -s_6 j_y^{(1)}, \end{aligned}$$
(33)
$$\begin{aligned}&\textstyle \frac{\partial _{0}{j_z^{(0)}}}{\partial {t}} + \frac{\partial {p_{zx}^{(0)}}}{\partial {x}} + \frac{\partial {p_{yz}^{(0)}}}{\partial {y}} + \frac{\partial {}}{\partial {z}} \left( \frac{10}{19} C + \frac{1}{57} e^{(0)} - \frac{1}{2} p_{xx}^{(0)} - \frac{1}{2} p_{ww}^{(0)} \right) = -s_8 j_z^{(1)}. \end{aligned}$$
(34)

By solving Eqs. (32), (33) and (34) for \(j_x^{(1)}\), \(j_y^{(1)}\) and \(j_z^{(1)}\), and substituting the results into Eq. (31), we obtain

$$\begin{aligned} \textstyle \frac{\partial _{1}{C}}{\partial {t}}&= \textstyle \frac{\partial {}}{\partial {x}} \left\{ \left( \frac{1}{s_4} - \frac{1}{2} \right) \left[ \frac{\partial _{0}{j_x^{(0)}{}}}{\partial {t}} + \frac{\partial {}}{\partial {x}} \left( \frac{10}{19} C + \frac{1}{57} e^{(0)} + p_{xx}^{(0)} \right) + \frac{\partial {p_{xy}^{(0)}}}{\partial {y}} + \frac{\partial {p_{zx}^{(0)}}}{\partial {z}} \right] \right\} \nonumber \\&\quad + \textstyle \frac{\partial {}}{\partial {y}} \left\{ \left( \frac{1}{s_6} \!-\! \frac{1}{2} \right) \left[ \frac{\partial _{0}{j_y^{(0)}}}{\partial {t}} \!+\! \frac{\partial {p_{xy}^{(0)}}}{\partial {x}} \!+\! \frac{\partial {}}{\partial {y}} \left( \frac{10}{19} C \!+\! \frac{1}{57} e^{(0)} \!-\! \frac{1}{2} p_{xx}^{(0)} \! +\! \frac{1}{2} p_{ww}^{(0)} \right) \!+\! \frac{\partial {p_{yz}^{(0)}}}{\partial {z}} \right] \right\} \nonumber \\&\quad + \textstyle \frac{\partial {}}{\partial {z}} \left\{ \left( \frac{1}{s_8} \!-\! \frac{1}{2} \right) \left[ \frac{\partial _{0}{j_z^{(0)}}}{\partial {t}}\! +\! \frac{\partial {p_{zx}^{(0)}}}{\partial {x}} \!+\! \frac{\partial {p_{yz}^{(0)}}}{\partial {y}} \!+\! \frac{\partial {}}{\partial {z}} \left( \frac{10}{19} C \!+\! \frac{1}{57} e^{(0)} {-} \frac{1}{2} p_{xx}^{(0)} \!-\! \frac{1}{2} p_{ww}^{(0)} \right) \right] \right\} .\nonumber \\ \end{aligned}$$
(35)

In order to model isotropic diffusion, mixed spatial quantities must vanish, thus \(p_{zx}^{(0)} = p_{yz}^{(0)} = p_{zx}^{(0)} = 0\). Furthermore, for the sake of symmetry, we set \(p_{xx}^{(0)} = p_{ww}^{(0)} = 0\) and use the same collision rates in all three Cartesian directions, \(s_4 = s_6 = s_8 := 1/\tau \). \(s_1=0\) is required to ensure mass conservation. All other elements of \(\mathsf{S}\) are free to choose, and they only affect numerical stability rather than mass conservation. The only remaining unknown parameter is \(e^{(0)}\). In order to obtain a similar expression for the relaxation time as the SRT-LB model, d’Humières et al. (2002) chose \(e^{(0)} = - 11C\) when applying the MRT-LB model to simulate a velocity field. However, this value makes our simulations unstable, which could be due to the fact that the momentum of neither the bacteria nor the chemoattractant is generally conserved. As a result, we choose \(e^{(0)} = - 1.5C\), which has the best stability in our parameter tests. Therefore,

$$\begin{aligned} \frac{\partial _{1}{C}}{\partial {t}} = \frac{1}{2} \left( \tau - \frac{1}{2} \right) \hat{{\varvec{\nabla }}}^2 C + \left( \tau - \frac{1}{2} \right) \hat{{\varvec{\nabla }}} \cdot \frac{\partial _{0}{\mathbf{j}^{(0)}}}{\partial {t}} . \end{aligned}$$
(36)

Like Hilpert (2005), we assume that the velocity fields, \(\hat{\mathbf{v}}\) and \(\hat{\mathbf{v}}_\mathrm{c}\), vary slowly in both time and space dimension, such that \(\partial _0 \hat{\mathbf{v}}/ \partial t \approx 0\), \(\partial _0 \hat{\mathbf{v}}_\mathrm{c}/ \partial t \approx 0\), \(\hat{{\varvec{\nabla }}} \cdot \hat{\mathbf{v}}\approx 0\), and \(\hat{{\varvec{\nabla }}} \cdot \hat{\mathbf{v}}_\mathrm{c}\approx 0\). We can then use Eq. (29) to rewrite the mixed derivative on the right-hand-side of Eq. (36), \(\hat{{\varvec{\nabla }}} \cdot \partial _0 \mathbf{j}^{(0)}/ \partial t = - ||\hat{\mathbf{v}}+\hat{\mathbf{v}}_\mathrm{c}||^2 C\). By multiplying Eq. (36) by \(\delta \) and adding the result to Eq. (29), we obtain the transport equation accurate to second order,

$$\begin{aligned} \frac{\partial {C}}{\partial {t}} + \hat{{\varvec{\nabla }}} \cdot C (\hat{\mathbf{v}}+ \hat{\mathbf{v}}_\mathrm{c}) = \hat{D} \hat{{\varvec{\nabla }}}^2 C - 2 \hat{D} ||\hat{\mathbf{v}}+\hat{\mathbf{v}}_\mathrm{c}||^2 \hat{{\varvec{\nabla }}}^2 C + O(\delta ^2) , \end{aligned}$$
(37)

where

$$\begin{aligned} \hat{D} = \delta \frac{1}{2} \left( \tau - \frac{1}{2} \right) . \end{aligned}$$
(38)

In LBM simulations, one needs to guarantee \(||\hat{\mathbf{v}}+\hat{\mathbf{v}}_\mathrm{c}|| \ll 1\), which can be achieved by choosing a suitable typical velocity to non-dimensionalize the transport equation. Otherwise, the simulations become numerical unstable. Therefore, the second term on the right-hand side of Eq. (37) is much smaller than the first term on the right-hand side and can be ignored. When \(\delta \rightarrow 0\), the LB model achieves solutions to the following dimensionless advection diffusion equation:

$$\begin{aligned} \frac{\partial {C}}{\partial {t}} + \hat{{\varvec{\nabla }}} \cdot C (\hat{\mathbf{v}}+ \hat{\mathbf{v}}_\mathrm{c}) = \hat{D} \hat{{\varvec{\nabla }}}^2 C , \end{aligned}$$
(39)

where \(\hat{D} = 1/\text{ Pe }\) as in Eqs. (4) and (5). We still need to make choices for \(\psi ^{(0)}\), \(q_{x,y,z}^{(0)}\) and \(m_{x,y,z}^{(0)}\). Like d’Humières et al. (2002), we choose \(\psi ^{(0)} = -C\), \(q_x^{(0)}=-7j_x^{(0)}{}/3\), \(q_y^{(0)}=-7j_y^{(0)}/3\), \(q_z^{(0)}=-7j_z^{(0)}/3\), and \(m_{x,y,z}^{(0)}=0\). For clarity, we display the entire equilibrium moment vector of the bacteria and the substrate:

$$\begin{aligned} \begin{array}{rcl} | m_b^0 \rangle &{} = &{} (\hat{b}, -1.5\hat{b}, {-}\hat{b},\hat{b} (\hat{v}_x {+} \hat{v}_{c,x}), -\frac{7}{3} \hat{b} (\hat{v}_x {+} \hat{v}_{c,x}), \hat{b} (\hat{v}_y + \hat{v}_{c,y}), -\frac{7}{3} \hat{b} (\hat{v}_y + \hat{v}_{c,y}), \\ &{} &{} \hat{b} (\hat{v}_z + \hat{v}_{c,z}), -\frac{7}{3} \hat{b} (\hat{v}_z + \hat{v}_{c,z}), 0,0,0,0,0,0,0,0,0,0)^T \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{rcl} | m_s^0 \rangle &{} = &{} (\hat{s}, -1.5\hat{s}, -\hat{s}, \hat{s} \hat{v}_x, -\frac{7}{3} \hat{s} \hat{v}_x, \hat{s} \hat{v}_y, -\frac{7}{3} \hat{s} \hat{v}_y, \hat{s} \hat{v}_z, -\frac{7}{3} \hat{s} \hat{v}_z,\\ &{}&{} 0,0,0,0,0,0,0,0,0,0)^T . \end{array} \end{aligned}$$

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Yan, Z., Hilpert, M. A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration, Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands. Bull Math Biol 76, 2449–2475 (2014). https://doi.org/10.1007/s11538-014-0020-1

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Keywords

  • Bacterial chemotaxis
  • Bacterial bands
  • Hydrodynamic dispersion
  • Lattice-Boltzmann method