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Thermodynamic Inhibition in a Biofilm Reactor with Suspended Bacteria

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Abstract

We formulate a biofilm reactor model with suspended bacteria that accounts for thermodynamic growth inhibition. The reactor model is a chemostat style model consisting of a single replenished growth promoting substrate, a single reaction product, suspended bacteria, and wall attached bacteria in the form of a bacterial biofilm. We present stability conditions for the washout equilibrium using standard techniques, demonstrating that analytical results are attainable even with the added complexity from thermodynamic inhibition. Furthermore, we numerically investigate the longterm behaviour. In the computational study, we investigate model behaviour for select parameters and two commonly used detachment functions. We investigate the effects of thermodynamic inhibition on the model and find that thermodynamic inhibition limits substrate utilization/production both inside the biofilm and inside the aqueous phase, resulting in less suspended bacteria and a thinner biofilm.

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Acknowledgements

This study was financially supported through an Ontario Graduate Scholarship and by a Highdale Farms - Arthur and Rosmarie Spoerri Scholarship in Natural Sciences awarded to HJG, an NSERC (CGS-M) Scholarship awarded to JMH, and an NSERC Discovery Grant (RGPIN-2019-05003) awarded to HJE.

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Correspondence to Harry J. Gaebler.

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Appendices

A Additional Information: Jacobian Entries

$$\begin{aligned} J_{11}&=-D-\frac{1}{Y}\frac{\partial g}{\partial S_1}(S_1,S_2)u-\frac{A}{V}\frac{\partial J_1}{\partial S_1}(\lambda ,S_1,S_2),\\ J_{12}&=-\frac{1}{Y}\frac{\partial g}{\partial S_2}(S_1,S_2)u-\frac{A}{V}\frac{\partial J_1}{\partial S_2}(\lambda ,S_1,S_2),\\ J_{13}&=-\frac{1}{Y}g(S_1,S_2),\\ J_{14}&=-\frac{A}{V}\frac{\partial J_1}{\partial \lambda }(\lambda ,S_1,S_2),\\ J_{21}&=\frac{1}{Y} \frac{\partial g}{\partial S_1}(S_1,S_2)u-\frac{A}{V}\frac{\partial J_2}{\partial S_1}(\lambda ,S_1,S_2),\\ J_{22}&=-D+\frac{1}{Y}\frac{\partial g}{\partial S_2}(S_1,S_2)u-\frac{A}{V}\frac{\partial J_2}{\partial S_2}(\lambda ,S_1,S_2), \\ J_{23}&=\frac{1}{Y}g(S_1,S_2),\\ J_{24}&=-\frac{A}{V}\frac{\partial J_2}{\partial \lambda }(\lambda ,S_1,S_2),\\ J_{31}&=\frac{\partial g}{\partial S_1}(S_1,S_2)u,\\ J_{32}&=\frac{\partial g}{\partial S_2}(S_1,S_2)u,\\ J_{33}&=g(S_1,S_2) -(D+k_u+\alpha ),\\ J_{34}&=\frac{A\rho }{V}\left( d(\lambda )+\lambda d'(\lambda ) \right) , \\ J_{41}&=\frac{Y}{\rho }\frac{\partial J_1}{\partial S_1}(\lambda ,S_1,S_2),\\ J_{42}&=\frac{Y}{\rho }\frac{\partial J_1}{\partial S_2}(\lambda ,S_1,S_2),\\ J_{43}&=\frac{\alpha V}{A \rho },\\ J_{44}&=\frac{Y}{\rho }\frac{\partial J_1}{\partial \lambda }(\lambda ,S_1,S_2)-k_{\lambda }-\left( d(\lambda )+\lambda d'(\lambda ) \right) . \end{aligned}$$

B Numerical Investigation of a Unique Asymptotically Stable Non-Trivial Equilibrium

To investigate the uniqueness and stability of a non-trivial equilibrium, we select a parameter set, given in Table 1, such that the washout equilibrium \(E^0=(S_1^\text {in},0,0,0)\) is unstable as per the conditions in Corollary 1. Initial conditions are fixed in the range

$$\begin{aligned} S_1(0)\in \left[ 0,S_1^\text {in}\right] , \quad S_2(0)=\left[ 0,S_1^\text {in}\right] ,\quad u(0)\in [0,0.01], \quad \lambda (0)\in [0,0.001], \end{aligned}$$

and we randomly sample 2000 sets of initial conditions and run the model (8) to steady state. Results are presented in Table 4. We find that for each state variable, the standard deviation is orders of magnitude smaller than the calculated mean, suggesting a unique non-trivial asymptotically stable equilibrium exists when the trivial equilibrium is unstable.

Table 4 Mean and standard deviation for interior steady-state values of 2000 model simulations with randomly sampled initial conditions

C A Biofilm Reactor Model with Suspended Bacteria and Monod Growth

The biofilm reactor model with suspended bacteria and without thermodynamic inhibition is given by

$$\begin{aligned} \left. \begin{aligned} \dot{S_1}&= D(S_1^{\text {in}}-S_1) - \frac{1}{Y}\frac{\mu S_1}{K_u+S_1}u - \frac{A}{V}j_1(\lambda ,S_1,S_2),\\ \dot{S_2}&= -DS_2 + \frac{1}{Y}\frac{\mu S_1}{K_u+S_1}u-\frac{A}{V}j_2(\lambda , S_1, S_2),\\ \dot{u}&= u\left( \frac{\mu S_1}{K_u+S_1}-D-k_u\right) + \frac{A}{V}\rho d(\lambda )\lambda - \alpha u,\\ \dot{\lambda }&= \frac{Y }{\rho }j_1(\lambda ,S_1,S_2)-\lambda k_\lambda +\frac{\alpha Vu}{A\rho }-d(\lambda )\lambda , \end{aligned} \right\} \end{aligned}$$
(20)

where

$$\begin{aligned} j_1(\lambda ,S_1,S_2)&:=D_{c_1}\frac{dc_1}{dz}(\lambda )={\left\{ \begin{array}{ll} \frac{\rho }{Y}\int \limits _0^{\lambda }\frac{\mu c_1(z)}{K_{\lambda }+c_1(z)}dz&{}if \quad \lambda>0,\\ 0 &{}if \quad \lambda =0. \end{array}\right. },\\ j_2(\lambda ,S_1,S_2)&:=D_{c_2}\frac{dc_2}{dz}(\lambda )={\left\{ \begin{array}{ll} -\frac{\rho }{Y}\int \limits _0^{\lambda }\frac{\mu c_1(z)}{K_{\lambda }+c_1(z)}dz&{}if \quad \lambda >0,\\ 0 &{}if \quad \lambda =0. \end{array}\right. }, \end{aligned}$$

and the concentration of substrates \(c_1=c_1(z)\) and \(c_2=c_2(z)\) inside the biofilm are given by the solution to

$$\begin{aligned} D_{c_1}\frac{d^2c_1}{dz^2}(z)&=\frac{\rho \mu }{Y}\frac{c_1}{K_\lambda +c_1},\qquad \frac{dc_1}{dz}(0)=0,\quad c_1(\lambda )=S_1, \end{aligned}$$
(21)
$$\begin{aligned} D_{c_2}\frac{d^2c_2}{dz^2}(z)&=-\frac{\rho \mu }{Y} \frac{c_1}{K_\lambda +c_1},\qquad \frac{dc_2}{dz}(0)=0,\quad c_2(\lambda )=S_2. \end{aligned}$$
(22)

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Gaebler, H.J., Hughes, J.M. & Eberl, H.J. Thermodynamic Inhibition in a Biofilm Reactor with Suspended Bacteria. Bull Math Biol 83, 10 (2021). https://doi.org/10.1007/s11538-020-00840-w

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