Numerical and experimental investigation of multi-species bacterial co-aggregation

Abstract

This paper deals with the mathematical modeling of bacterial co-aggregation and its numerical implementation in a FEM framework. Since the concept of co-aggregation refers to the physical binding between cells of different microbial species, a system composed of two species is considered in the modeling framework. The extension of the model to an arbitrary number of species is straightforward. In addition to two-species (multi-species growth) dynamics, the transport of a nutritional substance and the extent of co-aggregation are introduced into the model as the third and fourth primary variables. A phase-field modeling approach is employed to describe the co-aggregation between the two species. The mathematical model is three-dimensional and fully based on the continuum description of the problem without any need for discrete agents which are the key elements of the individual-based modeling approach. It is shown that the use of a phase-field-based model is equivalent to a particular form of classical diffusion-reaction systems. Unlike the so-called mixture models, the evolution of each component of the multi-species system is captured thanks to the inherent capability of phase-field modeling in treating systems consisting of distinct multi-phases. The details of numerical implementation in a FEM framework are also presented. Indeed, a new multi-field user element is developed and implemented in ANSYS for this multiphysics problem. Predictions of the model are compared with the experimental observations. By that, the versatility and applicability of the model and the numerical tool are well established.

Figure 1

Phase-field approach for a multi-species system, ${\varphi }_1$ and ${\varphi }_2$ represent species 1 and species 2 while $\alpha$ refers to the congregation state.

Figure 2

Double well potential $f(\varphi )$.

Figure 3

Two different patterns for development of biofilms. (a) Phase-contrast micrographs of colony morphology for Streptococcus Gordonii strain SPS$\_$885. The colony is generally circular and compact. A slightly undulate colony margin was observed at later time points. (b) Colony morphologies of Candida albicans strains SPS$\_$888 and SPS$\_$889. Colonies are rhizoid (branched and dendritic pattern) with irregular cores. Merging of colonies was observed for strain SPS$\_$889 at later time points. Scale bars, 100 $\mu m$.

Figure 4

Diffusion-limited growth for $r_D={10}^2$ (dendritic pattern) in the course of the time, from top to bottom.

Figure 5

Diffusion-unlimited growth for $r_D={10}^6$ (compact pattern) in the course of the time, from top to bottom.

Figure 6

Co-aggregates and biofilms. a–c Low magnification phase-contrast micrographs of cell suspensions with a magnified subsection showing bacterial cells. From left: Streptococcus gordonii strain $SPS\_017$, S. gordonii strain $SPS\_017$ mixed with Eikenella corrodens strain $SPS\_010$, and E. corrodens strain $SPS\_010$. Both strains originate from the same severe case of peri-implant disease. After 24 h of incubation large co-aggregates were observed in the mixture but not in monospecies suspensions. Image fragments at high magnification are included in the top left corners. d–e Confocal micrographs of 8-h biofilms representing monoculture or co-culture biofilms are shown in the same order as in a–c. Bigger aggregates were observed in mixed-biofilms compared to monospecies biofilms. Scale bars, $4\mathrm{\mu }\text{m}$ (a–c, small bar), $10\mathrm{\mu }\text{m}$ (a-c, large bar), and $15\mathrm{\mu }\text{m}$ (d–e).

Figure 7

Diffusion-limited growth for $r_D={10}^2$ in the course of the time, from top to bottom (see Supplementary Video 1).

Figure 8

Diffusion-unlimited growth for $r_D={10}^6$ in the course of the time, from top to bottom (see Supplementary Video 2).

Figure 9

Multispecies biofilm growth under different nutrient diffusivity.

Figure 10

Multispecies biofilm growth under the directional nutrient provision for $r_D={10}^6$.

Figure 11

Directional nutrient feeding from the top side in the presence of co-aggregation for $r_D={10}^6$ in the course of the time, from top to bottom (see Supplementary Video 3).

Figure 12

Directional nutrient feeding in the absence of co-aggregation for $r_D={10}^6$ in the course of the time, from top to bottom.