Prominent Effect of Soil Network Heterogeneity on Microbial Invasion

Using a network representation for real soil samples and mathematical models for microbial spread, we show that the structural heterogeneity of the soil habitat may have a very significant influence on the size of microbial invasions of the soil pore space. In particular, neglecting the soil structural heterogeneity may lead to a substantial underestimation of microbial invasion. Such effects are explained in terms of a crucial interplay between heterogeneity in microbial spread and heterogeneity in the topology of soil networks. The main influence of network topology on invasion is linked to the existence of long channels in soil networks that may act as bridges for transmission of microorganisms between distant parts of soil.

Figure 1

Soil samples and soil networks. (a) 2D sections of 3D-digital images of soil samples $N1$ and $P1$ (sample size: $3.5\times 3.5\times 3.5\mathrm{cm}$). The pore space and solid matrix are shown in white and grey colors, respectively. (b) Network representing the pore space of sample $N1$. The color of channels indicates their local cross-section area, $S(x)$, where $x$ gives the position along channels. The zoom on the right illustrates the definition of the arc length of channels, $L$, the node degree heterogeneity, and clustering (triangle 1-2-3) (cf. Table ). Both $L$ and $S(x)$ are dimensionless, i.e., scaled by the size of voxel in the 3D-digital image.

Figure 2

Invasion in soil networks. The vertical axis displays the density of colonization, $n_{\mathrm{C}}$, in network $N1$ averaged over ${10}^4$ invasions starting from different randomly chosen nodes. The horizontal axis shows the mean transmissibility of microbes, $⟨T⟩$, obtained by averaging over channels. Different curves correspond to invasions predicted by different models (and exploration controlled by $\alpha$ in model 4).

Figure 3

Bridging effect for edges in network $N1$. The color along the vertical direction gives the fraction of edges with given $L$ (i.e., fixed value on the horizontal axis) whose degree of bridging is larger than the value $B$ plotted along the vertical axis. For instance, the color inside the circle (triangle) indicates that the fraction of channels of length $L=10$ ($L=50$) with $B>0.9$, i.e., $l_{\mathrm{b}}>10$, is approximately $0.1$ ($0.7$). By definition, $B\in [0,1]$ but only values of $B>0.7$ are shown to highlight the variation in $B$ with $L$. Channels attached to border nodes of degree 1 are not included in the graph. See [20] for similar plots for other soil samples.

Figure 4

(a) Mean cross-section area of channels, $⟨\overline{S}{⟩}_{\mathrm{HT}}$, (b) mean arc length, $⟨L{⟩}_{\mathrm{HT}}$ and mean bridgeness, $⟨B{⟩}_{\mathrm{HT}}$, obtained by averaging over channels with high transmissibility (HT, $T>⟨T⟩$) in sample $N1$ for microbes with $⟨T⟩=0.5$. (c) Plots of the channels with $T>0.5$ colored according to their value of $L$.