Fig 1.
A conceptual diagram of the definition of a patch.
(A) A rough soil surface domain is discretised with hexagonal patches representing subdomains. The brown and blue colour scale indicate the homogenisation of the roughness and hydration condition for each patch, respectively. To obtain the characteristics of patches we assume that a patch consists of roughness elements, a conceptual water-retaining pyramid-shaped pore and a smooth surface region. It allows us to calculate the amount of water held on the rough surface from the capillarity and van der Waals adsorptive forces at a certain relative humidity. Roughness of a patch is characterised with two measures, a surface porosity Φ and a fractal dimension D. The hydration condition of the patch can be represented as the effective water film thickness w(ψm) as a function of the water matric potential ψm; Eq (2). (B) A rough surface domain would be comprised of various size of angular surface pores on the smooth surface. Surface porosity of a patch Φ determines the fraction of surface pores with respect to the patch domain (smooth surface+angular pores). (C) The fractal dimension D determines the size distribution of roughness elements for a patch, N(r). It follows a power-law with a fractal dimension D. In the model, we assumed that the surface pore is the shape of a square pyramid with base r and height r (Larger pores indicate deeper pits on the rough surface).
Fig 2.
A comparison of effective water film thickness distribution between smooth and rough surface domain.
Typical examples of rough surface domain are given as the effective water film thickness distribution at ψm = -3.6kPa. (A) a smooth surface domain (D = 1.2) and (B) a rough surface domain (D = 1.8) are provided for a comparison. To illustrate the role of the fractal dimension D for generating self-affine characteristics, the mean surface porosity of the domain is fixed as a constant for both cases (). A patch in the domain is assumed to be homogeneous inside for its roughness and hydration condition (effective water film thickness). However, to incorporate its roughness into the connectivity and tortuosity of the hydrological pathways in the patch, the global percolation probability of the domain and the local surface porosity Φi,j are considered to determine the residence time of microorganisms at the patch.
Fig 3.
The effective water film thickness of the rough surface domain as a single patch for different hydration conditions (expressed by the matric potential of the aqueous phase).
The surface porosity scales effective water film thickness when the fractal dimension is constant D = 1.4 (Dp = 2.4). When the surface porosity (Φ = 0.1) is low the model agrees with the experimental data of [97]. Here, we set the largest roughness element size rmax = 50 mm as a possible representation of the surface depression of the sample rock used in the experiment [97].
Fig 4.
Microbial locomotion in rough surface patch model.
(A) The mean flagellated swimming velocity on the surface with different surface porosities for different hydration conditions expressed by matric potential (bottom axis) and effective water film thickness when Φ = 0.4 (top axis). For comparison, we fixed the fractal dimension D = 1.8 and varied the surface porosity from 0.2 to 0.6. Measured values (red squares) from the work of [89], the mean microbial swimming velocity on the porous ceramic plate, show good agreement with the model when the surface porosity is about 0.4. Black dotted horizontal line indicates the onset of capillary force. The swimming speed at the bulk water is given as v0 = 14μm/s [85]. The roughness effect and the surface porosity reduce the mean swimming velocity to about 10μm/s even at the very wet case. (B) Heterogeneity of roughness patches on the domain can be mapped to the swimming velocity field for micro-organisms. Yellow-blue scale indicates the mean swimming speed. In a patch, the microbial locomotion follows the biased random walk following the probability to cross to adjacent patches, Eq (9). (C) The averaged minimum residence time at a patch (assuming patch size lp = 500μm) varies for different roughness measures, Eq (12). For a surface with constant fractal dimension, the averaged minimum residence time at a patch is higher when the surface porosity
is lower. The shaded area indicate the lower and upper values from 5 sample domains with the same mean roughness measures
.
Fig 5.
Microbial dispersion on rough surfaces.
(A) Simulated colony expansion of motile bacterial cells grown in a surface at ψm = −0.5kPa. The white-greyblue colour scale indicate the effective water film thickness distribution (blue = motile, white = sessile), Here, we did not use self-affine domain for the local surface porosity distribution to reflect the experimental setting of [35].). The initial nutrient concentration was given and the boundary condition was to maintain concentration at the boundary. (B) The time evolution of colony diameter (or the maximum microbial dispersion distance) is given from simulated results (-0.5, -1.0, and -3.0 kPa, these values were chosen to cover various hydration conditions to cover globally connected, locally connected, and fragmented habitats) and experimental results (-0.5, -1.2, and -3.6 kPa) for hydrated surfaces at three values of matric potential. Lines in different colours indicate simulated results. Filled symbols indicate the experimental results from [35].
Fig 6.
Colony expansion rates for analytical results, surface patch model results (three matric potentials were chosen to cover various hydration conditions representing globally connected, locally connected, and fragmented habitats), and experimental results are compared. Analytic results are calculated based on Eq (15). In analytical results, the average value of the surface porosity () is used as a representative value of the domain and the expansion rate becomes zero since the flagellated movement is disabled due to the capillary forces at about −2kPa. The simulation results of RSPM show non-zero colony expansion rate up to about −3kPa because of the heterogeneity of the domain.
Fig 7.
Examples of spatial patterns arising from different microbial consortia on rough surfaces for different hydration conditions.
On the top figures, schematics of trophic interactions are given (competition and mutualism). Two example interactions are simulated for wet case (ψm = −0.5 kPa) and dry case (ψm = −3.6 kPa). At t = 0 well-mixed two populations (50 each) are inoculated at the centre (marked as a yellow square) of the roughness domain (, and D = 1.2), Light blue indicates the distribution of aqueous habitats and blue and red dots indicate species 1 and 2, respectively. ni denotes the population of species i in the figure at the given time. The results show that different tropic interaction give a rise to different spatial organisations. For competitive interactions, we observe segregation between two species and altering the front line on the chemotactic band (A,C) while, for mutualistic interaction, the producer (Sp1) occupies the front line of the chemotactic band and the consumer (Sp2) follows (B,D). The spatial patterns are in qualitative agreement with the previous studies on model hydrated surfaces [101].
Fig 8.
Microbial population diversity and coexistence index analysis of rough surface patch model (D = 1.35 and , as an example case of sand) for a range of hydration conditions and associated aqueous phase connectivity.
(A) Time evolution of normalised Shannon Index on the nutrient limited surface at different values of matric potential considering a population with 50 different species (differentiated by their Monod growth parameters). (B) Relative abundance rank is plotted with the coexistence index following [110] when the population sizes reach to the steady state with nutrient limited condition (at t = 24hr). Different line colours indicate hydration conditions of the surface, ψm = −0.5kPa in blue (wet), ψm = −2.0kPa in yellow (intermediate), and ψm = −3.6kPa in red (dry). For each hydration condition, we tested two different inoculation schemes; (1) well-mixed population inoculations (shown in solid lines and empty symbols), and (2) random inoculation for the entire domain (shown in dashed lines and filled symbols). An example of aqueous habitat distribution is given for two different matric potential in Fig 8C. Typical microbial colony distributions for four different cases (wet-dry; mixed-random) at t = 18hr are shown in Fig 8D. White-grey-blue scale on the background show the the microbial swimming speed field (representing aqueous habitats) and circles in various colours at each patch represent the relative population sizes and colours indicate different species with different growth patterns according to the Monod Parameters, Ks and μmax, shown in the graph below. The scale bars indicate 10mm. Results are obtained under the competitive interaction over a single limiting nutrient among 50 different species. As the system desiccates, the microbial diversity (Shannon Index) becomes higher. It implies that species evenness is higher when system is dry and the coexistence index becomes larger than unity (marked as yellow region in 8B). Random inoculation of microbial cells exhibits higher diversity indices suggesting that pre-colonisation of slow-growing species derives benefits from a fragmented aqueous habitat.
Fig 9.
Roughness effects on microbial population diversity for different hydration conditions.
Normalised Shannon index, Eq (16), calculated for two domains representing different soil textures. Different fractal dimensions are assigned for sand (D = 1.35) and sand+slity-clay (D = 1.65). Simulated results are shown in red solid line and blue dashed line for sandy surface and sand (90%)+silty-clay (10%), respectively. The values were calculated from three different rough surface domains with same roughness parameters and the shaded areas represent 1 s.t.d. Model predictions of normalised Shannon index agree with the experimental data of [115] and show a decrease in microbial population evenness as the domain becomes wet (less negative matric potential values).