Fig 1.
Overview of the class of hybrid lattice-based bacterial growth models.
The tilted white field illustrates how bacterial agents (green disks) are constrained to a lattice (black dots). The behavior of the bacteria is determined by various properties and internal processes; the green box lists potential examples. The red lines indicate the Voronoi cells associated with the lattice sites, which define the neighborhood and are used to compute the dynamics of the continuous fields. The bacteria can be coupled to multiple fields, as illustrated by the orange, yellow, blue, and green slices. For example, the orange slice represents the concentration field of a nutrient that is consumed by the bacteria. Its dynamics are governed by a diffusion equation, for which the bacteria serve as sinks, here indicated as gray disks.
Fig 2.
Bacterial colony growth according to the minimal hybrid lattice-based model using a square lattice with nearest-neighbor interactions only.
The nearest neighbors on a square grid are as indicated by the blue arrows in the top-left inset. (a) Simulation with low initial nutrient concentration () where
is the minimum amount of nutrient that a bacterium needs to consume in order to divide. (b) Simulation with high initial nutrient concentration (
). The remaining parameter choices are specified in Table 1. The green shapes in the center of both figures show a single representative colony, where the color of each bacterium indicates the number of divisions in the lineage of the bacterium (central color bar). The maximum division number is 131 for colony (a) and 106 for colony (b). The outer ring shows a histogram for the distribution of normal vectors to the convex hulls fitted to colonies. The error bars show the standard error of the mean resulting from 500 independent realizations. The gray scale serves to guide the eye to regions of high incidence. The red circle provides the ideal, isotropic, distribution with value
. In both figures, the histograms indicate that the colonies tend to have a diamond shape.
Table 1.
Parameters and constants used in the minimal hybrid lattice-based model. Default values and their simulation units, and an estimate of simulation values in physical units are given. The maximum self consumption time is used to link the simulation time unit to the physical unit.
Fig 3.
Bacterial colony growth on a square lattice with various next nearest-neighbor interaction weights.
Simulations in the top row were all performed at high initial nutrient concentration, as in Fig 2b. The bottom row shows simulations performed at low initial nutrient concentration, as in Fig 2a. The neighborhood of a lattice site includes nearest and next-nearest neighbors as indicated in the bottom-right inset. The probability that a daughter cell is placed on a next nearest-neighbor site is weighted by , where
gives equal probability between placement on next-nearest and nearest-neighbor sites. The representation and parameter choices are otherwise the same as in Fig 2. Visual inspection suggests that there is no single choice for wd that can resolve discrete orientational symmetry in both growth regimes.
Fig 4.
Bacterial colony growth on a square lattice with next nearest-neighbor interactions.
The neighborhood of a lattice site is indicated by the blue arrows in the top-left inset. The weight of growth into the diagonal next nearest-neighbor sites was set to . The selection of parameters and representation is otherwise the same as in Fig 2. (a) For the low initial nutrient concentration the maximum division number is 129. (b) The maximum division number is 86 for the high initial nutrient concentration. In both figures, the histograms demonstrate that the shapes of the colonies are affected by the symmetries of the lattice.
Fig 5.
Three methods of generating pseudo-random coordinates for a disordered lattice.
The red dots indicate the randomly drawn lattice sites and the orange lines indicate the associated Voronoi tessellation in all figures. For the (a) restricted and (b) redrawn Vectorizable Random Lattice (VRL) proposed by Moukarzel and Herrmann [49], and Tucker [50] respectively, the gray squares indicate the reference cells. Each reference cell contains exactly one lattice site. The blue double arrow indicates the tuneable minimum separation between lattice sites l0. In (a) the minimal separation l0 is ensured by randomly placing the lattice sites inside the subdomains of the reference cell indicated as green squares. Our new method (c) uses snapshots of an off-lattice fluid of bidisperse soft disks to generate the disordered grid.
Fig 6.
Vectorizable random lattices have orientational order.
The top row shows the probability density functions (PDFs) of the distances between neighbors (blue) and the volumes of the Voronoi cells associated to the lattice sites (orange). The bottom row gives the pair distribution functions of the lattice sites in polar coordinates on a symmetrical log-scale. For the two vectorizable random lattices (VRLs), we chose l0 = 0.5.
Fig 7.
Solutions to a continuous bacterial growth model on the vectorizable random lattices show discrete orientational symmetry, while solutions on the fluid-derived lattice do not.
The Kitsunezaki model [19] was solved on the three disordered lattices with parameters D0 = 0.1 and . The initial condition for the density of active motile bacteria b was a disk of radius 5 with b = 1 and b = 0 everywhere else. The nutrient density field n was initialized with n = 1 everywhere. The minimum distance between lattice sites in the VRLs l0 was set to 0.5. The central figures show a single representative result of
where s is the density of inactive bacteria. The outer rings show the distribution of normal vectors to the fitted convex hulls as in Fig 2 based on 500 (independent) realizations.
Fig 8.
The hybrid model with various disordered lattices does not display lattice artifacts.
The top row shows simulations with high initial nutrient concentration (c0 = 3) and the bottom row shows simulations with low initial nutrient concentrations (c0 = 0.7). The minimum separation between lattice sites in the VRLs l0 was set to 0.5. The selection of parameters and representation is otherwise the same as in Fig 2.
Fig 9.
Including adhesion between cells can cause lattice artifacts to reappear in some realizations of the disordered lattice.
The adhesion strength per unit contact area between cells was set to J = 4. The central figures (green) show a representative final colony shape. The representation of the outer rings constructed from 1000 repetitions is as in Fig 2.
Fig 10.
Computation time shows power-law scaling of our hybrid bacterial growth model with the number of lattice sites.
The scaling was determined both for simulations with high initial nutrient concentration (blue dots) and for simulations with low initial nutrient concentration (orange dots). The connecting lines are to guide the eye. The exponent of the power law fit is (dashed line). The longest running simulation consisted of 3,034,566 lattice sites.
Fig 11.
The hybrid on-lattice model can be computed considerably faster than off-lattice methods.
Computation time of the hybrid lattice-based model (orange dots) is compared with an agent-based off-lattice simulation of a similar system (blue dots). The connecting lines are to guide the eye. Colony size is plotted against computation time during a single simulation. A power law was fitted to both curves (dashed lines). The exponent fitted to the off-lattice curve was and the exponent fitted to the on-lattice curve was
. The off-lattice simulation was performed using the iDynoMiCS software package [22] with parameters chosen as in Ref [13] and with bulk limiting nutrient concentration
g/L. The parameters used in the hybrid lattice-based simulation are initial nutrient concentration c0 = 3.0 and the remaining parameters as in Table 1. The insets show the final colonies where half the initial bacteria were colored red and the other half blue. Color was inherited from parent to daughter cells. The final configuration of the off-lattice simulation contains 41,697 agents and took approximately 11 hours to compute. The final configuration of our hybrid on-lattice simulation contains 42,015 agents and took approximately 19 seconds to compute.