To model the morphogenesis of rod-shaped bacterial micro-colony, several individual-based models have been proposed in the biophysical literature. When studying the shape of micro-colonies, most models present interaction forces such as attraction or filial link. In this article, we propose a model where the bacteria interact only through non-overlapping constraints. We consider the asymmetry of the bacteria, and its influence on the friction with the substrate. Besides, we consider asymmetry in the mass distribution of the bacteria along their length. These two new modelling assumptions allow us to retrieve mechanical behaviours of micro-colony growth without the need of interaction such as attraction. We compare our model to various sets of experiments, discuss our results, and propose several quantifiers to compare model to data in a systematic way.
Citation: Marie Doumic, Sophie Hecht, Diane Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6873-6908. doi: 10.3934/mbe.2020356
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Abstract
To model the morphogenesis of rod-shaped bacterial micro-colony, several individual-based models have been proposed in the biophysical literature. When studying the shape of micro-colonies, most models present interaction forces such as attraction or filial link. In this article, we propose a model where the bacteria interact only through non-overlapping constraints. We consider the asymmetry of the bacteria, and its influence on the friction with the substrate. Besides, we consider asymmetry in the mass distribution of the bacteria along their length. These two new modelling assumptions allow us to retrieve mechanical behaviours of micro-colony growth without the need of interaction such as attraction. We compare our model to various sets of experiments, discuss our results, and propose several quantifiers to compare model to data in a systematic way.
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Marie Doumic, Sophie Hecht, Diane Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6873-6908. doi: 10.3934/mbe.2020356
Marie Doumic, Sophie Hecht, Diane Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6873-6908. doi: 10.3934/mbe.2020356
Figure 2. Representation of the interaction between two bacteria i and j with an overlapping hi,j. The bacteria are represented by grey spherocylinders. The two balls by which we approximate the bacteria for the repulsive force are drawn in blue
Figure 3. Representation of the division of a bacterium i into two daughter bacteria i and j. The mother cell is of length li and the two daughter cells are of length l′i=l′j=0.5li−d0. The angle of the daughters are disrupted by dθi and dθj
Figure 4. Dataset 1: From left to right: Distributions of the increment length, lenght, and growth rate for 10 initial configurations. The experimental distribution are plotted in red and the numerical simulation distributions are plotted in blue
Figure 5. Dataset 2: From left to right: Distributions of the increment length, lenght and growth rate for 10 initial configurations. The experimental distribution are plotted in red and the numerical simulation distributions are plotted in blue
Figure 6. Dataset 3: From left to right: Distributions of the increment length, length and growth rate for 10 initial configurations. The experimental distributions are plotted in red and the numerical simulation distributions are plotted in blue
Figure 7. Plot of a colony from the experimental datasets: Dataset 1 (a), Dataset 2 (b) and Dataset 3 (c), at times corresponding to: four cell colonies (1), colony composed of N=40 cells (2), colony at time t=250 min. All the colonies are in the four-cell array organisation. The colours of the bacteria are determined by their orientations
Figure 8. Plot of the colony for A=1 (a) and A=0.4 (b) at t=250 min. The color of the bacteria are given by their angle from the horizontal axis
Figure 9. Evolution of the aspect ratio αR (a), the density (b) and the local order quantifier λ (c) as functions of the area of the colony, and of the distribution of the angle at division (d), for different values of A: A=1, A=0.8, A=0.6, A=0.4, A=0.2
Figure 10. Plot of the colony for α=0.5 (left) and α=0.9 (right) at t=70 min (which correspond to the moment where the colony is composed of four cells). The color of the bacteria are given by their angle from the horizontal axis
Figure 11. Evolution of the aspect ratio αR (a), the local order quantifier λ (b) and the density (c) as functions of the area of the colony, and of the distribution of the angle at division (d) for different values of α: (α,Tα)=(0.5,∞), (α,Tα)=(0.6,∞), (α,Tα)=(0.75,∞), (α,Tα)=(0.9,∞) and (α,Tα)=(0.9,12)
Figure 12. Evolution of the aspect ratio αR (a), the local order quantifier λ (b) and the density (c) as functions of the area of the colony, and of the distribution of the angle at division (d) for different values of Θ: Θ=10−5, Θ=10−3, Θ=10−1
Figure 13. Dataset 1: plots of the aspect ratio αR (a), the local order quantifier λ (b) and the density (c) as functions of the area of the colony, and of the distribution of the angle at division (d) for the experimental data (grey dashed curve), and numerical simulations for the case 1 (blue curve), case 2 (red curve), case 3 (yellow curve) and case 4 (purple curve). The plots of the numerical data are averaged over 10 simulations
Figure 14. Dataset 1: Plot of simulation at time t=53min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figure can be compare to Figure 7(1) Panels (a)
Figure 15. Dataset 1: Plot of simulation at time t=200min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figure can be compare to Figure 7(2) Panel (a)
Figure 16. Dataset 2: plots of the aspect ratio αR (a), the local order quantifier λ (b) and the density (c) as functions of the area of the colony, and of the distribution of the angle at division (d) for the experimental data (grey dashed curve), and numerical simulations for the case 1 (blue curve), case 2 (red curve), case 3 (yellow curve) and case 4 (purple curve). The plots of the numerical data are average over 10 simulations
Figure 17. Dataset 2: Plot of simulation at time t=111min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figures can be compare to Figure 7(1) Panel (b)
Figure 18. Dataset 2: Plot of simulation at time t=200min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figure can be compared to Figure 7(3) Panel (b)
Figure 19. Dataset 3: Plots of the aspect ratio αR (a), the local order quantifier λ (b) and the density (c) as functions of the area of the colony for the experimental data (grey dashed curve), and numerical simulations for the case 1 (blue curve), case 2 (red curve), case 3 (yellow curve) and case 4 (purple curve). The plots of the numerical data are averaged over 5 simulations
Figure 20. Dataset 3: Plot of simulation at time t=111min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figure can be compare to Figure 7 Panel (c)
Figure 21. Dataset 3: Plot of simulation at time t=200min for Case 1 (a), Case 2 (b), Case 3 (c) and Case 4 (d). The colors of the bacteria are given by their orientation. These figures can be compared to Figure 7(1) Panel (c)