Initial Model Parameter Estimation
The initial parameter settings for λA, λV, Vt, At, and J (see Eq. 2 in Supplement) and the predefined spatial relationships defined the reference model (model 1, Table 1, Fig. 2A). Starting from these settings, the contact energy coefficients (J) between NP, MM, UB cells and the matrix were modified, yielding the adhesion-based models 2 and 8 (Table 1). The purpose of this modification was to identify the range of contact energy coefficients required to induce cell aggregations, and to explore the aggregation behaviour of NP and MM cells in the CC3D model space (Figs 2B/C) (Andasari et al., 2012; Combes et al., 2016; Osborne et al., 2017; Swat et al., 2009).
Table 1. Model parameters with pre-set values applied in 3D simulations.
Parameter
|
Pre-set value
|
Models
|
Ref.
|
Target volume ( )
|
375 x 10-18 m3
|
1-8
|
this work, (Combes et al., 2016)
|
Lambda volume ( )
|
20 x 109 kgm-4s-2
|
1-8
|
(Osborne et al., 2017; Swat et al., 2009)
|
Target surface area ( )
|
312 x 10-12m2
|
1-8
|
this work, (Combes et al., 2016)
|
Surface lambda )
|
0.1 x 10-3 kgm-2s-2
|
1-8
|
(Osborne et al., 2017; Swat et al., 2009)
|
Time step (MCS)
|
0.017 h
|
1-8
|
(Osborne et al., 2017; Swat et al., 2009)
|
Surface temperature (T)
|
20
|
1-8
|
(Osborne et al., 2017; Swat et al., 2009)
|
Contact energy coefficient (J)
|
(all cells): 5 x 10-15 kgs-2
(NP&MM): 13 x 10-15 kgs-2
|
1,3-4;
2, 5-7, 8*
|
(Osborne et al., 2017; Swat et al., 2009)
|
Chemoattractant secretion (S)
|
3 DU/s
|
3-7
|
(Osborne et al., 2017; Swat et al., 2009)
|
Chemotaxis lambda ( )
|
100 x 10-27 kgm2s-2
|
3-7
|
(Osborne et al., 2017; Swat et al., 2009)
|
Global diffusion constant (D)
|
1.0 x 10-12 m2s-1
|
3-7
|
(Brown, 2011; Osborne et al., 2017; Swat et al., 2009)
|
Global decay constant ( )
|
1.0 x 10-7s
|
3-7
|
(Osborne et al., 2017; Swat et al., 2009)
|
*see Table 2 for description of multiple adhesion interfaces in model 8.
Next, the impact of chemotaxis on cell patterning was investigated, assuming secretion of chemoattractants by UB cells and/or NP cells themselves. To that end, parameters related to chemoattractant secretion (S),, diffusion (D),, degradation (γ),, and chemotaxis strength (λCL) were introduced (see Eqs 2, 3 in Supplement). To derive plausible value ranges for these parameters and explore the effect of chemotaxis on cell clustering, NP cells (model 4; Fig. 2D) or UB cells (model 3; Fig. 2E) were assumed as alternative sources of chemoattractant secretion. Finally, the combined effects of the contact energies and chemotaxis were considered in the remaining models (5,6,7; Figs 2F-H).
Simulated Pattern Formation
The eight model variants yielded distinctly different final cell patterns (Fig. 2). The initial conditions and parameters had only a limited effect to the final cell patterns (Fig S1, S2). Therefore, the following description refers to the model patterns obtained with optimized simulations using random initial cell distribution and no pre-formed chemoattractant gradient.
The reference model yielded a cell pattern without coherent clustering over time (Fig. 2A). The models involving chemoattractant secretion by NP cells with and without cell adhesion differences resulted in cell aggregates between the UB tips without adherence to the UB surface (Figs 2D/F). In the models assuming adhesion differences between MM and NP cells but no chemotaxis, streak- or ball-like clusters emerged throughout the inter-UB area (Figs 2B/C). In the models involving UB cell chemoattractant secretion, NP cells aggregated along the UB surface, with preference to the corner regions (Figs 2E/G/H).
Directed migration and preferential aggregation of NP cells in the UB corner was observed with the models 3, 5, and 7 (Figs 2E/G/H), resembling the formation of PTA in the corner region during nephrogenesis (Fig. 2C). The most consistent formation of NP cell clusters resembling PTAs was observed with model 7, which involves chemoattractant secretion by both UB and NP cells and stronger adhesion between NP and MM cells relative to other models (Figs 2H, 3B/E/H). The video of the model (7) behaviour can be viewed online (Tikka, 2019d).
Experimental Studies
The results of the cell movement analysis performed on the explant culture model experiments of Combes et al. (Combes et al., 2016; Lawlor et al., 2019; Lefevre et al., 2017) and our own experiments with a dissociation-reaggregation kidney organoid model are provided in (Figs 4–6).
The observed overall NP cell speed averages were 0.15±0.02 µm/min in the explant cultures, and 0.13±0.01 µm/min in the kidney organoid MM cells.
While cell speeds in the explant culture fluctuated considerably more than in the kidney organoid experiments, in both experimental settings the two cell types moved at different rates depending on their location relative to the UB tip (Fig. 4). In the corner region, the average cell speeds were 0.16±0.02 µm/minin the explant culture experiments and 0.15±0.01 µm/min in the kidney organoid (Combes et al., 2016). Average speeds in the tip region were 0.12±0.01 µm/min in both experimental settings. The slower relative cell movement of both MM and NP cells around the UB tip region was also apparent when expressed as the tip-to-corner speed ratio, which was below 1 for most of the observation time (Fig. 5).
According to 2D SOM analysis, stable speeds in the explant culture data were 0.19±0.02 µm/min in the corner, and 0.14±0.01 µm/min in the tip region (Combes et al., 2016; Kohonen, 1982). 2D analysis was performed, because 3D data (z axis) of the explant culture experiments was found (regularly) skewed. Stable cell speeds of the kidney organoid data (calculated by 3D SOM analysis) were 0.25±0.02 µm/min in the corner and 0.18±0.02 µm/min in the tip region respectively. The different speed of motion of cells in the tip and corner regions is illustrated in the coloured speed contours of the transformed SOM plots (Fig. 6).
Estimation of Final Model Parameters with Particle Swarm Optimization and Resulting Cell Patterns
The parameter ranges for PSO were chosen according to the initial model parameter estimates and the additional PSO algorithms (detailed further in the appendix). The optimization procedure aimed at maximising the amount of NP cells at the UB surface while simultaneously aligning the NP cell speeds in the model to the cell speeds observed experimentally in the explant culture setting, as mentioned in the methods.
The final model parameter values obtained by the PSO technique are given in Table 2. The improvement of the models achieved by the application of PSO is illustrated by the Best Quality Values (Table 2; lower numbers indicating better quality). The best model quality was obtained for model 7, while the other models showed either substantially lower quality or a spuriously high quality without matching the experimental situation. This applied in particular for the NP secreting models 4 and 6 and the adhesion model 8, where cells did not aggregate towards the UB surface in the first place.
Table 2. Optimized parameter values for each model variant (1-8). Values are presented as: Random (Uniform), e.g. for NP_ADH: 7.9 (13). {*} 8_ADH_ADH assumed nine cell-cell adhesion interfaces: {1-3} ‘Wall (UB) and NP/MM/Medium’, {4} ‘NP and NP’, {5} NP and MM’, {6} ‘MM and MM’, and {7-9} ‘Medium and NP/MM/Medium’. The parameters of spatial relationships ( , , NO), except , and the ones not mentioned here were constant (see Table 1).
|
REF
(1)
|
ADH
(2)
|
UB
(3)
|
UB_ADH (4)
|
NP
(5)
|
NP_ADH (6)
|
UB_NP_ADH (7)
|
ADH_ADH
(8)
|
Chemoattractant
secretion rate (DU/s)
|
0
|
0
|
3
|
3
|
3
|
3
|
6.91
|
0
|
Chemotaxis lambda
(10-27; kgm2s-2)
|
0
|
0
|
4.22 (4.19)
|
43.9 (43.8)
|
67.94
(71.3)
|
74.5 (74.3)
|
1.41
|
0
|
Global diffusion
constant
(10-12; m2s-1)
|
0
|
0
|
1
|
1
|
1
|
1
|
1.83
|
0
|
Global decay constant (s)
|
0
|
0
|
1 x 10-7
|
1 x 10-7
|
0.002
|
0.002
|
4.69 x 10-7
|
0
|
Surface motility
(Potts "temperature") (DU)
|
10
|
25
|
25
|
25
|
25
|
25
|
49.8
|
25
|
Surface lambda
(10-3; kgm-2 s-2)
|
0.01
|
0.01
|
0.01
|
0.01
|
0.01
|
0.01
|
8.11
|
0.01
|
Adhesion differences between cell types
(10-15; kgs-2)
|
5.0 {5}
|
7.9 {5}
|
5.0 {5}
|
6.68 {5} (6.98 {5})
|
5.0 {5}
|
7.9 {5} (13 {5})
|
7.86 {5}
|
21.1 {1}, 25.2 {2},
1.75 (3), 0.50 {4},
33.3 {5}, 37.7 {6},
35.8 {7}, 29.7 {8},
1.57 {9}
(26.6 {1}, 47.8 {2},
29.5 (3), 0.50 {4},
20.5 {5}, 0.52 {6},
24.1 {7}, 1.54 {8},
38.1 {9})
|
PSO Best Quality Value
|
-61,269
|
-56,436
|
-201,902
|
-206,525
|
-202,545
|
-205,261
|
-213,787
|
-340,361
|
In the optimized models, the calculated total distance travelled by the NP cells during the simulation period was between 90–160 µm. The optimization procedure resulted in a more accentuated difference in corner and tip cell speeds in the models involving chemoattractant secretion by the UB (3, 5, 7 in Figs 5, S3). The different speed of motion of cells in the tip and corner regions is also illustrated for model 7 by the coloured speed contours of the transformed SOM plots (Fig. 6C). The tip-to-corner speed ratio of NP cells in these models decreased with time, aligning with the NP cell speed ratio observed towards the end of the explant culture experiments (Fig. 5B). By contrast, the MM cell speeds in the kidney organoid experiments were better matched by the optimized NP secreting models (4 and 6; Fig. 5D). In the secreting models 3–7 the overall speeds of the NP cells both in the tip and corner regions consistently exceeded those of the MM cells (0.13±0.03 v. 0.03±0.02 µm/min; Fig. 5). Correspondingly, NP cells in the kidney organoid experiments were enriched in the corner according to UB secreting models (3, 5, 7; Fig. 7B). On the other hand, MM cells in the explant culture experiments increased in the tip similarly as in the NP secreting models or vice versa in the corner (4, 6; Figs 7, S4).
The enrichment of the cells in different compartments is partly reflected by the relative tip-per-corner cell distances of around 0.66 in the explant culture experiments, which were closest to the UB secreting models (3, 7; Figs S4, S5) (Combes et al., 2016). However, the relative tip-per-corner cell distances in the kidney organoids were best recapitulated by the adhesion-based or NP secreting models (1, 2, 4, 6; Fig. S5).