A computational model of stem cells' internal mechanism to recapitulate spatial patterning and maintain the self-organized pattern in the homeostasis state

Abstract

The complex functioning of multi-cellular tissue development relies on proper cell production rates to replace dead or differentiated specialized cells. Stem cells are critical for tissue development and maintenance, as they produce specialized cells to meet the tissues' demands. In this study, we propose a computational model to investigate the stem cell's mechanism, which generates the appropriate proportion of specialized cells, and distributes them to their correct position to form and maintain the organized structure in the population through intercellular reactions. Our computational model focuses on early development, where the populations overall behavior is determined by stem cells and signaling molecules. The model does not include complicated factors such as movement of specialized cells or outside signaling sources. The results indicate that in our model, the stem cells can organize the population into a desired spatial pattern, which demonstrates their ability to self-organize as long as the corresponding leading signal is present. We also investigate the impact of stochasticity, which provides desired non-genetic diversity; however, it can also break the proper boundaries of the desired spatial pattern. We further examine the role of the death rate in maintaining the system's steady state. Overall, our study sheds light on the strategies employed by stem cells to organize specialized cells and maintain proper functionality. Our findings provide insight into the complex mechanisms involved in tissue development and maintenance, which could lead to new approaches in regenerative medicine and tissue engineering.

Figure 1

The Turing signal patterns of $S_l$, generated by Gillespie algorithm. From right to left, $(d_l,d_a)=(5e3,1e5),(1e4,2e5),(2.5e5,5e6),(1e6,2e7)$, and from top to bottom, $s_{\mathit{upper}}=5,10,30$, and $s_{\mathit{lower}}=0$.

Figure 2

The initial and final state of the system in the absence of “leading” signal. Each pixel indicates one individual cell, S in cyan, P in green, (A) in yellow, and (B) in red, an empty position (EP) in blue, or an out-of-dish space (OD) in gray.

Figure 3

The presentation of static “leading” signal ($S_l$) (first, and third columns) and their corresponding patterns formation (second, and fourth columns) in the population. In the leading signal patterns, “Red” represents the value of 250, while “Magenta” represents the 0 value. In the second and fourth columns, as it is shown in the colorbar, each pixel indicates one individual cell, one individual cell, S in cyan, P in green, A in yellow, and B in red, an empty position (EP) in blue, or an out-of-dish space (OD) in gray.

Figure 4

The system behavior in the face of dynamic “leading” signals.The first row represents the Gaussian, spot, stripe, and reversed-spot patterns as the “leading” signals, $S_l$. The values of the signaling molecules are scaled in the range of [0, 250] to be in agreement with the values of static leading signals. The second row demonstrates the system state after 50 divisions (of progenitor cells). Each pixel in the second row can potentially demonstrates S, P, A, B cell types, an empty position or an out-of-dish space as shown by the colorbar in the second row. The third row shows the abundance of four cell types in the population through the simulation.

Figure 5

The system behaviour in the face of dynamic “leading” signals, $S_l$ with spot pattern. A. The first and second rows represent the system state after the 50 and 100 divisions (of progenitor cells), respectively. The first and second columns represent two different formulations of death rate: ${\gamma }_i=\frac{n_i}{n_i^{\ast }}\ast v_g$, and ${\gamma }_i=\sqrt[3]{\frac{n_i}{n_i^{\ast }}}\ast v_g$, where $i\in A,B$. As it is shown in the colorbar, each pixel can indicate one individual cell, S in cyan, P in green, A in yellow, and B in red, an empty position (EP) in blue, or an out-of-dish space (OD) in gray. The third row shows the abundance of four cell types through the simulation. B. The template and penalty filters of spot signal pattern. C. The scoring traces corresponding to the first and second columns in panel A. It indicates how the pattern in population in different states of the system through time is aligned with the desired initial signalling pattern.

Figure 6

The system behaviour in the face of dynamic “leading” signals, $S_l$ with Gaussian, reversed-spot and stripe patterns. The first and second rows represent the template matrices and the system state after 100 divisions (of progenitor cells), respectively. As it is shown in the colorbar, each pixel can indicate one individual cell, S in cyan, P in green, A in yellow, and B in red, an empty position (EP) in blue, or an out-of-dish space (OD) in gray. Here, ${\gamma }_i=\sqrt[3]{\frac{n_i}{n_i^{\ast }}}\ast v_g$, where $i\in A,B$. The third row shows the abundance of four cell types through the simulation. The fourth row demonstrates the scoring traces to evaluate the pattern maintenance in the population. It indicates how the pattern in population in different states of the system through time is aligned with the desired initial signalling pattern.

Figure 7

The system behabior in the face of Gaussian “leading” signal pattern, and in the absence of physical dish borders. As it is shown in the colorbar, each pixel can indicate one individual cell, S in cyan, P in green, A in yellow, and B in red, an empty position (EP) in blue, or an out-of-dish space (OD) in gray.