Model-Based Prediction of an Effective Adhesion Parameter Guiding Multi-Type Cell Segregation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Differential Migration Model
- Lattice: The PCA works on a squared lattice with periodic boundaries and we set throughout our numerical analysis.
- Cells: To emulate biological cells, every lattice site is assigned a cell type of all possible cell types in the system. In this way each cell regardless of its type, has approximately the same size and occupies exactly one lattice site. The number of cells of each type remains constant during the sorting process.
- Configurations: A configuration of the lattice S represents the state of the model at time t. It belongs to the set of all possible configurations .
- Migration: A configuration is changed by a cell position switch involving two adjacent lattice sites x and y with :Position switches between two cells of the same type do not change the configuration and are therefore neglected. Thus, neighboring cells situated at lattice sites x and y only switch their positions if they are heterotypic, i.e., .
- Differential adhesion: We assume that stronger bonds to neighboring cells hinder cell motility. Accordingly, a cell switch occurs with rate . The rate depends on the parameters and , which represent the binding strengths between cells at lattice sites x and y to positions from the von-Neumann-1 neighborhoods and , see Figure 1. The von-Neumann-1 neighborhood of a lattice site corresponds to all neighboring lattice sites with Manhattan distance one. The cell switch rate of the two adjacent lattice sites with and is as follows:
2.2. Order Indicator
3. Results
3.1. Cell System Parameters for Two Cell Types
3.2. The Effective Adhesion Parameter for Arbitrary Number of Cell Types
3.3. Numerical Evidence for the Effective Adhesion Parameter
3.3.1. The Asymptotic Level of Cell Segregation Depends on the Effective Adhesion Parameter
3.3.2. Estimating the Effective Adhesion Parameter Using Statistical Learning Methods
3.4. The Impact of Interfacial Tension, Adhesion or Repulsion on Cell Segregation
4. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. PCA Implementation
Algorithm A1 The PCA Algorithm. |
|
Appendix A.1. initializeLattice
Appendix A.2. initializeSupportDataStructure
Appendix A.3. checkSimulationRunCondition
Appendix A.4. getOneRandomCellSwitch
Appendix A.5. getLatticeSitesAssociatedToCellSwitch
Appendix A.6. interchangeLatticeOccupationsAtSites
Appendix A.7. getAffectedLatticeSites
Appendix A.8. getNeighborsOfLatticeSite
Appendix A.9. getCellSwitchRate
Appendix A.10. checkLatticeSitesAreHeterotypic
Appendix A.11. updateSdsEntryForLatticeSiteWithData
Appendix B. Order Indicator
2-Cell-Type Systems | 3-Cell-Type Systems | |
---|---|---|
1200 | 1175 | |
50 | 0 |
Appendix C. Numerical Evidence for the Effective Adhesion Parameter with Extended Cell-Type Number
(a) 4-cell-type systems | Model | |||
---|---|---|---|---|
Accuracy | ||||
SVM | 1.0000 | −0.6553 | 0.8022 | 0.9615 |
Logit | 1.0000 | −0.6625 | 0.7330 | 0.9610 |
Theoretical prediction | 1 | −2/3 | 0 | |
(b) 5-cell-type systems | Model | |||
Accuracy | ||||
SVM | 1.0000 | −0.5030 | 0.5889 | 0.9555 |
Logit | 1.0000 | −0.4913 | 0.5992 | 0.9565 |
Theoretical prediction | 1 | −0.5 | 0 |
Appendix D. The Impact of the Effective Adhesion Parameter on the Convergence Speed
Appendix E. Statistical Learning Methods
- SVM: used algorithm implementation “SVC”
- SVM: kernel “linear”
- Logit: used training algorithm/solver “lbfgs”
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(a) 2-cell-type systems | Model | |||
---|---|---|---|---|
Accuracy | ||||
SVM | 1.0000 | −1.9764 | −0.0374 | 0.9965 |
Logit | 1.0000 | −1.9763 | −0.0455 | 0.9965 |
Theoretical prediction | 1 | −2 | 0 | |
(b) 3-cell-type systems | Model | |||
Accuracy | ||||
SVM | 1.0000 | −0.9994 | 0.2835 | 0.9895 |
Logit | 1.0000 | −0.9935 | 0.2624 | 0.9900 |
Theoretical prediction | 1 | −1 | 0 |
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Rossbach, P.; Böhme, H.-J.; Lange, S.; Voss-Böhme, A. Model-Based Prediction of an Effective Adhesion Parameter Guiding Multi-Type Cell Segregation. Entropy 2021, 23, 1378. https://doi.org/10.3390/e23111378
Rossbach P, Böhme H-J, Lange S, Voss-Böhme A. Model-Based Prediction of an Effective Adhesion Parameter Guiding Multi-Type Cell Segregation. Entropy. 2021; 23(11):1378. https://doi.org/10.3390/e23111378
Chicago/Turabian StyleRossbach, Philipp, Hans-Joachim Böhme, Steffen Lange, and Anja Voss-Böhme. 2021. "Model-Based Prediction of an Effective Adhesion Parameter Guiding Multi-Type Cell Segregation" Entropy 23, no. 11: 1378. https://doi.org/10.3390/e23111378