By Carles Falcó
The study of biological tissues requires a deep understanding of the interactions between cells, tissues, and organs. While traditional approaches often examine tissues in isolation, cells exist within a dynamic environment that shapes their overall form and function through interactions with other cells and tissues. Our research team combined data from recent experiments with mathematical modeling and Bayesian inference to closely examine the complex dynamics of epithelial tissues [1].
Our work is based on a set of experiments by members of Daniel Cohen’s lab at Princeton University [3], who are studying the dynamic behavior of expanding and colliding epithelial tissues (see Animation 1). Their efforts revealed the complex patterns of growth and shape that arise when monolayers of epithelial cells with diverse initial geometries collide, ultimately providing insight into the interactions of tissues with different shapes. The Cohen Lab aims to design living materials that are composed of cells by understanding the main design principles that drive the self-assembly of tissue composites.
Animation 1. Growth and collision of epithelial tissues with different initial geometries. Animation courtesy of [3].
To describe these experiments, we utilized minimal models with as few parameters as possible. We selected the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) model and the Porous-Fisher model, which both describe the evolution of cell density \(\rho(x,t)\) in the tissue as a function of time and space. The Fisher-KPP model assumes random motion of cells and yields a linear diffusion term
\[\partial_t\rho+D\Delta\rho+r\rho(1-\rho/K).\]
In contrast, the Porous-Fisher model assumes that cell movement is guided by pressure gradients within a population:
\[\partial_t\rho=D\nabla \cdot (\rho \textrm{P}(\rho))+r\rho(1-\rho/K),\]
where the function \(\textrm{P}(\rho)\) represents the pressure that cells exert on each other. For the sake of simplicity, we explored the case in which population pressure increases linearly with density, i.e., \(\textrm{P}(\rho) \sim \rho\). The Fisher-KPP and Porous-Fisher models both assume that cells proliferate via logistic growth, as epithelial tissues undergo a contact inhibition of proliferation in which cell cycling decreases as cell density increases [4].
Figure 1. A combination of Bayesian inference and mathematical modeling describes epithelial tissue growth. 1a. Experimental results for the dynamics of freely expanding epithelial monolayers [2]. 1b. Posterior distributions for the parameters of the Fisher-Kolmogorov-Petrovsky-Piskunov model. 1c. Comparison between model predictions (solid line) and data (dashed line). Figure adapted from [1] and [2].
Both models present three parameters: a diffusivity coefficient \(D\) that relates to the speed at which cells move in the tissue, an intrinsic growth rate \(r\), and a carrying or saturating density \(K\) at which cells inhibit proliferation. To determine these parameters, we combined Bayesian inference techniques with data from a “simpler” set of experiments from the Cohen Lab that studied the dynamics of freely expanding epithelial monolayers [2] (see Figure 1a). This inference approach is able to fit the models to the experimental density profiles and quantify the associated uncertainty in the parameters through a posterior probability distribution. In our case, we can identify all of the models’ parameters with a very small variance (see Figure 1b). Interestingly, both the Fisher-KPP and the Porous-Fisher model can accurately describe the data with minimal differences between the two model predictions (see Figure 1c).
Figure 2. Comparison with experiments from the Cohen Lab at Princeton University [3]. The lower panels provide the density profiles along the black dashed lines in the upper panels. Note that the Fisher-Kolmogorov-Petrovsky-Piskunov model predicts the mixing of the two tissues. Figure courtesy of [1].
What happens when we use these calibrated models to predict the behavior of multi-tissue collisions? And how do these predictions compare to the Cohen Lab’s experimental results [3]? Although the Fisher-KPP model provides an overall satisfactory description of tissue collision dynamics, it fails near the boundaries. The assumption of random motion is responsible for this discrepancy, which—contrary to experimental observations—results in unrealistic regions of mixing between the colliding tissues. The Porous-Fisher model, however, accounts for population pressure and thus offers a more accurate representation that yields improved agreement with experimental measurements. Figure 2 visualizes the binary collisions and highlights the disparities between the two models.
Figure 3. The self-assembly of complex composite sheets to reproduce a tri-tissue tessellation inspired by M.C. Escher’s artwork. Figure adapted from [3].
Next, we investigated more complex collisions between epithelial monolayers by allowing multiple sheets to freely expand and eventually collide. A comparison between our findings and the Cohen Lab’s experiments [3] revealed that the Porous-Fisher model effectively captured these complex collisions by accurately considering the population pressure at the tissue boundaries. We then explored this model’s ability to uncover the design principles that underlie the self-assembly of tissue tessellations. Notably, the Porous-Fisher model successfully replicated the self-assembly of a tri-tissue tessellation that was inspired by the work of Dutch artist M.C. Escher (see Figure 3 and Animation 2).
Animation 2. Model prediction from experimental initial conditions to reproduce the tri-tissue tessellation in Figure 3. Animation courtesy of [1].
To conclude, our combination of minimal mathematical modeling, Bayesian inference, and experimental data illuminated the complexities of epithelial tissue growth in the context of tissue self-assembly. Our approach provides a systematic method to precisely measure and predict complex tissue configurations — with wide-ranging applications in the design of tissue composites and the broader field of tissue engineering. This method brings us closer to the final goal of designing and constructing materials that are composed of living cells, ultimately catalyzing the creation of truly “living” materials.
Carles Falcó delivered a minisymposium presentation on this research at the 2023 SIAM Conference on Computational Science and Engineering, which took place in Amsterdam, the Netherlands, earlier this year.
References
[1] Falcó, C., Cohen, D.J., Carrillo, J.A., & Baker, R.E. (2023). Quantifying tissue growth, shape and collision via continuum models and Bayesian inference. J. R. Soc. Interface, 20(204), 20230184.
[2] Heinrich, M.A., Alert, R., LaChance, J.M., Zajdel, T.J., Košmrlj, A., & Cohen, D.J. (2020). Size-dependent patterns of cell proliferation and migration in freely-expanding epithelia. Elife, 9, e58945.
[3] Heinrich, M.A., Alert, R., Wolf, A.E., Košmrlj, A., & Cohen, D.J. (2022). Self-assembly of tessellated tissue sheets by expansion and collision. Nat. Commun., 13(1), 4026.
[4] Puliafito, A., Hufnagel, L., Neveu, P., Streichan, S., Sigal, A., Fygenson, D.K., & Shraiman, B.I. (2012). Collective and single cell behavior in epithelial contact inhibition. Proc. Natl. Acad. Sci., 109(3), 739-744.
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Carles Falcó is a third-year Ph.D. student in mathematical biology at the University of Oxford and a fellow of the La Caixa Foundation. His research combines tools from interacting particle systems, partial differential equations, and Bayesian inference to understand complex phenomena at the cell and tissue level. |