Conference Agenda

.

We present a phase field model describing cell diffusion in a soft tissue, with particular attention to their active behavior in sensing nearby cells and the mechanical properties of the surrounding environment to guide their movement. The model derivation is framed within the principles of power balance and micro-balance expenditure, and, in principle, it has a general applicability to a population of mesenchymal cells moving inside a visco-elastic environment and “activated” by several chemo-mechanical cues [1]

A central aspect of the model relates the derivation of the cell population chemical potential: it is made of by the superposition of the entropic energy - homogeneous convex free energy - and the active potential induced by the microforces, responsible for triggering the instability process through the spinodal decomposition dynamics. The active component of the chemical potential is adherent to the phenomenological law firstly proposed in [2]. Additionally, the microforce balance is enhanced by the power conjugate associated with variations in the concentration gradient, orienting the cell diffusion toward a stationary limit pattern during a metastable dynamics [1,3]. This vector field could potentially model any directional cue or bias characterizing the interaction between cells and the surrounding elastic environment, making the cell activity at the microscale to emerge on the tissue mechanics.

In testing the descriptivity and the predictivity of this active cell population dynamics model, we apply it yo the case study of the hepatic fibrosis: a pathological process characterized by excessive deposition of extracellular matrix proteins in the liver, typically as a response to chronic injury caused by factors such as viral infections, alcohol abuse, or metabolic disorders. Over time, this scarring can disrupt normal liver architecture and function, leading to abnormal stiffening of the tissue and impairment of blood perfusion. The myofibroblasts are the activated mesenchymal cells responsible for extracellular matrix production during the over mentioned process. The progression stages of the pathology characterize for regular patterns deposition within the hepatic lobule - the fundamental functional unit of hepatic tissue - which are successfully replicated by our FEM simulation, with appropriate constitutive choices of the active microforce and microcouple fields.

References

• [1] F. Recrosi, A. Tatone, G. Tomassetti, “Driving forces in cell migration and pattern formtion”, arXiv, Soft Condensed Matter, arXiv:2410.22273,(2024).
• [2] G.F. Oster, J.D. Murray, A.K. Harris,”Mechanical aspects of mesenchymal morphogenesis”, J. Embryol. exp. Morph. 78 83–125. (1983).
• [3] F. Recrosi, R. Repetto, A. Tatone, G. Tomassetti ”Mechanical Model of Fiber Morphogenesis in the Liver”, Proceedings of XXIV AIMETA Conference, Lecture Notes in Mechanical Engineering. Springer, (2019).

Introduction

Cell growth and proliferation reduces under mechanical loading and confinement [1], but the mechanisms underlying this behaviour are not understood. Cells are subject to a volume checkpoint that restricts proliferation of small cells [2], and recent evidence further suggests that dilution of key cell-cycle inhibitors may govern this behaviour. In this study, we propose a biophysical model that couples transcriptional-translational kinetics with the hydromechanics of cell growth to investigate the feedback between cycle inhibitors, biomolecule synthesis, and mechanical loading in regulating cell growth and division.

Methods

Cell fluid volume is controlled by a balance between hydrostatic pressure and osmotic pressure arising from cytoplasmic solutes [3]. We first developed a mathematical model to predict mRNA and protein production, linking free amino acid availability to translation. Synthesised biomolecules in turn increase osmotic pressure to facilitate cell cycle growth, with volume changes arising from water flux further influenced by membrane tension and external mechanical loading. Synthesis of charged proteins additionally influences ion transport, with feedback to cellular osmotic pressure. We implemented this model within PhysiCell, an open source agent-based modelling framework to analyse discrete cell-cell interactions. This framework was further coupled with a custom finite element solver to describe contact between cells and surrounding hydrogel and matrix.

Results and Discussion

Our analyses reveal that biomolecule synthesis during the cell cycle increases osmotic pressure to drive cell growth. Free amino acids, mRNA, and proteins govern cell volume by modulating both intracellular osmotic pressure and membrane potential. With cell growth, a key cycle inhibitor Whi5 is diluted in agreement with experimental observations [4]. Sufficient dilution facilitates G1/S phase progression, with our model characterising the volume dependency of yeast cell division. However, when cells experience sufficient external loading, our model predicts that growth reduces due to increased hydrostatic pressure and cell division is restricted by insufficient Whi5 dilution and macromolecular crowding. Simulations also suggest that smaller cell birth size leads to an increased G1 phase duration, as observed by Schmoller et al [4]. Extension to the agent-based framework revealed how compaction arises from cell-cell adhesion, amplifying growth-induced pressure experienced by proliferative cells. Simulations reveal that mechanical feedback between surrounding hydrogel deformation and contacting cells further suppresses fluid intake and the rate of cell growth, underpinned by insufficient Whi5 dilution and reduced biomolecule synthesis in agreement with experimental reports [1]. This framework will next be mapped to understanding stress-dependent tissue and tumour growth, where the protein Rb may play a similar role to Whi5 [5]. Thus, leveraging insight from suppression of yeast growth can shed light on the role of cycle inhibitors on tumour progression and identify potential therapeutic targets.

References

1. Alric et al. (2022), Nature Physics, 18.

2. Varsano et al. (2017), Cell Reports, 20

3. McEvoy et al. (2020), Nat Commun, 11.

4. Schmoller et al. (2015), Nature, 526.

5. Zatulovskiy et al. (2020), Science 369.

Heart morphogenesis is mechanically driven by inhomogeneous and anisotropic growth patterns, which involve buckling and twists of an initially tubular shape [1]. Genetic mutations and growth defects in this complex but robust process may yield aberrant shapes which may have detrimental consequences in life expectancy or even fatal effects. For instance, greb1l mutants lack sufficient growth in heart outflow track and are absent of torsional growth, which is believed to cause crisscross malformations [2].

Despite recent advances in microscopy data and determination of heart geometry during its morphogenesis, it is still difficult to predict and quantify the growth patterns that are responsible of tis final shape. This is due to the variability of the shape, and the presence of anisotropies. For this reason, computational techniques that can estimate the growth distribution in different embryos become extremely useful to determine differences and growth defects.

In this work, we present a methodology for computing growth distributions from a series of three-dimensional snapshots [3]. By resorting to a finite element discretization of our domain, and a piecewise constant distribution of orthotropic growth tensor, we formulate an inverse problem that aims at computing a set of optimal patterns that match the resulting shape in a least square sense. Additionally, we also include the unknown boundary conditions as part of our inverse problem, given the impossibility to accurate measure boundary tractions.

Due to the multiplicity of the solutions, the optimization is regularized and iteratively solved for reducing the effects of the regularization parameters. We demonstrate through a series of synthetic examples that the growth can be correctly reproduced, and also apply the methodology to a series of heart subdomains. We also show that the optimal conditions of the inverse problem have a Hamiltonian structure, which can be exploited for the numerical solution of the problem. We finally also show that the same problem can be employed to predict contractility patterns in organism locomotion [4].

REFERENCES

[1] Le Garrec et al. A predictive model of asymmetric morphogenesis from 3D reconstructions of mouse heart looping dynamics. eLife 2017;6:e28951. DOI: https://doi.org/10.7554/eLife.28951

[2] S Bernheim, A Borgel, JF Le Garrec, E Perthame, A Desgrange, C Michel, L Guillemot, S Sart, CN Baroud, W Krezel, F Raimondi, D Bonnet, S Zaffran, L Houyel, SM Meilhac. Identification of Greb1l as a genetic determinant of crisscross heart in mice showing torsion of the heart tube by shortage of progenitor cells., Dev Cell, 58(21), 2023.

[3] C Olivesi, JJ Muñoz. Inverse analysis for the computation of growth and boundary conditions in elastic bodies. Comp. Mechanics, 2024. doi.org/10.1007/s00466-024-02546-5

[4] A Bijalwan and JJ Muñoz. Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates. Comp. Meth. App. Mech. and Eng., 420:116697, 2024.

To form new sprouts during angiogenesis, endothelial cells must coordinate their migration through biophysical and biomechanical signaling between each other and the micro-environment. A relatively simple model of angiogenesis is intersegmental vessel (ISV) formation in zebrafish. While various molecular, cellular, and mechanical factors coordinate ISV pathfinding between the somites, the specific contributions of ECM components remain incompletely understood. Here we hypothesize that guidance through ECM molecules laid down in the intersomitic space confines a process of self-organized pattern formation to the intersomitic space. We combine theoretical and experimental approaches in the zebrafish to investigate this hypothesis. Firstly, we developed a hybrid mathematical model to study the effect of ECM mechanics on a self-organized mechanicsm of endothelial network formation. A network of fibers represents a generic ECM, which is simulated using a coarse-grained mass-spring system, while endothelial network formation is modeled with a Cellular Potts Model (CPM) and partial-differential equation model for intercellular signaling through small cytokines. This framework was extended to incorporate ISV-specific factors such as VEGF and semaphorin signalin, cell polarization, and integrin-based mechano-sensitive of adhesion of cells to the ECM. In absence of an ECM, our model predicts the formation of endothelial network-like structures, as in our previous work. However, in presence of an ECM, the sprouts of consisting of endothelial cells migrate along high concentrations of the ECM. Thus the model predicts that if the concentration of ECM were reduced in the intersomitic space, the endothelial cells should organize into network-like structures. To investigate the contributions of the ECM to ISV formation in the zebrafish, we employed morpholino-mediated gene knockdown of ECM components in endothelial cell- and ECM-tagged zebrafish lines, coupled with high-resolution laser scanning confocal microscopy. Simultaneous knock-down of one type of ECM protein delays ISV sprouting during the first six hours; however, vessel development eventually proceeds normally, albeit with reduced cell proliferation and minor hypersprouting events. Notably, simultaneous knockdown of an additional ECM protein resulted in aberrant vessel pathfinding, leading to disorganized, incomplete ISV development, resembling endothelial network patterns formed through self-organization by endothelial cells in in silico and in vitro. Furthermore, we observed that endothelial cells interact with and migrate along laminin- and fibronectin-rich pathways, as revealed by zebrafish reporter lines, further underscoring the role of these ECM components in guiding vessel formation. Our simulations suggest that ECM stiffness may significantly influence endothelial cell migration, with cells potentially traveling further along VEGF gradients on stiff ECM compared to soft ECM under moderate VEGF sensitivity, providing a potential explanation for the observation in ECM component knockdowns. These results indicate that ECM-regulated tip cell migration could be a key determinant of ISV growth speed and patterning. All in all, by integrating experimental and theoretical approaches, our study suggests that an interplay between endothelial self-organization and ECM-guided pathfinding contributes to ISV formation.