AA1 – Life Science



Mathematical Modelling of Cellular Self-Organization on Stimuli Responsive Extracellular Matrix

Project Heads

Sara Checa (Charité), Ansgar Petersen (Charité), Barbara Wagner (WIAS)

Project Members

André H. Erhardt (WIAS)

Project Duration

01.01.2021 − 31.12.2021

Located at



Biological background: Cell-cell mechanical interactions as well as mechanical interactions between single cells with their surrounding extracellular matrix (ECM) are key in many biological processes, such as tissue regeneration, angiogenesis, cancer and tissue morphogenesis in general. The ECM is a network consisting of extracellular macromolecules and minerals, such as collagen, enzymes, glycoproteins and hydroxyapatite that provide structural and biochemical support to surrounding cells. It is known that cells probe their environment by imparting (their internally generated) traction forces to the ECM and that by pulling on the fibres of the matrix they are able to create a tensioned ECM network. To date, how these temporal interactions between cells and the ECM influence cellular organisation and tissue patterning remains unknown. Although the most advanced models have included the fibrillar organisation of the extracellular matrix, the ability of the cells to remodel the extracellular matrix and to change the mechanical and viscoelastic properties of the ECM and how these affect further cellular behaviour remains unknown.


The goal: Our main goal is to unravel the complex physical interplay between active cell contraction, ECM deformation and cell-induced ECM remodeling and its role on temporal cellular self-organisation and tissue patterning. To achieve this, we aim to develop mathematical models to simulate the physical remodelling of the ECM as a result of cellular traction forces.


The mathematical model: We will model the ECM as multiphase model for two species:

(i) Elastic network consisting of permanent network (glycosaminoglycans), and temporal (breakable) network (fibronectin/collagen fibers),

(ii) incompressible solvent: J = 1+Cv, where v denotes the volume of a solvent molecule and C its concentration, while the deformation gradient is given by \mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}} and J=\det(\mathbf{F}),

combined with a continuity equation \dot{C}+\nabla_0\cdot\mathbf{j}_0=0, where \mathbf{j}_0 is the nominal flux and \mathbf{j}_0=J\mathbf{F}^{-1}\mathbf{j} and macro force balance \nabla\cdot \mathbf{T}=0. In addition, a dissipation inequality provides \mathbf{j},\mathbf{T}, etc., and the free boundary is responsive to temperature, applied stress and deformation and breakage of crosslinks, as applied by the cell’s traction forces (vertical and horizontal).


We combine agent-based models for cellular self-organization with continuum models for the extracellular matrix, which they remodel. This is used to capture the feed-back mechanism that link local cell-matrix interactions to tissue-scale patterning guided by experimental data.


Applications and motivation: Our studies will have a wealth of further important applications, such as impact of ECM remodelling on the cellular behaviour, in connection with heart failure, where it is essential to prevent adverse ventricular remodelling and restore organ functionality in affected patients. In addition, electrophysiological changes during heart failure involve amongst others ion channel remodelling, alterations in calcium handling and remodelling of the extracellular matrix. One example for such an electrophysiological change are so-called early afterdepolarisations, which are an important cause of lethal ventricular arrhythmias in long QT syndromes and heart failure. Moreover, ECM density for instance regulates the formation of tumour spheroids through cell migration. Furthermore, it has been shown that the internal architecture of macroporous biomaterials can be utilised to control ECM patterning in bone tissue defect scenarios.


External Website

Related Publications

  1. E. Borgiani, G. N. Duda, S. Checa. Multiscale Modeling of Bone Healing: Toward a Systems Biology Approach. Front. Physiol., 8:287, 2019.
  2. E. Brauer, E. Lippens, O. Klein, G. Nebrich, S. Schreivogel, G. Korus, G. N. Duda, and A. Petersen. Collagen fibrils mechanically contribute to tissue contraction in an in vitro wound healing scenario. Adv. Sci., 6:1801780, 2019.
  3. G. L. Celora, M. G. Hennessy, A. Münch, S. L. Waters and B. Wagner. A kinetic model of a polyelectrolyte gel undergoing phase separation. doi 10.20347/WIAS.PREPRINT.2802, 2020.

  4. G. L. Celora, M. G. Hennessy, A. Münch, S. L. Waters and B. Wagner. Spinodal decomposition and collapse of a polyelectrolyte gel. doi 10.20347/WIAS.PREPRINT.2731, 2020.

  5. S. Checa, and P. J. Prendergast. A mechanobiological model for tissue differentiation that includes angiogenesis: a lattice-based modeling approach. Ann. Biomed. Eng. 37:129–145, 2009.
  6. S. Checa, M. K. Rausch, A. Petersen, E. Kuhl, and G. N. Duda. The emergence of extracellular matrix mechanics and cell traction forces as important regulators of cellular self-organization. Biomech. Model. Mechanobiol., 14:1–13, 2015.
  7. A. H. Erhardt. Early afterdepolarisations induced by an enhancement in the calcium current. Processes, 7(1):20, 2019.
  8. A. H. Erhardt, and S. Solem. Bifurcation analysis of a modified cardiac cell model. SIAM J. Appl. Dyn. Syst., 21(1), 231–247, 2022.
  9. A. H. Erhardt, and S. Solem. On complex dynamics in a Purkinje and a ventricular cardiac cell model. Commun. Nonlinear Sci. Numer. Simul., 93, 2021.
  10. M. G. Hennessy, G. L. Celora, A. Münch, S. L. Waters and B. Wagner.  Asymptotic study of the electric double layer at the interface of a polyelectrolyte gel and solvent bath. doi 10.20347/WIAS.PREPRINT.2751, 2020.

  11. M. G. Hennessy, A. Münch, and B. Wagner. Phase separation in swelling and deswelling hydrogels with a free boundary. Phys. Rev. E, 101:032501, 2020.
  12. C. Kühn, and S. Checa. Computational Modeling to Quantify the Contributions of VEGFR1, VEGFR2, and Lateral Inhibition in Sprouting Angiogenesis. Front. Physiol., 10:288, 2019.
  13. A. Petersen, A. Princ, G. Korus, and et al. A biomaterial with a channel-like pore architecture induces endochondral healing of bone defects. Nat. Commun., 9(1):4430, 2018.

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