IBCell

an Immersed Boundary model of a Cell

Intra-, inter- and extra-cellular elements included in the immersed boundary model of a eukaryotic cell:

Morphological changes in the cell undergoing certain life processes:

cell properties:
(1) cell growth
(2) cell division
(3) cell apoptosis
(4) cell necrosis
(5) cell epithelial polarisation
(6) cell movement
(7) cell directinal sensing

Modelling approaches and computer simulations of individual cell life processes:

Cell growth and division in 2D
The process of cell proliferation is started by introducing point sources inside the cell and the balancing sinks outside the cell to transport the fluid from the extracellualr space into the cell. The fluid created by sources-sinks dublets causes the cell growth by pushing cell boundaries and by increasing cell area. The sources are inactivated when the cell doubles its area. At this time the contractile ring is created by introducing contractile forces on the opposite sides of the cell boundary. This causes division of the cell into two daughter cells of approximately equal areas.

Cell growth and division in 3D
The same algorithm as in 2D case is used here, however, the contractile ring is formed by short linear springs acting along the cell equator.

Orientation of the axis of cell division
The orientation of cell division depends on the cell shape and its location within the cell cluster.

In a dividing unpolarised cell (a dark grey cell) two new nuclei are located along the cells longest axis (a thin line joining two nuclei) and the contractile ring (a dark dashed line) is placed between them orthogonally to this axis in such a way as to split the cell into two approximately equal parts.

A polarised cell (a dark grey cell) acquires only one orientation of the axis of cell division (a dark dashed line) that is orthogonal to the part of cell membrane that is in contact with the ECM. Two new daughter nuclei are then placed orthogonal to the axis of cell division giving rise to two lumenal cells.

Cell apoptosis in 3D
An apoptotic cell undergoes very characteristic changes in its morphology, including detachment from the neighbouring cells, shrinkage of the cell volume, collapse of the cytoskeleton and alternations in the cell surface resulting in a bubbling appearance.

Cell apoptosis in 2D
The cell apoptotic death is modelled by a gradual shrinkage of the cell area due to release of the fluid to the outside environment. This is achieved first by disassembling all cell adherens junctions and blocking them from further cell-cell contacts and by placing the sink-source doublets along the membrane of the whole cell with sinks located inside the cell and the balancing sources outside. The resulting fluid flow causes the cell boundary to collapse and the cell area is gradually reduced until it reaches a prescribed minimal value, that we assume to be 15% of the initial cell area. At this time, the cell is considered to be dead. The movie shows three dying cells (red).

Epithelial cell polarisation in 3D
The mature epithelial cells acquire three different cell membrane domains: an unbounded apical side facing free space inside the epithelium (lumen) and two bounded sides that are in direct contact with either the extracellular matrix (a basal side) or the membranes of other cells (lateral sides). Cell polarity is determined by recognising lateral and basal sides and by developingthe apical side and tight junctions along its lateral sides.

Epithelial cell polarisation in 2D
Cell epithelial polarisation is modelled as a multi-step process via the development of three distinct membrane domains: the basal domain is determined by sensing contact with the extracellular matrix; the lateral side is formend between two outer neighbouring cells; the lateral connections are transformed into tight junctions sealing two neighbouring cells; The cell develops its free apical side by disassembling all adherens junctions expressed on the side opposite to the basal domain and disjoint from the lateral domains.

Cell chemotactic movement
Cells can inspect their local microenvironment, sense concentrations of external factors in their vicinity and move towards the highest concentration of diffusive factors (chemotaxis) or up a gradient of substrate-bound chemoattractants (haptotaxis). Cell directional movement in response to gradients of chemoattractants is modelled by introducing drag motility forces defined at all points on the cell boundary (that play the role of chemotactic receptors) pointing toward the highest concentration of the stimulus in cell microenvironment.


More detail on particular cell life processes, their numerical implementation within the immersed boundary framework, and computatinal simulatins can be found in:

• K.A. Rejniak, "An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development", Journal of Theoretical Biology, 2007, 247:186-204. available on: http://dx.doi.org/10.1016/j.jtbi.2007.02.019

• K.A. Rejniak, "Modelling the Development of Complex Tissues using Individual Viscoelastic Cells", chapter IV.3 of the book "Single-Cell-Based Models in Biology and Medicine" edited by A.R.A. Anderson, M.A.J. Chaplain, K.A.Rejniak, Birkhauser-Verlag, Mathematics and Biosciences in Interaction (MBI) series, 2007, ISBN 978-3-7643-8101-1.

• K.A. Rejniak, A.R.A. Anderson, "A computational study of the development of epithelial acini: I. Sufficient conditions for the formation of a hollow structure", Bulletin of Mathematical Biology, 2008, 70(3):677-712. available on: http://dx.doi.org/10.1007/s11538-007-9274-1