ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

4. MODEL
When cells adhere on a substrate, they spread, contract and generate actin stress fibres
between their adhesion sites 3 . If the surface of the substrate is convex or flat, the cell
follows the geometry, even while contracting. On a concave surface however, the actin
fibres joining the extremities of the cell contract and pull the membrane away from the
scaffolds. A cell can therefore be represented by a chord joining two points of the
surface. Locally, the resulting interfacial motion δ is the distance between the substrate
and cell membrane (Fig.1A) and can be derived as a function of local geometry via
simple geometrical considerations.
Fig.1: The chord representation of a cell justifies the curvature-driven behaviour of
tissue growth. A simple geometrical construction can explain the evolution of the
interface observed during in vitro tissue culture.
A chord representing a cell of size l is attached to a surface with a local radius of
1
curvature R = . In this case:
κ
2
2 ⎛ l ⎞
R = + R
(Eq.3)
( ) 2
−δ
⎜ ⎟
⎝ 2 ⎠
and therefore:
⎛
2 ⎞
( ) ⎜ ⎛ l ⎞
δ l,
R = R 1−
1−
⎟
⎜ ⎜
⎟ (Eq.4)
2 ⎟
⎝ ⎝ 4R
⎠ ⎠
When R >> l , a series expansion at the first order of δ ( l,
R)
when R → ∞ gives:
2
l
δ ( l , κ ) = κ
(Eq.5)
8
Thereby, local interfacial motion resulting from the deposition of a single cell is shown
to be proportional to local curvature, and the equivalence with the curvature-driven
growth description proposed by Rumpler etal 9 and implemented in this work is verified
3
when r = l . Such a model for cell deposition can be extended to tissue growth.
2
Chords assembled along a curved interface represent the deposition of a collection of
cells and define a new surface where other cells will sit on.
By demonstrating the quantitative equivalence between curvature-controlled tissue
growth and the interfacial motion due to tensile chords laid down on a curved surface
(Fig.1B), we link the cellular response to geometry with the macroscopic pattern of