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

We implemented our individual-based model (IBM) for cell expansion in the software DEMeter++
[5]. DEMeter++ offers a generic platform for particle-based simulations, such as the discrete element
method and smoothed particle hydrodynamics. Cells are considered as particles that interact
by physical forces. At each time-step, the forces are integrated for each particle, its resulting
velocity is calculated and the cell positions are updated accordingly.
3.1.1 Equation of motion
Central to the individual-based model is the equation of motion. Since yeast lives in a low
Reynolds number environment, their inertia can be neglected. Therefore we obtain a first-order
equation of motion for each cell i that needs to be solved for the velocity, similar to [3]:
Fi ∑ j + Fi,budding + Fi,Brownian = Γiwvi + ∑ Γi j(vi −v j), (1)
jnni
jnni
with at the left hand side the mechanical forces between next neighboring cells jnni, the biological
force of cell division (budding) and a force representing Brownian motion. The viscous terms are
at the right hand side of this equation: the viscous drag force due to suspension in liquid, Γiwvi and
the cell-cell friction forces between contacting cells jnni.
We use the conjugate gradient method to obtain a coupled solution for the velocities. Next, a
forward Euler scheme updates the cell positions asxi(t + dt) =xi(t) +vi(t + dt) · dt.
3.1.2 Contact Mechanics
The crucial step in “kin-recognition” happens when two yeast cells come into close contact. There
are abundant models in the literature to capture the potential between two cells in close contact,
see e.g. [6, 7, 1, 3, 8]. The simplest would be a harmonic potential, i.e. a linear spring which
is at rest for a non-zero apparent overlap δ of the cells. Hertz showed that the repulsive forces
between solid, elastic bodies are actually non-linear. This repulsive potential can be modified to
take adhesion forces into account. Typically, it is assumed that the energy released by “binding” of
adhesion-molecules is constant, and that those molecules are relatively evenly distributed. Then,
the total energy released by the contact is proportional to the area of contact, thus yielding for the
modified Hertz-potential:
V (δ) = − 8δ 5 2
ˆR + σ · πδ ˆR. (2)
15Ê
We use the common definitions for
Ê =
1 − ν 2 i
Ei
+ 1 − ν2 j
E j
−1
and ˆR
1
= +
Ri
1
−1 R j
with Ei and E j being the Young’s moduli, νi and ν j the Poisson numbers and Ri and R j the radii
of cells i and j, respectively. In equation (2), σ is defined as “surface energy per unit area” and is
therefore the measure for cell adhesion. It can also be interpreted as average number of adhesionmolecules
per area multiplied with their respective adhesion-energy.
Interestingly, the adhesive part of the interaction-force due to this model for cell-cell contact does
not directly depend on the overlap anymore:
F(δ) = − 4
3Ê δ 3
2 ˆR + σπ ˆR (3)
The more complex JKR-potential (after [7]) yields for small apparent overlaps and the high
Young’s moduli of the yeast cells the same forces as the modified Hertz-model, which performs
slightly faster in the implementation and is therefore used.