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

on the actual necessary time needed to neutralize harmful bacteria.
3. THE HYPOTHESES OF THE MATHEMATICAL MODEL
In this section, we summarize the mathematical model that is introduced and discussed
in more detail in [1]. We consider a hypothetical cell that is spherical in its nonactivated,
equilibrium state. This cell is allowed to deform and to move in an extracellular
environment. This environment could be a gel-like medium. Further, we
consider bacterial sources that are allowed to move through the domain or to stay at a
fixed position. Each bacterium secrets a chemical into the medium, which diffuses
through the extra-cellular environment and cell, and subsequently attracts the cell. The
bacteria are modeled as point sources and the concentrations are assumed to be small.
This justifies the use of linear diffusion from point sources, hereby enabling the use of a
superposition of Green's Functions for the description of the concentration of the
chemo-attractant. Hence, no discretization methods like the finite-element method are
used in our model.
The cells sense the concentration of the bacterially released chemical at the discrete
points on the cell boundary if the concentration exceeds a certain threshold value. If the
concentration is above this threshold value, then the point at which the chemical is
sensed moves towards the gradient of the concentration, at a speed proportional to the
local concentration gradient of secreted chemicals, with mobility constant β. The
mobility of the points on the cell surface makes the cell deform and move towards the
bacterium that releases the chemical.
Next to the ability of the cell to deform and move, the cell also exerts and experiences a
driving force to deform back to its equilibrium shape. This driving force is modeled via
hypothetical springs that are linked between all points and the cell centre (i.e. the
nucleus, which is modeled as a point here), and between neighboring points on the cell
surface, see Fig. 1 for a two-dimensional representation. Hence upon vanishing of all
bacterial sources, the cell shape transforms back to the original spherical geometry
where all points on the cell surface have the same mutual distance as initially, in which
the cell (and hence its centre) possibly has migrated as a result of the bacterial sources
that were