Mathematical modelling of biolm growth based
on a cubic spline representation
Erik Alpkvist, Niels Chr. Overgaard
Stefan Gustafsson, Anders Heyden
Applied Mathematics Group
School of Technology and Society
Planctonic bacteria that attach to a surface form a colony, referred to as a biolm. Biolms tend to build heterogeneous structures whose geometry is very complex. This rises the following question: which rule governs the development of such structures. Some likely candidates are: (a) The interaction of surround-ing liquid. (b) Cell-to-cell signalsurround-ing (also known as quorum senssurround-ing). (c) The dynamics of transport and consumption of nutrients.
The model presented here focuses on the aspect (c); the biolm is considered as a continuum whose growth is governed by the concentration of the nutrient. The assumption made in this model is that new biomass is distributed in the direction of the nutrient concentration gradient and thus increasing the whole colony's nutrient uptake in an eective way.
To be more precise, the biolm and the liquid are supposed to occupy a xed connected domain Ω in R2 (the plane). The boundary of Ω is divided into two
parts ∂Ω = Γ1
Γ2. A part of Γ1 contains the surface (substratum). Γ2 is the
contact interface to a large store of nutrients.
The part of Ω which is occupied by the biolm, at the time t, is denoted
Bt, and we assume that the biolm density is uniform (in fact=1) throughout
Bt. The surrounding liquid Ω\Bt is assumed to be in hydrostatic equilibrium,
so that the role of the liquid is just to provide a medium through which the nutrients are transported from the resovoir at Γ2 to the biolm.
Let c = c(x, t) be the concentration at x ∈ Ω and time t ≥ 0. For the trans-port and consumption of the nutrients we use a semi-linear diusion equation.
D∆c − ct= f (c, Bt) in Ω ∂c ∂n = 0 on Γ1 c = g on Γ2 (1)
Where D > 0 is the diusivity, which is assumed to be the same in all of Ω, and g = g(x) ≥ 0 is the (xed) concentration for x ∈ Γ2. The function f, which
models the nutrient uptake by the biolm, follows the so called Monod kinetics:
f (c, Bt) =
k2+c in Ω
0 otherwise. (2) Now, the biolm growth is much slower than the diusion of nutrients, so we may, at each time t, use the steady state solution of (1) to model the distribution of the nutrients: That is, for any xed time t, c is taken to be the solution of the semi-linear elliptic PDE:
D∆c = f (c, Bt)
+ Boundary cond. from (1) (3) The interface ∂Bt(the boundary of Btrelative to Ω) is represented by cubix
splines. Let pi(t), i = 1, · · · , N be the control points for the spline at the time t.
The growth of the biolm, in a small time increment ∆t, is achieved by pushing the control points along the graident of the stationary solution of 3:
pi(t + ∆t) = λ∇c(pi(t))∆t + pt(t), i = 1, · · · , N. λ = (RB tf (c) dx)/( R ∂Bt∇c · n ds) (4)
We denote the components of Bt by Λi and thus we have Bt=
i=1Λi for n
such disconnected domains in Bt. The total increase of biomass in a domain
Λiper unit of time will be given by
Λif (c) dx = λi
∂Λi∇c · n dsand thus our
choice of λi for a each Λi is given.
At time t = 0 the domain B0 is chosen to be a nite union of half circles
placed randomly with the center on the substratum.
The model simulations show typical biolm shapes with channels and towers resembling mushrooms. Considering the spreading algorithm (4) our conclusion is that these structures are favourable for biomass nutrient availability in the colony.
Figure 1: Simulations in a quadratic domain with substratum at the bottom. Top left corner: The initial domain. Top right corner: 25 iterations. Bottom: 50 iterations.