Mathematical analysis of the reaction diffusion processes occuring within a biofilm: submitted as a 3 paper thesis for the degree of Masters of Philosophy in biotechnology at Massey University in New Zealand

Abstract

This thesis investigates the behaviour of the solutions to mass transfer type reaction-diffusion equations within a biological film around a spherical particle, around a rotating cylinder and on a slab. In general these biological films may be found in any biological system and as examples of systems with spherical, cylindrical and slab geometry we examine biofilms in a fluidised bed biofilm reactor, biological growth around a rotating cylinder and on a rotating biological disk contactor.

These equations model concentrations of substrates within a single biofilm as a function of position and time.

In dimensionless coordinates , these equations have the form

= _1_ l (xa-1 QY) - 4/ F(y) a-1 ax ax

(P)

X

for a = 1,2,3 being geometries of a slab, cylinder and sphere respectively and \w-• ti hai-a

F(y) may correspond to F0,F1 Fn or Fmm and where '

FO = 1 corresponds to zero order kinetics F1 = y corresponds to first order kinetics Fn =yn corresponds to nth order kinetics where n is an integer and

F Y corresponds to Michaelis-Menten reaction kinetics mm 1 + y

The dependent variable y corresponds to concentration and the independent variables x and t correspond to distance and time respectively.

The model parameters are saturation parameter , which describes the concentration and Thiele modulus <1>, which is the ratio of reaction to diffusion coefficients. The boundary conditions are

(a,t)=O and y(1,t)=1 for all time >0

Here a is the internal boundary, a parameter that corresponds in spherical geometries to the ratio of the radius of a support media to the total radius of the bioparticle, in cylindrical geometries to the radius of a cylinder without biofilm to the total radius of the cylinder with biofilm and in slab geometry to be the ratio of the inactive region of diffusion to the thickness of biofilm measured from the centre of the slab. This may vary but for our purposes we shall take it to be a constant.

The major part of this thesis is concerned with the solution to the steady state associated with problem (P).

Using the maximum principle and, methods of upper and lower solutions and standard topologocal results from non-linear analysis, existence , uniqueness and monotonicity results are obtained.

In particular it is shown that the steady state problem has a unique solution

for all values

,.,n,.,f C "\ ,-,.f -1-h'"' .f,-,,.,..,..,

1 above in the geometries slab, cylinder and sphere.

Liiv ,v1111

It is also shown that if F(0) = 0, F(y) 0, the unique solution of the steady state problem associated with (P) is strictly greater than zero. This is indeed true for nth order and Michaelis-Menten kinetics. For the zero order case F(0) * 0 implies that our solution to the steady state problem associated with (P) could become negative. Having a negative concentration is not a physical reality and we impose a third boundary condition that redefines a in terms of Thiele modulus, <j>.

We also show that our solution to the steady state problem associated with problem (P) is monotonically decreasing in <P and monotonically increasing in f3 for all geometries and all orders. This may be generalised to all time for the unsteady state case.

The final part of this thesis considers the equation of the existence of solutions to problem (P) which are defined for all time . Such existence results are derived and approximate analytical solutions are obtained for some special cases ; but for general regions , only partial answers are obtained.