A new mathematical model for chemotactic bacterialcolony growth

E. Alpkvist, N. Chr. Overgaard, S. Gustafsson and A. Heyden

School of Technology and Society, Malm University, SE-20506 Malm, Sweden

Abstract A new continuum model for the growth of a single species biofilm is proposed. The geometry ofthe biofilm is described by the interface between the biomass and the surrounding liquid. Nutrient transportis given by the solution of a semi-linear Poisson equation. In this model we study the morphology of achemotactic bacterial colony, which grows in the direction of increasing nutrient concentration. Numericalsimulations using the level set method and finite difference schemes are presented. The results show richheterogeneous morphology.Keywords Biofilm; continuum model; diffusion; level set method; mass conservation; simulations

IntroductionBiofilm morphology has been a very active area of research ever since the rich geometricalstructures of biofilms were first observed around 1990 using confocal scanning lasermicroscopy. The interest in biofilm morphology has increased the demand for mathe-matical models to aid the understanding of experimental results and (ideally) to accuratelypredict the behaviour of biofilms.

Among the earliest models for biofilm growth is the one proposed by Wanner and Gujer(1986). Their model is essentially one-dimensional, so even though it can predict theamount of biomass correctly, it cannot account for the spatial irregularities in real biofilms.Since then several biofilm models, all of which are capable of producing spatially heteroge-neous solutions, have been put forth. Eberl et al. (2001) have proposed a continuum model,where both the nutrient and the biomass are described by scalar fields which satisfy a sys-tem of partial differential equations of the reaction-diffusion type. Picioreanu et al. (1999)use a model where the nutrient concentration satisfies a reaction-diffusion equation and thegrowth of the biofilm is modelled by cellular automata (see Kreft et al., 1998). It hasrecently been proposed by Dockery and Klapper (2001) that the biofilm should be modelledas a viscous fluid. The transport of the nutrients is still given by a reaction-diffusionequation.

The complex morphology of biofilms raises the following question: which rule governsthe development of such structures? Some candidates are: (a) the interaction with the sur-rounding liquid, (b) cell-to-cell signalling, and (c) the dynamics of transport and consump-tion of nutrients. It is most likely due to a combination of these mechanisms. To devise amodel for biofilm growth which includes all of these factors is probably very difficult. Tofully understand biofilms one should also take care to estimate the material properties of thebiomass and incorporate them into the model, see Klapper et al. (2001). The model pro-posed in this paper focuses entirely on mechanism (c).

The modelThe following model intends to describe an idealized single species bacterial colony whosepreferred growth, induced by chemotaxis, is in the direction of greater nutrient concentra-tions. This is the same assumption used in Ben Jacobs et al. (1994, page 41). The model can

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be formulated in any number of spatial dimensions, but our simulations are carried out inthe two-dimensional case. For accuracy, we therefore only present the two-dimensionalversion of our model.

We consider a system (liquid and biofilm) which is contained in an open rectangle:

= {x = (x1, x2): 0 < x1 < W, 0 < x2 < H}

For later use, we divide the boundary of into two parts = 12, where 1 = {x:x2 = 0} {x:x1 = 0} {x:x1 = W} consists of the bottom and the sides, and 2 = {x:x2 = H} is the topof . The biofilm is attached to the bottom of the rectangle, i.e. the set {x:x2 = 0} is the sub-stratum. We may imagine the sides of to be in contact with similar systems, or to consistof walls that are impermeable to the nutrients, so that there will be no net transport of nutri-ents through 1. We also assume that 2 is the interface to a large reservoir of nutrients,where the concentration of the nutrient is held at a fixed positive value.

Since the biofilm consists of a single species of bacteria, we assume that it may be ade-quately described by its density = (x, t), x , t 0. We furthermore assume that thedensity can only assume two distinct values: zero and 0 0. Thus, if Bt denotes the subsetof which is occupied by the biofilm at time t 0, then

(1)

This assumption is essential to our model, and it implies that the biofilm growth is com-pletely described by tracking the boundary Bt of the biofilm at any time t 0.

The transport of nutrients

The bacteria in our model are assumed to consume a single kind of nutrient, whose concen-tration (measured in mass per unit volume) is given by a non-negative scalar function c =c(x, t), x , t 0. The nutrients are transported through the system by diffusion, and areconsumed at a rate f = f(c, ) within the biofilm. Thus c must satisfy a reaction-diffusionequation of the form:

(2)

where D = D(c, ) is the diffusivity. In our calculations it is assumed that D = D0 is constantthroughout . The nutrient uptake is assumed to follow the so called Monod kinetics:

where k1, k2 > 0 are constants. Since the biofilm growth is much slower than the diffusionprocess, we may, at any time t 0, use the steady state solution of (2) to model the distribu-tion of the nutrient in the system. Thus, for a given biofilm density , we take c to be thesolution of the semi-linear Poisson equation:

(3)

The boundary condition c/n = 0 means that there is no (net) flux of nutrients through thebottom and the sides of the domain. The condition c = c0 corresponds to the assumption thatour system is in contact with a large reservoir of nutrients at 2.

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( , ) ,

\t x

Bt=

0

0for

for B .t

x

x

=( ) ( , ),D c ct

f c

f c k ck c

( , ) , =+

1

2

D c f cc

c c

=

=

( , ),

.

in ,/ = 0 on

on

1

2

n

0

Biofilm growth

The growth of the biofilm is assumed to induce a flow of biomass in the system. This volu-metric flow is modelled by a time-varying vector field u = u(x, t) defined in . The con-sumed nutrients are transformed into biomass through a function describing the substrateuptake rate q. We assume conservation of mass in our model, expressed by the equation ofcontinuity:

(4)

We also assume the flow to be in the direction of the gradient of the nutrients:

u = gc (5)

where g = g(x, t) is a scalar function. We express this assumption verbally by saying that thebiofilm grows in the direction of increasing nutrient concentration. The rate of uptake isassumed to satisfy the linear function

q(f) = f,

where 0 < < 1. An important point to our model is that it is possible to choose g in (5) suchthat the equation of continuity (4) holds. In fact, if we recall the assumption in (1), that 0 in Bt, then we conclude that /t = 0 inside Bt. Hence it follows from (3) and (4) that

0(gc) = f(c, 0) = D0c in Bt.

This equation is clearly satisfied if we take g = D0/0. This argument shows that we haveconservation of mass if we choose the flow field to be

(6)

MethodsTo compute the growth of the biofilm Bt in a small time increment t, we first compute cfrom (3), using the density given in (1) as input. Then we propagate the boundary Btalong the vector ut, with u given by (6), to obtain the points of Bt+t. How this is achievedin practice is the content of this section.

The level set method

To get a handy description of Bt we use the so called level set method introduced by Osher andSethian (1988). We give a brief outline of the theory which is adapted to the problem at hand.For a more complete introduction to the level set method the reader should consult one of therecent monographs on the subject, such as Sethian (1996), or Osher and Fedkiw (2003).

By a level set function for Bt we mean a continuous function = (x, t) such that

Bt = {x ; (x, t) < 0}.

In particular we have (x, t) = 0 for x Bt and t 0. The biofilm growth is now encoded inthe time evolution t (, t) of the level set function. Suppose we have a fictitious point x =x(t) which follows the boundary Bt. Then (x, (t), t) = 0 for all t, so if we differentiate withrespect to time we find that

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t q f c+ =( ) ( ( , )).u

u = D

c0

0.

In our model a point x Bt moves with the velocity u(x, t) = (D0/0) c (x, t). If we sub-stitute the velocity field u for the particle velocity x(t) in the equation above, then we getthe following partial differential equation, which is the level set equation for biofilmgrowth:

(7)

The initial condition for (7) is (,0) = 0 (), where 0 is determined by the initial biofilm B0.

Numerical methods

The rectangular computational domain is discretized with a Cartesian grid. The indicesfor the grid points are i = 1,. . .,N and j = 1,. . .,N for the x1 and x2 spatial dimension with agrid point distance of h. At time t = 0 we randomize a distribution of biomass at the gridpoints on the substratum. For each time tn with tn+1 = tn + k we find the nutrient concentra-tion by a second-order finite difference scheme combined with a Gauss-Newton method forthe nonlinearity in (3). We construct the initial condition for (7) using the signed Euclidiandistance from every point to the interface of a randomized distribution of biomass B0 on thesubstratum. Now we solve (7) by an explicit Euler scheme

The discretizations for the spatial derivatives of for a point xi,j are performed using a first-order forward and backward difference technique:

for the spatial dimension x2 the discretization is chosen in a similar manner. The ratiobetween k and h should be limited, this limit is known as the CFL number or the Courantnumber and is determined by the maximal flow of information. Choosing k to satisfy thiscondition is a necessary condition for stability of our numerical scheme. We need to ensure| | 1 to make smooth enough to approximate its spatial derivatives within somedegree of accuracy. For this reason it is always advisable to reinitialize the level set func-tion between the time steps. For further reading on this reinitialization procedure seeSussman et al. (1994).

Results and discussionFor all simulations a stopping criterion is implemented. This criterion is fulfilled when thedistance between the top boundary 2 and some point in Bt is less then a fifth of the height of. Without stopping criteria our model would eventually grow outside of its computationalbox.

We see from Figure 1 that the model generates typical biofilm structures such as chan-nels and mushroom-like fingerings. Although not seen in this simulation the model some-times tends to develop holes. Until this model is compared to experimental data onepossible confirmation is comparison with existing models. Picioreanu et al. (1999)

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0 = + ( ( ), ) ( ) ( ( ), ).x x xt t t t t t

t t

Dt c t( , ) ( , ) ( , )x x x+ =0

00

i j

ni jn

n n

kD

c, ,

.

+

= 1

0

0

grouped a number of parameters to a nondimensional quantity G = W2 f(c0)/(D0c0). With anincreasing G-number several other models show the morphological structure of the biofilmbecoming more heterogeneous. The influence of high and low G-number for our model wasinvestigated. Presented here are results varying the uptake rate k1. With an increasing k1 theconsumption of nutrient will increase and thus make nutrient availability in Bt drop. Somesimulations with varying k1 for the same initial state were performed on a [100 100] gridwith other parameters set equal to 1.

For our simulations an increasing G-number shows the same impact as other models onthe simulated biofilm morphology, as seen in Figure 2.

ConclusionsWe have presented a new continuum model for a single species chemotactic bacterialcolony growth. Simulations show that the proposed model admits spatially heterogeneoussolutions with fingerings. Given the rule for biomass growth in our model (growth is in thedirection of greater nutrient concentrations) such structures seem to be favourable for nutri-ent availability. Moreover, the predicted morphology becomes more heterogeneous forlower nutrient availability. This is in agreement with previous models.

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Figure 1 Simulation of an evolving biofilm. Level sets for biomass and iso-concentration contours of thenutrient are shown. Simulated at [170 170] grid points. All parameter values set to 1. From the top left tothe bottom right iteration number [25, 100, 200, 286]

Figure 2 Level sets for biomass and iso-concentration contours of the nutrient are shown. Simulations ofthe same initial state with varying parameter k1 with other parameters kept equal to 1. From left to right k1 =0.1, 1, 10

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