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Ecological Modelling 222 (2011) 2552– 2560

Contents lists available at ScienceDirect

Ecological Modelling

journa l h o me pa g e: www.elsev ier .com/ locate /eco lmodel

ssessing biodegradation benefits from dispersal networks

homas Banitza,∗, Ingo Fetzerb, Karin Johsta, Lukas Y. Wickb, Hauke Harmsb, Karin Franka

Department of Ecological Modelling, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, GermanyDepartment of Environmental Microbiology, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, Germany

r t i c l e i n f o

rticle history:eceived 14 January 2010eceived in revised form 16 June 2010ccepted 14 July 2010vailable online 11 August 2010

eywords:acterial modelioremediationrganic pollutantsioavailabilityungal networks

a b s t r a c t

The performance of biodegradation of organic pollutants in soil often depends on abiotic conditions andthe bioavailability of these pollutants to degrading bacteria. In this context, bacterial dispersal is an essen-tial aspect. Recent studies on the potential promotion of bacterial dispersal by fungal hyphae raised theidea of specifically applying fungal networks to accelerate bacterial degradation processes in situ. Ourobjective is to investigate these processes and their performance via simulation modelling and addressthe following questions: (1) Under what abiotic conditions can dispersal networks significantly improvebacterial degradation? and (2) To what extent does the spatial configuration of the networks influencethe degradation performance? To answer these questions, we developed a spatially explicit bacterialcolony model, which is applied to controlled laboratory experiments with Pseudomonas putida G7 organ-isms as a case study. Using this model, we analyzed degradation performance in response to differentenvironmental scenarios and showed that conditions of limited bacterial dispersal also limit degradation

performance. Under such conditions, dispersal networks have the highest potential for improving thebioavailability of pollutants to bacteria. We also found that degradation performance significantly varieswith the spatial configuration of the dispersal networks applied and the time horizon over which perfor-mance is assessed. Regarding future practical applications, our results suggest that (1) fungal networksmay dramatically improve initially adverse conditions for biodegradation of pollutants in soil, and (2)

cture

the network’s spatial stru

. Introduction

Biodegradation in polluted soils is an important field of environ-ental management. Its performance, however, depends on both

he abiotic conditions and the spatiotemporal interactions of theollutants to be degraded and the degrading bacteria. In particular,he bioavailability of soil-bound pollutants is highly relevant foroth the possibility and the duration of biodegradation (cf. Harmsnd Wick, 2006; Semple et al., 2007). Fungal hyphae, with waterlled slime cover around them, can enhance the bioavailability byobilizing pollutant-degrading bacteria in soil, as it was demon-

trated by experimental studies (Kohlmeier et al., 2005; Wick et al.,007). These studies suggest that fungal hyphae can act as disper-al networks, increasing the performance of biodegradation in situ.his in turn is pivotal for developing more energy efficient and envi-onmentally friendly biotechnologies for remediation of pollutedoil sites. Such technologies require a comprehensive understand-

ng of the dynamics, functioning and services of soil–bacteria–fungicosystems. Particularly, when applying dispersal networks, thempact of both their spatial configuration and the abiotic conditions

∗ Corresponding author. Tel.: +49 341 2351708; fax: +49 341 2351473.E-mail address: thomas.banitz@ufz.de (T. Banitz).

304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2010.07.005

and accessibility are decisive for the success of such tasks.© 2010 Elsevier B.V. All rights reserved.

on biodegradation performance have been rarely studied. Thesefactors, however, may be essential for efficient biodegradation.

In the present study, we investigate these factors in a consumer-resource system consisting of bacteria as consumer and organicpollutants as resource. We study the impact of different disper-sal networks with various spatial configurations and under variousabiotic conditions on the system’s spatiotemporal dynamics, withparticular focus on the performance of bacterial degradation. Forthis purpose we use an integrative approach including simulationmodelling and an experimental case study.

In a first step, we develop a process-based, spatially explicitcomputer simulation model. The model differs from existingindividual-based bacterial models (e.g. Ginovart et al., 2002a; Kreftet al., 2001) and reaction-diffusion models (e.g. Kawasaki et al.,1997; Mimura et al., 2000), as it considers energy requirementsfor bacterial dispersal, and includes ecophysiological rules for allo-cating the bacterial energy uptake to the processes maintenance,dispersal and growth. Moreover, the spatiotemporal dynamics ofresources are explicitly taken into account, and dispersal networkscan be incorporated in terms of versatile spatial configurations.

Furthermore, we perform a case study by conducting well-defined laboratory experiments with bacterial colonies on agar andmonitor their growth in time and in dependence on both abioticconditions and dispersal network configurations. The qualitative

T. Banitz et al. / Ecological Modelling 222 (2011) 2552– 2560 2553

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ig. 1. Spatial patterns of bacterial colonies on agar plates. The grey shading indica3 h under 0.1 g/l initial glucose concentration and 3 g/l agar concentration. (a) Exlucose concentration and 5 g/l agar concentration, with a dispersal network of fou

nd quantitative comparisons of model output patterns to thesexperimental observations allow for determining necessary pro-esses and parameter values in the model.

In order to gain a mechanistic understanding of the system’spatiotemporal dynamics, we use the model to investigate scenar-os which are not experimentally feasible, due to their complexpatial configurations. Thus, we are able to make predictions ofiodegradation performance in response to dispersal network con-gurations and abiotic conditions. We show that bacterial dispersal,nd thus pollutant uptake and degradation, strongly depend onbiotic conditions, and that dispersal networks can efficientlymprove degradation performance, especially under otherwisenfavourable conditions.

. Materials and methods

.1. Case study – laboratory experiments

.1.1. Organism and culture conditionsPseudomonas putida PpG7 (NAH7) bacteria (Dunn and Gunsalus,

973) and glucose (Fluka, Switzerland) were used as model organ-sm and model substrate, respectively. Bacterial colonies wererown at a constant temperature of 30 ◦C in Petri dishes on minimaledium agar (MMA; Wick et al., 2001) under various abiotic condi-

ions in terms of agar and glucose concentrations. Both the softnessf the agar, which governed the dispersal of bacteria, and the initialoncentration of glucose, which was the sole energy source, were

aried (0.1 and 1 g/l glucose concentration; 3, 4 and 5 g/l agar con-entration). The applied conditions ranged from ‘swimming agar’below 3.5 g/l agar concentration) to ‘swarming agar’ (above 3.5 g/lgar concentration; cf. Harshey, 2003).

cterial concentrations, increasing from black (no bacteria) to white. (a and b) Afterental result. (b) Simulation model result. (c and d) After 66 h under 0.1 g/l initial

fibres. (c) Experimental result. (d) Simulation model result.

2.1.2. ObservationsEach agar plate was inoculated in the centre with approximately

6 × 107 cells pregrown on glucose and the colonies were observedfor 66 h by hourly image scanning (e.g. Fig. 1a) using a commercialflatbed scanner (HP Jetset 7400c). From these images, the total areaof the bacterial colonies was calculated with image analysis soft-ware ImageJ (Rasband, 1997). Thus, subsequent image sequencesprovided the change of colony areas as a function of time for eachcombination of the two varied abiotic parameters (Fig. 2). Theexperiments were replicated four times.

2.1.3. Dispersal networksIn order to simulate the effects of dispersal networks, we used

disposable polymer coated glass fibres based on Mayer et al. (2000)as easily handled substitutes for fungal networks. Glass fibresmimic the attributes of fungal hyphae, as thin films of water emergearound the glass fibres and provide dispersal corridors for the bac-teria. They were placed on MMA, and experiments with 0.1 g/linitial glucose concentration and 3, 4 and 5 g/l agar concentrationwere conducted (e.g. Fig. 1c).

2.2. Simulation model

2.2.1. OverviewThe purpose of our ecological model is to describe resource-

consumer-systems consisting of bacteria degrading organic

pollutants, in order to analyze their spatiotemporal dynamics andassess their degradation performance in response to various dis-persal networks and under various abiotic conditions. Therefore,the model spatially explicitly describes the dynamics of bacterial

2554 T. Banitz et al. / Ecological Modelling 222 (2011) 2552– 2560

Fig. 2. Total area of bacterial colonies plotted versus time. Columns show different initial glucose concentrations, increasing from left to right, rows show different agarconcentrations, increasing from top to bottom (see graph titles). Four experimental replicates for each scenario are plotted as grey crosses, simulation model outcomes areplotted as solid black curves (see legend). Note that at approximately 60 cm2 the size of the agar plates is reached.

cr

wtDeacsb

olony growth and substrate depletion, using the following set ofeaction-diffusion equations (see Table 1 for units):

∂Cx

∂t= ∇ (Dx(Cx, Cs)∇Cx) + (q(Cs)YG − a − d(Cx, Cs)) Cx, (1)

∂Cs

∂t= Ds∇2Cs − q(Cs)Cx, (2)

∂Cy

∂t= max (0, a − q(Cs)YG) Cx, (3)

here Cx is the concentration of active bacteria, Cy is the concen-ration of inactive bacteria, and Cs is the concentration of substrate.x is the bacterial diffusion coefficient, which depends on thenvironmental conditions (agar concentration, dispersal network

vailable), and varies with Cx and Cs. Ds is the constant diffusionoefficient of substrate. The bacterial reaction term includes theubstrate uptake rate q, the growth yield YG, and the losses ofiomass due to maintenance a and dispersal d.

According to this system, the explicit spatiotemporal dynam-ics of concentrations of bacteria and substrate are approximatedwith a finite difference method, inspired by the BacSim model(Kreft et al., 1998), on a 2-dimensional circular grid with reflec-tive boundaries, which represents an 88 mm diameter agar plate.From a process-based point of view, one simulation time step �tcomprises the following sequence of processes: substrate uptakeby bacteria, uptake allocation, bacterial dispersal, bacterial repro-duction and growth, and substrate diffusion (see Appendix A for adetailed description of these submodels). Fig. 1b shows an exampleof a bacterial colony simulated with the model.

2.2.2. Dispersal networksIn the simulation model, high diffusivity corridors were imple-

mented through a high bacterial diffusion coefficient in thecorresponding grid cells (cf. Appendix A.3; Table 2). The value ofthe bacterial diffusion coefficient along the dispersal networks wasdetermined by qualitative comparison of the simulation model

T. Banitz et al. / Ecological Modelling 222 (2011) 2552– 2560 2555

Table 1List of symbols.

Symbol Description Values Unitsa

Bacterial dispersal reduction factor –∂ Partial derivative symbol – –�t Time step 30–60 s� Gradient in space – mm−1

�2 Laplace operator in space – mm−2

�i,j,teff

Effective bacterial growth rate in cell (i,j) at time t h−1

a Maintenance biomass loss h−1

Ca Agar concentration g l−1

Cs Substrate concentration gs mm−2

Ci,j,ts Substrate concentration in cell (i,j) at time t gs mm−2

Cx Bacterial concentration (dry mass) gx mm−2

Ci,j,tx Bacterial concentration in cell (i,j) at time t gx mm−2

Ci,j,tx Weighted average of bacterial concentrations in nbhi,j gx mm−2

Cy Inactive bacterial concentration (dry mass) gy mm−2

Ci,j,ty Inactive bacterial concentration in cell (i,j) at time t gy mm−2

cl Grid cell side length 1 mmd Dispersal biomass loss h−1

di,j,t Dispersal consumption rate in cell (i,j) at time t gs gx−1 h−1

di,j,teff

Effective dispersal consumption rate in cell (i,j) at time t gs gx−1 h−1

Dx Bacterial diffusion coefficient mm2 h−1

Di,j,tx Bacterial diffusion coefficient in cell (i,j) at time t mm2 h−1

Di,jx,max Maximum bacterial diffusion coefficient in cell (i,j) mm2 h−1

H Harmonic mean – –i,k Spatial coordinates (index in x-direction) –j,l Spatial coordinates (index in y-direction) –nbhij 9 point neighbourhood of cell (i,j) –q Substrate uptake rate gs gx

−1 h−1

qi,j,t Substrate uptake rate in cell (i,j) at time t gs gx−1 h−1

qi,j,teff

Effective uptake rate for growth in cell (i,j) at time t gs gx−1 h−1

t Time h (k,l)

dry bi

oe

TM

wk,l Weight for diffusion between cell (i,j) and cell

a gs, grams of substrate; gx, grams of dry biomass of active bacteria; gy, grams of

utcome (e.g. Fig. 1d) to the observations from the laboratoryxperiments with glass fibres (e.g. Fig. 1c; cf. Section 2.1.3).

able 2odel parameters.

Parameter Symbol Values

Minimum dispersal fraction �min 0.5b

0.05c

Maximum effective growth rate �effmax 0.4142

Maintenance rate a 0.0005

Strategy thresholds c0 0.0191

c1 0.1083c2 0.5327

Dispersal reduction limite Cx,� 0.125 × 10−6

Maximum dispersal concentratione Cx,max 0.25 × 10−6

Maximum dispersal consumption rate dmax 0.0059

Substrate diffusion coefficient Ds 2.33

Maximum bacterial diffusion coefficient Dx,max 6.54f

2.41g

0.33h

144i

Monod half-saturation constant Ks 0.09

Maximum substrate uptake rate qmax 0.6908

Growth yield coefficient YG 0.6

Fraction of protein mass in dry biomass 0.5

Fraction of dry biomass in wet biomass 0.25

a gs, grams of substrate; gx, grams of dry biomass.b On swimming agar (agar concentration < 3.5 g/l).c On swarming agar (agar concentration > 3.5 g/l).d cf. Section 2.3.e Not on dispersal networks.f 3 g/l agar concentration.g 4 g/l agar concentration.h 5 g/l agar concentration.i On dispersal networks.

–

omass of inactive bacteria.

2.3. Model parameterization according to the case study

The maximum effective growth rate �effmax of P. putida PpG7 bac-

teria was approximated for growth on liquid minimal medium in

Unitsa Source

– Qualitative fit to experiments, (cf. Golding et al., 1998)

h−1 Own measurementd

gs gx−1 h−1 Parameter optimizationd

gs gx−1 h−1 Parameter optimizationd

gx mm−2 Qualitative fit to experiments, (cf. Golding et al., 1998)gx mm−2 Qualitative fit to experiments, (cf. Golding et al., 1998)gs gx

−1 h−1 Parameter optimizationd

mm2 h−1 Zhang and Fang (2005)mm2 h−1 Parameter optimizationd,f ,g ,h, qualitative fit to experimentsi

gs l−1 Approximated from literature (cf. Hardy et al., 1993)gs gx

−1 h−1 Calculated from Eq. (A.17)gx gs

−1 Approximated from literature (cf. Isken et al., 1999)––

2 odelling 222 (2011) 2552– 2560

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Fig. 3. Simulated consumption of glucose under 0.1 g/l initial glucose concentration,plotted versus time. No dispersal network – solid curves with squares, Network 1 –dotted curves with triangles, Network 2 – dashed curves with circles, Network 3 –dash-dotted curves with diamonds (see legend). The images on the right show thecorresponding dispersal network configurations (white grids) and simulated spatialcolony patterns after 100 h. The grey shading indicates bacterial concentrations,increasing from black (no bacteria) to white. (a) Agar concentration 3 g/l. (b) Agar

556 T. Banitz et al. / Ecological M

resence of 2 g/l glucose (Fluka, Switzerland) as sole energy sources described earlier by Wick et al. (2001). Moreover, the modelontains a number of parameters, which were approximated fromiterature (cf. Table 2). Following the approach of pattern-oriented

odelling (Grimm et al., 2005), the remaining parameter valuesere indirectly determined by optimizing the fit of the model

utput to a particular pattern observed from the laboratory experi-ents – the total area covered by the bacterial colony as a function

f time. For this purpose the Shuffled Complex Evolution Metropolislgorithm (Vrugt et al., 2003), a Markov Chain Monte Carlo method,as used. Fig. 2 shows the model output, using the resulting param-

ter values, in comparison to the experimental output from thease study. Table 2 gives an overview of the model parameters andalues applied.

.4. Simulating biodegradation performance in response toispersal networks

To analyze bacterial degradation performance as a function ofbiotic conditions, time and dispersal network configurations, weonitored the amount of glucose consumed by bacteria in the

imulation model. According to the experimental setup in the par-icular case study considered, we varied the abiotic conditions inerms of agar concentrations from 3 to 5 g/l. The initial glucoseoncentration was set to 0.1 g/l, at which results are representa-ive for a wide range of glucose concentrations (cf. Fig. 2). Wepplied dispersal networks with grid-like structures as shown inig. 3. By simulating the combined dynamics of bacteria and glucosever a total period of 100 h without and with different dispersaletworks, we analyzed, if and how such networks may changehe biodegradation performance. We varied the coverage of theetworks and the networks’ position relative to the bacteria inocu-

um to test, whether these factors change the simulation resultsignificantly.

. Simulation results

Fig. 3 shows the biodegradation performance in response to dis-ersal networks, measured in terms of glucose consumption duringacterial colony growth, as a function of abiotic conditions (differ-nt agar concentrations, compare graphs a, b and c), time (x-axis),nd dispersal network configurations (different line types in theraphs, see legend and colony images) over a total period of 100 h.

First, Fig. 3 highlights the impact of varying abiotic conditions,n terms of different agar concentrations. The results without dis-ersal networks (solid curves with squares, see legend) clearly

ndicate a decrease in biodegradation performance with an increas-ng concentration of agar, and so with an increasingly limitedacterial dispersal. This model outcome is robust to a wide range ofhanges in glucose concentration. While the addition of dispersaletworks generally leads to minor improvements under high dis-ersal conditions (low agar concentration, Fig. 3a), they are foundo significantly improve glucose consumption under low dispersalonditions (high agar concentration, Fig. 3c).

Second, the graphs in Fig. 3 demonstrate the importance of spec-fying the time horizon over which biodegradation performance isssessed. For instance, as depicted in Fig. 3a, there are significantifferences among the four simulated network configurations after0 h, but no differences occur after 100 h. Whereas in Fig. 3c, threef the four scenarios show the same performance after 40 h, butignificant differences after 100 h.

Third, the impact of different dispersal network configurationsdifferent line types, see legend) on glucose consumption, in com-arison to consumption without dispersal networks, is visualized.dding a grid-like dispersal network on half of the plate’s area (Net-

concentration 4 g/l. (c) Agar concentration 5 g/l.

work 1) lets the bacteria disperse faster on the plates and hence,they do degrade faster than without the dispersal network. Thiseffect gets stronger when the network covers almost the wholeplate (Network 2) or when the initial distance between the net-work’s branches and the bacteria inoculum is reduced (Network

3). The differences in degradation performance, resulting from thefour different dispersal network scenarios presented, increase withan increasing concentration of agar and vary over time.

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Taking everything into account, these results are a strong indi-ation that, in addition to the presence of dispersal networks, threeactors will affect biodegradation performance in reality: the abioticonditions, the time horizon over which performance is assessed,nd the structure and location of the dispersal networks.

. Discussion

In this study, we present a spatially explicit bacterial colonyodel, which simulates the spatiotemporal dynamics of pollutant-

egrading bacteria, growing on agar plates with and withoutarious dispersal networks and under various abiotic conditions.

number of studies have modelled bacterial colony growth ongar plates using reaction-diffusion equations (Golding et al., 1998;awasaki et al., 1997; Kessler and Levine, 1998; Kitsunezaki, 1997;imura et al., 2000), reaction-diffusion equations coupled with a

ydrodynamic equation (Lega and Passot, 2003, 2004), individual-ased modelling (Ben-Jacob et al., 1994; Ginovart et al., 2002b;reft et al., 1998; Krone et al., 2007), or individual-based mod-lling coupled with reaction-diffusion equations (Golding et al.,999). Selected aspects of these bacterial studies have been adaptedo our model. For instance, we used finite difference algorithmso approximate the solutions of a set of reaction-diffusion equa-ions for the concentrations of bacteria and organic substrate (seeppendices A.3 and A.5), and we used the Monod kinetic function

see Appendix A.1). However, our model differs from existing stud-es in two aspects. First, we implemented ecological concepts ofesource allocation (e.g. Brown et al., 2004; Ernest et al., 2003;ohst et al., 2008) as we explicitly consider division of consumernergy uptake into fractions for bacterial maintenance, dispersalnd growth (see Appendix A.2). Second, our model is first andoremost focused on the degradation of organic pollutants, and itserformance in response to dispersal networks and abiotic condi-ions. This ecosystem service of bacteria in soil is of high relevanceWhitman et al., 1998), as polluted soils are known to severelynfluence human food production, health, recreation or even repro-uction (Ehlers and Luthy, 2003; Harms and Bosma, 1997).

The modelling approach has major advantages for investigat-ng the bacterial colony dynamics: it permits to simulate a largeumber of scenarios with different conditions, it provides a mech-nistic understanding of the ecosystem service of interest – theacterial degradation of organic pollutants, and it allows for directlyeasuring the performance of this ecosystem service – in terms of

he simulated glucose consumption. This is unfeasible in laboratoryxperiments, as indicator compounds added to make the glucoseonsumption quantifiable bear the risk of modifying the bacterialehaviour.

We identified three important factors influencing the perfor-ance of bacterial degradation in response to dispersal networks:

he abiotic conditions, the time horizon, and the dispersal net-ork’s spatial configuration.

.1. The abiotic conditions

The agar concentration was identified as a key factor limitinghe biodegradation performance. The more fluid the medium is,he easier the flagellated bacteria can disperse, and the faster theollutants can be accessed and, hence, become bioavailable foregradation. This is in accordance with the general dependencef bacterial motility on moist conditions (Harshey, 2003; Schroll etl., 2006).

We have also shown that the abiotic conditions are essential forhe relative improvement of biodegradation performance, whichispersal networks can cause: The more difficult it is for the bac-eria to disperse, the more crucial the dispersal networks are for

ing 222 (2011) 2552– 2560 2557

reaching remote areas. This does not imply that the biodegrada-tion under unfavourable abiotic conditions will outperform thebiodegradation under favourable abiotic conditions (cf. Fig. 3),but that dispersal networks have the potential to compensate forthe detrimental effects unfavourable abiotic conditions have onbiodegradation.

4.2. The time horizon

The time horizon considered proved to be an essential aspect forassessing the ecosystem service’s performance. Our results revealthat contrasting findings can arise when not specifying this timehorizon (cf. Fig. 3). Hence, when assessing the performance ofbiodegradation in response to different abiotic conditions and dis-persal networks, one should always be aware of the time horizon’simpact.

4.3. The dispersal network’s spatial configuration

The results show that the dispersal network’s ability to accel-erate pollutant degradation is highly sensitive to its spatialconfiguration. Both the network coverage and the initial distancefrom the bacteria inoculum to the network appear to be importantcriteria for determining the potential improvement of biodegrada-tion performance.

We used an integrative approach combining the simulationmodel with laboratory experiments. In a case study, P. putidaG7 colonies were grown under varying well-defined abiotic con-ditions. Contrary to studies with other bacterial strains (Bacillussubtilis (Ohgiwari et al., 1992; Wakita et al., 1994), Pseudomonasaeruginosa (Köhler et al., 2000; Rashid and Kornberg, 2000),Mycobacterium gilvum (Fredslund et al., 2008)), in which multi-ple morphological growth patterns under various conditions couldbe identified, we observed a homogeneous circular colony growthover the whole range of abiotic conditions examined, when no dis-persal networks were applied.

Our model shows a sound accordance with real bacterialcolonies. It is able to reproduce the observed bacterial behaviourqualitatively, in terms of the colony patterns (e.g. Fig. 1a and b),and quantitatively, in terms of the increase of colony areas overtime (Fig. 2). We also included simple dispersal networks in thelaboratory experiments. The model qualitatively reproduces thecorresponding colony patterns too (e.g. Fig. 1c and d), in particularas it was designed to incorporate bacterial diffusion coefficients,which vary temporally and spatially in structured environments(see Appendix A.3).

5. Outlook – potential of fungal networks for acceleratingbiodegradation in soil

Bioremediation of soil-bound pollutants relies on the presenceof pollutant-degrading bacteria, optimal physical and chemicalconditions for their activity, and the bioavailability of pollutants.Particularly the latter is often achieved by homogenization of thepolluted matrix by mechanical treatment of soil. Our results, bycontrast, suggest that fungal networks have the potential for largelyimproving the bioavailability of pollutants to degrading bacte-ria in soil, without excavation and mechanical treatment. This isespecially valid for otherwise unfavourable abiotic conditions forbacterial degradation. We are convinced that in many cases pro-

moting the growth of fungal networks will be more accomplishableand energy efficient for bioremediation in situ, than modifying theabiotic conditions of the whole site. However, for assessing theprospective degradation benefits of fungal networks, both their

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558 T. Banitz et al. / Ecological M

patial structure and accessibility need to be considered, and theime horizon has to be taken into account.

cknowledgements

This study contributes to the research program ‘Terrestrialnvironment’ of the Helmholtz Association and was kindly sup-orted by the Helmholtz Impulse and Networking Fund throughhe Helmholtz Interdisciplinary Graduate School for Environmen-al Research (HIGRADE). The authors wish to thank Rita Remer andana Reichenbach for skilled technical help, and Daniela Inkrot,usann Pleger and Birgit Würz for performing bacterial dispersalssays. Thomas Banitz wishes to thank Cristian Picioreanu and theU Delft biofilm modelling group for facilitating a research visit androviding comprehensive support in model development.

ppendix A. Submodels

Note that the description and units of all symbols are given inables 1 and 2.

.1. Substrate uptake

In agreement with other bacterial models (e.g. Grijspeerdt etl., 2005; Kreft et al., 2001; Panikov, 1996; Picioreanu et al., 2007;chuler, 2005; Xavier and Foster, 2007) the Monod kinetic functionMonod, 1949) is used for calculating the substrate uptake in celli,j) at time t:

i,j,t = qmaxCi,j,t

s

Ks + Ci,j,ts

, (A.1)

here the maximum uptake rate qmax is calculated according to Eq.A.17) below. The corresponding change in substrate concentrations

Ci,j,t+1s − Ci,j,t

s

�t

∣∣∣∣react

= −qi,j,tCi,j,tx . (A.2)

.2. Uptake allocation

The substrate uptake qi,j,t is divided into fractions for main-enance, dispersal and growth. The fraction for maintenance a isonstant. The fraction for dispersal (dispersal consumption rate)s defined by an uptake allocation strategy in dependence on themount of substrate uptake:

˜ i,j,t = dmax

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0 qi,j,t ≤ c0

qi,j,t − c0

c1 − c0c0 < qi,j,t ≤ c1

1 c1 < qi,j,t ≤ c2

qi,j,t − qmax

c2 − qmaxotherwise

, (A.3)

ith: c0 ≤ c1 ≤ c2, c0 ≥ a, c1 ≥ a + dmax, c2 ≤ qmax.This strat-gy includes a linear increase of dispersal from 0 to dmax forubstrate uptake between c0 and c1, a constant maximum dispersaletween c1 and c2, and a linear decrease from dmax to 0 between c2nd qmax. If the substrate uptake is below the maintenance, the bac-eria will not be able to disperse. The artificial parameters c0, c1 and2 were optimized according to experimental data (cf. Section 2.3).s the bacterial dispersal depends on the bacterial concentration

oo (see Appendix A.3 below), the effective dispersal consumptionate di,j,t

effis given by:

˜ i,j,teff

= ˛di,j,t . (A.4)

ing 222 (2011) 2552– 2560

The fraction of uptake not used for maintenance and dispersalremains as effective uptake for bacterial growth:

qi,j,teff

= qi,j,t − a − di,j,teff

(A.5)

A.3. Bacterial dispersal

The bacterial dispersal depends on the bacterial concentrationin cell (i,j) at time t, which is incorporated by factor (cf. Eq. (A.4)):

= min

(1, �min + (1 − �min)

Ci,j,tx

Cx,�,

Cx,max

Ci,j,tx

). (A.6)

This factor can reduce the bacterial dispersal in two ways. First,dispersal is reduced at low bacterial concentrations (below Cx,�). Onswimming agar (agar concentration below 3.5 g/l) bacteria secretematerials which make the liquid more suitable for swimming(Golding et al., 1998). Hence, a higher concentration of bacteriafacilitates the dispersal. On swarming agar (agar concentrationabove 3.5 g/l) the dependence on concentration is even stronger,as in order to disperse, the bacteria need to produce a lubricationfluid (Cohen et al., 1999; Golding et al., 1998), often also referredto as wetting agents (Bees et al., 2000; Harshey, 2003; Matsuyamaand Nakagawa, 1996). That is why the minimum bacterial dispersalfraction �min is set to almost zero for swarming agar (see Table 2for parameter values).

Second, the maximum bacterial dispersal concentration Cx,max

accounts for the decreasing bacterial velocity at high concentra-tions in a colony due to an increasing probability of collisionsbetween bacteria (Golding et al., 1998). If the concentration of bac-teria in a grid cell Ci,j,t

x exceeds this threshold, only the fraction ofbacteria below this threshold can disperse.

As concentration dependent reduction of dispersal along dis-persal networks does not occur, is a constant and equals 1 in thiscase.

The dispersal of bacteria is modelled as diffusion. For this pur-pose the finite difference approximation from the BacSim model(Kreft et al., 1998) is adapted to our model. The original diffusionalgorithm is

Ci,j,t+1x − Ci,j,t

x

�t

∣∣∣∣diff

= Dx

cl2Ci,j,t

x , (A.7)

where Ci,j,tx is the weighted average of bacterial concentrations Ck,l,t

x

in the 9 point neighbourhood of a regarded grid cell (i,j):

nbhi,j ={

(k, l) : (i − 1 ≤ k ≤ i + 1) ∧ (j − 1 ≤ l ≤ j + 1)}

, (A.8)

and cl is the side length of one grid cell. As in the BacSim model, theweights wk,l according to the following stencil:

∀(k, l) ∈ nbhi,j : wk,l =

⎧⎪⎨⎪⎩

120

(k /= i) ∧ (l /= j)

−1 (k, l) = (i, j)15

otherwise

. (A.9)

Thus, Eq. (A.7) reads:

Ci,j,t+1x − Ci,j,t

x

�t

∣∣∣∣diff

= Dx

cl2

∑(k, l) ∈ nbhi,j

(k, l) /= (i, j)

wk,l

(Ck,l,t

x − Ci,j,tx

). (A.10)

In our system, the bacterial dispersal is related to the substrate

uptake–the more uptake is available for dispersal (see AppendixA.2) the wider bacteria can diffuse. This ecology based conceptrequires diffusion coefficients varying in time and space, whichdepend on the effective dispersal consumption rate di,j,t

eff(cf. Eq.

odell

(r

D

TicwDcsi

H

Too(

Ishn

A

bTib

�

wtiab

w0s

yt

Y

a

q

T. Banitz et al. / Ecological M

A.4)), related to the potential maximum dispersal consumptionate dmax:

i,j,tx =

di,j,teff

dmaxDi,j

x,max. (A.11)

he maximum diffusion coefficient Di,jx,max depends on location as

t is either the standard diffusion coefficient (depending on agaroncentration Ca) or the high diffusion coefficient on dispersal net-orks. Adapting to spatiotemporally varying diffusion coefficientsi,j,tx , the approximation algorithm for the changes in bacterial con-entration needs to be modified. With regard to the finite differencecheme, the effective diffusivity between two neighbour grid cellss given by the harmonic mean of their diffusion coefficients:

(Di,j,tx , Dk,l,t

x ) = 2Di,j,tx Dk,l,t

x

Di,j,tx + Dk,l,t

x

. (A.12)

he harmonic mean is used, because it provides the average velocityver a distance, when half the distance is travelled with one and thether half with another velocity. Inserting these diffusivities (Eq.A.12)) into Eq. (A.10) results in

Ci,j,t+1x −Ci,j,t

x

�t

∣∣∣∣diff

= 1cl2

∑(k,l) ∈ nbhi,j

(k, l) /= (i, j)

wk,l

(Ck,l,t

x −Ci,j,tx

)H(Di,j,t

x , Dk,l,tx ).

(A.13)

n addition to varying diffusion coefficients, which depend on sub-trate uptake, this algorithm allows for the incorporation of veryigh diffusion coefficients for grid cells, which belong to dispersaletworks.

.4. Bacterial reproduction and growth

The processes of reproduction and growth are not distinguishedut modelled simultaneously as the growth of bacterial biomass.his growth is related to the substrate uptake too – the more uptakes available for growth (see Appendices A.1 and A.2) the more theacteria can grow. The effective bacterial growth rate is given by

i,j,teff

= �effmax

qi,j,teff

qmax − a. (A.14)

here �effmax is the maximum effective growth rate, and qmax − a is

he potential maximum effective uptake rate, calculated by assum-ng maximum uptake, which is exclusively used for maintenancend growth. Hence, the change in biomass through growth is giveny

Ci,j,t+1x − Ci,j,t

x

�t

∣∣∣∣react

= �i,j,teff

Ci,j,tx , (A.15)

hich can be negative, if the maintenance rate is not met (i.e. qi,j,teff

<

). In this case bacterial cells become inactive – they do not consumeubstrate, disperse, or grow any more but do remain in the system.

To determine the maximum uptake rate qmax we use the growthield coefficient corrected for maintenance, which only considershe substrate consumption for growth (cf. van Bodegom, 2007):

G =�i,j,t

eff

qi,j,teff

(A.16)

nd, by inserting Eq. (A.14), get:

max = �effmax

YG+ a. (A.17)

ing 222 (2011) 2552– 2560 2559

A.5. Substrate diffusion

The substrate diffusion is modelled with the same approxima-tion algorithm as the bacterial dispersal above (cf. Appendix A.3),but with a spatially and temporally invariant diffusion coefficient.Hence, the finite difference equation is (cf. Eqs. (A.7) and (A.10)):

Ci,j,t+1s − Ci,j,t

s

�t

∣∣∣∣diff

= Ds

cl2

∑(k,l) ∈ nbhi,j

wk,lCk,l,ts . (A.18)

A.6. Summary

Summing up all processes, we can write the following discreteequations which correspond to the reaction-diffusion model givenby Eqs. (1)–(3) (cf. Section 2.2.1):

Ci,j,t+1x − Ci,j,t

x

�t

∣∣∣∣diff +react

= 1cl2

∑(k,l) ∈ nbhi,j

(k, l) /= (i, j)

wk,l

(Ck,l,t

x − Ci,j,tx

)

× H(Di,j,tx , Dk,l,t

x ) + qi,j,teff

YGCi,j,tx , (A.19)

Ci,j,t+1s − Ci,j,t

s

�t

∣∣∣∣diff +react

= Ds

cl2

∑(k,l) ∈ nbhi,j

wk,lCk,l,ts − qi,j,tCi,j,t

x (A.20)

Ci,j,t+1y − Ci,j,t

y

�t

∣∣∣∣∣react

= max(

0, −qi,j,teff

)YGCi,j,t

x . (A.21)

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ing 222 (2011) 2552– 2560

Thomas Banitz studied Mathematics in Germany andLithuania and received the diploma degree in 2004 andthe M.S. degree in 2007, both from the Leipzig Universityof Applied Sciences, Germany. He joined the Departmentof Ecological Modelling at the Helmholtz Centre for Envi-ronmental Research – UFZ in 2006 and started his PhDwork on modelling the dynamics of microbial degradationprocesses in 2007.

Ingo Fetzer, ecologist, works as post-doctoral researcherat the Department of Environmental Microbiology atthe Helmholtz Centre for Environmental Research – UFZsince 2005. He studied at the Universities of Tromsø,Longyearbyen (Norway), MGU Moscow (Russia) and Bre-men (Germany). His expertise is related to microbialinteractions (competition, mutualism, facilitation) andpopulation dynamics, assessment of tropical interactionnetworks, application of microbial systems for evaluationof evolutionary and common (macro-)ecological concepts.His past research covered the importance of biodiversity,structure and function for resilience of microbial com-munities, and the development of tools and models for

detection of community shifts.

Karin Johst studied Physics at the Technical Universityof Dresden, Germany. She received her PhD in theoreticalchemical physics and worked in this research field at theInstitute of Isotope and Radiation Research, Leipzig. Since1992 she has been working at the Department of Ecologi-cal Modelling at the Helmholtz Centre for EnvironmentalResearch – UFZ. Her areas of research include ecologicaland ecological–economic modelling with focus on popula-tion and community dynamics in dynamic and disturbedlandscapes and adaptation strategies to climate change.In these fields, she published more than 50 peer-reviewedresearch articles, works in international research projects,and supervises PhD students.

Lukas Y. Wick, environmental chemist/microbiologist,is group leader at the Department of EnvironmentalMicrobiology at the Helmholtz Centre for EnvironmentalResearch – UFZ. After his PhD in chemistry (University ofBasel) and post-doctoral stays at the Massachusetts Insti-tute of Technology (MIT), the Swiss Federal Institute forEnvironmental Science and Technology (EAWAG), and theSwiss Federal Institute of Technology Lausanne (EPFL) hejoined the UFZ in 2004. He has a track record of over 55internationally peer-reviewed research articles and bookchapters. His expertise is related to microbial dispersalin the subsurface and the ecology and prediction of thebiodegradation of subsurface contaminants.

Hauke Harms is head of the Department of EnvironmentalMicrobiology at the Helmholtz Centre for EnvironmentalResearch – UFZ and professor at the University of Leipzig.After his PhD in microbiology (Hamburg University) hecontinued as a fellow of the European EnvironmentalResearch Organisation at Wageningen University, TheNetherlands, and as research group leader at the SwissFederal Institute for Environmental Science and Technol-ogy (EAWAG). In 1998 he became Assistant Professor atthe Swiss Federal Institute of Technology Lausanne (EPFL)before he moved to Leipzig in 2004. He is co-author of 160refereed journal articles and book chapters in the fields ofmicrobial ecology and biotechnology.

Karin Frank is head of the Department of Ecological Mod-elling at the Helmholtz Centre for Environmental Research– UFZ and professor at the University of Osnabrück. Shehas a PhD in Mathematics (University of Leipzig) and ahabilitation in Applied Systems Sciences (University ofOsnabrück). Since 1992, she is working in the field of