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Ecological Modelling 247 (2012) 1– 10

Contents lists available at SciVerse ScienceDirect

Ecological Modelling

jo ur n al homep ag e: www.elsev ier .com/ locate /eco lmodel

A combination of cellular automata and agent-based models for simulating theroot surface colonization by bacteria

Adrialy L. Mucia, Milko A. Jorquerab,∗, Ándres I. Ávilac, Zed Rengeld, David E. Crowleye,María de la Luz Morab

a Magister en Ciencias de la Ingeniería, Mención Modelación Matemática, Universidad de La Frontera, Temuco, Chileb Center of Plant, Soil Interaction and Natural Resources Biotechnology, Scientific and Technological Bioresource Nucleus, Universidad de La Frontera, Avenida Francisco Salazar 01145,Temuco, Chilec Departamento de Ingeniería Matemática, Centro de Modelación y Computación Científica, Universidad de La Frontera, Temuco, Chiled School of Earth and Environment, University of Western Australia, Australiae Department of Environmental Sciences, University of California, Riverside, USA

a r t i c l e i n f o

Article history:Received 21 November 2011Received in revised form 26 July 2012Accepted 30 July 2012

Keywords:AgricultureBiofertilizerPhosphate solubilizationPlant–bacteria interactionsPlant growth-promoting rhizobacteriaRhizoplane

a b s t r a c t

Models of root colonization by bacteria facilitate conceptual and practical understanding of fundamentalprocesses that influence plant health, mineral nutrition, and stress tolerance. In this study, we exploredthe use of cellular automata and agent-based models to simulate the primary colonization of roots bybacteria as determined by selected parameters related to bacterial growth (nutrient uptake, fitness, repro-duction and starvation) and environmental constraints (space, nutrient depletion, and pH). The resultswere then compared with observations from experiments examining the colonization of plant roots by aGFP – strain of Escherichia coli. The latter experiments were conducted with plants grown in agar mediumcontaining calcium phosphate that allowed visualization of bacterial distribution (aggregates and abun-dance) and phosphate solubilization at root microsites. The numerical models revealed outcomes fordiverse numerical scenarios, which agreed with the in vivo data and provided a basic framework fordescribing bacterial colonization of plant roots. Further efforts will be required to evaluate factors affect-ing the competence and ecology of bacterial communities at rhizosphere microsites, but offer promisefor the development of precise predictive models with practical applications for agriculture.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Microorganisms that colonize plant roots have a profound influ-ence on plant fitness, facilitating both nutrient cycling and directstimulation of root growth through the release of plant hormonesinto the rhizosphere. Current models of the process by whichmicrobial communities become established on plant roots describea successional development of different communities along theroot axes that are driven by spatial and temporal variations inthe quantities and composition of organic substances that aredeposited into the rhizosphere (Lugtenberg et al., 2001; Marschneret al., 2011). Initial colonization of newly produced root tissues thatare largely sterile begins with growth of microorganisms that aredeposited along with root cap cells and polysaccharides into thesoil macropores. Subsequently, as new root tissue undergoes celldivision and expansion in the zone of elongation, relatively largeamounts of labile carbon are released into the rhizosphere behind

∗ Corresponding author. Tel.: +56 45 325467; fax: +56 45 325053.E-mail address: mjorquera@ufro.cl (M.A. Jorquera).

the root apices. This in turn stimulates the proliferation of bacteriathat were deposited by the root cap along with other opportunis-tic bacteria that inhabit the macropores or that move to the rootvia chemotaxis. These primary colonizers cause further chemicalchanges in the rhizosphere by secretion of organic acid anions, pro-tons and CO2 that alter the niche selectivity of the rhizosphere andthat function to prime organic matter degradation and mobilizationof iron and phosphorus (Marschner et al., 2011). Finally as the rootcontinues to push forward through the soil pores, the roots matureand slough off the cortex tissues, which consist of more recalci-trant materials including cellulose, pectins, and phospholipids. Asthese substances become depleted, microbial communities alongthe older root parts undergo a final succession as the original popu-lations turnover, leaving species that are generally adapted to acrowded, oligotrophic environment in which remaining cells ofprimary colonizers undergo extreme starvation (Marschner andCrowley, 1996). Each of these distinct zones provides unique set ofconditions that selectively affect the growth of different microor-ganisms and distinct microbial community structures (Yang andCrowley, 2000). Other factors that shape the final composition ofthe rhizosphere microbial community include the soil texture, pH,

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2 A.L. Muci et al. / Ecological Modelling 247 (2012) 1– 10

and organic matter, and differences in exudate composition relatedto differences among plant species, variations in plant nutritionalstatus, and seasonal effects related to plant developmental stage.This results in a complex system in which it is difficult to sepa-rate out the effects of different variables and their interactions onmicrobial species distribution in the rhizosphere, but which canbe modeled by tracking the growth and fate of individual bacterialstrains using molecular markers or reporter gene constructs.

Along with tracking individual strains, computer models offeranother powerful tool for simulating the rhizosphere colonizationprocess and could be helpful in predicting the effects of envi-ronmental variables on plant–microbial interactions (see review:Luster et al., 2009). To date, various numerical models based ondifferential equations have been employed to simulate plant–soilinteractions, water and nutrient uptake by roots, carbon flow inthe rhizosphere, and survival of plant growth-promoting rhizobac-teria (PGPR) in the rhizosphere (Toal et al., 2000; Roose and Fowler,2004; Strigul and Kravchenko, 2006; Roose and Schnepf, 2008).These include two-dimensional and three-dimensional models thatdescribe diffusional transport and bacterial movement on the rhi-zoplane (Scott et al., 1995) as well as the interactions betweenroot systems and water and nutrient (nitrate) transport and uptakein the rhizosphere (Dunbabin et al., 2002). With the develop-ment of low-cost, high-power computing, another novel approachthat merits investigation is the use of cellular automata that candescribe simple iterative patterns such as bacterial reproduction.Examples from the literature include descriptions of processessuch as the spread of influenza, tumor growth, and tissue for-mation (Beauchemin et al., 2005; Mallet and De Pillis, 2006;Rozante et al., 2007). Another complementary approach is the useof agent-based models, which are now commonly in social science,economy and finance (LeBaron, 2000; Tesfatsion, 2003; Gilbert,2008) for simulating the actions and interactions of autonomousagents (either individually or collectively) and their effects on thewhole systems. Agent-based modeling approach has already beenused in some areas of ecology and biology (Mansury et al., 2002;Bousquet and LePage, 2004; Tang et al., 2007); for example, to studypatterns that form during the development of Anthrax colonies(Krawczyk et al., 2003). Regarding the plant rhizosphere, com-bining cellular automata and agent-based model approaches maybe particularly useful for examining spatial patterns of bacterialdistributions on the root surfaces in relation to the factors that con-trol microbial growth rates. This combination of the two modelingapproaches does not appear to have been explored yet for studyingplant–microbe interactions in the rhizosphere.

Many bacterial species in the rhizosphere have been studiedfor their contribution to plant health by (i) controlling phy-topathogens and pests, (ii) producing plant growth regulators thatstimulate root growth, thereby enhancing water and nutrient-use efficiency, and (iii) increasing nutrient availability throughthe production of enzymes and organic acid anions that increasenitrogen, phosphorus, and trace metal bioavailability to plants(Martínez-Viveros et al., 2010). In this research, our objective wasto develop the mathematical foundation for a model combining thecellular automata and agent-based model approaches to simulatethe distribution of bacteria on the root surface as determined byselected bacterial and environmental parameters. To validate themodel under simplified conditions, we carried out in vitro stud-ies to validate the root colonization patterns by a single agent, inthis case a strain of Escherichia coli that expressed green fluores-cence protein, enabling visualization of the colonization processby epifluorescence microscopy. For this purpose, we used an agarmedium containing water-insoluble calcium phosphate that couldbe dissolved by acidification and formation of organic acid anioncomplexes. Visualization of phosphate solubilization is therebyobserved as the formation of clearing zones in the agar medium.

We then explore the correspondence between visual observationsand the results obtained by numerical simulation modeling underdifferent scenarios.

2. Model development

2.1. Rhizoplane model (R)

The cellular automata was a triplet where R = (G, SV, T), G is a two-dimensional cylindrical grid of square cells, SV is the set of statusvariables, and T was the summary function for a set of transitionrules.

2.1.1. Automata grid (G)The rhizoplane was modeled as an elongating cylinder following

the typical behavior of an actively growing primary root. It wasrepresented as a two-dimensional square automata grid (Fig. 1a)G = {(x, y): x = 1, 2, . . ., n and y = 1, 2, . . ., m} of sizes n and m, wherex was the vertical direction related to depth and y the horizontaldirection related to width. The growth is represented by changingn at each time change. Each grid cell (x, y) was bounded by a setof neighboring cells (Vxy). To model the cylinder, we consideredperiodical neighboring cells in the y-direction, which means thatthe (x, m) cell is neighbor to the (x, 1) cell. All the inner rows hadeight neighbors. Also, on the top row x = 1, each cell had just fiveneighbors. To model the root tip, we connected each cell with thecell in front with respect to the cylinder geometry. On the bottomrow x = n, each cell had five neighbors, and the across neighbor wasgiven by the cell (n, y + �m/2�), where �m/2� represented the closerinteger to �m/2�.

2.1.2. Status variables (SV)Each grid cell was associated with 5 status variables describ-

ing the soil properties represented by SVxy = (Nxy, Pxy, Bxy, Fxy, Axy),where

(a) Nxy represents the amount of available nutrients in a cell, withvalues 0, 1, 2, . . ., Nmax, where 0 means no nutrients and Nmax isthe maximum nutrient amount.

(b) Pxy represents the rhizoplane pH with values between 3 and 8.(c) Bxy represents the bacterial number with a limit of Bmax, with

Bxy ≤ Bmax.(d) Fxy represents the root growth on the rhizoplane, where Fxy = 1

means the grid cell is occupied by root and Fxy = 0 means thegrid cell is occupied by soil.

(e) Axy are bacterial parameters introduced by time changes,Axy = (BNxy, BDxy, NCxy), where BNxy contains the daughter cellsof the bacteria after division, BDxy contains dead bacteria, andNCxy are nutrients consumed by bacteria.

2.1.3. Transition rules (T)Each status variable (SV) described above was governed by a

transition rule, which is updated at each iteration of the model run.The set of rules is given by T = (TN, TP, TB, TF, TA) as follows:

(a) TN updates the available nutrients Nxy at each cell. This functiondepends on availability of nutrients (Nxy), which is reduced bythe nutrients consumed by the bacteria (NCxy) at each iterationTN(Nxy, NCxy) = Nxy − NCxy.

(b) TP updates the pH depending on the current cell pH Pxy andthe bacterial number Bxy present in neighboring cells Vxy. Thisparameter was based on the presumed rate of acidification ofmicrosites due to secretion of protons by bacteria that colonized

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Fig. 1. (a) Schematic presentation of automata grid (G) used for the modeling of rhizoplane as a cylinder. (b) Time–course of clover root growth used for modeling of G. (c)Representation of clover root growth as G. Yellow: root (active G cells). Green: soil (inactive G cells). (For interpretation of the references to color in this figure legend, thereader is referred to the web version of the article.)

the rhizosphere. This transition rule is described in scenario 1(Section 3.2).

(c) TB updates the bacterial number Bxy at each grid cell, therebyrelating the current bacterial number Bxy, newborn bacte-ria BNxy and dead bacteria BDxy at each iteration TB(Bxy, BNxy,BDxy) = Bxy + BNxy − BDxy.

(d) TF activates the cell (x, y) when the root grows. This functiondepends on time and position of the cell. The growth is modeledby L(t) = R0(1 − eUt), where L is length (cm), t is time (d), andR0 (cm) and U (1/d) are constants depending on plant species(Fig. 1b and c). Thus, TF = 1 if x < L(t); otherwise, TF = 0. Thus, thegrid parameter n (see Section 2.1.1) corresponds to the discreteversion of L(t).

(e) TA represents the update in bacterial parameters after eachiteration that affects environmental properties. The update isperformed externally depending on the decisions of agents inthe cell (x, y).

2.2. Bacterial model (B)

The bacteria take information from the environment, process it,and then they take some action on the environment. At each step,we assume that individual active bacteria either continue to acquirenutrients until they reach the internal level required for division,or they reproduce and further colonize the rhizoplane. The bacte-rial reproduction depends on space (bacterial cell number in grids),nutrients (including organic acid anions exuded by roots), and thepH level (decreased by proton exudation that may accompany exu-dation of organic acid anions by roots) in the cell. Growth ratesdecrease as pH decreases below the optimum. If the environmen-tal conditions are suitable, the bacterium reproduces, then adjusts

its fitness level, and then either places a new bacterium in a neigh-boring grid cell, or enter the stasis, or with a continuing decline inpH, enter the death phase and die. At each step, the environmentalchanges that were performed by the bacteria were updated on thestatus variables, and the process was reiterated.

As the model was generated to provide only the basic processes,the discrete model had the following simplifying assumptions:

(a) The relevant abiotic parameters are space, nutrients and pH.(b) The bacteria are static on the rhizoplane.(c) One bacterial species was represented in the model; hence,

there is no competition or any other interaction with otherbacterial species on the rhizoplane.

(d) The bacteria carry out only one action per time iteration and,when the environmental conditions are favorable, reproductionoccurs.

(e) The bacterium has maximum number of iterations tstasis beforedying.

The agent-based model B = 〈AG, R〉 consisted of AG = {Ag1, Ag2,. . ., AgS} (the finite set of agents) and the environment R as givenin Section 2.1. The maximum number of bacteria was defined byS = n × m × Bmax. Each bacterium was represented as Agi placed on a(x, y) cell in G. The behavior of each agent was defined by Ag = (ABMV,TR), which depends on the neighborhood in R.

2.2.1. Agent-based model variables (ABMV)The ABMV = (Ac, P, F) considered the action that the bacteria per-

formed, a set of perceptions from the environment and a fitnesslevel for survival.

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(a) Action (Ac) represents the set of actions taken by the agent onthe environment, Ac is a value from reproduction Rep, feedingA or death D, which was modified by the rule Action at eachiteration step

(b) Perceptions P = (N, CC, APR) are obtained from environment byeach agent and they are1. N is the nutrient level required by each bacterium to survive.

If N = 1, nutrient is low bacteria go into a starvation statusand can no longer divide. If N = 2, nutrient is acceptable andbacterium goes into a stationary phase, but retains the abilityto divide again. If N = 3, nutrient is high, the bacterium feedsand its fitness increases toward the threshold required forcell division.

2. CC represents the amount of nutrients consumed by eachbacterium depending on the nutrient in the cell Nxy and theamount of nutrients consumed by bacteria under optimalcondition CCmax.

3. APR = (a(xn, yn)) indicates conditions for reproduction, a = 0corresponds to unsuitable conditions, and a = 1 representproper conditions for reproduction. The new bacteria willbe located at a neighboring cell (xnyn) and the choice willdepend on the new environmental conditions.

(c) Fitness level F = (f, t) represents the bacteria fitness f to reproduce,and t is the bacteria current stasis time between divisions, whenthe fitness level is lower than threshold �.

2.2.2. Transition rules for agent-based modelThe rules that cause changes in each agent-based model variable

were represented by TR = (vision, next, action). At this stage, it wasimportant to establish the sequential order of the transition rules:first we updated the perceptions of each bacterium, after whichthe fitness level was updated, finally leading to an action state (celldivision, stasis, or death).

(a) Vision updated perceptions P = (N, CC, APR) depending on theenvironmental status SV. Considering nutrients Nxy and thebacterial number Bxy, the average nutrient amount in each bac-terium Nbxy = Nxy/Bxy is computed to change the perception by

N =

⎧⎪⎨⎪⎩3 if Nbxy > ˛,

2 if ̨ > Nbxy > ˇ,

1 if ̌ > Nbxy,

where ̨ and ̌ are the upper and the lower threshold of theavailable nutrients per bacterium. If a bacterium can feed, thenCC = min(Nbxy, CCmax). If there are no nutrients, CC = 0. The newlocation depended on environmental conditions, and the possi-ble choices were the set PC = {(u, v) ∈ V: Buv < Bmax and Puv[pH1,pH2]}, where [pH1, pH2] is the pH range within which bacte-ria can live and also the cell occupancy. If there were no cells,a = 0 (unsuitable conditions for reproduction). If there was morethan one grid cell where new bacterial cell could be placed, wechoose the coordinates of the cell with the largest average nutri-ent content among PC candidates, which is given by the argmaxfunction

(xn, yn) = argmax(u,v) ∈ PC

(Nuv

Buv

)In case of ties, the new bacterium was randomly placed.

Finally a = 1 (conditions for reproduction).

(b) Next updated the fitness level F = (fi + 1, ti + 1) of the agent basedon the new perceptions a and N, and the old fitness level (fi, ti).The new status fi + 1 changed according to

fi+1 =

⎧⎨⎩fi

2a = 1 and fi > �

fi + Kn, fi < �

where � is defined as the fitness level for reproducing. If fi > �and the environment is suitable a = 1, regardless of the nutri-ent level N the bacterium reproduces, and the fitness is splitbetween the old and the new bacterium. If the fitness fi < � , itchanged Kn depending on the nutrient level N, which is −K1 ifN = 1, 0 if N = 2, and K2 if N = 2. In this case, the environmentalcondition was not used.

For the starvation time ti + 1, if the fitness fi + 1 < �, thenti + 1 = ti + 1. If ti + 1 < tstasis bacteria went into the stationary phaseuntil it overcome the � threshold, or else it died if it reaches thetstasis limit time.

(c) Action takes the fitness level F and perception P for selecting anew action Ac at ti + 1

Action =

⎧⎨⎩Rep a = 1 and fi > �, reproduction

A feeding

D ti = tstasis, death

Each action represented a change in the environment statusvariables. If the action was Rep, the bacteria placed a new daugh-ter cell in (xn, yn) with an initial fitness level of fi/2 and increasedby one the population number Bxnyn. If the action is A, bacte-ria consume CC nutrients, thus decreasing Nxy in the cell. If theaction is D, the bacterium was eliminated from its cell, and thebacterial population number Bxnyn decreased by one.

2.3. Dynamics of the model

The conceptual diagram of the model is shown in Fig. 2. Thesteps are described as follows:

(a) First, we start the environment loading the status variablesSVxy = (Nxy, Pxy, Bxy, Fxy, Axy), where Nxy, Pxy, Bxy are given data,Fxy is active just for the top of the grid, and Axy has only zerovalues for each parameter.

(b) Second, for each bacterium located in an active cell, we set upthe variables Ac, P = (N, CC, APR), and F = (f, t). Notice that weneed the status variables from environment to set up bacterialvariables.

(c) Third, we start the dynamics by iterating time depending uponthe reproduction time of the bacteria1. First, the model computes transition rules for each bacterium

in a sequential order: Vision for updating the perceptionP = (N, CC, APR), Next for updating the fitness level F = (f, t),and Action for updating its effect on the environment Ac.Once all bacteria are updated, bacterial parameters of theenvironment Axy = (BNxy, BDxy, NCxy) are updated.

2. Next, the environmental rules are computed, and the statusvariables are updated. Notice that they are independent ofeach other.

(d) The simulation stops after a number of time steps.

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Fig. 2. Schematic diagram of cellular automata and agent-based model interaction. Environment offers pH and nutrients, and it counts the number of bacteria. Agents readenvironmental parameters (bacterial number, nutrient level, G cells number, nutrient consumed and pH), perform an action, modify perceptions from the environment, andupdate the fitness level. Next, environment updates the status variables using bacterial changes, and the process iterates.

2.4. Effect of parameters on rhizoplane colonization

The model parameters described above are directly related tocolonization characteristics. For environmental parameters, Nmax

and Bmax, they depend on the modeled root.For the bacterial parameters, the thresholds ̨ and � modify the

cell division rate as follows:

(a) If ̨ decreases, the bacteria require lower nutrient concentra-tions to maintain or increase the fitness.

(b) If ̨ increases, the bacteria require more nutrients to keep theirfitness status. If ̨ decreases, the bacteria survive under lownutrient conditions without their fitness being affected.

(c) If the ability to reproduce � is increased or decreased, repro-duction will be enhanced or restricted.

The parameters � and tstasis are related to the longevity of bacte-ria on the rhizoplane. Thus,

(a) If � increases, the bacterium requires a higher fitness level tokeep its activity and it does not go into death phase under lownutrient conditions. If � decreases, the bacteria keep their activ-ity under low fitness status. To avoid having bacteria enter thedeath phase after reproduction, this parameter is also ruled by2� < � .

(b) The tstasis parameter is linked to the bacterial ability to continueactivity under low nutrient conditions without dying. If tstasisincreases, the bacteria live longer, but if tstasis decreases, thebacteria die rapidly.

Regarding interaction between the pH of the growth mediumand bacteria, it is known that bacteria (particularly in the pres-ence of poorly soluble P compounds) can release protons and lowerthe pH at the microsites (Marschner et al., 2011; Richardson andSimpson, 2011). In addition, acidification reduces growth and activ-ity of bacteria (Marschner et al., 2011). Thus, in an unbufferedsystem, we assume that grid cells with higher density and activ-ity of bacteria will have a lower pH. See also scenario 1 (Section

3.1). The pH parameters pH1 and pH2 define cells that were fittedfor successful colonization of roots by the hypothetical bacteriummodeled here.

2.5. Model analysis

The model was programmed in Matlab® (version 7.0;MathWork®), and analyses were performed using a personal com-puter containing an Intel® CoreTM2 Duo processor, 2.2 GHz harddisk and 2 GB RAM. The model analysis is summarized in Fig. 2.

In this study, the root growth was based on observations fromexperiments in which clover roots (Trifolium pratense) were col-onized by the bacterium, E. coli. In these experiments, the cloverroots reached an average of 5 cm length, with 0.2 cm diameter after7 days. Considering a G automata grid of 104 �m2, n = 500, m = 60,a total of 30,000 cells were required to represent the clover root.In the root growth formula (see Section 2.1.3 and Fig. 1b and c),the parameters used were U = 0.151 and R = 7.68. The dimensionsused to describe the bacteria assumed a 1-�m diameter, whichwas estimated by light microscopy and plate counting. The gen-eration time for E. coli under optimal conditions is 20 min (Powelland Errington, 1963), which after 7 days, corresponds to T = 504generations (iterations).

2.6. Input and output of the model

The input parameters required are shown in Table 1. Theiteration number represents the temporal progress during rhizo-plane colonization. The number of cells was determined by n × m(lines × columns). The amount of available nutrients in a particularcell (x, y) was between 0 and Nmax, where 0 means no nutrients andNmax is the maximum nutrient amount. The reproduction success ofindividual bacterium is represented by a positive value for � . Wheneach bacterium reaches the � fitness level, the bacterium proceedsto reproduce and colonize the rhizoplane. At each iteration, thebacterial population number per active grid cell is registered andrepresented by a specific color. After reproduction, the nutrient andspace parameters in the cell are updated.

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Table 1Parameters used in model and input files.

Parameter Description Scenario 1 Scenario 2 Scenario 3

Bacterial parameterststasis Time limit for bacterial starvation 10 40 40t Iteration number (bacterial generations) 300 300 300˛ Upper limit of nutrients required for feeding bacteria 10 10 10ˇ Lower limit of nutrients required for feeding bacteria 5 5 5� Fitness level for reproduction 6 6 6K1 and K2 Increase and decrease in the fitness level 4 4 4� Threshold for bacterial starvation 2 2 2f0 Internal status 0a 0 0

Environmental parametersBmax Maximum bacteria number in a cell 400 40 40Nmax Maximum nutrient level in a cell 5000, 10,000 and 15,000 8000 1000b

n Automata vertical cell number 125 125 125m Automata horizontal cell number 15 15 15pH1 and pH2 Upper and lower pH limits for bacterial survival 4.5–6.5 4.5–6.5 4.5–6.5CCmax The amount of nutrients consumed by bacteria under optimal conditions 5 5 5

Root growth Yes No YesNutrientsc Homogeneous Homogeneous Heterogeneous

Input filesNutrient file Initial nutrient concentrations on the rhizoplanepH file Initial levels of pH on the rhizoplaneBacteria file Initial distribution of bacteria on the rhizoplane

a Values represent initial condition of f0.b Periodically increased according to 2 × Bmax .c Distribution in automata grid (G).

As one of the model outputs, the program generates an AVImovie showing the progress of bacterial colonization. The modelalso generates graphs showing the growth of active and starvedbacteria on the rhizoplane.

3. In silico simulations

3.1. Scenario 1

We evaluated the effect of the cell initial nutrient amount onbacterial population density on the root and the concomitant influ-ence of bacterial activity on pH. We designed an artificial scenariowhere the initial pH was randomly distributed over the grid cellsrepresenting the rhizoplane. Bacterial colonization was simulatedusing the following initial parameters: T = 300, ˛ = 10, ˇ = 5, � = 6,� = 2, tstasis = 10, Bmax = 400, CCmax = 5, K1 and K2 = 4, pH1 = 4.5 andpH2 = 6.5. The root length was 0.4 mm, and one bacterium per cellwas placed in each of the upper four rows of G with an internalstatus f0 = 0 to initially simulate feeding. The pH in individual gridcells (Pxy) decreased by 0.1 if there were enough bacteria to colonizefour cells, with a low pH limit of pH1. In this scenario, the simula-tion was performed with a homogeneous distribution of nutrientsin the whole grid. The goal of this scenario was to evaluate the effectof three different initial nutrient amounts (Nmax = 5000, 10,000 or15,000).

3.2. Scenario 2

We next considered the effect of colonization along an ini-tially sterile root to simulate the classical bacterial growth curve(Zwietering et al., 1990). The parameters associated with the Gautomata and bacteria were as described for scenario 1, excepttstasis and Bmax were both set at 40. In this scenario, there was noroot growth and the iterations were performed with the wholegrid. The nutritional amount was Nmax = 8000 distributed homoge-nously throughout the rhizoplane. As in scenario 1, the bacteriawere placed in each grid cell with an internal status f0 = 0.

3.3. Scenario 3

This scenario evaluated bacterial colonization on the rhizoplaneunder conditions in which there were pulses of nutrients into therhizosphere that occur naturally when exudates are released ina diurnal pattern reaching a maximum shortly after the onset ofphotosynthesis. In this scenario, the nutrient amounts in the cellswere increased and updated every 18 iterations (equivalent to 6 h).To avoid initial nutrient saturation on the rhizoplane, the start-ing nutrient amounts was set at Nmax = 1000 and then periodicallyincreased according to 2 × Bmax. Changes in pH were included asa limiting factor for bacterial growth, and the remaining parame-ters associated with the G grid and bacteria were the same as thosedescribed above for scenarios 1 and 2.

3.4. Simulation videos

The numerical simulation videos can be freely downloaded fromthe following URLs:

Scenario 1: http://dl.dropbox.com/u/47300928/Scenario1Nmax 5000.avi

http://dl.dropbox.com/u/47300928/Scenario1 Nmax 10000.avihttp://dl.dropbox.com/u/47300928/Scenario1 Nmax 15000.avi

Scenario 2: http://dl.dropbox.com/u/47300928/Scenario2 withoutgrowth.avi

Scenario 3: http://dl.dropbox.com/u/47300928/Scenario3 ActNut.avi

4. Experimental data

4.1. Selection of growth medium, host plant and bacteria

Assays in vitro were carried out to visualize the rhizoplane colo-nization by bacteria under phosphorus (P) limited conditions. TheP limitation was chosen because it allowed the visualization of pH

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Fig. 3. Results of bacterial growth and bacterial colonization on the rhizoplane by using the model under scenarios 1 (a and b), 2 (c and d), and 3 (e and f). In a, c and e graphs,black and red bars represent soil (inactive G cells) and root (active G cells), respectively; blue and yellow-orange dots represent higher and lower bacterial density on therhizoplane, respectively. In b, d and f graphs, orange and green lines represent the number of active and starved bacterial cells, respectively. The proximal (older) part of rootis at the top of the a, c and e graphs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

changes that are mediated by the proton exudation. In soils, plantsalso can moderate the pH of the rhizosphere in response to irondeficiency, and by release of hydrogen or hydroxyl ions that helpmaintain an electrochemical balance during uptake of ammonia ornitrate nitrogen, respectively.

The solubilization of calcium phosphate by biological activitywas revealed by appearance of clear zones. The medium usedwas National Botanical Research Institute’s phosphate growth agar(NBRIP; in g L−1: 10 d-glucose, 5 Ca-phosphate, 5 MgCl2·6H2O,0.25MgSO4·7H2O, 0.2 KCl, 0.1 (NH4)2SO4, and 15 agar) that con-tains tricalcium phosphate [Ca3(PO4)2] as a sole P source (Nautiyal,1999).

Four different host plants were examined in preliminary in vitrostudies, including linseed (Linum usitatissimum), two species oflupin (Lupinus albus and L. angustifolius), and clover (T. pratense).The seeds were disinfected by ethanol (70% (v/v) for 1 min) andhypochlorite (0.8% (v/v) for 15 min) and germinated in a growthchamber for 2–3 days at 20 ◦C. Clover was selected on the basis

of its quick root growth and formation of a main cylindrical rootin NBRIP (Fig. 4). Bacterial colonization of the roots was exam-ined using a strain of E. coli (Jorquera et al., 2006) that had beengenetically modified to express green fluorescence protein (GFP),thereby allowing easy visualization of the colonization process byepifluorescence microscopy.

4.2. Colonization assays

The clover seeds were disinfected as described above andimmersed in a bacterial suspension containing 108 CFU mL−1 for2 h to inoculate the seeds (Fig. 5a). The seeds were then germinatedat 20 ◦C for 2 days. Experimental units consisted of modified Petriplates containing NBRIP agar with a slit in the side that allowedthe roots to be inserted into the medium, and the plant shoots togrow in the atmosphere. The plates were incubated at 20 ◦C for 10days (Fig. 4a–c). Phosphorus solubilization patterns were visuallydocumented along the root axes (Fig. 4d).

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Fig. 4. Clover growth in colonization assays on NBRIP agar, after 4 (a), 6 (b) and 10 days (c). Clear zones (arrows) in NBRIP agar (d) were used as an indicator of phosphatesolubilization by biological activity.

4.3. Epifluorescence microscopy

Seed and root samples were taken during the colonizationassays and observed directly with an epifluorescence microscope(Olympus BX-41, Japan), equipped with a 100-W mercury lamp anda B2A filter for blue (GFP) excitation. Microscopic images were cap-tured using a CCD camera (Q-Imaging MicroPublisher, 3.3 MPixels,Canada).

5. Results and discussion

Discrete models have been used for modeling bacterial col-onization of surfaces, mainly based on cellular automata whereenvironmental properties are fixed. The main disadvantage ofthis simplification is the lack of feedback describing how bac-terial activity incrementally changes the environment as thebacteria grow. Our model will consider the dynamic changesin the rhizosphere and bacterial growth rates as an interactiveprocess.

Cellular automata were used to simulate the rhizoplane, withthe assumptions that there was no direct interaction amongautomata, but that there were indirect interactions between neigh-boring bacterial cells. This model facilitated reproduction of theheterogeneity in soil properties with respect to nutrients, pH, bac-terial number, and root growth. These properties change over timeby transition rules. Notice that we assume that all properties wereaffected to varying extents by bacteria.

On the other hand, bacteria possess their own dynamics inwhich their growth and death rates depend on environmentalconditions. To model the complex behavior of the bacteria, we con-sidered an agent-based model, which does not take into accountspatial relations, but is more general than cellular automata forincluding environmental properties. Agent-based model thus takesinto account changes in bacterial growth rates as affected by

nutrient amount, the depletion of nutrients with growth, the effectsof bacterial nutritional status on their pH lowering (or organic acidproduction) activity, and at each iteration leads to an update on thestatus of individual bacteria to divide, go into the stasis, or enteran irreversible death phase. The dynamic changes were modeledby first changing the properties in relation to environmental sta-tus, and next by changing the parameters related to fitness andthresholds at which they changed their physiological status. Thus,the dynamics of the interactions are represented as an integratedsystem based on both cellular automata and agent-based modelmodels.

In relation to in silico simulations, at each of the tested nutri-ent amounts in numerical scenario 1, bacterial colonization wasmainly governed by differential nutrient availability in the rootelongation zone. This is in agreement with Hartmann et al. (2009)and Buddrus-Schiemann et al. (2010) who showed that the pop-ulation of root-colonizing bacteria increases in proportion to thequantities of root exudates that are released. Here, we also con-sider that the interactions of the plant host with microorganismsmay be further affected by differences in the amounts and types ofexudates that are released into the rhizosphere (Marschner et al.,2011). This simulation showed a similar curve describing the num-ber of active and starved bacteria over time (Fig. 3b). The occurrenceof local minima for bacterial growth was attributed to the effectsof starting the simulation with random pH values in the cells. Werecognized that this is an artificial scenario, which was examinedonly to validate the model. In reality, studies have demonstrateddistinct patterns in the spatial distribution of pH along roots (Fanget al., 2007), which could be considered in further analyses withthis model. As expected, results of the run showed that a largernumber of active bacteria were maintained at the highest nutrientamount (Nmax = 15,000) (Fig. 3a). In contrast, at a low initial nutri-ent amount (Nmax = 5000), a large number of bacteria entered intostasis and starvation (data not shown).

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Fig. 5. (a) Clover seed inoculated with E. coli expressing green fluorescent protein (magnification 4×). (b and c) Root growth showing formation of bacterial aggregates(arrows) on the rhizoplane (magnification 10× in ‘b’ and 4× in ‘c’). (d) Arrow shows the bacterial aggregates on the rhizoplane at the clear zone in agar around upper (older)root region (magnification 100×). Root surface shows red autofluorescence under blue excitation. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of the article.)

The bacterial colonization pattern attained in scenario 2 isshown in Fig. 3c. This scenario was performed to demonstratethat under certain conditions the model may simulate the typi-cal bacterial growth curve that consists of a log phase, followed bya stationary phase, and then a death phase. The results showedheterogeneous colonization of the rhizoplane. The formation ofbacterial aggregates on the root is a phenomenon commonlyobserved under natural conditions, and this behavior has beenmainly attributed to heterogeneous availability of nutrients in therhizosphere via root exudation at cell junctions (Hartmann et al.,2009; Buddrus-Schiemann et al., 2010). The growth curve clearlyshowed the lag, exponential, stationary and death stages (Fig. 3d).This is a typical bacterial growth curve that has also been used formodeling (Zwietering et al., 1990). In addition, scenario 2 showedthat when the availability of nutrients decreased throughout therhizoplane, the number of active bacteria decreased, whereas thenumber of starved bacteria increased. In nature, it is known thatstarvation induces changes in the physiological status of gram-negative bacteria that enables them to survive in stasis for extendedperiods without nutrients (Lappin-Scott and Costerton, 1990). Inthis state, the dormant but viable bacteria can respond to increasesin nutrient availability and resume active growth.

In scenario 3, the results revealed a greater population densityin the upper (older) zone of the rhizoplane (Fig. 3e). However,bacterial aggregates also formed in the distal elongation zone asin scenario 1. These results are in agreement with prior studieson the colonization of tomato and barley roots by Pseudomonas inwhich there was dense colonization of the older, basal root parts,and likewise, dense colonization of the root cap (Gamalero et al.,2004; Buddrus-Schiemann et al., 2010). In the study presentedhere, the number of active and starved bacteria (Fig. 3f) followed

the same pattern and showed steep increases and decreases duringiterations. As colonization progressed, there were greater numbersof starved bacteria as nutrients became depleted in the denselycolonized locations on the roots.

Regarding root colonization assays, microscopic observationsrevealed heterogeneous colonization and aggregates of E. colicells on seeds and the rhizoplane (Fig. 5a and b). As discussedpreviously, it has commonly been reported that bacteria do nothomogenously colonize the roots, but instead form aggregatesof microcolonies that densely colonize the older root parts andthe root cap (Gamalero et al., 2004; Buddrus-Schiemann et al.,2010). Roots grow by expansion of the subapical region thatresides a few millimeters behind the root tip. These areas initiallybegin as near sterile as the surfaces of the new root cells expandlongitudinally. The zone of elongation is also the site of maximumnutrient availability for microorganisms as root exudates arereleased from the elongating cortex tissue. These sites are thuscolonized by bacteria until the population reaches a density atwhich they are predicted to consume all the nutrients as rapidlyas they are released from the roots (Crowley and Gries, 1994).Later, as the root tip is pushed forward, the primary colonizingbacteria that initially inhabited the maturing zone of elongationenter into a state of extreme starvation (Marschner and Crowley,1996; Lin and Crowley, 2001) and the community compositionchanges to more oligotrophic species that are better adaptedto starvation (Yang and Crowley, 2000). The rhizosphere soilsolution in this zone consist primarily of bacterial productsthat accumulate from the fermentation of more labile carbon,namely reduced organic acid anions such as acetate, lactate andbutyrate, which comprise up to 90% of the organic substances in therhizosphere (Koo et al., 2006). In nature, these low-energy products

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are ultimately degraded by secondary fermenters that completethe degradation process via syntrophic methanogens and acetate-oxidizing bacteria. In our experiments, the system was muchsimpler, and the in vitro experiments showed only the transitionsin the abundance and activity of E. coli (Fig. 5c and d), along withaccumulation of organic acid anions derived from this organism.

Phosphate solubilization in the rhizosphere can be achieved bymany organic acid anions, including lactate and succinate, whichare the main organic acid anions produced by E. coli (Warneckeand Gill, 2005). Because our model system was axenic and lackeda secondary fermenting community, the accumulation of organicacid anions along the older root parts and root tips (revealed byformation of clear halos) were in agreement with the model thatexamined rhizosphere dynamics for a single organism (Fig. 4d).Dense colonization by E. coli cells was also observed at the root cap(Fig. 5b and c), along with concomitant solubilization of phosphate.Future explorations of this process with the model would need toconsider rhizosphere dynamics for communities comprising sec-ondary fermenters that could degrade the organic acid anions andaffect their accumulation. It is generally estimated that about 50%of the bacterial species in soils, as well as many fungi, have theability to solubilize phosphate, representing an essential functionof the rhizosphere microbial community.

6. Conclusions

The automata cellular and agent-based models were success-fully combined to model vertical and horizontal colonization ofthe rhizoplane by bacteria under different numerical scenarios.The formation of bacterial aggregates on the rhizoplane as wellas increased abundance of bacteria on the rhizoplane zone withincreased nutrient availability was also simulated and predicted byusing several bacterial (feeding, starvation and reproduction) andenvironmental (space, nutrient content and pH) parameters. Theexperimental assays showed the coexistence of diverse coloniza-tion scenarios during plant growth, and confirmed the rhizoplaneas a complex environment.

It is noteworthy that the present model is the first approachto analyzing experimental and theoretical parameter values in therhizosphere based on one bacterial species only. Future researchshould be focused on evaluating the competence and ecology of var-ious bacterial species in the rhizoplane microsites and developingmore accurate predictive models that can stimulate new investiga-tions with practical applications for agricultural biotechnology.

Acknowledgements

This study was supported by FONDECYT no. 11080159 andMEC-CONICYT no. 80110001. Crowley acknowledges support fromBARD project US-4264-09. We also thank professors Dr. MarcoGubitoso from Instituto de Matemática e Estatística of Universidadede São Paulo, Brazil (IME-USP) and Dr. Anahí Gajardo from Depar-tamento de Ingeniería Matemática of Universidad de Concepción,Chile (DIM-UDEC) for useful discussions about the model.

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