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Journal of Theoretical Biology 248 (2007) 448–459

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Individual-based model and simulation of Plasmodium falciparuminfected erythrocyte in vitro cultures

Jordi Ferrera,�, Jaume Vidalb, Clara Pratsa, Joaquim Vallsa, Esperanza Herrerosb,Daniel Lopeza, Antoni Giroa, Domingo Gargallob

aDepartament de Fısica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Escola Superior d’Agricultura de Barcelona,

Campus del Baix Llobregat, Avda. del Canal Olımpic s/n, 08029 Castelldefels, SpainbDrug Discovery Biology Group, Diseases of the Developing World Center, From GlaxoSimthKline, R&D, Tres Cantos, Madrid, Spain

Received 23 November 2006; received in revised form 21 May 2007; accepted 24 May 2007

Available online 2 June 2007

Abstract

Malaria is still one of the most fatal diseases in the world. Development of an effective treatment or vaccine requires the cultivation of

the parasite that causes it: Plasmodium falciparum. Several methods for in vitro cultivation of P. falciparum infected erythrocytes have

been successfully developed and described in the last 30 years. Some problems arising from the current harvests are the low parasitaemia

and daily human supervision requirements. The lack of a suitable model for global culture behavior makes the assay of new

methodologies a costly and tenuous task. In this paper we present a model and simulation tool for these systems. We use the INDividual

DIScrete SIMulation protocol (INDISIM) to qualitatively reproduce the temporal evolution of the erythrocyte and merozoite

populations. Whole system dynamics are inferred by setting the rules of behavior for each individual red blood cell, such as the nutrient

uptake, metabolism and infection processes, as well as the properties and rules for the culture medium: composition, diffusion and

external manipulation. We set the individual description parameters according to the values in published data, and allow population

heterogeneity. Cells are arranged in a three-dimensional grid and the study is focused on the geometric constraints and physical design of

experimental sets. Several published experimental cultures have been reproduced with computer simulations of this model, showing that

the observed experimental behavior can be explained by means of individual interactions and statistical laws.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Individual-based model; Plasmodium falciparum; Simulation; Merozoite; Erythrocyte; In vitro culture; Malaria

1. Introduction

Malaria is still one of the most fatal diseases in theworld, and it continues to be a great burden for the affectedareas (WHO, 2005). Pathologic symptoms in humans arehigh fever and anemia, among others, and these appear dueto blood infestation caused by a protozoan parasite.Plasmodium falciparum is one of the most deadly species.The sexual part of the parasite cycle takes place inside thehuman erythrocyte (RBC, red blood cell). This cycle beginswith the invasion of the RBC by a single parasite formcalled merozoite and lasts 48 approximately hours. Duringthe cycle, the infected RBC changes its physiology and

e front matter r 2007 Elsevier Ltd. All rights reserved.

i.2007.05.030

ing author. Tel.: +3493 552 11 44; fax: +3493 552 10 01.

ess: Jordi.Ferrer-Savall@upc.edu (J. Ferrer).

ultrastructure, and four stages of the infection cycle areusually distinguished: ring stage, tropozoite stage, schizontstage and fragmenter stage. At the end of the parasite cyclethe RBC dies and lyses, releasing from 5 to 30 merozoitesthat may invade other RBCs. In vitro cultivation of theerythrocytic stages of P. falciparum has been widely andusefully applied in nearly every aspect of research onmalaria; it is essential for the development of an effectivemalaria treatment or vaccine. The current methods in usefor in vitro propagation of the parasites are based on thecandle-jar method, which was first developed by Trager andJensen (1976). This method keeps RBCs incubated at 38 1Cin an artificial culture medium mixed with human serum ina low oxygen atmosphere. It demands daily humansupervision, as the medium must be replaced, andsubcultures are required twice a week to limit culture

ARTICLE IN PRESSJ. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459 449

infection rates. There is a demand for improving thecurrent culturing techniques, and modeling the wholeculture system behavior is taking the first steps in thisdirection (Jamshidi and Palsson, 2006).

Large-scale dynamic modeling of cell populations andsubsequent computer simulations may provide clues for acorrect description of our culture system (Henson, 2003), asit has thrown light over the global behavior of manymicrobial communities, such as bacteria and yeasts(Ginovart et al., 2002b, c, 2006; Prats et al., 2006). Manymathematical models for describing the dynamics of in vivo

P. falciparum infected human RBC populations have beencarried out, regarding the human immune response (Hos-chen et al., 2000; McKenzie and Bossert, 1997) and thedifferential behavior among variants through a singleinfection. They have also proved to be useful for describingspecific features of both in vivo and in vitro systems, such asthe effect of temperature variation on the infected cellspopulation (Kwiatkowski and Nowak, 1991) and theassessment of cross-resistance to antimalarial treatments(Noedl et al., 2001). However, models based on amathematical description of the population dynamics ofinfected RBC in vitro cultures, comprised a set of differentialequations to be applied to the whole system, have notbeen found.

Individual-based Models (IbM) are bottom-up ap-proaches to the culture systems: they start from descriptionat an individual level to infer global behavior. By usingIbM, we can study different aspects of the individual modelseparately, as well as their cross-effects on populationevolution. IbM simulations have been shown to be usefulfor reproducing and describing the behavior of complexsystems such as microbial communities in batch cultures(Kreft et al., 1998), trophic relations in an ecosystem(Grimm, 1999), and the spreading of malarial epidemics(Gu et al., 2003), among others. INDividual DIScreteSIMulation (INDISIM) is the IbM methodology used toperform the simulation experiments of the present model(Ginovart et al., 2002a).

Our work is part of a project designed to make adescription of the population dynamics of the asexual cycleof the parasite in the human blood in the mesoscopic scale.The ultimate goal of the project is to improve currentexperimental culture protocols.

In this paper we present a model that includes both RBCand merozoite populations in a three-dimensional spacegrid. Its suitability for simulating in vitro cultures has beenchecked by reproducing the experimental results obtainedin several classic published experiments. The preservationof P. falciparum infected RBCs for 64 h performed byPavanand et al. (1974) was used to calibrate our model,and the continuous culture of infected RBCs for 48 and 54days performed by Jensen and Trager (1976) was used tovalidate it. We addressed our study to the spatialcharacteristics of the cultures and the evolution of theparasitaemia (the relative population of infected RBC tototal RBC population). Specifically we have dealt with a

collection of RBCs settled on the bottom of in vitro staticcultures, composing the sediment called the hematocritlayer. This model is being used as an aid for theexperimental task performed by the Drug DiscoveryBiology Group (DDBG), Diseases of the DevelopingWorld Center, from GlaxoSimthKline, R&D, Tres Cantos,Madrid.An outlined description of the IbM is presented below,

in line with the ‘‘Overview–Design concepts–Details’’(ODD) standard protocol (Grimm et al., 2006). TheOverview of the model consists of three elements: Purpose(Section 2.1), State variables and scales (Section 2.2) andProcess overview and scheduling (Section 2.3). DesignConcepts are described in Section 2.4. This section presentssome general concepts underlying the design of the modeland the strategic considerations taken into account to dealwith the complexity of the system. Next, some details thatwere omitted in the overview are specified. Initial values forthe characteristic variables and parameters of the modelare presented in Section 2.5, and the external constraintsintroduced in our model are depicted in Section 2.6.Section 2.7 describes in more detail the submodels thatrepresent the processes listed in Section 2.3. Section 3 isaddressed to the particular results obtained by thesimulation experiments. Section 3.1 presents the experi-mental data and sets in real systems, while the input data tocarry out the corresponding virtual experiments arepresented in Section 3.2. Section 3.3 presents the simulationresults. Section 4 expounds an analysis of the results andthe model, and Section 5 concludes assessing the outcomesof this work.

2. Description of the model

2.1. Purpose

The first aim of the current work is to uncover thephysical processes that determine the evolution of healthyand infected RBC populations within an in vitro culture ofP. falciparum infected cells. In particular, it tackles theeffect of the limitations on the diffusion of importantsolutes for the RBC metabolism, and the role of the processof extracellular parasite spreading through the hematocritlayer. We also focus on the macroscopic physical andgeometric constraints to which experimental designs aresubjected, and on the effect of external manipulationprotocols on the population dynamics. The ultimate goalof our work is to improve current experimental protocols.

2.2. State variables and scales

The model proposes two entities: individual RBCs andthe culture environment. The description of the culturesystem is discrete both in time and space: all actions occurat constant intervals called time steps, and all entities arecharacterized by a set of integer numbers. Individuals arespread over a whole number of spatial cells that form

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a three-dimensional grid. Each spatial cell describes theculture environment locally.

A set of variables determines the state of the system at agiven moment (t). Each individual RBC (j) is described bythe vector: Ej[ei(t)]; (i ¼ 1:k), which contains k statevariables: age, time in culture, location in the space model,metabolic state, post-invasion time, and infection stage.Each RBC has a spatial extent or fixed volume, and maynot overlap with other individuals, nor with the externalform of the parasite.

The culture environment is described locally: our modeldivides the space into a three-dimensional grid composedof finite-sized cubic spatial cells, called microcells. A vectorSXYZ containing a set of q characteristic state variables,SXYZ[sm(t)] (m ¼ 1:q), is defined for each microcell, withX,Y and Z being the coordinates of the cell in the spatialgrid. The local variables si(t) are: number of RBC, numberof merozoites, number of units of glucose, number of unitsof lactate, and fraction of the microcell volume thatcorresponds to free culture medium (the portion of thespatial cell which is not occupied by RBCs or merozoites).Merozoites are considered to be a local property of thespatial cells and they are not individually distinguished.They have a fixed common volume. We consider fivedifferent types of merozoites, with regard to their time inthe extracellular culture medium. Metabolite units (glucoseand lactate) are also regarded as variables that characterizethe culture environment locally and are not considered tobe individual entities. Fig. 1 shows a depiction of thesystem model and entities.

The time step was set to 6min. This value is smallenough to accurately describe the spreading of merozoites.At the same time, simulation of long-term experimentalcultures (30 days’ cultivation) can be carried out in 10,000iterations, a number that is computationally feasible inuseful time.

We set the characteristic length of the model to 5 mm.This corresponds to the average radius of a real RBC andrepresents the side length of microcells. Splitting the spacemodel into microcells, and specifically choosing thischaracteristic length, facilitates the building and processingof individual actions. It also allows an explicit description

Fig. 1. (a) Side view of an experimental static culture, according to the MR4 pr

(iv) modeled fraction of the culture system. (b) Spatial model of the hema tocrit

RBC (ring stage), (ix) empty microcell.

of the process of merozoites spreading, which is central inthis model. Our simulations represent just a small fractionof the experimental culture system: we monitor from30,000 to 3� 105 RBCs, and we cover volumes that rangefrom 0.06 to 1.2mm3. We consider this system big enoughto be representative of the whole system, and appropriatefor a statistical analysis.Our simulation space is a three-dimensional representa-

tion of an average fraction of the culture system, whichcovers a section of the hematocrit layer (Fig. 1). We imposeperiodic boundary conditions through its lateral borders.We build specific upper and lower boundary conditions tomaintain the vertical asymmetry that appears in realcultures, as well as the observed structure of thehematocrit. The lower plane of our simulated spacecorresponds to the bottom of the culture plate, so it is aclosed boundary. The upper plane of our grid correspondsto the border between the free culture medium and thehematocrit layer, and is an open boundary.

2.3. Process overview and scheduling

The model proceeds in discrete 6-min time steps. At eachtime step, all actions on the culture system occursequentially and build the configuration of the next stepscenario. Firstly, the actions performed by the individualstake place; afterward the program locally carries out theactions on the culture medium. While performing eachaction, individual and local variables involved with theprocess are updated straightaway. The high-level variablesof the system, such as populations and spatial configura-tion, are updated at the end of each time step, and theoutput information is registered. The process is repeateduntil simulation ends.Each individual must carry out the following series of

actions: motion, uptake and metabolism, infection andmortality. The order in which individuals act is chosenrandomly at each time step, to avoid the effects of arbitrarylisting and thus simulating simultaneous activity.

Motion: Each RBC has a certain probability of movinginto a lower first neighbor spatial cell. In contrast,individual location is haphazardly changed at each time

otocol: i) low oxygen atmosphere, (ii) free culture medium, (iii) hematocrit,

. (c) Top view: (v) culture medium, (vi) RBC, (vii) merozoite, (viii) infected

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step for each RBC when performing suspended culturesimulations.

Uptake and metabolism: Nutrient uptake is fulfilleddepending on its availability in the microcell, whilemetabolic outcome is assessed and metabolic state isupdated for each RBC.

Infection: For a healthy RBC, this process lies inassessing whether it is invaded by a merozoite. For aninfected RBC, the evolution of the infection cycle of theparasite is assessed. At the end of the cycle, the cell dies andreturns several merozoites to the medium (see Section 2.7).

Mortality: At the end of its sequence of actions, RBCendurance or death is tested.

After every RBC has acted, the processes concerning themedium are taken into account. Actions on the medium aredivided in two modules processed as arranged: systemresponse and experimental protocols.

System response: This is assessed locally and consists ofthe diffusion of metabolites and the spreading and aging ofmerozoites. Simulation space is covered by processing theactions on each microcell in a random order set at eachtime step. For each cell, we evaluate the transportphenomena among its first neighbor microcells.

Experimental protocols: These are external manipulationson the system as a whole and are performed globally. Theyconsist of agitation, medium renewal and subculturing.Agitation is the shuffling of all the individuals andredistribution of the locally defined variables, while mediumrenewal is the resetting of glucose and lactate concentrationsto their initial values, and the removal of all the merozoites.Subculturing consists of the extraction of a certain fraction ofmedium and RBCs and the incorporation of the equivalentamount of fresh medium and healthy RBCs.

An overview of the described model is summarized inTable 1.

2.4. Design concepts

Emergence: Population dynamics and structure emergefrom the behavior of individuals, but individual behavior ispreset according to empirical rules and modulated by thecourse of the infection through the culture system.Evolution of the parasitaemia also emerges from theindividual’s life and infection cycles. Further, it stronglydepends on the spreading process of merozoites, which isrepresented by empirical rules.

Sensing: Individuals are assumed to know all theircharacteristics and they apply the rules according to them.Individuals sense the state of the culture environmentlocally, and act according to the state of their own spatialcell and first neighbor microcells.

Interactions: Two kinds of interactions are modeledimplicitly: struggle for nutrients and contamination withwaste. One interaction, contagion, is explicitly modeled. Itis mediated by merozoites.

Stochasticity: Randomness is introduced in the para-meters that rule the stochastic processes such as diffusion

of solutes or spreading of the parasite. It is also presentwhen setting the individual initial state variables, to takeinto account the statistical variability and biologicaldiversity. Normal distributions centered on the publisheddata values and with the corresponding variability are usedin this case. It is also introduced when setting the lists ofaction for both individuals and microcells, in order toavoid the effect of sequential turns of action.

Observation: At each time step, relevant variables such asoutput and information concerning the state of system arestatistically analyzed. These variables include: age structureamong individuals, post-invasion age structure for infectedRBCs, percentage of infected RBCs or parasitaemia, localconcentration of solutes and spatial distribution of RBCsand merozoites.

2.5. Initialization

Cultures are initialized with homogeneous distributionsof RBCs, infected RBCs and nutrients. No waste particlesor merozoites are initially distributed. Dimensions of thespatial grid, initial parasitaemia, total number of RBCs,and age structure of the inoculum are specifically set toreproduce experimental behaviors. These will be presentedin Section 3, in addition to other characteristic values, suchas boundary conditions and parameters controlling theexperimental protocols, and value used as seed to generaterandom numbers.

2.6. Input

Two external processes drive the model system dy-namics: discrete external manipulation of the whole system,representing the experimental protocols (Section 3.2) andcontinuous external input of nutrients through the upperboundary layer. Free culture medium is considered as acontinuous source of metabolites during the culturingprocess. For this reason, the upper plane of our simulatedgrid will always have the initial concentration values fornutrient and waste, no matter what the concentrationgradient in the lower planes is.

2.7. Submodels

2.7.1. Individual actions

Motion: RBCs are randomly shuffled at each time step insuspended cultures, and have a sedimentation rate in staticcultures. This rate is represented as a probability of fallinginto a lower first-neighbor spatial cell, if there is enoughroom available in the destination.

Uptake and metabolism: RBCs require a constantexchange of substances with their surrounding medium,in order to maintain their structure. The current model forthe individual uptake and metabolism processes minimallydepicts the glycolitic pathways of the RBC. Two metabolickey substances have been taken into account: nutrientand waste, related numerically to glucose and lactate,

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Table 1

Overview of processes, parameters and default average values for parameters of INDISIM-RBC

Model entities Variable Value Source

Red bood cells (RBC) Individual parameters

Individuals Volume 83 mm3 Jung et al. (1975)

Actions at each time step Density 1.1 g/ml Linderkamp et al. (1993)

Motion

Uptake and metabolism

Infection (invasion of healthy

RBCs, four-stage cycle, and

lysis)

Viability

Individual characteristics

Identity number -

Metabolic requirements (H) 25 glucose units/ts Joshi and Palsson (1989b)

(I) 2500 glucose units/ts Kirk (2001)

Infection-stage dependant accidental

death probability per time step

(H) 0 Increasing fragility of the infected

forms (Shjerman et al., 2004). Specific

values are set to adjust experimental

data for each assay

(R) 0

(T) �10�4–10�3

(S) �10�3

(F) �10�3–10�2

Total infection cycle 48 h ¼ 480 ts Perlmann and Troye-Blomberg (2002)

Duration of infection cycle and stages (R) 18 h ¼ 180 ts

(T) 17 h ¼ 170 ts

(S) 7 h ¼ 70 ts

(F) 6 h ¼ 60 ts

Maximum time for infection

susceptibility

5 days ¼ 1200 ts -

Maximum value of invasion

probability for healthy RBCs

0.8 Schuster (2002)

Individual variables

Time in culture 0–5 days ¼ 0–1200 ts

Spatial position -

Infection state H,R,S,T,F

Infection age 0–48 h ¼ 0–480 ts

Metabolic state index 0–3

Local environment Parameters

Microcells Microcell volume 125mm3

Actions at each time step Merozoite volume 2 mm3 Perlmann and Troye-Blomberg (2002)

Motion: diffusion and spreading

of merozoites.

Experimental protocols

Local Variables

Fraction volume of culture medium 0–1; Average ¼ 0.35

Number of nutrient (glucose) units (Initial value) 11.500 (RPMI 1640)

Number of waste (lactate) units (Initial value) 0

Number of merozoites (Initial value) 0

Abbreviations: H, healthy RBC; I, infected RBC; R, ring-stage form; T, tropozoite-stage form; S, schizont-stage form; F, fragmenter-stage form; ts, time

step. Solute unit, 105 molecules of solute. Dimensionless parameters are either probabilities or indexes with a qualitative meaning.

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459452

respectively. At each time step, each RBC can take up avariable amount of the nutrient available in its spatial celland also releases an amount of waste that is twice theuptake (Joshi and Palsson, 1989a, b, 1990a, b). If theuptake is not enough to fulfill the RBC’s maintenancerequirements, the metabolic state index of the cellincreases, which means greater requirements for the nexttime step. This index indicates the metabolic stress that theRBC bears, and exceeding its threshold value causes thedeath of the RBC. Individual upkeep requirements are setamong the population according to a Gaussian distributionaround a fixed mean value /US. For healthy RBCs, itis maintained constant through time: /UhealthyS ¼ 20glucose units/time-step ¼ 2� 106 molecules/minute (Junget al., 1975).

For infected RBCs, /US varies through the infectionprocess (Dame et al., 1996; Kirk, 2001; Lew et al., 2003)and depends on the post-invasion age (t) in the following

manner: /US ¼ /UhealthyS � f(t)

f ðtÞ ¼

1; totR ðRing stageÞ;

2ðt� tR � 50Þ; tRototR þ 50 ðTropozoite stageÞ;

100; totF � 50 ðSchizont stageÞ;

minð1; 100�2ðt�tF þ50ÞÞ; tF �50ototC ðFragmenter stageÞ;

8>>>><>>>>:

where t is given in simulation time steps, tR is the durationof the ring stage of the parasite cycle, tF is the durationuntil reaching the fragmenter stage, and tC is the totalduration of the infection cycle. The function f (t is depictedin Fig. 2(a).

Infection; invasion of healthy RBCs: Every healthy RBCmight be infected by a merozoite located at the samemicrocell. Merozoites make their attempts to invade inrandom order and succeed with a probability pinf. Theinvasion susceptibility pinf is an individual characteristic ofthe RBC that decreases with RBC age (tage) and time in

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Fig. 2. (a) Post-invasion time (t) evolution of individual metabolic requirements for infected RBCs. f(t): multiplier factor. tR: end of the ring stage of

the infection cycle. tF: beginning of the fragmenter stage of the infection cycle. tC: end of the infection cycle, lysis of the RBC and egress of merozoites.

(b) Evolution of the individual infection susceptibility. t, time in culture. pmax, maximum invasion probability, for young RBCs. pmin, minimum

invasion probability, for old RBCs, tms, maximum time of infection susceptibility; tma, maximum individual age.

Table 2

Effective diffusivities of glucose and lactate

Solute Through the free-

medium layer

Through the hematocrit layer

Glucose D0 ¼ 9.3� 10�6 cm2/s Deff ¼ 0.34 �D0 ¼ 3.16� 10�6 cm2/s

Lactate D0 ¼ 1.45� 10�5 cm2/s Deff ¼ 0.18 �D0 ¼ 2.61� 10�6 cm2/s

D0 corresponds to diffusivities in the free culture medium (Jou, 1985) and

Deff correspond to the obtained effective diffusivity through an hetero-

geneous medium composed by spheres with a fraction of volume

occupancy equal to 0.63 (Stewart, 1998).

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459 453

culture (t) (Pasvol et al., 1980; Shjerman et al., 2004):

pinf ðtÞ ¼ pmax �tage

tma

ðpmax � pminÞ

� �OðtÞ,

where OðtÞ ¼1; totms;

e�t=tms ; t4tms;

(

where pmax and pmin are the maximum and minimum invasionprobabilities for young and old erythrocytes, respectively, tma

is the RBC’s maximum age and tms is the RBC’s maximumtime in culture for invasion susceptibility. This dependence isshown in Fig. 2(b). It is important to point out that eachinvaded RBC becomes infected at the end of the time step.This allows multiple invasion of RBCs, as has been reportedin experimental in vitro cultures (Simpson et al., 1999).However, host cells may support no more than threemerozoites, according to the rules imposed by the model.

Infection; egress of merozoites: During the last 14 h of itsintracellular cycle, the parasite replicates its DNA andproduces several merozoites through a mitotic process.Many theoretical models, reflecting a wide range ofopinions, describe the egress of merozoites at the end ofan infection cycle (Lew, 2001; Wickham, 2001); recentlypublished experimental results indicate that merozoiterelease is an explosive event (Lew, 2005).

In this model, 8, 16 or 32 merozoites egress from aninfected RBC at the end of its infection cycle. The numberof released merozoites is given as 2k, where k is propor-tional to the length of the last stage of the infection cycle.Each released merozoite is dumped into the RBC microcell,or into any of the nearest neighboring spatial cells, and willremain in the culture medium until it invades another RBCor dies due to degradation.

Mortality: At the end of the individual turn of action,the viability of the RBC is assessed by checking its age,time in culture, metabolic stress index, lysis after theinfection cycle and accidental death probability, whichdepends on the infection stage.

2.7.2. Actions concerning the culture medium

Parasite motion: Once in the extracellular medium,merozoites cannot migrate to other locations on their

own. Parasite motile systems are only useful for invading ahealthy RBC, after having contacted it, and diffusion dueto Brownian motion is negligible at this scale. However,sedimentation due to gravity in combination with diffusionis a non-negligible mode of transport for merozoitesthrough the hematocrit (Jou, 1985). The spreading ofmerozoites is implemented with a downwards-biasedrandom motion for each extracellular parasite, performedfor every merozoite at each time step and limited by theroom available at the destination microcell. Merozoitesmay endure for four time steps once in the culture medium.

Diffusion of solutes: The number of substance units ineach spatial cell is continuously updated due to thebiological activity, so its spatial distribution is nothomogeneous. At the end of each time step, diffusion isallowed, to make concentration gradients smoother. Thepresent model evaluates the concentration of solutes ateach spatial cell and compares it with the concentration ateach of the surrounding microcells. Excess solutes arepartially shared out, as far as is allowed by the diffusive-ness of the solute. Diffusiveness of glucose and lactatethrough H2O and effective diffusiveness through thehematocrit layer have been measured at the culturingtemperature of 37 1C; their accepted values can be found inmany reviews (Stewart, 1998), and are shown in Table 2.The order of evaluation of spatial cells is shuffled at eachtime step, in order to avoid introducing anisotropies in thediffusion process.

Experimental protocols: External manipulations onthe systems are taken into account by managing the

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environmental conditions according to the experimentalprotocols published in the references. These specific inputson the system are described below.

3. Simulation experiments

3.1. Reference real experiments

In this paper we refer to many culture methods andexperimental assemblies outlined below.

We have set the geometric characteristics of our modelaccording to these experimental displays and compare ouroutput data to some of their measured macroscopiccharacteristics, through the various protocols.

The first reference is the experiment of Pavanand et al.(1974) for preserving P. falciparum in culture for more thana complete cycle of infection. Preservation of parasitizedhuman RBCs with negligible loss in parasite viability isachieved for 64 h in a closed static culture system. RBCsremain placed in thin layers at the bottom of a sterileplastic tray. They are incubated at 37 1C in a mixture ofculturing medium and human sera, with no mediumrenewal. The experimental conditions in this set ofexperiments are quite different from the ones used in laterworks: concentration of the hematocrit is 9.36% in volume,with 0.2ml being the total volume of the culturing aliquots.The depth of the hematocrit layer in the culture is notspecified. The culture medium is also slightly differentin this first preservation experiment: initial glucoseconcentration is 2 g/l (Diggs et al., 1971). The use of atrial culture medium, which has been improved in laterassays, and the lack of a suitable culture atmospheresuggest a high mortality among RBCs in this experimentalsystem.

Initial parasitaemia is approximately 1%. Three formsare distinguished by their infection stages: rings, tropo-zoites and schizonts, and infected RBC counts are givenas%. The experimental data sets from this paper were usedto calibrate our model.

The second reference is the experiment carried out byTrager and Jensen (1976). Two different experimental datasets are used, corresponding to different protocols. In thefirst one, a continuous static culture of P. falciparum

infected RBCs is maintained for 48 days in a semiopensystem with a continuous flow of culture medium. Thesecond protocol achieves the preservation of parasitizedRBCs for more than three months in a static culture. Inthis case the system is closed and there is a discrete dailymedium renewal. This method is best known as the candle-

jar method and is the one still in use nowadays, with slightmodifications such as the introduction of incubators tocontrol the composition of the culture atmosphere. Bothprotocols incubate RBCs at 38 1C under an atmospherewith low oxygen, use the same blood source, parasitestrain, and culture medium preparation, and maintainsimilar spatial conditions: 1ml total culture mixture, with5% hematocrit in flat bottomed vials. The height of the

hematocrit layer is set to 0.15mm. Initial concentrations ofglucose and lactate are 2.667 and 0mM, respectively(culture medium: RPMI 1640). According to the firstprotocol, the extraction and refilling rate of culturemedium is 50ml/day and medium renewal is performedcontinuously. With the candle-jar method, a completeexchange of the culturing medium is performed daily. Themedium renewal is carefully executed, so no disturbance ofthe RBC population is expected.In these long-term experiments, a continual renewal of

the RBCs in culture is needed due to their degradation inculture and the exhaustion of healthy RBCs susceptible toinvasion as the infection spreads. This renewal is known assubculturing. Subcultures are performed every three orfour days, and they consist of the dilution by volume of thewhole culture medium in a similar medium containing onlyhealthy RBCs. Depending on the parasitaemia, three-fold,3.5-fold or four-fold dilutions are effected, this last onebeing the most usual. Some of the subculturings areindicated in the experiment, while others are not preciselyspecified but may be deduced from the indicated dilutionshown in the experimental data sets. Four forms of theasexual intraerythrocytic stages of the parasite are distin-guished in this experiment: rings, tropozoites, schizontforms with two nuclei and fragmenters (schizont formswith more than two nuclei). Additional information on theexperimental protocol can be found in ‘‘Methods in CellBiology’’ (Trager, 1994).The third source of experimental data is a series of

preservation experiments testing different types of culturemedium performed by the ‘‘Drug Discovery BiologyGroup, Diseases of the Developing World Center, Glaxo-SimthKline R&D’’. These cultures were handled accordingto the MR4 protocol (MR4, 2004), in order to test thecommercial culture mediums provided by different com-panies. The dimensions of the cultures were: 20ml totalculture mixture, 5% by volume consisting of hematocrit,which forms a layer that measures 0.2mm in height.Experimental data do not distinguish different intra-erythrocytic stages of the parasite.

3.2. Specifications for simulation experiments. Input data

Simulation experiments try to describe the particularbehavior of the above-mentioned culture systems. All theresults here presented correspond to simulations of staticcultures, where RBCs have settled, forming the hematocritlayer. After a transitory phase, the concentration of RBCsamong the hematocrit layer is the same in every simulation.We performed our simulations many times, and modifiedthe seeds to generate pseudo-random-number chainsthat we used to introduce noise, heterogeneity andvariability among the population. We set the input datato minimize the differences between the experimentalresults and the simulation output. Specific values for theparameters used to carry out the simulations are presentedin Table 3.

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Table 3

Simulation input values corresponding to reference experimental displays

Input data Pavanand et al.

(1974)

Trager and Jensen (1976) DDBG

Continuous

flow

Candle-jar

method

Synchronous Asynchronous

Initial values

Simulation duration 800 ts 12000 ts 6000 ts 1000 ts 2000 ts

Number of microcells 40� 40� 50 50� 50� 50 50� 50� 50 50� 50� 50 50� 50� 50

Size of microcells (side) 6mm 5mm 5mm 5 mm 5 mmInitial number of RBCs 30 000 40 000 30 000 30 000 30 000

Initial parasitaemia 1.20% 3% 1.30% 0.95% 1.30%

Mean of initial post-invasion time distribution 40 ts 215 ts 20 ts 300 ts 20 ts

Experimental protocol

Medium renewal period - 1 ts 1 day 1 day 1 day

Medium renewal rate - 20% 100% 100% 100%

Subculture period - 3–4 days 3–4 days 3–4 days 3–4 days

Subculture rate - 66%–75% 66%–75% 66%–75% 66%–75%

Parasite

Maximum and minimum infection probabilities for healthy RBCs

(pmin; pmax)

0; 0.8 0; 0.8 0; 0.85 0; 0.85 0.05; 0.9

Fragility of infected stages Pd(T) ¼ 0.005 Pd(S) ¼ 0.006 Pd(S) ¼ 0.005 Pd(T) ¼ 0.001 Pd(T) ¼ 0.0001

Pd(S) ¼ 0.025 Pd(F) ¼ 0.01 Pd(F) ¼ 0.012 Pd(S) ¼ 0.007 Pd(S) ¼ 0.005

Pd(F) ¼ 0.01 Pd(F) ¼ 0.002

Cont. flow: stands for continuous flow culture, line A in the experiment. Candle-jar method stands for line B in the experiment. This protocol is still in use.

Abbreviations: ts, time step; pmax, maximum invasion susceptibility, for healthy young RBCs; pmin, minimum invasion susceptibility for healthy old RBCs;

Pd(j) ¼ death probability per time step, for RBCs belonging to the j infection stage. T: tropozoite stage, S: schizont stage, F: fragmenter stage.

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459 455

3.3. Results

The current state of our work allows us to present amodel that is capable of reproducing the real behavior ofexperimental culture systems, based on a bottom-upapproach and a schematic, yet consistent, individualdescription.

Calibration of the program: Reproduction of a short-term culture that preserves malaria infected RBCs in vitro.The values for pdeath, pinf, tms, and the initial post-invasiontime distribution for infected RBCs are set to adjustexperimental data. We present the set initial values inTables 1 and 2. We show the simulation results in Fig. 3.The simulation results qualitatively reproduce the experi-mental behavior.

Validation of the program: Reproduction of long-term in

vitro cultures of parasitized RBCs. We maintained thevalues for most of the parameters, except those specified inTable 2. Fig. 4 shows the simulation output data comparedto the results of the two experimental culture methodspublished by Trager and Jensen (1976), a continuouslyrefreshed static culture (a–e), and a candle-jar method

culture (f–j). The simulation results reproduce the experi-mental data in terms of qualitative behavior and order ofmagnitude.

With slight modifications in the values of the parameters,we can reproduce the dynamics of many different realculture systems based on the candle-jar method, specificallythose performed by DDBG, in their routine protocols.Fig. 5 shows the simulation output data compared to

experimental in vitro cultures carried out under thestandard MR4 protocol conditions (MR4, 2004), wherewe again find a strong correlation between simulation andexperimental results.

Observed qualitative behavior: We found that glucose andlactate concentrations are not the limiting factor for theglobal behavior of in vitro cultures, at least whileperforming the cultivation under the usual experimentalconditions. Nor is there is any local limitation on thespread of the infection due to the diffusion limitations ofthese substances. The spreading of merozoites and invasionsusceptibility of RBC are thresholds for the evolution ofthe parasitaemia in static cultures. Once the localtransmission of the parasite is secured, the cultural limitingfactor is found to be the geometric long-term constraintson the spreading of merozoites, which depends on themacroscopic features of the experimental devices.

4. Discussion

We have developed a model that is able to reproduce themacroscopic behavior of a complex system such as in vitro

cultures.Guideline criteria for modeling: This is a first model to

approach the description of a complex system at amesoscopic scale. The criteria for accepting this modelare: (i) simplicity and consistency in the derivation of theparameters and processes, and directness of their relationwith the microscopic individual properties from which they

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Fig. 3. Dynamic behavior of simulations compared to experimental results published by Pavanand et al. (1974). Blue dots represent experimental results.

Black small dots represent simulation results. (a) Total parasite count. (b) Ring-stage parasite count. (c) Tropozoite-stage parasite count. (d) Schizont-

stage parasite count.

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459456

stem, and (ii) the likelihood of simulation outcomes toexperimental results.

These criteria are difficult to evaluate quantitatively, butthis does not mean that they cannot be assessed. Directnessof the model can be qualitatively measured by taking intoaccount (a) which parameters must be set in order to adjustsimulation results, and (b) to what extent the parametersthat govern the model are related to measurable individualproperties. The greater the number of ad hoc setparameters, the closer the model is to an artifact with nopractical applicability. The importance of parameters indetermining the system dynamics must be also taken intoaccount. We do this by evaluating the sensitivity of thesimulation outcomes to the variation in the ad hoc setparameters.

We want our simulations to qualitatively reproduce thereal behavior of experimental cultures. This means that wedemand that our model reproduce the dynamics of realsystems under similar conditions, and also keep theoutcomes at the same order of magnitude as in experi-mental systems.

Analysis of the model: Some values for the parametersand input variables that govern our model are taken fromthe accepted values found in literature, and some have beenfixed ad hoc to fit experimental results. Our model has twokinds of ad hoc set parameters: those concerning generalobserved behaviors of the culture systems, and thoserelated to the particular behavior of some cultures.The first kind of ad hoc parameters includes the

maximum time in culture for RBC survival, and theinvasion susceptibility. Both parameters suggest that theremight be some underlying mechanisms that govern thebehavior of the cultures, which have not yet been isolatedor described. IbM could provide appropriate tools touncover and determine these mechanisms.The second category of preset parameters consists of the

mortality for each infection stage, and the maximuminvasion probability for healthy RBCs. These parametersrefer to individual characteristics of a specific parasitestrain and blood sample under specific conditions, whichare different for each of the reproduced experiments. Weare currently working on an experimental assembly to

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Fig. 4. Output data from simulations compared to experimental results published by Trager and Jensen (1976). Blue dots represent experimental results.

Black small dots represent simulation results. (a–e): Continuous flow experimental method. (f–j): Candle-jar experimental method. (a,f) Total parasite

count. (b,g) Ring-stage parasite count. (c,h) Tropozoite-stage parasite count. (d,i) Schizont-stage parasite count. (e,j) Fragmenter-stage parasite count.

Graphs show similar dynamics although some punctual different data.

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459 457

indirectly measure them for technological applications. Thesensitivity of the simulation outcomes to the maximum andminimum invasion susceptibility of healthy RBCs is small:variations above 5% of the current values are required toobtain qualitatively different behaviors from the simulated

cultures. But the system sensitivity to individual fragility issignificant, and slight variations in the accidental deathprobabilities per time step produce completely differentbehaviors. A quantitative analysis of the sensitivity andsoundness of all the parameters used in the model should

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Fig. 5. Simulation output data compared to experimental results obtained from two different commercial culture mediums, performed according to the

MR4 protocol Week culture. Blue dots represent experimental results. Black small dots represent simulation results. (a) Sample of a synchronous culture.

(b) Sample of an asynchronous culture.

J. Ferrer et al. / Journal of Theoretical Biology 248 (2007) 448–459458

be performed in depth when dealing with a specific parasitestrain under controlled and well-known conditions, and itshould be addressed to predict the system response tospecific external manipulations. Meanwhile, such anexhaustive analysis would hardly contribute to deepenthe comprehension of the culture systems. The currentversion of the model was built to establish a usefulmethodology to describe in vitro cultures of P. falciparum

infected erythrocytes. No attempt was made to obtainspecific results for direct technological applications.

5. Conclusion

Microbial communities are well known at two levels ofdescription. Firstly, these systems are well described at acellular and molecular level; many publications havecontributed towards a thorough characterization of thehuman red blood cell and its interrelation with the malariaparasite. On the other hand, the system has also beenwidely studied at the macroscopic level. As a matter of fact,most experiments are carried out involving a great numberof cells. However, there is a gap between these two levels ofdescription. This bottom-up model is a working tool; it isfar from being consummate and does not pretend to becomprehensive. Nevertheless, it offers a start towardbuilding bridges between the macroscopic and microscopicapproaches. The aim of this work, then, is to improveholistic knowledge of the system.

Acknowledgments

The financial support of the European Social Fund andDURSI-Generalitat de Catalunya 2006FIC-336 and thePlan Nacional I+D+i of the Ministerio de Educacion yCiencia CGL 2004-01144 is acknowledged.

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