Consequences of celltocell Pglycoprotein transfer on acquired multidrug resistance in breast cancer: a cell population dynamics model
 Jennifer Pasquier†^{1},
 Pierre Magal†^{2},
 Céline BoulangéLecomte^{1},
 Glenn Webb^{3} and
 Frank Le Foll^{1}Email author
DOI: 10.1186/1745615065
© Pasquier et al; licensee BioMed Central Ltd. 2011
Received: 8 April 2010
Accepted: 26 January 2011
Published: 26 January 2011
Abstract
Background
Cancer is a proliferation disease affecting a genetically unstable cell population, in which molecular alterations can be somatically inherited by genetic, epigenetic or extragenetic transmission processes, leading to a cooperation of neoplastic cells within tumoural tissue. The efflux protein Pglycoprotein (Pgp) is overexpressed in many cancer cells and has known capacity to confer multidrug resistance to cytotoxic therapies. Recently, celltocell Pgp transfers have been shown. Herein, we combine experimental evidence and a mathematical model to examine the consequences of an intercellular Pgp trafficking in the extragenetic transfer of multidrug resistance from resistant to sensitive cell subpopulations.
Methodology and Principal Findings
We report celltocell transfers of functional Pgp in cocultures of a Pgp overexpressing human breast cancer MCF7 cell variant, selected for its resistance towards doxorubicin, with the parental sensitive cell line. We found that Pgp as well as efflux activity distribution are progressively reorganized over time in cocultures analyzed by flow cytometry. A mathematical model based on a Boltzmann type integropartial differential equation structured by a continuum variable corresponding to Pgp activity describes the cell populations in coculture. The mathematical model elucidates the population elements in the experimental data, specifically, the initial proportions, the proliferative growth rates, and the transfer rates of Pgp in the sensitive and resistant subpopulations.
Conclusions
We confirmed celltocell transfer of functional Pgp. The transfer process depends on the gradient of Pgp expression in the donorrecipient cell interactions, as they evolve over time. Extragenetically acquired drug resistance is an additional aptitude of neoplastic cells which has implications in the diagnostic value of Pgp expression and in the design of chemotherapy regimens.
Reviewers
This article was reviewed by Leonid Hanin, Anna MarciniakCzochra and Marek Kimmel.
Introduction
Current view on cancer
In essence, cancer is a proliferation disease affecting a genetically unstable cell population. The somatic mutation theory, which is the prevailing paradigm, stipulates that cancer arises from a stepwise accumulation of changes in genes that progressively drives subclonal neoplastic cells to evolve independently from the others, escaping from proliferation control and competing for space and resources and, finally, to kill the host. The key feature is that the cancerpromoting changes are intrinsic molecular events that are i) inheritable by daughter cells and ii) selectable, in the sense that they confer selective advantage in the cell population environment [1, 2]. The essential set of aptitudes that a cell must acquire to become malignant includes selfsufficiency of growth factors, resistance to death signals, evasion from apoptosis, limitless replicative potential, sustained angiogenesis, and proclivity for tissue invasion and metastasis [2]. It is also widely recognized that genetic instability results in the creation of diverse daughter cells, capable of interaction with the cell microenvironment, conferring to the tumour the properties of a heterogeneous tissue [3].
Within such tumour tissue, molecular alterations are considered to be somatically inherited by genetic, epigenetic (traits that are not dependent on the primary sequence of DNA) or extragenetic (posttranslational protein modifications that initiate and support positive feedback loops) transmission processes [1], leading to a cooperation of neoplastic cells [4].
Role of Pglycoprotein in breast cancer
Despite decades of research, breast cancer remains a major public health issue. Worldwide, breast cancer represents 23% of overall female cancers and accounts for 40,000 deaths each year in the United States alone. Currently, the lifetime risk of developing a breast cancer for women born in the United States is 1 in 8. For metastatic forms of breast cancers, the 5year relative survival is only 26% [5] and complete remission has been estimated as low as 3.1% after this period [6]. Response to firstline chemotherapies span from 30 to 65% and are followed, after a period varying from 6 to 10 months, by disease progression [7]. In fact, resistance to chemotherapy is believed to cause treatment failure in 90% of metastatic breast cancer patients [8].
Resistance to chemotherapy is related to the overall mechanisms that are involved in a decrease of drug efficacy against tumours [9]. Among these factors, proteins lowering the intracellular concentration of chemotherapeutics and belonging to the ATPbinding Cassette (ABC) transporters are wellknown for their specific responsibility [10]. These membrane proteins are characterized by their ability to efflux a large panel of both chemically and functionally unrelated compounds comprising potent cytotoxics currently used in chemotherapeutic treatments. Consequently, tumours overexpressing this type of energydependent pump have been very early identified on the basis of their MultiDrug Resistance (MDR) phenotype [11]. The Pglycoprotein (Pgp) was the first drugefflux protein characterized [12]. With the growing number of molecularly cloned ABC transporters (48 different genes up to now have been identified in the human genome), a rational nomenclature have been proposed. Metaanalysis [13] or immunochemistry studies [14] have determined that approximately 40% of all breast cancer tumours express ABCB1/MDR1 coded Pgp. ABCB1/MDR1 gene expression has a prognostic value for cell resistance to anticancer drugs [15, 16] and for treatment failure [17].
Genetic versusextragenetic resistance transmission
Until now, the development of multidrug resistance in neoplastic cells was explained as the consequences of two main mechanisms. On one hand, cells natively expressing drugefflux proteins retain their phenotype throughout the process of malignant transformation. On the other hand, in nonexpressing cells, chemotherapeutics have been shown first to induce Pgp expression and, second, to exert a selection of resistant cells during the course of chemotherapy [18–21]. Therefore, Pgp induction was considered to only depend on the cell type and/or the previous history of exposure to cytotoxics. Biomathematical models have been proposed to describe the kinetics of Pgp induction as a function of tissue cytotoxic concentration [22] with the objective of adapting the time course of drug administration to overcome as much as possible multidrug resistance.
Recently however, extragenetic transmissions of multidrug resistance have been reported. These unexpected events all involve direct transfers of Pgp to ABCB1/MDR1 nonexpressing recipient cells from various multidrug resistance donors, namely adherent cell lines in vitro and in vivo [23], stromal cells isolated from patients suffering of ovarian cancers [24] and lymphoblastic leukemia cells in suspension [25]. These findings actually add a new modality of resistance appearance, and possibly spreading, in a population of tumour cells. Theoretically, celltocell Pgp transfers in conditions of antineoplastic treatment could confer a significant advantage to sensitive cells, keeping them alive long enough to produce their own Pgp copies, under the induction process, and thus, to resist chemotherapy. However, the real dynamics of extragenetic acquired resistance in a growing population of cancer cells has remained, until now, unknown. Herein, we propose a biomathematical model, derived from a previous analysis [26], supported by experimental parameters and involving processes of i) cell proliferation and death, ii) Pgp induction and degradation and iii) celltocell Pgp transfer, to help answer this question. A parallel study in progress will present experimental data on mechanics of transfer [27]. The present model was supplied by new data obtained in vitro from MCF7 human breast adenocarcinoma, a cell line that has previously been shown to be capable of intercellular Pgp transfer [23]. The main objectives of the present work were to investigate the influence of both cancer cell line intrinsic factors (growth rate, initial proportions of sensitive and resistant cells, and Pgp transfer rate), as well as treatment management parameters (i.e. cycles of drug administration), on overall multidrug resistance, both genetically and extragenetically transmitted.
Results
MCF7 proliferation and resistance
In 2005, Levchenko et al. have shown that several drug sensitive cell lines, including the wellestablished human breast adenocarcinoma cells MCF7, are able to acquire multidrug resistance extragenetically, via a direct intercellular transfer of the resistance protein Pgp from a derived mdr1transfected cell line. In the present work, the consequences of Pgp transfers in MCF7 were studied by using, as a Pgp donor, a cell line variant obtained by selection for resistance to doxorubicine (Doxo) and therefore called MCF7/Doxo. MCF7/Doxo have previously been shown to overexpress Pgp, to be resistant to doxorubicin and to efflux Pgpsubstrate fluorescent probes in a verapamil and cyclosporine Asensitive manner [28].
Transfers of Pgp and efflux activity in cocultures
To investigate whether Pgp transfers generate a functional drug resistance, the ability of cocultured cells to efflux calceinAM, a Pgp substrate probe which is a precursor of the fluorescent dye calcein, was assayed again by flow cytometry. In these experiments, cell fluorescence accumulation is therefore an inverse function of Pgp activity. Symmetrically to what observed with membrane Pgp content, the peak of fluorescence corresponding to initially sensitive cells, with a high calcein concentration, was also progressively shifted, but to the left, towards regions of less fluorescence accumulation (Figure 3B). The fluorescence values decreased from 2553 ± 227 (n = 9) at day 0 to 322 ± 31 (n = 21, mean ± S.E.M.) at day 6 of cocultures. These cells therefore acquired an aptitude to efflux the fluorescent dye. However, by contrast with Figure 3A, the activity distribution is additionally characterized by the appearance of a third subpopulation displaying an intermediate efflux activity. This middle peak increased in amplitude with time to reach 32.5 ± 1.6% of the cells at day 6 of cocultures (n = 21, mean ± S.E.M.). Concurrently, the peak amplitudes corresponding to the subpopulations of MCF7 with higher and, especially, lower calcein concentrations slightly declined. Interestingly, the total mass of activity (Figure 3D) also remained constant with time.
Taken together, these results strongly suggest the occurrence of a mechanism of transfer i) involving only a small fraction of the membrane content in protein of the donor cells but conferring a significant efflux activity to the recipient cells and ii) exhibiting a continuous exchange mode simply dependent on the gradient, i.e. the net difference in Pgp expression levels separating the cells under concern. In order to test these hypotheses and to evaluate their consequences in the dynamics of multidrug resistance within a cell culture, a mathematical transfer model was developed.
Estimation of transfer parameters
To quantify the transfer of Pgp activity, which is assumed to result in a gain of resistance, we have used a mathematical model, presented in the material and methods section and derived from Hinow et al. (2009). In the absence of cytotoxics and according to Figure 1, the growth rate is independent of the Pgp content. Herein, the growth rate was thus assumed to be identical for MCF7 and MCF7/Doxo in coculture (ρ = 0.63 day^{1}).
The transfer rate τ, the transfer efficiency f, and the thresholds δ_{min} < δ_{max} were estimated by using intensive parameter estimation computations. The conditioned flow cytometry activity distribution (see material and methods) at day 0 was used as an input for the numerical simulation of the model.
List of the model parameters, their significations, values and symbols
Symbol  Interpretation  Value  Units  Method 

ρ  Growth rate of cells in absence of drug  0.63  day^{1}  measured 
c  Drug concentration  0 or 1  μM  set 
τ  Rate of transfer of Pgp activity  0.4  day^{1}  fitted 
F  Activity transfer efficiency  0.2    fitted 
δ_{min}  Minimum threshold for transfers  10^{1.4}  fluorescence unit  fitted 
δ_{max}  Maximum threshold for transfers  10^{1.9}  fluorescence unit  fitted 
Fitted parameters seem to indicate that some limitations, or constraints, occurred for transfers between cells in our experimental conditions. First, the transfers were not homogeneous over the whole assortment of activities expressed by cells within the cocultures. In this respect, a threshold δ_{min} distinct from 0 suggests that transfers did not occur or were not detected when Pgp activities of the cells were to close. In addition, since the threshold δ_{max} is not equal to the maximal value 10^{4}, transfers did not arise either in case of extreme activity differences. This implies the occurrence of a permissible range of activities governing the optimal transfers between donor and recipient cells. Moreover, a transfer rate of τ = 0.4 means that transfer events occurred in average every 1/ τ = 2.5 days for each cell in our experiments. Finally, the quantity of transferred activity globally corresponds to a fraction f = 0.2 of the difference between the activity of the donor and recipient cell.
Consequences of resistance activity transfers on model response to chemotherapeutic cycles in cocultures
Fast acquisition of Pgp activity by doxorubicinsensitive MCF7 in a population of growing tumoural cells may be of first relevance to determine the dynamics of response to chemotherapy. Thus, to test the significance of Pgp transfers within a coculture of sensitive and resistant cells, simulations were carried out by using our mathematical model of activity transfers, fed with biological data.
Herein, the simulations were built with basic parameters determined in vitro, in particular the growth rates of monolayers and the transfer parameters. Therefore, cell proliferation was artificially high in abstraction of any physiological environment and the growth curves obtained by simulation should not be considered as data clinically exploitable. Likewise, practical aspects of treatment administration involving pharmacokinetics of doxorubicin (such as elimination halflife, drug clearance, volumes of distribution) were ignored. The simulated treatment was simplified into a bolus of 1 μM doxorubicin that reaches the maximum concentration in the extracellular compartment with no delay and maintains the same level over 10 days. Then, the treatment was interrupted for 4 days (i.e. doxorubicin concentration is zero). The treatment cycled every 14 days for 4 cycles.
In the absence of transfer (Figure 8A&8C), resistant cells grew linearly, and represented after few days the majority of the cells in the cocultures. The number of sensitive cells decreased during the treatment and increased during the interruption phases. In these simulations, the initial distribution of sensitive cells was based on low efflux activities analyzed by flow cytometry. As a consequence, the population of sensitive cells did not correspond to a single ray but was rather represented by a bell curve shape encompassing a spectrum of efflux activities, from zero to several arbitrary fluorescent units. Consequently, from a cycle to another, the sensitive cells having the less negative growth rates were selected by the treatment. For that reason, the population of sensitive cells decreased to a minimum and, then, increased again by growing during treatment interruptions while the growth rates converged to zero during the doxorubicin phases.
In the presence of transfers (Figure 8B&8D), results differed from above and were greatly influenced by the initial number of resistant cells in the inoculum. In fact, with transfers during treatment, growth rates of resistant and sensitive cells converged to an intermediate value, due to a loss of activity by resistant combined with a gain of activity by sensitive cells. With initial conditions corresponding to 0.1% or 1% of resistant cells, after an initial phase during the first treatment, the growth rates of the cell population tend to become analogous. The main observation is that transfers significantly decreased the growth rate of the whole cell population exposed to cytotoxics, especially when the coculture is initiated with a relative large proportion (1%) of resistant cells.
Discussion
In the present work, we observe effective transfers of Pgp and efflux activity from a Pgp overexpressing MCF7 human breast cancer cell line variant, selected for its resistance towards doxorubicin, to the parental sensitive cell line. Our results thus confirm the work of Levchenko et al. (2005) conducted in vivo and also in vitro on several adherent cell lines, including MDR1transfected MCF7 [24]. Functional transfers of Pgp have also been described very recently between leukemic cells cultured in suspension [25]. In addition, the possibility for a malignant cell of inheriting Pgp from a normal cell was shown by Rafii et al. in a study where sensitive ovarian cancer cells acquired Pgp by interaction with particular stromal cells isolated from the peritoneal cavity of patients[24]. To our best knowledge, the present work and the three above cited references are the first reports describing extragenetic direct celltocell transfers of Pgp. Among these lines, it should be mentioned that an intercellular transfer of the drug resistance phenotype was first reported indirectly in a pioneering study of Frankfurt et al. [30].
The present data highly reinforce the fact that the Pglycoprotein, a 170 kDa polypeptide consisting of 1280 amino acids that spans 12 times the cell membrane [31], can be exchanged from a donor cell to a recipient one, keeping its ability to efflux drugs and, thus, conferring the MDR phenotype. In addition to the discovery of the ability, for cells, to transfer huge integral membrane proteins, these important findings also reveal the occurrence of an additional extragenetic mechanism to acquired chemoresistance, in the absence of pressure selection, in vitro. The possible occurrence of such Pgp transfers in vivo may be crucial to understand Pgp expression in breast cancers. In this context, it should be remarked that data concerning Pgp expression in human breast tumours are known to suffer of considerable variability across studies, as revealed by a compendium based on 31 articles compiling results obtained mainly by immunodetection methods onto 1,200 breast carcinomas [13]. The discrepancies in the literature were ascribed in a large extent to a lack of rigorous methodology, preparation of samples as well as antiPgp antibodies varying from a study to another [32, 33]. Some immunohistological studies carried out in normal breast collected from healthy donors failed to find Pgp expression [34, 35]. However, by using a more sensitive approach, namely RTPCR research for Pgp transcripts, low but significant levels of Pgp mRNAs have been detected in normal tissues adjacent to chemotherapynaïve tumours [36]. Importantly, low basal levels of Pgp efflux activities have been shown to be sufficient enough to contribute extensively to drug resistance, in vitro [37]. In this context, the levels of Pgp activities heritable by Pgp transfers and found in the present study are largely compatible with a resistance increase in vivo. The timecourse of Pgp transfers, detectable between the third and the fourth day in coculture, is also short enough to overcome the effects of cytotoxics used in chemotherapy.
The particular spatial organization of MCF7 in cocultures has certainly severe consequences on the contact surfaces between sensitive and resistant cells. As seen under microscopic observation, cells were not organized as a homogeneous mixture of wellseparated cell variants, but rather in delimited islets of sensitive settling aside from resistant cells. As a result, transfers between the most separated populations in term of chemoresistance (full sensitive versus full resistant cells) may occur only at the frontier of islets in case of celltocell contactdependent transfers [23, 24], keeping in mind that transfers can also arise between cells of intermediate Pgp levels. From this, it is assumed that the transfer process is a rather complicated phenomenon in which spatial structures bring limitations. Such spatialconstraint or regulated Pgp transfers may provide an additional rationale to the heterogeneity of Pgp expression previously observed in studies carried out on heterogeneous tumour samples [13, 33]. Additional studies focused on mechanisms of Pgp transfers are needed to go further concerning that point.
As a matter of fact, our approach to quantify the parameters of transfers by using biomathematical modelling, points out some constraints, revealed by the parameters fitted to the biological data. The mathematical model has been constructed to i) be applicable to a population of interacting cells in proliferation, ii) to consider the cell population as a continuum density structured by the quantity p of transferable Pgp activity and iii) to have rules governing transfer. The rules of the transfer process have been delineated and discussed in detail in a companion paper [26]. The better fit was obtained by using the activity transferred from resistant to sensitive cells, undirected and directed shifts being neglected. Herein, we consider that Pgp activity is a better estimation of therapy resistance as compared to Pgp expression, because activity is directly responsible for resistance to cytotoxic drugs. Moreover, this choice allows us to use directly the model to simulate the response of a coculture to chemotherapeutic cycles (see below). Nevertheless, a correlation between Pgp levels and activity has been described in several cell lines [15, 38, 39]. Parameters fitting revealed quantitative aspects of Pgp activity transfer, in particular a transfer rate τ corresponding to one event every 2.5 days, a value to be compared to the doubling time of MCF7 close to 1 day. In addition, the calculation of the average activity and the average level of Pgp expression over the time in a coculture showed that theses values remain stable throughout the experiments. We conclude that there is neither significant gain nor loss of efflux activity and Pgp expression during the coculture experiments. As a consequence, the changes in the fluorescence distributions (for both Pgp and activity) can be attributed to transfers, which are not rare phenomena compared to the cell cycle duration. However, in a culture of growing cells transferring Pgp, the topology of the exchange network is constantly changing. This aspect is not taken into consideration in the transfer model which does not adapt rules of transfer as a function of time. The estimation of the transfer rate τ is therefore a coarse estimation that should be reevaluated in future works including spatial dynamics.
The model gives also a transfer efficiency f = 20%, corresponding to the amount of activity transferred each time with regard to the absolute activity difference between the donor and recipient cells, and two transfer thresholds, that can be viewed as bottom and top activities delimiting authorized transfers. These parameters can be related to the life span of potential transfer vectors, that are constructed and destroyed for instance, or to the size of single Pgp cargos. Concerning this point, it has been shown that high density Pgp microdomains alter local properties of lipid membrane environment and promote Pgp clustering [40]. In this respect, biological data as well as transfer parameter f suggest an organized quantal transfer of Pgp, in which the minimum unit of exchange would probably consist in a cluster of several copies of the transporter. Several intercellular carriers have been proposed to be involved in Pgp transfers. In adherent cells lines, close contacts between donor and recipient cells have been shown to be required [23, 24]. In the other, in a model of liquid tumour, Pgp containing microparticles released by resistant cells and bound to sensitive cells, have been isolated [25]. The possibility that both mechanisms actually coexist in our cell line is examined in an independent study [27].
The model was used to simulate and to quantify the consequences of Pgp activity transfers on the cell distribution within a growing coculture during exposure cycles to cytotoxics. More precisely, the model was fed with biological data corresponding to sensitivity of MCF7 variants to doxorubicin. Doxorubicin is among the preferred agents used in chemotherapy regimens for preoperative breast tumour reduction, adjuvant treatment after a first line surgical lumpectomy/mastectomy and radiotherapy of invasive breast carcinoma or systemic treatment of recurrent or metastatic breast cancers [41]. Doxorubicin is administrated alone or in combinations with other cytotoxics. In the present simulations, the concentration of doxorubicin was set at 1 μM, a value corresponding to the weekly supplementation of MCF7/Doxo culture medium. This concentration is also close to the levels of doxorubicin found in tumours under chemotherapy, reported to vary in the 100,1 μM range depending on the time separating the administration of the drug to the measures [42–44].
The major consequence of Pgp activity transfers is that subpopulations of sensitive and resistant cells were no longer compartmentalized. Via Pgp activity transfers, flows of cells depopulate and repopulate resistant and sensitive cells. This was obvious once the second cycle of treatment began, where the population of sensitive cells increased by adjunction of cells coming from resistant cells, which lose Pgp activity. Conversely, sensitive cells acquired Pgp activity and, thus, developed progressively positive growth rates during the treatment phases. When the number of donor cells was increased up to 1%, the resulting exchanges of Pgp caused a significant loss of activity for the resistant cells. Acquisition of Pgp activity was therefore slowed down for sensitive cells. As a result, in that case, the numbers of sensitive and resistant cells within the coculture converged (became close) with time and the overall growth rate of the cell population was reduced (compared to all the other situations), giving a coculture with fewer cells at the end of the four chemotherapeutic cycles. In our model, the gain of Pgp is central for survival and growth of non Pgp expressing cells whereas the loss of Pgp by resistant cells has predominant effects on the overall growth rate of the simulated tumour. We also conclude that the initial proportion of resistant cells in a population has a great influence on the growth of the whole population in the presence of transfer.
The present study sheds new light on fundamental features underlying extragenetic acquisition of multidrug resistance in vitro, with a particular attention given to quantitative aspects. Globally, Pgp transfers led to an integration of the responses to the drug across the cell population and a 'collective' phenotype different than the sum of parts. However, MCF7/Doxo represents an extreme case of Pgp overexpression and fast growing cells. Future work should be conducted with cells expressing intermediate amounts of efflux activity, since transfers have been demonstrated to be dependent on the Pgp levels [23]. Moreover, an improved mathematical model taking into account evolving 2D or 3D spatial considerations has to be developed together. Physiopathological implications of celltocell protein transfers may be crucial, especially if other membrane proteins and various cell types are concerned.
Material and Methods
Cell lines
The cell lines used in the present study were the wildtype drugsensitive human breast adenocarcinoma MCF7, purchased at the American Type Culture Collection (LGC standards, Molsheim, France), and a multidrug resistant MCF7/Doxo variant, kindly provided by Pr. J.P. Marie (Hôtel Dieu, Paris, France). The cells were grown in RPMI 1640 medium supplemented with 5% heatinactivated fetal bovine serum, 2 mM Lglutamine, and 1% antibiotic/antimycotic solution. Doxorubicin (1 μM) was added to the culture medium for the maintenance of the multidrug resistant phenotype of MCF7/doxo cells. Cultured cells were incubated at 37°C under a watersaturated 95% air5% CO_{2} atmosphere. Exposure of MCF7/Doxo to the Pgp inhibitors verapamil or cyclosporine A dosedependently abolished both resistance to doxorubicin and Pgp activity in MCF7/doxo [28].
For proliferation assays, cells were plated at 10^{5} per 60 X 15 mm tissue culture dish with or without doxorubicin. Cells were dissociated by treatment with trypsin/EDTA and then counted in a Malassez chamber. Cell proliferation was followed every 12 hours during six days.
Analysis of Pgp expression and activity by flow cytometry
Cultured or cocultured MCF7 were trypsinresuspended and washed with HBSS before analysis. For intercellular Pglycoprotein transfer studies, Pgp was labelled using a phycoerythrin (PE)conjugated UIC2 mAb (Beckman Coulter France, Villepinte, France). The fluorescent light (FL) was quantified using a Cell Lab Quanta SC MPL flow cytometer (Beckman Coulter) equipped with a 22 mW 488 nm excitation laser. Voltage settings of photomultipliers were not modified throughout the experiments. Each analysis consisted in a record of 10 000 events, triggered on electronic volume (EV) as primary parameter, according to a particle diameter exceeding 8 μm. Red fluorescence was measured in FL2 channel (log scale) through a 575 nm band pass emission filter. More than 93.1 ± 0.4% (mean ± standard error of the mean) of gated events exhibited a FL2 > 1 for MCF7/Doxo.
To study Pgp activity, resuspended cells were loaded with 0.25 μM calcein acetoxymethylester (Invitrogen Life Technologies, Carlsbad, CA) in RPMI for 15 min at 37°C in the dark. Green FL was quantified via the FL1 channel (log scale) through a 525 nm band pas filter. Controls of MCF7, MCF7/Doxo and extemporaneous mixtures of 50:50 MCF7: MCF7/doxo were analysed before cocultures, in all experiments.
Analysis of Pgp by flow cytometry transfer in tagged MCF7
75%confluent sensitive MCF7 were loaded with 25 μM of the cellpermeant reactive fluorescent dye CellTracker Blue 4chloromethyl6,8disulforo7hydroxucoumarin (Invitrogen Life Technologies, Carlsbad, CA) for 2 hour at 37°C in 25 cm² culture dishes. The cells are then washed twice with HBSS and covered with fresh complete culture medium. Either CellTracker Blue loaded sensitive MCF7 (ctbMCF7), or MCF7/doxo, or a carefully homogenized 50:50 mixture of ctbMCF7:MCF7/doxo were plated at the desired density on 60 mm culture dishes and analysed by flow cytometry after labelling using the PEconjugated UIC2 mAb, as described before.
Flow cytometry data conditioning and normalization
Accurate fitting of the mathematical model to flow cytometry allevents histograms required raw data noise reduction. Thus, listmode data were segregated into binned logarithmic histograms, with a bin width of 0.05 in log scale of arbitrary fluorescence units. The distributions were normalized, by scaling to an integral area of 1.0, in order to obtain a density of probability.
Mathematical model of Pgp activity transfer
In this section, we describe a mathematical model that has been developed and used to describe the consequences of a celltocell Pgp protein transfer in terms of multidrug resistance activity. Herein, the resistance activity is viewed as the ability of a cell to efflux cytotoxics, which is considered to be correlated to the number of Pgp copies within the cell membrane.
Our modeling approach views the transfer of Pgp proteins between cells as a stochastic process in which individual transfer events are represented in a continuum population density function u(t,p), where t is time and p is the quantity of Pgp on individual cell membranes. The density function u(t,p) is the solution of an integropartial differential equation of Boltzmann type, which describes an evolving population behavior as an average of individual transfer events governed by specified rules. The aggregate transfer behavior is modeled as an integral operator, with a kernel specified by the transfer rules, acting on the continuum density. We assume individual transfer events arise randomly (Rule A1), occur at a specified average rate (Rule A2), and are delimited by parameters specific to allowable quantities of Pgp transferred between paired cells (Rules A3 and A4) (see below). The large number of cells and proteins involved justifies our assumption that a continuum population density function describes the dynamic transfer processes.
Rules for the transfer model:

(A1) The probability that a pair of individuals is involved in a transfer event is independent of their Pgp values, and the pairing is chosen randomly from all individuals.

(A2) The time between two transfer events follows an exponential law with mean τ1 > 0 (alternatively, τ is the rate of transfer per unit time).

(A3) Let 0 < f < 1 (we call f the transfer efficiency). A transfer between two individuals, one with value Pgp1 and the other with value Pgp2, results in a transfer of the fraction f of their difference f Pgp1  Pgp2 gained by the smaller and lost by the larger of the two.

(A4) An exchange takes place only if the difference of the transfer pair Pgp1 and Pgp2 in assumption (A3) is > δ (we call δ ∈ (0, 1) the transfer threshold).
We decompose the construction of the model by considering separately the process of i) cell proliferation and death, ii) Pgp induction and degradation rate, both, at the cell and the population level, and iii) Pgp transfer between cells. We consider p the Pgp efflux activity of a cell, functionally measured by the loss of calcein fluorescence for a resistant cell compared to a sensitive one, and we introduce u(t,p) the density of cells at time t having a Pgp activity p. In the sequel, for a given cell, the fluorescence p is assumed to be a function of Pgp expression on the cell surface. This function is strictly increasing.
where $\overline{\varphi}$ extends φ by 0 outside of the interval (1,10^{4}). The main idea in the construction of the transfer operator T is to derive the probability of a recipient partner to acquire a level p of Pgp in a transfer event from all possible donor partners. To describe a transfer event we use the following rule: if a cell C_{1} and a cell C_{2} have, respectively, a quantity p_{ 1 } and p_{ 2 } of Pgp activity before transfer, then, after transfer, C_{1} (respectively, C_{2}) will have an activity p_{ 1 }f(p_{ 1 }p_{ 2 })(p_{ 2 }p_{ 1 }) (respectively, p_{ 2 }f(p_{ 1 }p_{ 2 })(p_{ 1 }p_{ 2 })). So the fraction transferred is f(p_{ 1 }p_{ 2 }), which depends on the absolute value of the difference between p_{ 1 } and p_{ 2 } (the distance between the Pgp activities of C_{ 1 } and C_{ 2 } ). The model is used with constraints that correspond to a admissible interval (δ_{ min, } δ_{ max } ) for p_{ 1 }p_{ 2 } to allow transfer. As a consequence, when p_{ 1 }p_{ 2 } < δ_{ min } or p_{ 1 }p_{ 2 } > δ_{ max } there is no transfer (i.e. f(p_{ 1 }p_{ 2 }) = 0) and otherwise a constant fraction is transferred (i.e. f(p_{ 1 }p_{ 2 }) = σ). Therefore, the parameter 0 < σ < 1 is called the transfer efficiency, and 0 ≤ δ_{ min } < δ_{ max } are called the transfer thresholds.
In the above model, the cell distribution of Pgp is assumed to be homogeneous in space. In practice the spatial distribution of Pgp can play an important role in the transfer process. Here we neglect this aspect. This problem will be investigated elsewhere.
Reviewers' Comments
Reviewer 1: Leonid Hanin, Idaho State University, Department of Mathematics, United States of America
This is an important and wellwritten paper that deals with the efflux Pglycoproteins (Pgp), exchange of Pgp between cells including a mathematical model of this process, and the role of the Pgp transfer in bringing about extragenetic resistance to chemotherapeutic treatment of breast cancer.
1. General Comments
1. I think more biological discussion of Pgp activity and the nature of Pgp transfer is warranted. In particular, what is the time scale for transfer processes and is the level of cellular Pgp activity heritable?
Quantitative values, relatively to arbitrary fluorescent units given by flow cytometry, are now added in the results in term of heritable cellular protein and activity, as well. This question and the point concerning the time scale for transfers are also discussed more extensively.
2. To put formation of chemotherapeutic drug resistance into a more practical perspective, it would be helpful to mention whether normal breast tissue cells display Pgp based efflux activity and whether Pgp exchange may occur between normal and malignant cells.
The one and only article showing a possibility of Pgp transfers between normal and cancer cells is the report of Rafii et al. (2008) detailed at the beginning of the discussion. The possible basal Pgp activites in normal breast and its consequences in terms of MDR are discussed p18.
3. Pp. 78. The mathematical model is introduced too abruptly. I believe the authors should write a paragraph or two about underlying general assumptions, the "big" stochastic picture behind their model, explain the formula for the transfer operator including the role of normalization in more detail, and give stochastic interpretation of coefficient τ.
A part concerning data conditioning and normalization has been added p7 in Methods section. We have also modified the presentation of the model, recalling the precise assumptions for the transfer process.
4. P. 8. The paper deals with the model of Pgp transfer with ε = h = 0. Given the predominantly biological readership of the journal, I think it would be better to omit these terms in the model and refer the reader to paper [20] for a more general model accounting for diffusion and drift.
We agree with the referee, and we modified the system accordingly to the comment.
5. Paper [20] establishes the existence, under certain simplifying assumptions, of the limiting equilibrium distribution of Pgp activity. What is it in a slightly more general case of two thresholds considered in the present work (in the absence of cell proliferation)? How fast is this equilibrium reached? Is this relaxation process slower or faster than cell cycling?
The two threshold case was in fact also considered in the paper by Hinow et al. Up to now the authors did not study the speed of convergences to the equilibrium distributions (which is indeed a difficult question). In the present model the cell cycle is simplified but includes the growth rate of cells (neglecting the oscillation induced by the cell cycle). This problem will be considered in future works.
6. What are the relative effects of Pgp transfer and cell proliferation on chemotherapeutic activity of DOXO? One can compare survival of twocompartment tumor cell population exposed to DOXO under two scenarios: (1) no cell proliferation, just Pgp exchange and (2) cell proliferation alone assuming equilibrium distribution of Pgp activity. This would give some idea about the net effect of Pgp transfer.
Indeed we did not consider the point (1) since in cancer the case with no cell proliferation does not arise. Point (2) corresponds to Figure 8A&8C, where cell growth of a twocompartment tumor cell population was plotted without transfer.
7. It is not enough to say that model parameters were "estimated by using intensive parameter estimation computations". Which experimental data were used for model fitting? How was the method of least squares implemented? What weights were used, if any? Also, what are the confidence limits or standard deviations of model parameters? If the confidence intervals are too wide the reported parameter estimates may be misleading.
This point has been detailed.
8. References [22–44]46] were never mentioned in the text. Am I missing a supplement to the paper?
Corrected.
2. Technical Comments
1. P. 7, last line. "... fluorescence p is assumed to be a function of Pgp expression on the cell surface." It is important to emphasize that this function is strictly increasing.
OK, this has been added.
2. P. 9, lines 67 from below. Shouldn't the sign of the transfer terms be plus in both cases? Otherwise, if p1 < p2 then the cell with smaller activity loses it even further while the cell with larger activity gains it.
The signs are correct in the paper, and the part has been carefully explained in the companion paper of Hinow et al.
3. P. 9. What is the biological basis of the two thresholds, δmin and δmax?
The mentioned thresholds are a conjecture introduced to investigate whether biological data contained constraints in the transferred quantity when fitted with the proposed model. See answer to general comment 3.
4. P. 14, lines 67 and p. 18, lines 23. Please explain in more detail the interpretation of coefficient τ. Is Pgp transfer thought of as some kind of Poisson process? See also general comment 3.
Yes, the time between transfers is assumed to follow an exponential law. This part is now contained in the paper; see assumption (A2) page 8.
5. P. 15, lines 57, Figure 8B and p. 26, legend to Figure 8B. Where does the function ρ(p) displayed in Figure 8B come from? Is it determined experimentally?
This function has been established empirically given that it is known that resistance increases as a function of efflux activity which depends on the expression of Pgp (see ref 15, 39 and 40). However, there is no precise study linking the number of Pgp copies at the membrane and the efflux activity. It has to be considered that Pgp pumping is regulated in the minute range, at least by the phosphorylation state of the protein. As a consequence for a given cell at a particular time, a heterogeneous population of Pgps, with pumping activities varying from 0 to a maximal rate, can coexist in the membrane.
6. P. 24, legend to Figure 3, lines 12 from below. Where are these "vertical solid lines"?
It was corresponding to an older version of the figure. This sentence has been removed.
7. P. 25, legend to Figure 5. Please explain the role and selection of quadrants in more detail.
This whole legend has been rewritten for clarification.
8. P. 25, legend to Figure 6. What are the cell volumes matched to? Give the matching volumes or a reference.
This whole legend has been rewritten for clarification. This particular issue is developed in the result section p12 lines 15 from below
9. P. 27. Check the estimates for δmin and δmax. Shouldn't they be 1.4 and 1.9? What are the units?
It is correct in the paper. The confusion comes from the fact that we used for numerical simulation log(p) in Figure 6(as well as for the numerical simulations) while in Table 1the result is expressed only using p. Figure 6has been corrected.
We thank the referee to point out this problem!
1. P.35, Figure 7. Fluorescence counts are reported on the log scale. Were parameters estimated on the same scale? Changing scale may affect the estimates of model parameters.
See the answer of point 9 above.
3. Stylistic Comments
1. P. 5, line 10. The paper denoted [ref] is not specified.
This work is submitted. The given reference is related to an oral communication based on the study.
2. Use a uniform name for the cell line MCF7/DOXO. In some places in the text it is called MCF7/doxo, see e.g. pp. 6 and 7.
We harmonized the name of cell line to MCF7/Doxo throughout the text
3. P. 6, line 2 from below. What is S.E.M.?
It is the standard error of the mean, now specified in the text.
4. P. 8. The role and nature of coefficient (or function) h was never explained.
All coefficients, including coefficient h, are exposed in table 1.
The function h has been removed.
5. P. 8, line 7 from below. "describe" should be "describes".
Ok, this has been changed
6. P. 8, last line. Delete "of" in "transfer of operator T".
Ok, this has been deleted
7. P. 14, line 6 from below. What is "volumes of distribution"?
The volume of distribution is a pharmacological concept used to quantify the distribution of a drug between plasma, extracellular fluid, intracellular fluid and other compartments of the body after administration. This concept is useful for studies carried out in vivo.
8. P. 18, lines 45 from below. The phrase "biological data as well as modelized transfer parameters suggest an organized quantal trogocytosis of Pgp" is very obscure. What is the meaning of the words "modelized" and "quantal"?
The sentence as been modified as following: In this respect, biological data as well as transfer parameter f suggest an organized quantal transfer of Pgp, in which the minimum unit of exchange would probably consist in a cluster of several copies of the transporter.
9. P. 19, line 4. Shouldn't "administrated" be changed to "administered"?
Ok, this has been changed.
10. P. 19, lines 78. What does it mean "separating the injection to the measures"?
In some studies, the effective doxorubicin concentration in the tumor environment is followed after intravenous administration of a dose of the drug. Given that doxorubicin is distributed in several physiological compartments, metabolized and excreted (pharmacokinetics), its concentration in the tumor decreases over time and is all the more low than the time separating the i.v. infusion to the measure is high. The sentence has been slightly modified.
11. P. 20, ref 5. Volume number and pages are missing.
Ok, Volume number and pages have been added.
12. P. 25, line 9. DE should be DF.
Ok, this has been changed.
13. P. 25, line 10. Change "analysis" to "analyses".
Ok, this has been changed.
The recommendations of the referees were most helpful to us in the improvement of this manuscript.
Reviewer 2: Anna MarciniakCzochra, University of Heidelberg, Institute of Applied Mathematics, Heidelberg, Germany
The paper is devoted to an interdisciplinary study of the dynamics of cancer cell drug resistance induced by the expression of the drug efflux protein, Pgp. A mathematical model is developed to investigate and validate the hypothesis of celltocell transfer of resistance. The model describes the dynamics of cancer cell population structured by the level of Pgp activity corresponding to the resistance level. It is expressed in the form of a diffusion  drift equation with a nonlocal reaction term describing the transfer between cell subpopulations having different levels of Pgp activity. The model is calibrated and validated based on in vitro experiments with cocultures of drugresistant and drugsensitive human breast carcinoma cells. The model is interesting and shows the evolution of the resistance structure in cancer cell population. Model predictions are compared to experimental observations and biologically relevant conclusions are drawn.
I recommend the manuscript for publication in Biology Direct after minor revisions.
I think that for completeness of paper it would be good to clarify:
1. What is the function h(p)?
This question is related to a comment of Reviewer 1: "The paper deals with the model of Pgp transfer with ε = h = 0. Given the predominantly biological readership of the journal, I think it would be better to omit these terms in the model and refer the reader to paper [20] for a more general model accounting for diffusion and drift".
We agree with both referees, and we modified the system accordingly to the comment.
2.Why is spatially homogeneous approach correct? Are there any mixing conditions imposed in the experimental setting? Or it is just a first approach ? Do the authors expect that including spatial effects will influence the results? It would be good to comment on this issue, when model is introduced.
Of course, this is a first approach. From a biological point of view, there is a lot to do in order to give an underlying mechanism for Pgp transfers and several directions have to be explored. Another manuscript will be submitted soon concerning mechanistic. A comment as been added p10 on this issue.
3.How important is the "diffusion" process included in the model? How large is
ε? Figure 7 showing that there exist no cells with some small levels of Pgp
activity suggests that such diffusion process is very small, if any.
The term ε has been removed
Minor remarks
1. The quality of Figures 4, 6, 8, 9 is not satisfactory, in particular fonts are too small.
Quality of mentioned figures has been enhanced.
Figure 9 is now Figure 8
2. The description of Figure 3 is confusing; what is (C) and (D)?
It was corresponding to an older version of the figure. This whole legend has been rewritten for clarification.
3. On p.5 the reference is missing
This work is submitted. The given reference is related to an oral communication based on the study.
4. P. 11, line 8 from below: it is written that the peak of activity is shifted to the left towards the region of higher activity. Is it correct? It seems that "left" means "lower activity".
The description is correct. In fact, in these experiments, the cells were loaded with a fluorescent dye that is a substrate of the efflux pump protein. As a consequence, cells expressing a low activity level accumulate intracellular fluorescence and are located in the right part of the graph after analysis, in region of high fluorescence light (FL). On the contrary, cells exhibiting higher activity levels actually efflux the dye and are shifted to the left, a region corresponding to less intracellular fluorescence and thus to higher pumping activity. This is given p12, line 5 from below in the revised version of the manuscript.
Reviewer 3: Marek Kimmel, Houston, United States of America
The paper concerns the important mechanism of horizontal transfer of agents of multiple drug resistance among cancer cells. Based on their own experiments and published data, the authors designed a mathematical model kinetics of which are consistent with the observations. The mathematical model, in the form of a Boltzmantype equation, will serve as a realistic component for models of chemotherapy. The current version of the paper has been improved based on referees' remarks.
Notes
Declarations
Acknowledgements
This work was supported by grants of the federation FED 4116 SCALE (SCiences Appliquées à L'Environnement). Jennifer Pasquier was a recipient for a fellowship from the Conseil Regional de HauteNormandie The authors are indebted to Pr. JeanPierre Marie (Hôtel Dieu, Paris, France) for providing MCF7/Doxo and PSC833.
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.