Mathematical models of the transitions between endocrine therapy responsive and resistant states in breast cancer

, 20140206, published 7 May 201411 2014 J. R. Soc. Interface Chun Chen, William T. Baumann, Jianhua Xing, Lingling Xu, Robert Clarke and John J. Tyson therapy responsive and resistant states in breast cancerMathematical models of the transitions between endocrine

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ResearchCite this article: Chen C, Baumann WT, XingJ, Xu L, Clarke R, Tyson JJ. 2014 Mathematicalmodels of the transitions between endocrinetherapy responsive and resistant states inbreast cancer. J. R. Soc. Interface 11: 20140206.http://dx.doi.org/10.1098/rsif.2014.0206

Received: 26 February 2014Accepted: 15 April 2014

Subject Areas:computational biology, systems biology,biomathematics

Keywords:breast cancer, endocrine resistance, potentiallandscape, sequential treatment,intermittent treatment

Author for correspondence:John J. Tysone-mail: [email protected]

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rsif.2014.0206 orvia http://rsif.royalsocietypublishing.org.

Mathematical models of the transitionsbetween endocrine therapy responsiveand resistant states in breast cancerChun Chen1, William T. Baumann2, Jianhua Xing3, Lingling Xu4, Robert Clarke5

and John J. Tyson3

1Graduate Program in Genetics, Bioinformatics and Computational Biology, 2Department of Electrical andComputer Engineering, 3Department of Biological Sciences, and 4Department of Physics, Virginia PolytechnicInstitute and State University, Blacksburg, VA 24061, USA5Lombardi Comprehensive Cancer Center, Georgetown University School of Medicine, Washington, DC 20057, USA

Endocrine therapy, targeting the oestrogen receptor pathway, is the mostcommon treatment for oestrogen receptor-positive breast cancers. Unfortu-nately, these tumours frequently develop resistance to endocrine therapies.Among the strategies to treat resistant tumours are sequential treatment(in which second-line drugs are used to gain additional responses) andintermittent treatment (in which a ‘drug holiday’ is imposed betweentreatments). To gain a more rigorous understanding of the mechanismsunderlying these strategies, we present a mathematical model that capturesthe transitions among three different, experimentally observed, oestrogen-sensitivity phenotypes in breast cancer (sensitive, hypersensitive andindependent). To provide a global view of the transitions between thesephenotypes, we compute the potential landscape associated with themodel. We show how this oestrogen response landscape can be reshapedby population selection, which is a crucial force in promoting acquiredresistance. Techniques from statistical physics are used to create apopulation-level state-transition model from the cellular-level model. Wethen illustrate how this population-level model can be used to analyse andoptimize sequential and intermittent oestrogen-deprivation protocols forbreast cancer. The approach used in this study is general and can also beapplied to investigate treatment strategies for other types of cancer.

1. IntroductionBreast cancer remains one of the most common cancers in women. Approxi-mately 70% of breast cancers express oestrogen receptor-a (ERa), which is amajor driver of survival and proliferation in breast cancer cells [1]. Endocrinetherapies, targeting the ERa pathway, have become routine and effective treat-ments for breast cancer [2]. Three types of endocrine therapies are commonlyused in breast cancer: (i) aromatase inhibitors (AIs) to deprive cells of 17b-oestradiol (E2, the primary oestrogen in breast tumours), by reducing E2biosynthesis, so as to remove the ligand of ERa; (ii) selective oestrogen receptormodulators (SERMs, e.g. tamoxifen), to inhibit ERa binding with E2; and(iii) selective oestrogen receptor downregulators (SERDs, e.g. Faslodex), to inhibitE2 binding and target the ERa protein for degradation. Despite their success inreducing the annual death rate from breast cancer by one-third over the pastdecades, resistance to endocrine therapies often develops, creating a fundamentalproblem in breast cancer treatment [2].

The development of resistance to endocrine therapy has often been thought of asa progressive, step-wise phenomenon whereby cancer cells gradually shift from anE2-dependent, responsive phenotype to a less-responsive phenotype, and ulti-mately to an E2-independent phenotype [3,4]. This progression implies thatendocrine treatments should be sequenced to take advantage of the cancer cell’s

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oestrogen-responsiveness state. Pre-clinical and clinical obser-vations suggest that breast cancer with acquired resistancemay respond to the second- and third-line endocrine treatments(figure 1a). For example, a proportion of breast cancers that havedeveloped resistance to tamoxifen can still be inhibited by AIs,and some AI-resistant breast cancers can still benefit from Faslo-dex [3]. Nonetheless, cross-resistance among these therapiesoccurs, and both the frequency and duration of responses tothe second- and third-line therapies are often less than thoseachieved with a first line intervention. Various combinations ofconsecutive anti-oestrogen treatments are still under clinicalinvestigation [6]. In addition, giving breast cancer patientsa ‘drug holiday’ has been suggested to deal with acquired endo-crine resistance [7–9] (figure 1a). The duration of treatment andthe interval between treatments are often critical parameters forthe effectiveness of any intermittent treatment regimen. However,a rigorous understanding of how breast cancer patients canbenefit from intermittent endocrine treatment is still lacking, pre-venting us from optimizing treatment strategies. Optimization ofsequential treatment faces a similar problem [10].

In this initial study of devising a time-varying therapy toprevent the onset of resistance, we consider only the effect ofE2-deprivation therapy for simplicity. A mathematical modelis built to account for the different E2-responsiveness pheno-types observed in breast cancer cells and used to explore thetransitions from endocrine responsiveness to endocrine resist-ance under sequential and intermittent treatment.

It is widely reported that there are three phenotypes (states)of breast cancer cells related to E2 sensitivity [3,5] (figure 1b).

Both clinical observations [11] and cell line studies [12] suggestthat breast cancer cells, which are E2 sensitive initially(figure 1b, left), can adapt to E2 deprivation by entering anE2-hypersensitive state, for which a non-physiologically lowconcentration of E2 is sufficient to support cell proliferation(figure 1b, middle). Prolonged and profound E2 deprivationcan lead to an E2-independent state [3,5] (figure 1b, right).

Notably, not all cells make the state transitions necessary forproliferation. Rather, the E2 sensitivity progression describedearlier most probably represents somatic evolution, wherebycells that switch to the hypersensitive or independent states out-grow their disadvantaged peers. The diversity of cell phenotypesthat is necessary for somatic evolution can be caused by genomicor non-genomic heterogeneity [13]. In this paper, we consideronly non-genomic heterogeneity, whose occurrence is supportedby experiments that demonstrate the reversibility of E2-hyper-sensitive and -independent phenotypes in breast cancer cellsduring two years of E2 manipulation in vitro [14]. We do notaddress the issue of genomic heterogeneity due to mutation,which probably occurs over a much longer timescale.

In physics, an intuitively appealing way to visualize thedynamics of a system is to view the system as a particlemoving on a surface defined by a ‘potential energy’. In twodimensions, this surface can be plotted as a landscape wherethe particle moves downhill and falls into basins. When thelandscape contains multiple basins of attraction, there are mul-tiple stable states where the particle can remain indefinitely. Ifthe system is subjected to random external forces, it is possiblefor the system to transition from one basin of attraction to

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Figure 1. Breast cancer shifts between endocrine responsive and resistant states. (a) Schematic summary of the consequences of single, sequential and intermittentendocrine treatments in breast cancer. Yellow denotes responsive, red resistant and grey the treatment-withdrawn states. (b) Schematic of the transitions between differentE2 sensitivity phenotypes in breast cancer. Left, E2-sensitive state; middle, E2-hypersensitive state; right, E2-independent state. Blue cells in the middle panel indicate a lowproliferation index (PI), while red cells indicate a high PI. The bottom panel shows the dose – response curves of the PI to E2 levels (from TRACE to LOW to HIGH) for thecorresponding E2 sensitivity states. The transitions are reversible when the conditions on the arrows are applied. (c) Wiring diagram based on the three activation modes ofERa and their crosstalk with GFR signalling. E2ER, E2-liganded ERa; ERM, membrane ERa; ERP, phosphorylated ERa; GFR, growth factor receptors; barbed arrows,activation; blunt arrows, inhibition. (d ) The levels of ERM, GFR and ERP for the three different E2 sensitivity states, as summarized in [3,5].

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another, with important implications for phenotypic plasticity.Inspired by this idea C. Waddington proposed describing cellfate decisions by the dynamics of a control system on an‘epigenetic landscape’ [15]. The basins of attraction were (inWaddington’s view) stable cell fates, and the paths that cellstake across the landscape corresponded to the differentiationevents of cells in a developing organism. In recent years, theability to compute a potential landscape from an underlyingmathematical model has been used to visualize a variety ofcell fate decisions [16–22]. While the imagery of Waddington’slandscape has been used to discuss the fate of cancer cellsduring the development of resistance to therapy [18,21], littlehas been done to quantitatively map this landscape from amathematical model of the underlying signalling network ofcancer cells.

Based on the mathematical, cell-level model that accountsfor the three E2 sensitivity states observed in breast cancercells, we compute the potential landscape, which we referto as the oestrogen response landscape. This landscape pro-vides a visual picture of how breast cancer cells changephenotypes in response to variations in protein expression(noise) and varying environmental conditions (E2 withdra-wal therapy). We also evaluate the effects of populationselection in reshaping the oestrogen response landscape.In addition, we apply a multiscale modelling strategy toestimate the transition probabilities in a simple population-level, state-transition model, based on the minimum actionpath (MAP) between basins of attraction on the oestrogenresponse landscape [19,23]. By using this state-transitionmodel, we investigate the population dynamics of breastcancer cells under both sequential and intermittent treatment.The model shows that a proper treatment sequence isimportant to achieve maximum response and to avoidcross-resistance. The quantitative model also allows us tosearch for an optimal protocol for intermittent treatment toovercome acquired endocrine resistance. Thus, our studyprovides a methodology for exploring breast cancer treat-ment strategies grounded in cellular-level, biologicallybased, mathematical models.

2. Material and methods2.1. Three oestrogen receptor-a activation modes

and crosstalk with growth factor receptorsThe ERa pathway in breast cancer cells has at least three distinctbranches [2,24–26]: (i) the E2 branch, where ERa bound to E2becomes an active transcription factor of genes supporting cellsurvival and proliferation; (ii) the phosphorylation branch,where ERa becomes an active transcription factor subsequentto phosphorylation at multiple sites (including Ser118, 104,106 and 167) in its AF1 domain and (iii) the membranebranch, where membrane-bound ERa mediates the activationof the MAPK and PI3K/AKT pathways (the ‘non-genomic’role of ERa, since it is not acting as a transcription factor inthis branch) [2]. These three branches of the ERa pathway,encompassing both the genomic and non-genomic roles ofERa, crosstalk with each other and with the growth factor recep-tor (GFR) signalling network (figure 1c) [27,28]. We consider thefollowing four crosstalk mechanisms: (i) E2-liganded ERa (E2ERin figure 1c) inhibits expression of both GFR [29,30] and mem-brane ERa [31] (ERM in figure 1c); (ii) both GFR and ERMactivate downstream kinases that phosphorylate ERa [2,24](ERP in figure 1c); (iii) ERM has positive interactions with

GFR signalling at different levels [2,24]; (iv) both ERM andGFR are involved in auto-activation loops at the signallingand/or epigenetic levels [5,31–33]. Experimental data indicatethat the different E2 sensitivities in breast cancer cells are deter-mined by the balance of the three ERa signalling branches(E2ER, ERP and ERM) and GFR signalling [3,5]. In particular,E2-sensitive cells depend on E2ER for proliferation, whereasE2-hypersensitive cells depend upon active ERM, which contrib-utes to a high level of ERP. In E2-independent cells, proliferationdepends on high GFR activity, which also produces a highlevel of ERP. These experimental observations are summarizedin figure 1d.

We create a phenomenological model that of necessity doesnot attempt to incorporate complex mechanistic details, manyof which are inadequately known, but rather to reproducethe experimental observations summarized in figure 1d usingthe simplified signalling network of figure 1c. Since ERP activitydoes not feedback on production or destruction of other speciesin the model, and its state is determined by the state of ERMand GFR, we do not include ERP in our dynamical model. Inaddition, we assume the binding of E2 to ERa is sufficiently fastthat the amount of E2ER can be modelled as an algebraic functionof the E2 level. Hence, we model only the dynamics of ERM andGFR, incorporating the influences shown in figure 1c. Proliferationis modelled by using figure 1d and the activity levels of ERM andGFR to determine a cell’s E2-sensitivity state. The cell is thenassigned a proliferation index (PI), as a function of E2, based onthis sensitivity as in figure 1b (see also figure 2a,b). This procedurereproduces the experimental observations that E2-sensitive cellsgrow in high levels of E2, but die in low or trace levels, whereasE2-hypersensitive cells grow in high or low E2 concentrations,but die in trace levels, and E2-independent cells grow under allE2 conditions. The numerical rates of growth and death, however,are not based on experiment.

2.2. Model implementationOur mathematical model of the influence diagram in figure 1cuses stochastic differential equations (SDEs) to capture both thedeterministic effects of species’ interactions in a signalling net-work and the random effects of intrinsic and extrinsic sourcesof noise [34–37]. Let Xi be the name of the ith species in our inter-action diagram (E2ER, ERM and GFR). Letting [Xi] ¼ activity ofXi and xi ¼ log10[Xi], we write the SDEs for our model as

dxi

dt¼ Fi þ zi(t), where i ¼ 1, 2, 3; (2:1)

where

Fi ¼ gi # [H(Wi)$ xi], where H(Wi) ¼1

1þ e$Wiand

Wi ¼ vi þX3

j¼1

vi,j # xj: (2:2)

We assume that all concentrations in our model have a dynami-cal range of about 10-fold, so, on a log10 scale, the xi’s varybetween 0 and 1. The parameters gi determine the rates atwhich the variables xi approach their time-varying ‘target’values, xi ¼ H(Wi). Note that H(Wi) is a sigmoidal function thatvaries between 0 and 1, with a value of 1/2 when Wi ¼ 0. Wi isthe net activation or inhibition on species Xi, and the leadingterm, vi, determines whether Xi is activated or inhibited whenthere are no regulatory signals impinging on Xi from any speciesin the network. In the summation, vij is the strength of the inter-action from Xj to Xi (vij . 0 for activation and ,0 for inhibition).zi(t) is a temporally uncorrelated, statistically independentGaussian white noise process with

kz(t)z(t0)l ¼ 2 #Di # d(t$ t0), (2:3)


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where kzi(t)l ¼ 0 and Di characterizes the magnitude of therandom fluctuations on each species.

The details of our model are provided in the electronic sup-plementary tables. Electronic supplementary material, table S1defines the variables of our model, table S2 lists the preciseequations of the model and table S3 records the parametervalues that we used for our simulations. In deriving the modelequations, we applied a quasi-equilibrium approximation toERE2, because the binding of E2–ER occurs on a much fastertimescale than variations in ERM and GFR, which also mayhave transcriptional, possibly epigenetic, regulation. Hence,only ERM and GFR are described by SDEs. We also assumedthat ERM and GFR are subjected to similar random fluctuations,so D1 ¼ D2 ¼ D in equation (2.3). We used MATLAB (v. 7.9.0) toperform simulations.

Our modelling is done at a phenomenological level;consequently, none of our parameters are directly related to mea-surable physical rate constants of the system. Phenomenologicalmodelling is both reasonable and necessary, because the rate con-stants for most of the interactions in the model are unknown. Theparameters are assigned to match the phenotypic properties ofthe ERa–GFR control system. Initially, the parameters for thedeterministic part of the model were chosen to ensure thatthe system has attractors corresponding to the E2-sensitive,-hypersensitive and -independent states (figure 2a). For the sto-chastic simulations, we chose the noise parameter, D ¼ 0.005,to ensure that the system’s transitions occurred in a manner simi-lar to experimental observations. To model selection pressures,we chose the shape of the PI curves for cells in each state(figure 2b) to capture qualitatively the experimentally observed

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Figure 2. E2 levels and selection pressure remodel the oestrogen response landscape. (a) The state space of the system is divided into four regions: R1 (E2-sensitivestate), R2 (E2-hypersensitive state) and R3 þ R4 (E2-independent states). (b) PI as a function of oestrogen level (E2) for cells in R1 (cyan), R2 (green) and R3 þ R4

(blue). (c,d ) Oestrogen response landscape at different E2 levels are computed using U ¼ 2ln(Pss), either without (c) or with (d ) selection pressure. The elevationdecreases from red to blue. White balls are put in the basins with lowest U in each plot. [E2] ¼ 102, 100 and 1022, corresponding to HIGH, LOW and TRACE levels ofE2 in figure 1b. The phase planes are plotted below the surfaces and the yellow balls represent the fixed-point attractors located at intersections of the ERM and GFRnullclines. The ERM nullcline (red curve) is the locus of points for which ERM ¼ H(WERM), and the GFR nullcline (green curve) is the locus of points for whichGFR ¼ H(WGFR). In (c) cells in all states are assumed to have the same PI (PI ¼ 0 for all [E2]), thus cells are not under selection pressure. In (d ) cells proliferate ordie according to the PI curves in (b). PI . 0 means cell growth outpaces cell death; PI ¼ 0 means a balance of cell growth and death; PI , 0 means cell deathoverwhelms cell growth. ERM, activity of membrane ERa; GFR, activity of growth factor receptors.


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growth properties of E2-sensitive, -hypersensitive and -indepen-dent cells.

2.3. Mapping the oestrogen response landscapeA dynamical system whose state moves in the direction of thenegative gradient of a potential function can be convenientlyvisualized by plotting the potential energy landscape and view-ing the system state as a particle that moves downhill on thislandscape, coming to rest at a local minimum. While openbiological systems, such as the one we consider here, are notgoverned by the gradient of a potential function, the ideas in[17–20,38] can be used to compute a potential landscape usefulfor visualization. The basins of this landscape are regions ofhigh probability for finding the system, and the hills are regionsthat the system rarely visits.

We write the governing equation (2.1) of our breast cancermodel in vector form as

dxdt¼ F(x)þ z(t): (2:4)

where x ¼ x1x2

! ", with similar definitions of F and z. Let P(x,t) be

the probability distribution function for state x at time t,i.e. P(x,t)dx1dx2 ¼ probability of finding the system in the smallarea (x1, x1 þ dx1) % (x2, x2 þ dx2) at time t. The equationgoverning P(x,t) is

@P(x, t)@t

¼ $r # J(x, t), (2:5)

where r # J ¼ (@J1/@x1)þ (@J2/@x2) and

J(x, t) ¼ F(x)P(x, t)$DrP(x, t) (2:6)

is the probability flux for our system.We wait long enough for the probability density function,

and hence the flux, to reach a steady state, and then solveequation (2.6) for F(x) in terms of Pss(x) and Jss(x)

F(x) ¼ $DrU(x)þ Jss(x)Pss(x)

, (2:7)

where U(x) ¼ 2ln(Pss(x)). Thus, F can be expressed as the gradi-ent of a potential function U(x) plus a flux field. Although thesystem’s dynamics is not governed solely by the gradient ofU(x) but includes as well a ‘spiralling’ motion due to the fluxfield [15], the generalized potential U ¼ 2ln(Pss) still providesa useful visualization of the dynamics of the system. We referto U(x) as the oestrogen response landscape in this study.Because of how U(x) is related to Pss(x), basins of U(x) corre-spond to high probability states and hills to low probabilitystates. Details for solving the diffusion equations are providedin the electronic supplementary material, Text S1.

2.4. Cell state transition modelIn the small noise case, the system spends most of its time fluc-tuating around one of its m steady states (m ¼ 4 in this study)with occasional transitions between states. Hence, it is inefficientto track the states of cells in a large population by simulatingeach cell individually. To study state switching within a largepopulation of cells, we transform the cellular-level, multiplesteady-state model into a Markov state-transition model byusing the cellular model to compute the Markov state transitionrates. Hence, we divided the two-dimensional state space of thecellular-level model into four blocks, representing the four dis-crete phenotypes, which we refer to as ‘regions’ R1, R2, R3 andR4 in figure 2a. Similar to [39,40], we describe a population ofcells by a vector, v, of length m ¼ 4. The ith entry of the vectoris the number of cells in region Ri, and the total number ofcells in the population at time t is N(t) ¼

P4i¼1 vi(t). The

population vector v evolves according to the equationdv/dt ¼ A # v where A is an m % m transition rate matrix. Theoff-diagonal elements of A, Aij ¼ kij, are the transition rates fromregion j to region i, and the diagonal elements areAii ¼ ui $

Pj=i kij, where ui is the proliferation rate of cells in

region i. Thus, starting from a vector reflecting the initial popu-lations in R1, . . . , R4 at time 0, we can easily compute thetemporal evolution of the population (for details, see the elec-tronic supplementary material, Text S2). The average PI of thepopulation at each time step is then calculated byPI(t) ¼ (N(tþ Dt)$N(t))/ (Dt #N(t)). PI is the specific reproduc-tion rate of the population: PI . 0 signifies an expandingpopulation and PI , 0 signifies a diminishing population.

2.5. Minimum action pathTo estimate the transition rates kij among the different pheno-typic regions of the dynamical system, which we need for thepopulation-level model, we followed the approach in [19,23,41]based on applying the Wentzell–Freidlin theory [42] to equation(2.4) for the ‘small noise’ case. The key idea of this theory is thatthe most probable transition path from Rj at time 0 to Ri at time T,w&ij(t), t [ [0, T], minimizes the action functional (equation (2.8))over all possible paths

ST[wij] ¼12

ðT

0jwij# $F(wij)j

2dt: (2:8)

This optimal path is referred to as the MAP. We are interested inMAPs between fixed-point attractors (stable steady states) inregions, R1, . . . , R4. The MAPs connecting each pair of attractorswere computed numerically by applying the minimum actionmethod used in [23,41]. For T large enough to allow a minimumaction transition from the attractor in Rj to the attractor in Ri,ST[w&ij] becomes independent of T, which is the goal. In ourcase, T ¼ 40 is a reasonable choice: large enough but not toolarge (see the electronic supplementary material, figure S1).

The MAP w&ij is useful for our purposes because we find thatkij, the transition rate from state Rj to state Ri, is related to ST[w&ij]by the empirical equation

kij ¼ b # exp {$a # (ST[w&ij])n}, (2:9)

where the parameters a, b and n are estimated by fittingequation (2.9) to the results of direct Monte Carlo simulationsof the dynamic model, where the Langevin equation (2.4) isintegrated by the explicit method

xi(tþDt)¼ xi(t)þgi # [H(Wi)$xi(t)] #Dtþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 #Di #Dt

p#hi(t): (2:10)

Here the hi are independent, normal, random deviates withmean 0 and variance 1.

3. Results3.1. The oestrogen response landscapeBoth clinical observations and cell line studies suggest thatbreast cancer cells can shift between three different E2 sensi-tivity states [3,5] (figure 1b): E2-sensitive, -hypersensitive and-independent. In E2-sensitive cells, E2-liganded ERa (E2ER)is the major driver of pro-survival and pro-proliferationsignalling, whereas ERa in the cell membrane (ERM), phos-phorylated ERa (ERP) and GFRs are kept at low levels ofactivity. As summarized in [3,5], E2-hypersensitive cells areclosely associated with elevated ERa in the cell membrane(ERM), where ERa plays a non-genomic role to mediate theactivation of the MAPK and PI3K/AKT pathways. GFRs,such as IGF1R and HER2, are not necessarily overexpressed


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in E2-hypersensitive cells. A possible explanation for hyper-sensitivity is that ERa, phosphorylated by kinases in theupregulated pathway, may have increased affinity for E2. InE2-independent cells, however, increased GFR and possiblyother signalling pathways play pivotal roles, probably by activat-ing MAPK and PI3K/AKT that further phosphorylate ERa(ERP), driving E2-independent growth. These observations aresummarized in figure 1d. The activity levels of ERM andGFR determine the E2-sensitivity state of our model. Thus,of the four regions in figure 2a, R1 corresponds to E2-senstivity(GFR low, ERM low), R2 corresponds to E2-hypersensitivity(GFR low, ERM high), and R3 and R4 correspond to E2-independency (GFR high, ERM high or low).

As discussed in Material and methods, the parameters ofour model were chosen to reproduce the qualitative exper-imental data summarized in figure 1d [3,5]. The timescale ofthe model is not explicitly specified, but based on experimentalobservations we expect it to be on the order of weeks ormonths. The oestrogen response landscape computed fromour model is plotted in figure 2c for three different E2 con-ditions ([E2]¼ 102, 100 and 1022) corresponding to HIGH,LOW and TRACE amounts of E2. Varying the level of E2reshapes the landscape and can induce cells to transitionfrom one state to another. When E2 level is HIGH, the poten-tial landscape has only one dominant basin in R1, where bothERM and GFR are low (figure 2c, left), indicating an E2-sensitive state. Most cells will stay in this region. ReducingE2 to LOW level (figure 2c, middle) reshapes the landscape,deepening the basin in R2 and causing some cells to shiftfrom the E2-sensitive state to the E2-hypersensitive state.This transition allows the oestrogen receptor to become acti-vated at much lower E2 concentrations, essentially inducinga left-shift in the proliferation dose–response curve. Furtherdecreasing E2 to TRACE amounts creates a dominant basinin R4, the E2-independent state (figure 2c, right), and theE2-sensitive basin in R1 becomes very shallow. Both E2-hypersensitive and -independent cells will coexist in thiscondition, with a preference for the independent state.

In addition to cell state transitions (cellular adaptation) inresponse to a changing E2 environment, breast cancer cellsare also under selection pressure due to the different prolifer-ation rates that apply to each cell state in a given E2environment. To model this selection pressure, the statespace of the model was divided into four regions (R1, R2,R3 and R4 as before), with each region assigned a specificPI as a function of [E2] (figure 2b). Allowing cells in eachregion to proliferate at their assigned rates given the specifiedE2 concentration results in the landscapes in figure 2c (see theelectronic supplementary material, Text S1 for calculationdetails). Selection pressure deepens some basins and destroysothers (figure 2d ). In particular, for LOW values of E2,selection pressure causes the dominant basin to shift fromthe E2-sensitive state (R1) to the E2-hypersensitive state (R2).

Thus, our model reproduces the different cell statesobserved experimentally in response to E2 manipulations.The oestrogen response landscape we have computed here pro-vides a useful visualization of the global behaviours of thebreast cancer system with three different E2 sensitivity states.

3.2. A state-transition modelTo move from one basin to another, cell state transitionsrequire the system to cross a barrier. The transitions are

driven by intrinsic and extrinsic noise, with lower barrierscorresponding to more probable transitions. The most prob-able routes for transitions among the four regions on theoestrogen response landscape, for a specified value of E2,were computed by finding the paths that minimize theaction, a measure of the difficulty of moving from oneregion to another (see Material and methods), over all poss-ible paths that start and end in the desired regions. Anexample of MAPs on the oestrogen response landscape isshown in figure 3a. Interestingly, the route taken from onebasin to another is not the same as the route of the reversetransition, as seen by the non-overlapping yellow and redcurves in figure 3a. This divergence of MAPs is a consequenceof the non-gradient component of the flow field (equation(2.7)) [16,17,43].

The action along the MAP between two steady states infigure 3a can be used to estimate the transition probabilitybetween the two regions [19]. Thus, we again divide thestate space into four regions (R1–R4 in figure 3b) correspond-ing to the four cancer cell phenotypes. Each region has atmost one attractor. For various E2 levels, the transition ratesbetween pairs of regions were computed by Monte Carlosimulations of the stochastic model (electronic supple-mentary material, figure S2). In figure 3c, we plot thesetransition rates as functions of the action of the MAP betweenthe corresponding fixed points. As expected, the points alllie close to a single curve described by equation (2.9), witha ¼ 80.48, b ¼ 0.07197 and n ¼ 0.9805. Thus, for differentvalues of [E2] and any pair of fixed points we can computethe action, and then use equation (2.9) to estimate the tran-sition probability, without recourse to additional MonteCarlo simulations. The comparisons between transitionrates estimated from actions and from direct Monte Carlosimulations at different E2 levels are shown in figure 3d,e.Because the transition rates out of R3 are very much greaterthan the transition rates into R3 (k13 ' k31, k23 ' k32,k43 ' k34), the third block is rarely populated and canbe safely neglected. The transition rates between R1 and R4

are much smaller than their transition rates with R2

(k41 ( k21, k14 ( k24); hence, direct transitions between R1

and R4 can also be safely neglected.Given the estimated transition rates, we built a determi-

nistic model of phenotype transitions and simulated thegrowth of cancer cell populations under different E2 con-ditions. Figure 4a shows the steady-state proportion ofeach phenotype as a function of E2 level for the case ofno selection pressure (PI ¼ 0 for each phenotype). This dis-tribution of phenotypes is consistent with the landscapeplot in figure 2c. Note that when E2 is LOW ([E2] ¼ 1),although there is a small population of E2-hypersensitivecells (green curve), the majority of cells are still in the E2-sensitive state (blue curve). However, adding the effect ofpopulation selection significantly changes the outcome(figure 4b). As in figure 2d, the selective advantage of E2-hypersensitive cells in the LOW E2 environment substan-tially increases the proportion of E2-hypersensitive cells.The proportion of E2-independent cells under TRACE E2conditions ([E2] ¼ 1022) is greatly enriched (figure 4b,right). Corresponding steady-state probability distributions,Pss(x1,x2), in the state space of the cellular-level model areshown in figure 4c,d. Simulating the phenotype transitionmodel at the population level provides a substantialreduction in computation compared with Monte Carlo


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simulations of the cellular-level model. Consequently, wecan easily study the consequences of both sequential andintermittent treatments of breast cancer under a varietyof conditions.

3.3. Sequential treatment of breast cancerThe effects of sequential E2 deprivation on breast cancercells have been reported in both clinical and pre-clinical

settings [3,5,11]. For example, in pre-menopausal women,E2-dependent breast cancers often regress after surgicalremoval of the ovaries, which lowers circulating E2 from200 to 15 pg ml21. After an initial stage of regression, breastcancer often regrows. AIs, now given as second-line thera-pies, can induce additional tumor regression by furtherdecreasing E2 to less than 5 pg ml21 [11].

Creation of an additional ‘response window’ using asecond-line endocrine therapy can be captured by our state

R1–R2R2–R4

[E2][E2]

r2

r4r3

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R3 R4

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(b)

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fit

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nsiti

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te

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MAP r2 r4MAP r4 r2MC R2 R4MC R4 R2

(e)

R2R1

Figure 3. MAPs characterize state transitions on the oestrogen response landscape. (a) The oestrogen response landscape U ¼ 2ln(Pss) as a function of ERM(membrane ERa) and GFR (growth factor receptors), for [E2] ¼ 1021 and no selection pressure. White balls are put in basins. The red and yellow lines arethe estimated MAPs connecting different basins. The arrows indicate the directions of the transitions. Note that the red and yellow lines do not overlap.(b) The ERM – GFR phase plane (left panel, [E2] ¼ 1021 for illustration) is divided into four regions (marked by different colours). Each region represents acell phenotype in the state transition model (R1 – R4, right panel; green arrows represent reversible transitions). In each region, there is at most one fixed-point attractor. The fixed-point attractors (r1 –r4, light green dots) and first-order saddles (dark green dots) are located by finding the intersection points ofthe nullclines for ERM and GFR. The ERM nullcline (red curve) is the locus of points for which ERM ¼ H(WERM), and the GFR nullcline (green curve) is thelocus of points for which GFR ¼ H(WGFR). (c) Points represent transition rates, computed by direct Monte Carlo (MC) simulation, for transitions between regionsR1 and R2 (blue) and between regions R2 and R4 (red). ST [w&ij ] is the value of the action along the MAP between the relevant fixed points in the regions, evaluatedfor T ¼ 40. The dashed line is the best fit of equation (2.9) to the data (a ¼ 80.48, b ¼ 0.07197 and n ¼ 0.9805). (d,e) Compare the transition rates estimatedby equation (2.9) with the transition rates computed by direct Monte Carlo simulations, as functions of [E2].

0

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(d )

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10–2 1 102

10–2 1 102 10–2 1 102

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Figure 4. The state transition model approximates the dynamical behaviour of the oestrogen response landscape. (a,b) Right panels: the steady-state proportions ofcells in three of the E2 sensitivity states as a function of [E2] in the state-transition model (solid lines) and the cellular model (discrete markers) either withoutselection pressure (a) or with selection pressure (b). Left panels show the corresponding dose – response curves of PI to [E2] for cells in each state. Cyan, E2-sensitivestate (R1); green, E2-hypersensitive state (R2); blue, E2-independent state (R4). The other E2-independent state (R3) is not populated to any appreciable extent.(c,d ) The corresponding steady-state probability distributions (Pss) in the state space of the cellular level model are plotted as heat maps for [E2] ¼ 1022, 1 and102, for the case without selection (c) and with selection (d ). ERM, activity of membrane ERa; GFR, activity of growth factor receptors.


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transition model (figure 5a). To illustrate this, we assume aprimary therapy that can reduce the E2 level from HIGH toLOW, and a secondary therapy that can reduce the levelfrom HIGH to TRACE. Decreasing E2 in a step-wise fashion(figure 5a, top), by first applying the primary therapy andthen changing to the secondary therapy, results in sequentialtransitions of the cell population from the E2-sensitive to theE2-hypersensitive to the E2-independent state (figure 5a,middle). At each transition, cells in the previously favouredstate begin to die off while some make the transition tothe newly favoured state and begin to grow. Plotting theaverage population PI versus time clearly shows the tworesponse windows of negative average growth (yellowphases, figure 5a, bottom).

Notably, if the E2 steps are applied in the reverse order(figure 5b), corresponding to applying the secondary therapyfirst, the simulation shows no second response window. Thismight be viewed as an example of the phenomena of cross-resistance to endocrine therapies in breast cancer. In figure 5b,the green phase represents the cross-resistance of cells toendocrine treatment associated with LOW E2 when the cells

have been pretreated with (and gained resistance to) endocrinetreatment associated with TRACE levels of E2.

3.4. Intermittent treatment of breast cancerIntermittent E2 treatment to reverse acquired resistance andprovide a second response window is simulated in figure 5c.Shifting between two different endocrine therapies can alsoproduce a similar result (figure 5d). These simulation resultsprovide support for both pre-clinical and clinical observationsthat intermittent treatment may be better than continuoustreatment [7–9].

The state transition model can serve as a quantitative toolto find the optimum values for treatment duration, Ttreat,and treatment breaks, Tbreak. Figure 6a shows the results forTtreat ¼ 3000 and Tbreak ¼ 2000, where cell populationgrowth is reduced—but not stopped—by intermittent treat-ment. By contrast, intermittent treatment with Ttreat ¼ 1000and Tbreak ¼ 2000 (figure 6b) causes the total cell populationto stop growing, thereby controlling—but not eliminating—the cancer. Comparing these two cases, we note that shorter

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t (×103) t (×103)0 2 4 6 8 0 2 4 6 8

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8 0 4 12t (×103)

8

Figure 5. Sequential and intermittent treatments provide additional response windows to endocrine therapy. Different E2 withdrawal sequences are used to mimicdifferent endocrine treatment regimens: (a) reducing E2 from HIGH, to LOW, to TRACE level; (b) changing E2 from HIGH, to TRACE, to LOW level; (c) two identicalendocrine treatments are applied with a break in between; or (d ) two different endocrine therapies are applied with a break in between; time in arbitrary units. Theresulting changes in proportions of cells in different E2 sensitivity states are shown in the middle panels. Cyan, E2-sensitive state (R1); green, E2-hypersensitive state(R2); blue, E2-independent state (R4). The lower panels indicate how the average PI of the cell population changes during the course of the treatment. Yellow,responsive state; red, resistant state; green, cross-resistant state. The dotted horizontal line indicates no growth (PI ¼ 0).


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treatment durations sometimes may be more beneficial thanlonger treatments.

Using the model to explore the effects of the two keytreatment parameters on the expansion of the cancer cellpopulation, we plot in figure 6c the total number of cellsafter treatment as a function of Ttreat and Tbreak. Since thesimulation starts with 103 cells, any parameter valueswithin the 103-contour (superimposed on figure 6c) willcause cancer progression to be arrested or reversed. Theseresults point out the potential utility of a population-level,state-transition model grounded on a cellular-level, mechanisticmodel for optimizing cancer treatment protocols.

4. DiscussionAs a heterogeneous population of cells, cancer can respondby adaptation to many changing stressors and stimuli in itsenvironment. This plasticity can lead to acquired resistanceand drug failure, a fundamental problem in cancer treatment.One widely accepted explanation for cancer plasticity is thatcancer cells with advantageous mutations due to genomicinstability can be selected and enriched in a Darwinian-typeevolutionary process [44]. These mutations are importantand recent work has identified point mutations and trans-locations of the ERa gene (ESR1) in hormone-resistant andmetastatic breast cancer [45–47]. Our interest is in cases ofreversible resistance that have been observed in experimentsand clinical observations. Reversible resistance cannot beexplained by genetic mutation and likely occurs on a muchshorter timescale. An appealing alternative explanationfor this case is the cancer stem cell (CSC) model, whichuses the principles of developmental and stem cell biologyto explain the multiple cell states and phenotypes as

non-genomic sources for Darwinian selection [13]. Increasingevidence of non-genomic heterogeneity has been reported indifferent cancers, including breast cancer [39]. To determinethe efficacy of cancer treatments, there is a significant needto investigate mechanisms of phenotype transitions incancer cells, and to relate these transitions at the cellularlevel to the effects of drug selection at the population level.

Considering the three phenotypes of oestrogen (E2) sensi-tivity in breast cancer cells as an example of non-genomicheterogeneity, we modelled the crosstalk among the threemajor ERa signalling modes (E2ER, ERM and ERP) and theGFR pathway for cell survival and proliferation. The model,though necessarily high level, provides a qualitative matchto the progressive, step-wise phenomenon of shifting E2sensitivities and changing endocrine responsiveness inbreast cancer cells under E2 manipulation.

The complexity of cancer must also be considered. Cancercells may have as many steady states as there are physio-logical states, including a spectrum of cell phenotypesfrom different differentiation stages [48]: CSCs, intermediateprogenitor cells and fully differentiated ‘bulk’ cells. Suchphysiological phenotypes, and the transitions among them,have been widely reported in breast cancer. For example,the epithelial–mesenchymal plasticity of breast cancer createsa broad spectrum of phenotypes, whereby epithelial primarycancers become mesenchymal-like and re-epithelialize into asolid tumour at secondary, metastatic sites [49]. In addition,the recent molecular classification of breast cancers intosimple, fixed categories (luminal A, luminal B, HER2þ,basal-like, normal-like) is not robust and has neglectedthe heterogeneous and dynamic properties of cancer [50].Phenotypic transitions among stem-like, basal and luminalsubpopulations within a given breast cancer cell populationhave been reported in experiments [39]. Unfortunately,

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Figure 6. Different regimens of intermittent treatment can drive different response outcomes. The endocrine treatment consists of periodically switching E2 levelfrom HIGH to LOW and back to HIGH, with Ttreat ¼ duration at LOW E2 level and Tbreak ¼ duration at HIGH E2 level (time in arb. units). As indicated in the toppanels, two intermittent treatment regimens are simulated: (a) Ttreat ¼ 3000, Tbreak ¼ 2000 and (b) Ttreat ¼ 1000, Tbreak ¼ 2000. Both treatments change theproportion of cells in different E2 sensitivity states (second panels). Cyan, E2-sensitive state (R1); green, E2-hypersensitive state (R2); blue, E2-independent state(R4). The average PI of the cell population is shown in the third panel. Yellow, responsive state; red, resistant state. The dotted horizontal line indicates no growth(PI ¼ 0). The total number of cells N(t), is plotted in the fourth panel on a logarithmic scale. The grey interval is used for the calculations in (c). (c) We plot, as afunction of Ttreat and Tbreak, the average value of cell number klog10Nl over the interval T [ [2 % 104, 3 % 104]. The black and white dots indicate the cases in(a) and (b), respectively. Since we start from a population of 1000 breast cancer cells, any combination of Ttreat and Tbreak that puts the system within the dottedwhite contour line, log10N ¼ 3, will suppress cancer growth.


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current knowledge of the underlying molecular mechanismsfor these phenotypic transitions is limited. Nevertheless, thesources of non-genomic heterogeneity in breast cancer cellsare now under extensive study and will ultimately enablethe creation of enhanced models with more (meta)-stablesteady states corresponding to the various physiologicalstates of breast cancer cells.

Mathematical modelling can help understand the emer-gent behaviours of cancer physiology at different levels[51–56]. For example, several mathematical models havebeen developed to address the mechanisms by whichcancer cells develop resistance to treatment. While mostmodels attribute the source of Darwinian selection to raremutations [57,58], some have taken non-genetic heterogeneityinto account by modelling spontaneous transitions betweenCSCs and ‘bulk’ cells at the population level [59,60]. In thisstudy, we adopted a multiscale modelling strategy by firstcreating a biologically plausible, cellular-level model of tran-sitions among key breast cancer phenotypes. We then deriveda state transition model for population-level responses toendocrine therapies from the cellular-level model. This strat-egy enables a systematic understanding of how adaptationsto a changing environment can cooperate at both the cellularand population levels to create resistance to oestrogen-withdrawal therapies. Our modelling results indicate thatselection pressure can substantially change the shape ofthe oestrogen response landscape (figure 2c,d ) and changethe steady-state distribution of the state-transition model(figure 4a,b). The transition rate approximation we propose

provides a useful bridge between the molecular mechanismsof cellular phenotypic transitions and the phenomenologicalparameters that have been widely used to describe these tran-sitions in population-based models.

We applied the state transition model to understand theeffects of sequential and intermittent treatment that haveproved beneficial for breast and other cancers [3,5,7–9,61–63]. Many studies have focused on the benefits of thesestrategies in reducing side effects. Here we focused on anothermajor advantage, the potential of these strategies to reduce orreverse acquired resistance during treatment. The approachused in this study provides a potential path to investigate treat-ment strategies and improve treatment efficacy in breast andother cancers. The next step will be to move from a qualitativeto a quantitative model in breast cancer cell lines with thegoal of making and validating model predictions of theefficacy of therapy protocols. If optimized, time-varyingprotocols provide significant advantages in reducing resistanceand proliferation in cell lines, then the challenging task ofmoving these results to the clinic can begin.

Acknowledgements. We thank Jin Wang, Lei Zhang, Hang Zhang andXiaojun Tian for help and/or discussions.Funding statement. This work was supported in part by US NationalInstitutes of Health grant U54-CA149147 (to R.C., J.J.T. and W.T.B.),US National Science Foundation grant DMS-0969417 (to J.X.), and bya fellowship to C.C. provided by the Virginia Polytechnic Instituteand State University graduate program in Genetics, Bioinformaticsand Computational Biology.

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Mathematical models of the transitions between endocrine therapy responsive and resistant states in breast cancer