Quantitative constraint-based computational model of tumor-to-stroma coupling via lactate shuttle

Fabrizio Capuani,1 Daniele De Martino,1, 2 Enzo Marinari,1, 3, ∗ and Andrea De Martino1, 2, 4, ∗

1Dipartimento di Fisica, Sapienza Universita di Roma, Rome (Italy)2Center for Life Nano Science@Sapienza, Istituto Italiano di Tecnologia, Rome (Italy)

3INFN Sezione di Roma 1, Rome (Italy)4Soft and Living Matter Laboratory, Istituto di Nanotecnologia (CNR/NANOTEC), Consiglio Nazionale delle Ricerche, Rome (Italy)

Cancer cells utilize large amounts of ATP to sustain growth, relying primarily on non-oxidative, fermentativepathways for its production. In many types of cancers this leads, even in the presence of oxygen, to the secretionof carbon equivalents (usually in the form of lactate) in the cell’s surroundings, a feature known as the Warburgeffect. While the molecular basis of this phenomenon are still to be elucidated, it is clear that the spillingof energy resources contributes to creating a peculiar microenvironment for tumors, possibly characterized bya degree of toxicity. This suggests that mechanisms for recycling the fermentation products (e.g. a lactateshuttle) may be active, effectively inducing a mutually beneficial metabolic coupling between aberrant and non-aberrant cells. Here we analyze this scenario through a large-scale in silico metabolic model of interactinghuman cells. By going beyond the cell-autonomous description, we show that elementary physico-chemicalconstraints indeed favor the establishment of such a coupling under very broad conditions. The characterizationwe obtained by tuning the aberrant cell’s demand for ATP, amino-acids and fatty acids and/or the imbalancein nutrient partitioning provides quantitative support to the idea that synergistic multi-cell effects play a centralrole in cancer sustainment.

INTRODUCTION

At heart, a cell’s energetic problem consists in selectinghow to process nutrients (say, glucose molecules) into chem-ical energy (adenosine 5’-triphosphate, ATP) that will thenbe transduced into useful forms of mechanical or chemicalwork. Rapid cellular growth, in specific, requires high ratesof macromolecular biosynthesis and of energy production,which presupposes (a) fast ATP generation, and (b) tight con-trol of the cell’s redox state, i.e. that the ratio between thelevels of electron donors and acceptors stays in a range thatguarantees functionality. Most often, molecular oxygen is theprimary electron acceptor in cells, playing a central role in theelectron transfer chain (ETC) that constitutes the main ATP-producing mechanism in cells. When a glucose molecule en-ters the cell, it is normally metabolized by glycolysis, a highlyconserved reaction pathway that converts each glucose anaer-obically into two molecules of pyruvate, with the concomi-tant production of 2 ATPs. In presence of oxygen, cells canoperate the ETC, which begins with the conversion of pyru-vate into acetyl-coenzyme-A (acetyl-CoA). The reaction path-ways responsible for the subsequent production of ATP (andof many macromolecular precursors like amino-acids) are theTricarboxylic Acid (TCA) cycle and Oxidative Phosphoryla-tion (OXPHOS). These complex groups of reactions (roughly100 processes altogether in the bacterium E. coli) are able togenerate the largest energy yield in terms of molecules of ATPproduced per glucose molecule intaken (up to 36, adding tothe 2 given by glycolysis), and release carbon dioxide as awaste product. In absence of oxygen, however, cells cannotrely on the ETC and the ATP yield of glycolysis (2) is to agood approximation all the energy they can generate. In suchconditions, the pyruvate obtained from glycolysis is then re-

∗ Authors contributed equally

duced to other carbon compounds (e.g. acetate, ethanol, lac-tate) that are normally excreted in variable amounts. The con-version of pyruvate to lactate, is carried out by a single reac-tion catalyzed by the enzyme lactate dehydrogenase (LDH).

The energy-generating strategies just described are, in asense, the two extremes, and cells usually operate mixturesof the two even in presence of oxygen, leading to ATP yieldsbelow the theoretical maximum of 38 (typically around 30).However, fast proliferating cells normally display high ratesof glucose intake and produce ATP anaerobically even in thepresence of oxygen, thereby spilling potentially useful carbonand energy resources. A hint about why a large glucose in-flux may favor the use of lower-yield pathways is providedby the fact that processing high glucose fluxes via glycolysisrequires high rates of production of adenosine 5’-diphosphate(ADP) and of NAD+, via oxidation of NADH. The simplestway to convert NADH back into NAD+ is by reduction ofpyruvate to lactate via LDH. Therefore sustaining high ratesof glucose metabolization may imply lactate overflow. Thishowever seems to suggest that a cell with a large glucose in-take should always prefer to generate energy by glycolysis.Therefore, different constraints (physical, regulatory, thermo-dynamic, etc.) may be at work in the selection of a cell’s ener-getic strategy [1]. We note that recent high-throughput studiesof the compounds secreted by growing bacteria in controlledenvironments (the so-called exo-metabolome) uncovered that,besides the standard outputs of overflow metabolism, a pre-viously unsuspected diversity of molecules accompanies theexcretion of carbon equivalents [2].

Likewise, aerobic glycolysis with lactate overflow (a.k.a.Warburg effect) is found to occur in many types of cancers[3, 4], although it cannot be considered as a necessary signa-ture of malignancy [5]. In order to explain its predominanceat the molecular level, several ideas have been pursued, fromstructural or genetic abnormalities in the mitochondria [6], tothe roles played by the hypoxia-inducible factor (HIF) [7] andby a specific isoform of the glycolytic enzyme pyruvate ki-

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nase [8], to seemingly unrelated genetic events leading to theincrease of glucose transporters and the loss of growth con-trol in cancer cells [9]. The search for a molecular basis ofthe Warburg effect has given momentum to the idea that treat-ments targeting the simpler energy-producing apparatus of acell (as opposed to its complex and not entirely known ge-netic profile) carry a higher potential than gene-based thera-pies [10]. Unluckily, our understanding of cancer metabolismis still largely incomplete [11, 12], and research on the roleof metabolism in tumor genesis and progression is currentlyundergoing a major revival [13].

Cell-autonomous models of cellular metabolism based ongenome-scale reconstructions of the underlying network ofmetabolic reactions have been widely studied, proving effec-tive in explaining the origin of overflow metabolism in differ-ent contexts. Constraint-based approaches like Flux-BalanceAnalysis (FBA), for instance, have shed light on the roles thatdifferent types of constraints may play in driving the selectionof energetic strategies towards aerobic glycolysis. An interest-ing and intuitively appealing suggestion is put forward by themacromolecular crowding scenario [14], according to whichenzymes carrying out energy-efficient pathways can at mostoccupy a fraction of the intracellular volume (e.g. the mito-chondria in eukaryotes). Because reactions are assumed to beenzyme-limited, the activity of aerobic pathways will neces-sarily level out once all deputed enzymes work at full speed,while that of glycolysis may still increase. These models areindeed able to predict carbon excretion by unicellulars [15]as well as the Warburg effect in cancer cells [14, 16]. Manyissues however deserve further analysis.

One in particular concerns the fact that cellular waste prod-ucts such as lactate tend to deteriorate the extracellular envi-ronment, so that the sustainability of aerobic glycolysis de-pends crucially on the possibility that such a pollution is re-mediated within cancer’s microenvironment. This raises ques-tions about the role of tumor-stroma metabolic interactions.Tumors are universally characterized by a marked upregula-tion of glucose transporters [9, 17], which allows them tolargely surpass their non-aberrant (stromal) neighbors in thecompetition for nutrients. As a consequence, stromal cellsmust adjust their energetics in ways that have only started tobe analyzed [18].

Lactate exchanges are increasingly being recognized toplay an important role in this respect. In a scenario thathas been characterized by genetic signatures over the pastfew years, tumor-derived lactate can be taken up by glucose-starved cancer-associated fibroblasts (CAFs) [19], whichwould then both help their own survival and foster cancer’saberrant growth by sanitizing the environment. Here wepresent a quantitative in silico analysis of the above mecha-nism. Our results confirm that tumor-to-stroma lactate shut-tling appears as a robust consequence of basic physical andchemical constraints, in a manner that is largely independentof (i) the metabolic function (e.g. energy production, amino-acid synthesis, fatty acid synthesis, etc.) that aberrant cellsstrive to maximize and, perhaps more surprisingly, (ii) of thedegree to which they manage to maximize it. The core ofour conclusions can already be reached through a highly sim-

plified model of metabolism that captures the bare essentialfeatures of the mechanism. The intuition is then validated ona large-scale model of catabolism based on recent reconstruc-tions of the human reactome. After characterizing the networkat the cell-autonomous level, we consider a system formed bytwo such networks sharing the same glucose supply, focusingon the case in which one cell aims to maximize a growth-related objective function (e.g. energy production) with theother subject to a simple ATP maintenance constraint. Theemerging scenario is studied by tuning the ‘degree of max-imization’ by the aberrant cell through a rigorous samplingscheme that provides a statistically significant description ofall feasible metabolic phenotypes compatible with the con-straints. Lactate overflow and functional tumor-stroma cou-pling appears robustly upon maximizing production of ATPas well as of several biomass precursors.

It is important to stress that a tumor-to-stroma coupling,while compatible with the observation that the vast majorityof cancer cells display glycolytic metabolism, is not the onlylactate-based shuttling postulated to be relevant for tumor pro-gression (see e.g. [20]). Other mechanisms and their relationto the current study are outlined in the Discussion.

The remainder of the manuscript is organized as follows.The Results section begins with a brief summary of a styl-ized single-cell model of aerobic glycolysis presented in [14],where it is shown how a crowding constraint can divert ATPproduction from oxidative phosphorylation to fermentation.The simple model is then extended to two cells whereby theextent of lactate shuttle is computed analytically. A large scalemetabolic network of the human core metabolism (HCCN) isthen introduced. A rigorous sampling of the metabolisms pro-duced by HCCN confirms the possibility of cell coupling bymeans of lactate shuttle between a cancer cell and a stromalcell. The rigorous samplings allows us to compute the corre-lation coefficients among the fluxes of the cancer and stromalcell. In the Discussion section, experimental evidences forlactate shuttle are presented and, finally, physiological impli-cations and possible experimental validation of the results arediscussed.

RESULTS

Metabolic fluxes and crowding constraint in a minimalcell-autonomous model

Our starting point is the highly simplified cell-autonomousmodel defined in [14], in which lactate secretion is relatedto an ad hoc constraint limiting volume available for energyproduction in cells that maximize the ATP output flux. In par-ticular, when ATP demands exceed the volume allocated forATP production, the cell produces ATP also through fermen-tation. In the model of Vazquez et al., a cell is characterizedby a glucose intake flux UG,in, a glycolytic flux fglyc (yield-ing two pyruvate molecules per intaken glucose) and an over-

3

all mitochondrial OXPHOS flux fox that transforms pyruvateinto energy (a table summarizing the most relevant variablesdefined here is given as Table S1). Alternately, pyruvate canbe turned into lactate by lactate dehydrogenase (LDH), withflux fLDH. Lactate can then be expelled from the cell so asto avoid acidification. Mass balance and steady state condi-tions impose that metabolite levels do not change in time, sothat, for example, the net production and consumption fluxesof pyruvate must compensate, implying

2fglyc − fox − fLDH = 0 , (1)

(note that, to make the subsequent analysis simpler, our alge-bra differs slightly in numerical pre-factors from the one givenin [14]. The overall picture is however completely equivalent).The glycolytic flux is in turn limited by the flux UG that sup-plies glucose to the system, so that UG,in ≤ UG, while undermass balance UG,in = fglyc. The crowding constraint is im-plemented as Vglyc + Vox + VLDH ≤ VATP, where VATP rep-resents the cell volume devoted to ATP production, whereasVglyc, Vox, and VLDH are the volumes taken up by the gly-colytic, OXPHOS, and LDH enzymes, respectively. Introduc-ing the constants aglyc, aox, and aLDH representing the vol-ume occupied per unit of ATP production by glycolytic, OX-PHOS and LDH enzymes, respectively, this can be re-cast as[14]

aglycfglyc + aoxfox + aLDHfLDH ≤ ΦATP , (2)

where ΦATP is the volume fraction available for ATP produc-tion in each cell. Using (1), one obtains(aLDH +

aglyc2

)fLDH +

(aox +

aglyc2

)fox ≤ ΦATP . (3)

We shall adhere to [14] in employing the empirical estimatesaglyc ' 3× 10−3 (min/mM), aLDH ' 4.6× 10−4 (min/mM),aox ' 2 × 10−1 (min/mM), and ΦATP = 0.4. Equation (3)represents the “crowding constraint”, amounting to a globalconstraint to the overall flux of metabolites the cell can investin ATP production. To be consistent with [14], we derived thecrowding constraint on fluxes by limiting the volume fractiondevoted to energy production, but the constraint (3) might aswell be derived by limiting the amount of proteins (see, e.g.,[16]).

To assess the amount of ATP produced by these fluxes, weconsider that fglyc generates 2 ATPs and 2 pyruvates per glu-cose molecule, and that fox creates 18 ATP molecules fromeach pyruvate. The conversion of pyruvate to lactate does notgenerate any ATP and it only ensures that glycolysis can con-tinue by regenerating NADH to NAD. The overall flux of ATPproduction is finally given by

fATP = 2fglyc + 18fox . (4)

Vazquez et al’s results can be summarized as follows. Ne-glecting (3), ATP production is maximized when fglyc = UG,fox = 2fglyc = 2UG, and fLDH = 0, i.e. when all availableglucose is taken up by the cell and used in OXPHOS. Thissolution still holds if (3) is satisfied, i.e. as long as

(2aox + aglyc)UG ≤ ΦATP . (5)

If however UG > ΦATP/ (2aox + aglyc) ≡ uG, then thecellular resources available for high-yield energy productionpathways has been exhausted and ATP synthesis is no longerlimited by glucose but by intracellular resources (either vol-ume or proteins production capacity). Its maximization nowrequires that the glucose flux exceeding the possibility of pro-cessing by OXPHOS is diverted towards LDH. The maximumATP production flux and the corresponding lactate flux for thiscase can be obtained by inserting fLDH = 2UG − fox into (3)with the constraint saturated. One gets

fLDH =

{0 if UG ≤ uGαox(UG − uG) if UG > uG

, (6)

αox =2aox + aglycaox − aLDH

> 0

and

fox =

{2UG if UG ≤ uGαLDH(vG − UG) if UG > uG

, (7)

αLDH =2aLDH + aglycaox − aLDH

> 0

where vG = ΦATP/ (2aLDH + aglyc). Because of mass bal-ance, fLDH also equals the lactate excretion flux so that theWarburg effect is described by (6), which shows the existenceof a threshold glucose intake above which lactate is secreted.

Metabolic fluxes in a minimal model of two cells coupled via alactate shuttle

Let us now consider two replicas of Vazquez et al.’s celland assume that the two cells can interact via the exchangeof lactate, which, in particular, can be intaken and used asan alternative carbon source through reverse LDH (catalyzingthe conversion of lactate back to pyruvate). For sakes of sim-plicity, we distinguish between a donor cell (‘don’, which canonly excrete lactate) and an acceptor cell (‘acc’, which canboth excrete and import lactate) (see Fig. 1). In order to avoidstarvation, we furthermore impose that both cells generate aminimum flux fmin

ATP of ATP, i.e.

fATP,i ≥ fminATP , i ∈ {acc,don} (8)

where the ATP production flux is given by (4), namely

fATP,i = 2fglyc,i + 18fox,i i ∈ {acc,don} . (9)

As shown in the beginning of the previous section, whenfglyc = UG and fox = 2UG cells can extract the maximumATP from a single glucose molecule, i.e., fATP,i = 38UG.We observe that, in order to satisfy (8), the overall glucosesupply to the system composed of two cells should satisfy

UG ≥2fmin

ATP

38≡ 2uG,0 , (10)

4

FIG. 1. Schematic representation of the minimal model of twocells coupled via a lactate shuttle. Two cells (lactate donor andacceptor, respectively) share glucose as an energy source. Glucoseis partitioned according to the fluxes fglyc,don and fglyc,acc, whichconvert one internal glucose molecules to two pyruvate moleculesproducing two ATP molecules. Pyruvate can then undergo oxida-tive phosphorylation (with the irreversible fluxes fox,don and fox,acc,giving 36 more ATP molecules) or LDH (with fluxes fLDH,don andfLDH,acc, by which lactate (L) is obtained). Lactate is for simplic-ity assumed to be secreted upon production. If both cells producelactate, no coupling sets in, unless due to competition for nutrientsunder glucose limitation. If however the donor cell secretes lactate,the acceptor cell may intake it to replace glucose whenever its accessto the latter is limited (e.g. because the donor cell’s glucose intakeis large). In such cases a lactate shuttle will effectively couple themetabolisms of the two cells.

which defines uG,0 as the minimum glucose supply fluxthat can prevent cellular starvation. Denoting as UG,don theamount of glucose available to the donor cell, if the donor cellmaximizes ATP production, its flux organization will matchthe one found above for a single cell. It is useful to distin-guish two limiting cases:

(a) If the donor cell employs OXPHOS exclusively, whichhappens when 2uG,0 ≤ UG ≤ uG + uG,0, we haveUG,don = UG − uG,0 and UG,acc = uG,0. In otherwords, the donor cell takes up all of the available glu-cose except for the amount necessary for the survival ofthe acceptor cell. In these conditions, the acceptor cell’ssurvival is guaranteed by glucose availability.

(b) At the other extreme, if the acceptor cell can surviveeven using exclusively the lactate excreted by the donorcell, all glucose is intaken by the donor cell, so thatUG,don = UG and UG,acc = 0. This happens whenfLDH,don ≥ fmin

ATP/18, or

UG ≥fminATP

18αox+ uG . (11)

In intermediate situations, the amount of glucose intaken bythe acceptor cell gradually decreases from uG,0 to 0 as it in-creasingly uses lactate to produce pyruvate and thus ATP. Insuch cases, the oxidative pathway of the acceptor cell is fed

by both glucose and lactate fluxes as fox,acc = 2fglyc,acc +fLDH,don, which, via mass balance, implies (see (9))

fATP,acc = 38fglyc,acc+18fLDH,don = 38UG,acc+18fLDH,don .(12)

To obtain simpler expressions, it is useful to introduce theshorthands

uG,1 = uG,0 + uG , (13)

uG,2 =fminATP

18αox+ uG , (14)

and ∆UG for the amount of glucose rerouted from the accep-tor cell to the donor cell, so that

UG,acc = uG,0 −∆UG

UG,don = UG − uG,0 + ∆UG .(15)

Substituting UG,acc and imposing minimal ATP requirementin (12), we obtain

fLDH,don =38

18∆UG . (16)

Since the donor cell maximizes ATP production, we obtaina second expression for fLDH,don as a function of ∆UG bysubstituting UG,don: in (6)

fLDH,don = αox(UG,don−uG) = αox(UG−uG,0+∆UG−uG) .

This, combined with (16), determines the extra amount of glu-cose available to the donor cell

∆UG =αox

38/18− αox[UG − uG,0 − uG] . (17)

Expression (17) is valid if uG,1 < UG ≤ uG,2. For UG ≤uG,1 one has ∆UG = 0, while when UG > uG,2 ∆UG =uG,0. We can therefore summarize the fluxes for the donorcell as

fLDH,don =

0 if 2uG,0 ≤ UG ≤ uG,1

αox(UG + ∆UG − uG,1) if uG,1 < UG ≤ uG,2

αox(UG − uG) if UG > uG,2

(18)

and

fox,don =

2(UG − uG,0) if 2uG,0 ≤ UG ≤ uG,1

αLDH(vG − UG,don) if uG,1 < UG ≤ uG,2

αLDH(vG − UG) if UG > uG,2

.

(19)

Because the donor cell uses up most of the glucose, the ac-ceptor cell stays at the minimum ATP production rate fmin

ATPfor UG ≤ uG,2. In this regime, the fluxes of the acceptor cellare fixed and can be derived from the ones of the donor cell.One gets

fLDH,acc =

{0 if 2uG,0 ≤ UG ≤ uG,1

−fLDH,don if uG,1 < UG ≤ uG,2(20)

5

FIG. 2. Qualitative behaviour of the single-cell and of the donor-acceptor system for the coarse-grained model. (a) In the cell-autonomous model, lactate overflow occurs when the glucose intakeovercomes a threshold. Correspondingly, the flux through oxidativemetabolism increases until the threshold before slowly decreasingonce the crowding constraint is saturated and fermentation sets in.(b) In the donor/acceptor system, the donor behaves essentially as anautonomous cell and the acceptor adapts to it. For low glucose in-takes, it operates oxidative pathways at small rate. When the donorsaturates the crowding constraint excreting lactate, the acceptor im-ports it and uses it at a substrate to increase the flux through oxidativemetabolism.

and

fox,acc =

{2uG,0 if 2uG,0 ≤ UG ≤ uG,1

2UG,acc + fLDH,don if uG,1 < UG ≤ uG,2.

(21)

For large total glucose intakes, the donor cell excretes morelactate than necessary for bare survival of the acceptor cell,which can produce any amount of ATP as long as it is largerthan fmin

ATP. The metabolic state of the acceptor cell is there-fore not uniquely defined. When UG ≥ uG,2, for the fluxes ofthe acceptor cell one finds

−fLDH,don ≤ fLDH,acc ≤ −fminATP

38(22)

fox,acc = −fLDH,acc , (23)

while the ATP production is given by

fATP,acc = −18fLDH,acc . (24)

Note that fLDH,acc < 0 because the acceptor cell intakes lac-tate and reverses LDH.

The solution to this model is described pictorially in Fig. 2and discussed in Fig. 3, where we also show the feasible rangeof values of the fluxes of the acceptor cell.

In essence, this coarse-grained model suggests that an im-balance in the energetic demands of two cells can induce ametabolic coupling driven by a lactate shuttle from the high-to the low-demand cell, and provides a qualitative scenariofor how carbon utilization in both cells changes by tuning theoverall glucose supply. We shall now see that such a pic-ture is fully recovered within a large-scale model of humanmetabolism.

Metabolic fluxes and crowding constraint in a cell-autonomouslarge scale metabolic network

In order to characterize the exchange of carbon equivalentsamong interacting cells more in detail, we have analyzed ahuman metabolic network derived from the reconstructed hu-man reactome [21], to which we refer as the ‘Human CoreCatabolic Network’ (HCCN, see ‘Materials and Methods’).The HCCN is composed by 67 metabolites and 75 reactions(including uptake fluxes), 23 of which are reversible. Thecrowding constraint that accounts for finite cellular resourcesis represented as

aglycfHEX + aox(fPDH + fGLUN) + aLDHfLDH ≤ ΦATP

(25)where fHEX denotes the Hexokinase-1 flux, which irre-versibly channels glucose into glycolysis, fPDH denotes theflux through pyruvate dehydrogenase, by which pyruvate isdiverted into the TCA cycle and fGLUN denotes the mithocon-drial flux of nitrogen metabolism through glutaminase. Weexplore the space of flux patterns consistent with the con-straints by sampling solutions of the mass balance equationsSf = 0 (S denoting the stoichiometric matrix and f being aflux vector) such that flux vectors fdon of the donor cell ap-pear with probability

P (fdon) = A exp[βfATP,don] , (26)

where β ≥ 0 is a parameter and A is a normalization con-stant. The rationale is that, by tuning the value of β, we canpass from an unbiased sampling in which fATP,don takes onevery allowed value with uniform probabilty (β = 0) to onein which the donor cell maximizes its ATP output (β → ∞),thereby obtaining a complete and refined picture of how dif-ferent degrees of optimization by the donor cell impact theflux pattern of the acceptor cell. Solutions at the two extremescan be obtained with standard methods, as linear program-ming, but only our method allows us to sample realistic sub-optimal optimizations. In Fig. S6 we show how the ATP pro-duction increases as a function of β. In the following, as aprototype for sub-optimal optimization, we choose β = 5 thatcorresponds to roughly 70% of the maximum ATP production.Model and algorithmic details are discussed in ‘Materials andMethods’.

In Fig. 4a we show the ATP production of a single HCCNcell as a function of the available glucose. If ATP produc-tion is maximized (large β), one first encounters a regimewith a yield of roughly 30 ATPs per glucose molecule (to becompared with the theoretical value of 38) and then a regimewhere the yield is about 1.7 ATPs per glucose. ATP is pro-duced by OXPHOS in the former case, and by fermentationin the latter (see also Fig. 4b and Fig. 4c). As expected, net-works maximizing ATP shift their metabolic strategy at theglucose intake for which the resources available for ATP pro-duction by oxidation is exhausted. Note that for β = 5 oneobtains an ATP efficiency very close to optimal. Quite im-portantly and surprisingly, however, for β = 0 (i.e. whenno ATP maximization is performed) the emerging picture isqualitatively preserved, albeit with lower ATP yields. Indeed

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FIG. 3. Solution of the minimal model of two cells coupled via a lactate shuttle. A lactate donor cell that maximizes ATP production iscoupled to an acceptor cell. (a) Glucose intake for donor and acceptor as a function of total glucose supply. (b) ATP production as a function ofthe total glucose supply. The shaded area indicates that all values within that region are feasible. (c) Flux through fermentation fLDH (circles)and oxidative phosphorylation fox (crosses) as a function of the total glucose supply. (d) Fraction of ATP produced via fermentation (black)or via oxidative phosphorylation (red) in the donor cell as a function of the total ATP it produces.

solutions sampled at β = 0 appear to employ a mixture ofOXPHOS and fermentation even at low glucose intakes. It isremarkable that the resources-driven shift still occurs at uG,as in this case it corresponds to a largely suboptimal valueof ATP production. In other words, the HCCN might devotecellular resources to increase ATP production, but to do soit must explicitly be pursuing ATP maximization. This im-plies that the crossover from a high- to a low-yield strategy isa robust, embedded property of the network (and of the con-straints imposed) and not an exclusive characteristic of the ex-tremal solution that maximizes the ATP production.

In Fig. 4b, we focus on the fluxes through PDHm and LDH,that indicate whether pyruvate is directed towards OXPHOSor fermentation. We see that high ATP yields are obtainedby using OXPHOS exclusively. The typical network state atβ = 0 is less efficient at all glucose intakes, as it always di-verts a larger fraction of pyruvate to fermentation comparedto the ATP-maximizing cell. Finally, in Fig. 4c, we detail howglucose partitions between fermentation and OXPHOS. Thenetwork displays a reversal of glucose fate as the crowdingconstraint is saturated independently of whether ATP produc-tion is being maximized or not. The latter case however turnsout to be generically less efficient.

Metabolic fluxes in a large scale metabolic network of two cellscoupled via a lactate shuttle

We now consider two replicas of the HCCN and again dis-tinguish between a lactate acceptor and a donor cell. We alsoimpose that there is no external lactate source, and explorethe scenario where the donor optimizes ATP production to adegree tuned by the parameter β. Fig. 5a shows that, as ex-pected, an ATP-maximizing donor cell sequesters all of theavailable glucose except for the small amount required for theacceptor’s survival. This is reflected by the ATP productioncurves as a function of the total glucose supply displayed inFig. 5b. In such conditions, the acceptor’s ATP productionfluxes matches the minimum required for survival, i.e. fmin

ATP(which is set to be equal to 1 uG for simplicity) until the donorswitches to aerobic glycolysis, thereby excreting lactate. Thedonor, on the other hand, goes through an initial phase of ex-clusive OXPHOS use, followed by a switch to aerobic glycol-ysis when the crowding constraint does not allow for a furtherincrement of the mitochondrial flux. In Fig. 6a one indeedsees that glucose is mainly channeled to OXPHOS as long asthe crowding constraint allows for it. As soon as the latter issaturated, the donor diverts pyruvate to LDH, usefully regen-erating NAD from NADH and thus avoiding glycolysis halt.LDH generates lactate, which is then expelled and intaken bythe acceptor. As shown in Fig. 6b, this is accompanied by a

7

FIG. 4. An isolated HCCN maximizes the ATP yield by directingglucose to OXPHOS even in absence of an active ATP maximiza-tion. (a) Average ATP production as a function of the re-scaled av-erage glucose supply. (b) Average flux through LDH (crosses) andPDHm (circles) as a function of the re-scaled average glucose sup-ply. (c) Fraction of ATP produced via glycolysis (crosses) or viaOXPHOS (circles) as a function of the re-scaled total ATP produced.The flux through each pathway is re-scaled by half the amount of glu-cose intaken by the cell (because with one molecule of glucose cellsproduce two molecules of pyruvate). Curves describe the behaviourof an ATP-maximizing HCCN (black lines, β = 50), a loosely max-imizing donor (red lines, β = 5) or the result of a uniform samplingof the allowed flux states for a HCCN (blue lines, β = 0). Error barsfor the s.e.m. are smaller than symbols.

FIG. 5. When a lactate donor maximizes its ATP production itintakes most of the glucose supplied to the two-cell system. TheATP production by the acceptor cell increases in correspondenceto a decrease in efficiency of the donor’s metabolism. (a) Glu-cose intakes for two coupled cells as a function of the total glucoseavailable to the donor-acceptor pair. (b) ATP produced by the donor.Curves describe the behaviour obtained for two coupled HCCN cellswith an ATP-maximizing donor (black lines, β = 50), a looselymaximizing donor (red lines, β = 5) or for an unbiased sampling ofthe two-cell solution space (blue lines, β = 0). Error bars for thes.e.m. are smaller than symbols.

reversal of LDH in the acceptor: lactate is transformed intopyruvate that is then channeled to OXPHOS through PDHm.The acceptor can thus spare some cellular resources to pro-duce pyruvate, at the cost of becoming strongly dependent onthe donor for ATP production.

If one instead averages over all solutions without biasingfor ATP production by setting β = 0, the two cells producecomparable amounts of ATP and share glucose more evenly(see Fig. 5a). The fact that the donor turns out to employslightly more glucose than the acceptor is due to the intrin-sic asymmetry that is introduced by not letting the donor in-take lactate. This asymmetry is also sufficient to drive, evenwhen an unbiased statistical picture of the solution space isconsidered, a net lactate exchange as the typical behaviourBoth cells, however, produce a factor 6 (roughly) less ATPthan the maximum possible, given the glucose influx. Despitea similar overall ATP production profile, the internal fluxesof donor and acceptor differ significantly. While the donordistributes glucose almost evenly through LDH and PDHm,the acceptor can only generate ATP via OXPHOS, since itintakes lactate as an extra source of carbon. In Supplemen-tary Fig. S1, we show that two symmetric cells with unbiasedATP production show identical glucose intake, ATP produc-tion, and internal flux distribution. Even for symmetric cells,however, when the donor optimizes the ATP production, weobserve lactate shuttling between donor and acceptor cells, asillustrated again by Supplementary Fig. S1. In summary, formaximal ATP production (β → ∞) there is no difference be-tween ATP production and internal flux distributions of sym-metric and asymmetric cells-couples. For simplicity in theanalysis of the data, we used the asymmetric case where thedonor cell cannot intake lactate.

We observe that the fraction of ATP produced via oxidativephosphorylation utilized by the acceptor cell differs substan-tially from the one of the donor cell. In Fig. 4d we displaythe relative usage of the fermentative and the oxidative path-way and observe that, for low glucose supply, the acceptor cell

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FIG. 6. Oxidative and fermentative fluxes in a donor-acceptor system. (a)–(b) Average flux through LDH (circles) and PDHm (crosses) asa function of the average glucose supplied to the two-cell system. (c)–(d) Fraction of ATP produced via glycolysis (circles) or via oxidativephosphorylation (crosses) as a function of the total ATP produced by the donor cell. Pathways are normalized by the “pyruvate-equivalent”, i.e.by the sum of lactate and half-glucose intakes. Curves describe the behaviour obtained for two coupled HCCN cells with an ATP-maximizingdonor (black lines, β = 50), a loosely maximizing donor (red lines, β = 5) or for an unbiased sampling of the two-cell solution space (bluelines, β = 0). Error bars, which represents s.e.m., are smaller than symbol sizes.

only employs the oxidative pathway, independent of whetherthe donor cell maximizes ATP or not. When the donor cellmaximizes ATP production, it sequesters most glucose andthe acceptor cell is therefore forced to feed on lactate, whichcan only be usefully converted to ATP by means of OXPHOS.Conversely, when the donor cell does not maximize ATP pro-duction, it secretes a sizable amount of lactate because itsmetabolism is inefficient. The lactate acceptor, however, isalso mainly feeding on glucose (see Fig. 5a) and could in prin-ciple secrete lactate just like the donor cell. The reason why apurely oxidative metabolism is observed lies in the reversibil-ity of LDH. Although lactate intake by the acceptor is small,it suffices to force the pyruvate flux towards TCA and OX-PHOS, thereby making the acceptor more efficient than thedonor in energetic terms.

In order to assess the robustness of the occurrence of lactateoverflow metabolism under the imposed constraints, we havefurther analyzed the flux configurations in a dono/acceptorsystem in which the donor maximizes the production ofbiomass precursors. We show here (see Fig. 7) results ob-tained by maximizing the biomass defined as in Table S9,which essentially reproduce those described above. Such arobustness becomes less surprising in the light of the fact thatthe qualitative features of the switch from oxidative to non-oxidative metabolism are obtained even in an unbiased sam-

FIG. 7. Donor’s biomass flux and lactate overflow, and accep-tor’s lactate intake for two coupled HCCNs with a biomass-maximizing donor. The qualitative behavior obtained when thedonor is maximizing the ATP flux is reproduced in a more realis-tic case in which a biomass objective function is considered. SeeTable S9 for the biomass coefficients.

pling of the steady states. This observation suggets that sucha scenario is to a large degree embedded in the stoichiometryand in the main topological features of the underlying reactionnetwork.

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Correlation coefficients among metabolic fluxes of two cellscoupled by lactate shuttle under glucose limitation

To formally assess the extent of metabolic coupling be-tween an ATP-maximizing lactate donor and a lactate accep-tor, we computed the matrix of normalized Pearson correla-tion coefficients of each pair of fluxes in the solutions sampledfor different levels UG of the glucose supplied to the system.The Pearson coefficient between random variables X and Yis defined as r = cov(X,Y )/(σXσY ), where cov(X,Y ) de-notes their covariance and σX and σY stand for their respec-tive standard deviations. r ranges from −1 to 1 and quantifiesthe linear dependence of the two variables. More precisely,the linear correlation between variablesX and Y is more pos-itive the closer r is to 1, and more negative the closer r is to−1, while X and Y can be considered uncorrelated if r ' 0.For sakes of clarity, we have considered the correlations aris-ing in a system formed by an ATP-maximizing lactate donor(large β) and a lactate acceptor. For smaller values of β corre-lations get weaker while maintaing the same qualitative struc-ture. Fig. 8 displays three reduced correlation matrices (ob-tained for three different values of the overall glucose supply)where only a small subset of fluxes (each presentative of adifferent biologically relevant pathway) appears. Full matri-ces for three choices of UG are instead shown in Figs. S3–S5.

Intuitively, correlation patterns should depend strongly onUG. The essence of this dependence, which clearly emergesfrom Fig. 8, is that cross-correlations between donor and ac-ceptor build-up as the glucose supply increases, are maximalwhen the acceptor’s energetics depends on donor-derived lac-tate, and then decrease again rapidly to zero when the glucosesupply is large enough to sustain both cells.

If UG is low (sub-threshold for lactate overflow), theATP-maximizing donor necessarily outcompetes the accep-tor for the available glucose, establishing a degree of cross-correlations that is driven mainly by the unbalanced nutrientpartitioning. The donor indeed employs OXPHOS almost ex-clusively, while still leaking out a small amount of lactatewhich is taken up by the acceptor (i.e. there is no accumu-lation in the tissue). This suffices to establish the overall cor-relation among cells that can be seen in Fig. 8a.

As the glucose supply is increased further (above thresh-old for lactate overflow), the ATP-maximizing donor switchesto aerobic glycolysis and secretes lactate that is shuttled tothe acceptor in large amounts (viz. the strong negative cor-relation arising between the donor’s lactate outflux and theacceptor’s lactate influx). In such conditions, the overallcross-correlations increase, see Fig. 8b. In particular, gly-colytic fluxes in donor and acceptor anti-correlate, oxidativepathways weakly anti-correlate, while te donor’s biosyntheticpathways correlate weakly with the acceptor’s oxidative path-ways. Notice also that ATP flux in the donor weakly anticor-relates with every pathway in the acceptor’s metabolism.

At high enough UG, finally, cross-correlations decay asboth cells can rely entirely on glucose for their energetics.Because the maximum glucose uptake for an HCCN is given

FIG. 8. Restricted matrices of Pearson correlation coefficientsfor two coupled HCCN cells when the lactate donor maximizesthe ATP production and the overall glucose supply is large showsthat the two cells are not correlated. The intensity of the color rep-resents the magnitude of the correlation coefficient (see scale on theright hand side). The two cells can independently access glucose andinternal fluxes of the lactate acceptor and donor are essentially un-correlated. The four representative reactions displayed for each cellare the glucose influx (Glc, a proxy for glycolytic activity), LDH (aproxy for lactate overflow and exchange), PDH (a proxy for oxidativemetabolism) and the ATP production flux (ATP).

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by UmaxG = ΦATP/ (aG + 2aLDH) (this is determined by the

crowding constraint (25) when glucose is entirely channeledto fermentation, i.e. when fHEX = UG,don = fLDH/2), onemay assume that UG is “large enough” when it is larger than2Umax

G , as both cells would in this case have access to as muchglucose as they can intake. In this case, no donor-acceptorcorrelations should be expected. Fig. 8c shows that this is in-deed the case: intra-cellular correlations, described by the di-agonal blocks, greatly exceed inter-cellular ones (off-diagonalblocks).

In summary, when glucose availability is limited, the fast-growing donor necessarily outcompetes the acceptor for glu-cose and a network of cross-correlations is established be-tween the metabolic systems of the two cells. Metabolic path-ways in donor and acceptor correlated however most stronglywhen a lactate exchange from donor to acceptor is established,and decay again as the two cells become effectively decoupledin glucose-rich media. The build-up of a metabolic partner-ship is perhaps best seen by the fact that the donor’s glucoseinflux UG,don develops a correlation with the acceptor’s LDHflux at intermediate values of UG. Note also that UG,don cor-relates more and more strongly with the donor’s LDH flux asUG increases.

DISCUSSION

Experimental evidence on lactate shuttles in cancer

While lactate shuttling is a likely scenario in many physi-ological conditions, from muscle cells undergoing intense ac-tivity [22] to neuron-glia energetics [23–26], and has been re-ported to take place intra-cellularly at the mitochondrial andperoxisomal membrane [22], no direct measurement of inter-cellular lactate exchange in cancer exists. Strong indirect evi-dence, however, suggests that such a scenario is indeed plau-sible.

Indeed, tumor-stroma and tumor-tumor metabolic inter-actions are currently being characterized at various levels.The emerging picture increasingly suggests that cancer sus-tainment is a non-cell-autonomous phenomenon [27–30] andthat stromal cells might be potential targets for cancer treat-ment [31]. Lactate exchange in particular has been investi-gated both as fueling oxidative cancer cells and as support-ing non-aberrant cancer-associated cells [32, 33]. The lattercase corresponds to the model discussed here. Signatures oftumor-to-stroma lactate shuttle have been reported in terms ofthe over-expression of monocarboxylate lactate transportersjointly with increased PDH activity in cancer-associated fi-broblasts (CAFs) [19, 34]. Likewise, tumor-derived lactatehas been found to play a major signaling role (specifically,for the initiation of tumor angiogenesis) in vascular endothe-lial cells [35]. Inhibition of the lactate transporter MCT1 hastherefore been proposed as a possible anti-angiogenic strategy[36].

Taken together, these observations suggest that tumor cellsand their associated stromal and vessel cells can be seen asa collective, synergistic metabolic unit where each compart-ment carries out complementary functions reflected in their

energetic strategies. In such a microenvironment, aerobic stro-mal cells serve an essential ‘bioremediative’ role by remov-ing potentially toxic compounds, thereby reducing acidity andpositively feeding back on anaerobic cancer growth. The ex-istence of a consistent imbalance in energetic demands acrossdifferent compartments is crucial to establish this scenario.

Intercellular shuttling of lactate towards oxidative cancercells is known to occur in two distinct forms, namely eitherfrom non-oxidative (e.g. hypoxic) to oxidative tumor cells[36] or through the ‘reverse Warburg effect’, i.e. the onsetof aerobic glycolysis in CAFs [37–41]. The former case de-scribes the metabolic sysmbiosis that is established e.g. be-tween more and less hypoxic regions of a tumor, which puta-tively allows glucose to be delivered more efficiently to morehypoxic regions [42] (see [43] for a game-theoretic approachto modeling this kind of tumor-tumor interplay). In the lat-ter scenario, cancer cells induce oxidative stress in CAFs, re-sulting in CAFs switching to anaerobic glycolysis [44]. Thesecreted lactate is then imported by cancer cells for use in aer-obic pathways. This effect has indeed been observed in vitro[39, 41], and its relevance in vivo is currently under scrutiny[38]. Clearly, however, this type of shuttling is not necessarilytriggered by a strong imbalance in energetic demands betweenthe lactate donor and acceptor cells, and different constraintsin metabolic activity are likely to be required to understand itsorigin within genome-scale models.

It is worth noting that recently developed microfluidic plat-forms allow to probe the tumor microenvironment with un-precedented resolution and control over nutrient supply [45].More light will hopefully be shed on its activity as a functionalmetabolic assembly and on the role of lactate in specific.

Going beyond cancer, however, such studies highlight thefact that replacing the natural environment in which cells livewith a cell culture that is possibly optimized for the cell’sneeds might severely limit our ability to understand the ac-tual behavior of the system in vivo. On the other hand, theysuggest that cell-autonomous models may be unable to cap-ture some essential features of the energetics of cells: whencells employing different energetic demands share a limitedresource, a mutually beneficial microenvironment character-ized by a large-scale exchange of chemical species can be es-tablished. A thorough understanding of cellular growth strate-gies should take these aspects into account.

Conclusions

In this article we have studied the lactate-driven couplingthat is established between cells with different energetic re-quirements, showing that a lactate shuttle appears robustlyas a consequence of basic physico-chemical constraints. Ourmodel covers time scales over which the cellular metabolicnetworks have to adapt to the establishment of an imbalancein energy demands (as well as, for instance, in the levels ofmain carbon transporters), which are much shorter than thetime required to alter the nutrient availability profile signifi-cantly (e.g. via vascularization). We have illustrated the emer-gent metabolic interaction scenario in a simplified model that

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captures the essential fact that, for two such cells, recyclingfermentation products may be a good strategy to optimize cel-lular glucose usage. To reach these conclusions, we have con-sidered a human metabolic network model of cells interactingvia lactate shuttle and sampled its feasible metabolic states ac-cording to a prescribed probability distribution by which wecan tune the degree to which one of the two cells is pushingits ATP production. The scenario obtained in a highly sim-plified (but exactly solvable) model is fully retrieved in thiscase, where a thorough analysis of correlation patterns revealsfurther details of the cell-to-cell coupling.

Such a coupling would have serious physiological implica-tions. First, lactate recycling implies that lactate may accu-mulate in tissues only in late-stage tumors. Before neighbor-ing cells saturate their processing capabilities, the lactate ex-pelled by cancer cells would be taken in by non-aberrant cells,suggesting that localized lactate measurements might identifyearly-stage tumors. Second, lactate shuttling and recyclingsystems should be regarded as potential targets for treatment.

There are many aspects of the approach presented herethat can be further dissected. First is the nature of con-straints. From a physical point of view, imposing “volume”constraints of the type considered here is equivalent to impos-ing finite flux capacities on energy-efficient pathways. Thereare however strong indications that unicellular organisms ac-tively down-regulate energy-efficient pathways at high nutri-ent levels [46, 47]. The fact that cells invest energy and re-sources in silencing the synthesis of enzymes involved in aer-obic pathways suggests that the switch to an energy-inefficientpathway provides a real, physiological advantage for the cell.Its origin and nature are currently not understood, although theinclusion of additional “costs” due to regulation in metabolicmodels strongly suggests that the observed phenomenologycan be captured, at least to some degree, by accounting for thefact that a shift in energetic strategy requires a sizeable changeof a cell’s protein repertoire [1].

It is also important to note that the ETC in eukaryotestakes place in a separate cellular compartment (the mitochon-drion), at odds with prokaryotes like bacteria, where it oc-curs is the cell’s periplasm. We have entirely neglected thecomplications due to spatial organization, which is reasonableas long as one wants to focus on the ATP yield of energy-producing strategies. However, this aspect will become im-portant for higher-resolution modeling at genome scale. Like-wise, a more refined spatial modeling will allow for an in sil-ico analysis of a possible role of intra-cellular lactate shut-tling in cancer [22]. In brief, the key idea behind this is thatglycolysis-derived lactate could be employed directly as anadditional energy source for mitochondrial oxidation insteadof being excreted. Elementary biochemical properties of theLDH reaction indeed would appear to favour lactate (ratherthan pyruvate) production in the cytosol. Moreover, experi-mental facts indicate a possible role of intra-cellular lactateshuttling in cardiomyocites, neurons and skeletal muscle cellsfacing high energy demands [48]. Accounting for it in a com-putational scheme may help reveal novel details about possi-ble pathways of energy production during physical exercise orin rapidly growing cells.

Furtermore, while this work has focused on lactate as akey mediator of metabolic, energy-driven intercellular inter-actions, it is worth stressing that recent studies focusing onthe comprehensive analysis of the exometabolome of growingcells has revealed that metabolic interactions encompass manymore chemical species and is tightly coupled with the growthregime. In bacteria, for instance, released intermediates in-clude not only carbon equivalents (like acetate, pyruvate andethanol) but also amino acids and central metabolic interme-diates [like fructose-6-phosphate, glucose-6-phosphate, 2/3-phosphoglycerates, phosphoenolpyruvate, acetyl-CoA, citrateand α-ketoglutarate], higher amounts of which have beenfound to be consistently released when carbon availability ishigh, while intermediates are typically in-taken in carbon-limitation [2]. Similar interactions are now known to occurin cancer: leukaemia cells have indeed been recently shownto employ cysteine derived from bone-marrow stromal cellsas a means to fight oxidative stress [29]. As the scenario un-derpinning the establishment of such couplings is increasinglyelucidated, it will become possible to set up more refined andrealistic models to capture a greater extent of their physiolog-ical relevance.

Finally, it should be kept in mind that aerobic glycolysisin cancer cells, especially at later stages of oncogenic devel-opment, could be due to the fact that the tumor microenvi-ronment is hypoxic. The model we consider here addressesa faster time scale, over which the energy balance of neigh-bouring cells disrupts while the overall amount of resourcesavailable remains (roughly) unchanged, and suggests that atthese stages tumor-to-stroma metabolic interactions can pro-vide a mutually beneficial solution to the imbalance. It is alsoworth mentioning that cases are known in which malignantcells rely on increased oxidative phosphorylation rather thanon aerobic glycolysis for energy production (as found, for in-stance, in transformed human mesenchymal stem cells [5]).Taken together, these results suggest that cancer’s bioenerget-ics, starting with the aerobic glycolysis versus oxidative phos-phorylation “dilemma”, may be a dynamically modulated pro-cess that differs widely across tumor types [49]

The modeling approach discussed here, based on explor-ing the solution space induced by mass-balance equationsrather than on optimizing a prescribed objective function, isin our view the most suited to deal with multi-cell systemsin which extended cell-to-cell metabolic interactions are es-tablished by one cell’s deregulated metabolic demands. Thesampling method employed here is scalable and easy to im-plement, providing a highly promising tool for further stud-ies. It would in particular be important to analyze the emer-gence of these effects in full-fledged genome-scale models ofspecific cancers, as can be obtained e.g. by refining recon-structions of human metabolic networks with cancer expres-sion data. The full complexity of metabolic cell-to-cell in-teractions has only just started to be uncovered. Understand-ing its functional relevance by detailed in silico models mayprovide new insight into the mechanism of cancer progressionand further advance the search of specific, targeted treatments.It would be important to assess the relevance of lactate shut-tling in the context of tumors through experiments probing

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metabolism (or metabolite exchanges) directly, rather than bycollecting indirect evidence in the up- or down-regulation ofspecific transporters, and metabolic solution space samplingis known to provide useful keys for experiment design [53].Our results indicate at least two potential ways to obtain use-ful information. One possibility requires setting up an assaywhere the ability of non-aberrant cells to intake lactate canbe modulated in the presence of bona-fide lactate-secretingcancer cells. Such a modulation could be achieved, for in-stance, by either varying the relative concentration of donor-and acceptor-cells, or by changing the expression of lactateand glucose transporters in supporting cells. It is reasonable toexpect that, in such a setup, the extracellular acidification ratedue to the accumulation of lactate in the extracellular mediumshould negatively correlate with the lactate removal capacityof the supporting cells. On the other hand, results from ourstatistical sampling of the solution space suggest that differ-ent pathways (and hence, likely, the expression levels of dif-ferent enzymes) should be tuned in a specific way by chang-ing the overall glucose supply in a mixed culture shared byaberrant and non-aberrant cells. Monitoring expression lev-els in different, controlled nutrient conditions would thereforeprovide key validation to the coupling picture discussed here(and, more generally, to any cell-to-cell coupling scenario forcancer sustainment).

MATERIALS AND METHODS

Human core catabolic network for a single cell

We built a realistic metabolic network of ATP productionfrom the decompartmentalized human reactome Recon-1 [21]that we use throughout the manuscript. (While the more re-cent Recon-2 reactome [50] provides a higher resolution re-construction of human metabolism, the degree of detail pro-vided by Recon-1 more than suffices for our purposes.) TheHCCN includes the following Recon-1 pathways: Glycolysis,Pentose Phosphate Pathway, Citric Acid Cycle, and OxidativePhosphorylation. In addition, we included anabolic pathwaysfor the production of biomass precursors (amino acids andfatty acids) in an effective way. Details of the reconstructednetwork are given in Supporting Text S1. We then proceededby removing the leaves of the resulting network (a necessarypre-processing step required to implement the mass-balanceconstraints described below). To rid the model of a priori in-feasible loops, we resorted to the method described in [51].A single infeasible cycle was identified, which was fixed byleaving only one of the two isoforms of the isocitrate dehy-drogenase that uses NADP as cofactor.

The final HCCN we employed in this study is composedof 67 chemical species (listed in Table S2) and 75 reactions(listed in Tables S3 and S4), including the uptake/secretionof 11 metabolites (namely molecular oxygen, carbon dioxide,water, hydrogen, lactate, glucose, ammonia, phenylalanine,methionine, glutamine, and methyl group—which representsa generic methylation–whose bounds are fixed as describedin Table S5), and reactions consuming ATP, aminoacids and

palmitic acid. Among intracellular reactions, 23 are reversibleaccording to Recon-1’s thermodynamic assignments. We haveconsidered a medium with variable maximum glucose in-take, fixed maximum glutamine intake and unbounded oxygenavailability. Finally, ATP consumption presents a lower boundstanding for the minimum ATP consumption flux fmin

ATP nec-essary for cell survival (the bounds of all reactions are listedin Table S6 and S7).

Human core catabolic network for two coupled cells

In order to analyze the coupling of two identical cells, wesimply replicated the HCCN twice, the only difference ly-ing in the uptake reaction for lactate. In specific, the lac-tate donor can only secrete lactate (through an irreversiblereaction), while the acceptor can both excrete and importit. The coupling is ultimately determined by the partition-ing of glucose and by the shuttling of lactate. The formeris a shared resource, while the latter can be exchanged be-tween the two cells: both cells can produce lactate, but thereis no external source of lactate. We have included two fur-ther constraints for the sum of glucose and lactate flux ofthe donor-acceptor system, i.e., UG,don + UG,acc ≤ UG andULAC,don + ULAC,acc ≥ 0 (see Table S8). As for a singleHCCN, glucose can only be imported while lactate can onlybe excreted.

Crowding constraint for the human core catabolic network

The crowding constraint within the HCCN has been imple-mented as

aglycfHEX + aox(fPDH+fGLUN) + aLDHfLDH ≤ ΦATP

(27)where fHEX denotes the Hexokinase-1 flux, fPDH denotesthe flux through pyruvate dehydrogenase, by which pyruvateis diverted into the TCA cycle, and fGLUN denotes the fluxthrough glutaminase. For aglyc, aLDH and aPDH we use thenumerical values given in [14] for a coarse-grained model,since for unitary flux both fHEX and fox transport the sameamount of carbons present in glucose, which is exactly whatthe effective fluxes defined in [14] do.

Inequality (27) is valid for a cell that excretes lactate. Whenlactate can also be intaken, as occurs for the coupled HCCNcells, the flux through LDH can become negative. The moregeneral form of the crowding constraint we consider is there-fore

aglycfHEX + aox(fPDH + fGLUN) + aLDH|fLDH| ≤ ΦATP.(28)

For both the single HCCN and the two-HCCN models, weprovide as supplementary materials files containing the listsof metabolites and reactions (with bounds), and the explicitexpressions of each reaction in the HCCN in terms of inde-pendent variables, which provide a full characterization ofthe solution space polytope. The files can also be down-loaded from http://chimera.roma1.infn.it/SYSBIO, where an

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SBML file with the model and a text file with the stoichio-metric matrix of the single cell HCCN are also made avail-able. HCCN for a single cell has also been deposited inBioModels Database (biomodels.org) and assigned the identi-fier MODEL1506170000.

Representation of the solution space of constraint-basedmetabolic networks as a convex polytope

In mass-balance setups, a set of reaction fluxes f = {fr}(r = 1, . . . , N ) describes a non-equilibrium stationary state ifit satisfies the conditions

N∑r=1

Srmfr = 0 , m = 1, . . . ,M , (29)

with prescribed bounds f−r ≤ fr ≤ f+r . (Clearly, the zeroflux vector is excluded from these considerations.) Here,Srm denotes the stoichiometric index of chemical species m(m = 1, . . .M ) in reaction r. From a geometric viewpoint,(29) defines a convex polytope, while every point inside itrepresents a feasible flux configuration. In absence of criteriaallowing to select specific flux vectors from the solution space(e.g. via the maximization of an objective function), uniformsampling provides important information about average fluxesand correlations that not only describe the viable flux patternsin statistical terms, but may also give relevant guidelines fordesigning experiments [52]. When the dimension of the so-lution space is sufficiently small (typically around 10), MonteCarlo sampling, including straightforward rejection methods[53], can be applied. At higher dimensions, instead, exceedingcomputing times require the use of more advanced techniques.

Dimensional reduction of the solution space

We are interested in sampling the solution space of (29)with S = {Srm} given by the HCCN with the prescribedprobability measure defined in (26). Mass balance conditionsconstrain many of the 75 fluxes to be dependent on other vari-ables. To explore efficiently the space of viable flux config-urations, it is convenient to solve all such dependencies ana-lytically and then sample the (much smaller) space spannedby independent fluxes. To begin with, we transformed S to itsReduced Row Echelon Form (RREF) through Gauss-Jordanelimination , which can be implemented by standard pack-ages. Because the RREF of any matrix is unique, the RREFof the stoichiometric matrix applied to the reaction networkuniquely represents the dependent fluxes as a function of theindependent ones. The single-HCCN model turns out to haveonly 17 independent variables, while the two-HCCN versionhas 34. These numbers also represent the dimensionsD of therespective solution space polytopes.

Hit-and-run method allows to sample convex polytopesefficiently

The Hit-and-Run (HR) method [54, 55] can uniformly andefficiently sample a convex polytope, provided one starts froma point inside it. The problem of finding such a point can beeasily solved, e.g., via Motzkin relaxation [51, 56]. In brief,HR builds a Markov chain in two steps. First, starting froma point inside the polytope, a random direction is extracted.Second, along this direction a new point is chosen uniformlyat random inside the polytope. The scheme is then iteratedstarting from the new internal point. HR is efficient becauseonce the random direction is generated the search space is re-duced to a segment. In our implementation, we generate therandom direction using the Marsaglia-Bray method [57]. Topick a point at random along the segment, instead, we firstcompute the two intersections with the polytope and then ex-tract a uniform random number inside the segment identifiedby these two points on the line.

The Lovasz method speeds up the sampling of heterogeneouspolytopes

The above procedure, while fully exact, may howeverpresent a drawback in concrete cases. In fact, the mixingtime of HR in convex domains scales asO[D2(R/r)2] MonteCarlo steps, where D is the spatial dimension of the poly-tope while r and R are, respectively, the radius of the max-imum inscribed hypersphere and of the minimum circum-scribed hypersphere [58]. It is clear that the R/r factor canincrease convergence times dramatically for strongly hetero-geneous polytopes. This is precisely the case for the spaceof feasible steady states in a large-scale metabolic network,where flux distributions can be so heterogeneous as to span5 orders of magnitudes [59–61]. For the network analyzedhere, the ratio between the ellipsoid’s longest and smallestaxes turned out to be of order 104, leading to a conditionnumber of order R2/r2 ' 108, a value that would renderstraightforward HR too expensive in terms of CPU time. Tocircumvent this ill-conditioning problem, we have resorted toa standard pre-processing step that identifies the ellipsoid thatbest approximates the shape of the polytope [62]. Extractingvectors uniformly on such an ellipsoid guarantees that the di-rections along the longer axis are chosen more often, therebyremoving ill conditioning. In quantitative terms, such a pre-processing gives HR a mixing time of O[D7/2] Monte Carlosteps, thus implying a fully polynomial scaling with D. Afterpre-processing the slowest HR mode was found to decorre-late in an affordable time of O(104) Monte Carlo steps. Fur-thermore, the overall pre-processing time was negligible com-pared to the time spent for the actual solution space sampling,while the center of the ellipsoid provides an excellent startingpoint for the algorithm. For more details of this method werefer the reader to [63].

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A method for tuning the optimization of a given metabolic fluxfor constrained based metabolic networks

HR is easily modified to sample points according to anygiven flux distribution P (f) by imposing that the selectionof points along the segment takes place according to howthe distribution P projects onto the chosen direction ratherthan uniformly. To modulate P via a linear function L(f) ofthe fluxes, it is convenient to use the Boltzmann distributionP (f) ∝ exp [βL(f)], where β ≥ 0 is an interpolation pa-rameter. By varying β, one passes from an uniform sampling(corresponding to β = 0) to sampling flux configurations thatmaximize L(f) (corresponding to β � 1). For the HCCNnetwork studied here, the ATP production fATP is maximizedusing the above method with P (f) ∝ exp [βfATP ]. The valueof β for which fATP is effectively maximized can be deter-mined numerically. In Fig. S6 we plot the ATP production asa function of the β for five different values of the maximalglucose available for two coupled HCCN cells. One sees thatoptimal fATP is already achieved when β ' 50.

To produce the graphs presented in Figs. 4 through 8 andS1 through S6, we applied the Lovasz method to the reducedrow echelon forms of the HCCN networks and sampled the

corresponding regularized polytopes by means of the Hit-and-run method. For curves with β 6= 0, we used the Boltzmanweight presented above. The apply the Lovasz method andthe uniform and optimized HR sampling, we used C++ pro-grams that we developed and that can be freely downloaded athttp://chimera.roma1.infn.it/SYSBIO.

AUTHOR CONTRIBUTIONS

F.C. and D.D.M. performed the experiments. All authorsconceived and designed experiments, analyzed the data, andcontributed to the writing of the manuscript.

ACKNOWLEDGMENTS

Work supported by the DREAM Seed Project of the Ital-ian Institute of Technology, by the joint IIT/Sapienza LabNanomedicine, and by the PRIN project Statistical mechan-ics of disordered complex systems of the Italian Ministry ofUniversity and Research. The IIT Platform Computation isgratefully acknowledged.

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16

SUPPLEMENTARY TEXT

Text S1. Human Catabolic Core Network: biosynthetic reactions

Besides the major central carbon metabolic pathways, i.e. Glycolysis, Pentose Phosphate Pathway, Citric Acid Cycle, andOxidative Phosphorylation, our model includes most reactions of Glutamate metabolism as well as lumped reactions describingthe biosynthesis of all non-essential amino acids (i.e. the amino acids that can be synthesized by human cells) and palmitate, akey precursor of other fatty acids.

Amino acid biosynthetic processes can be divided roughly in two classes. The synthesis of tyrosine and cysteine requiresrespectively the essential amino acids phenylalanine and methionine. All other amino acids can instead be synthetized fromglutamine plus metabolites from the central carbon pathways. The processes leading to amino acid synthesis are of varioustypes, from simple single transaminase reactions to a complex chain of enzymatic reactions. In the latter case, by means of massbalance arguments we can lump together individual reaction steps to obtain effective reactions whose net output corresponds tothe desired amino acid(s). In all cases, our starting points were the reactions and pathways included in the reference metabolicnetwork reconstruction Recon-1 [21]. In the .xml file provided as Supporting Material, we include the list of the correspondingRecon-1 reactions that constitute the full pathway. In the following, we briefly outline these effective reactions.

• Glutamate synthesis: from glutamine via glutaminase (GLUN):

gln L + H2O→ glu L + NH4 (30)

• Alanine and aspartate synthesis: from glutamate through the transaminase reactions

akg + ala L↔ glu L + pyr (31)akg + asp L↔ glu L + oaa (32)

(akg = α-ketoglutarate ; pyr = pyruvate ; oaa = oxaloacetate)

• Proline synthesis: from glutamate via glutamate 5-kinase (GLU5K), glutamate-5-semialdehyde dehydrogenase (G5SD),L-glutamate 5-semialdehyde dehydratase (G5SAD) and pyrroline-5-carboxylate reductase (P5CR):

glu L + ATP + (2)NADPH + (2)H+ → pro L + ADP + Pi + H2O + (2)NADP (33)

• Serine synthesis: from glutamate via phosphoglycerate dehydrogenase (PGCD), phosphoserine transaminase (PSERT)and phosphoserine phosphatase (PSP):

3pg + NAD + H2O + glu L→ ser L + akg + NADH + Pi + H+ (34)

(3pg = 3-phosphoglyceric acid)

• Glycine synthesis: from serinevia glycine hydroxymethyltransferase (GHMT), methylenetetrahydrofolate dehydrogenase(MTHFD), methenyltetrahydrofolate cyclohydrolase (MTHFC) and formyltetrahydrofolate dehydrogenase (FDH):

ser L + H2O + (2)NADP→ gly + CO2 + (2)NADPH + (2)H+ (35)

• Arginine synthesis: from glutamate and aspartate via glutamate 5-kinase (GLU5K), glutamate-5-semialdehyde dehydroge-nase (G5SD), ornithine transaminase (ORNTA), carbamoyl-phosphate synthase (CBPS), ornithine carbamoyltransferase(OCBT), argininosuccinate synthase (ARGSS) and argininosuccinate lyase (ARGSL):

asp L + glu L + gln L + (5)ATP + NADPH + (3)H2O + CO2 →→ arg L + fum + akg + (5)ADP + (5)Pi + NADP + (5)H+ (36)

• Asparagine synthesis: from aspartate and glutamine via asparagine synthase (ASNS):

asp L + (2)ATP + gln L + (2)H2O→ (2)ADP + asn L + glu + (2)H+ + (2)Pi (37)

• Cysteine synthesis: from serine and methionine via methionine adenosyltransferase (METAT), adenosylhomocys-teinase (AHC), cystathionine beta-synthase (CBS), cystathionine gamma-lyase (CSE), 2-Oxobutanoate dehydrogenase,Propionyl-CoA carboxylase (PCCA) and methylmalonyl-CoA mutase (MMM):

met L + ser L + coa + (4)ATP + NAD + (4)H2O→→ cys L + succoa + (4)ADP + (4)Pi + (4)H+ + NADH + NH4 + CH3 (38)

(coa = coenzyme A ; succoa = succinyl-coa)

17

• Tyrosine synthesis: from phenylalanine via L-Phenylalanine hydroxylase, tetrahydrobiopterin oxidoreductase,tetrahydrobiopterin-4a-carbinolamine dehydratase and dihydropteridine reductase:

phe L + NADH + O2 + H+ → tyr L + NAD + H2O (39)

• Palmitic acid synthesis:

(8)accoa + (23)ATP + (14)NADPH + (17)H2O→→ palmitate + (8)coa + (14)NADP + (23)ADP + (23)Pi + (10)H+ (40)

(accoa = acetyl-coa)

In addition, we included the reactions for superoxyde reduction and FADH2 oxidation, that are necessary to correctly accountfor the redox state of the cell. For sakes of simplicity, we lumped together the reactions catalyzed by superoxide dismutase,gluthatione peroxidase and glutathione reductase to the single reaction

NADPH + O−2 + 2H+ → NADP + 2H2O . (41)

Likewise, we unified the two reactions that oxidate FADH2, thereby reducing ubiquinone by means of the Electron transportflavoprotein-ubiquinone oxidoreductase (ETF) to

FADH2 + Q10→ FAD + Q10H2 . (42)

Text S2. Maximizing the flux of precursors: results

In the main text, we show that lactate overflow and shuttle arises under both biomass and ATP maximization. Here, we showthe metabolic flux patterns obtained when in the lactate donor the production of single biomass precursors are maximized, toassess the robustness of the lactate overflow and shuttle. To limit the amino acid productions, we have assumed a fixed maximumtotal influx of glutamine, phenylalanine and methionine (equals to 2 mmol/(gDWh)) and a variable overall glucose supply.

All objective functions we tested yield a lactate overflow with lactate shuttle towards the acceptor, suggesting that the crowdingconstraint is responsible for the effect. In particular, we recover high levels of lactate shuttling for palmitate optimization andmoderate levels for the optimization of the production of amino acids. The only lactate shuttle truly related to an energeticimbalance is the one induced by palmitate optimization, while the others are similar in magnitude to the shuttling present inthe absence of any optimization (i.e. for β = 0). In Fig. S2, we display the production flux and lactate shuttling profiles as afunction of the total glucose supply for the two-cell system when the lactate donor maximizes palmitate, proline, cysteine, andserine, respectively. Other biosynthetic objective functions lead to similar features.

18

SUPPLEMENTARY FIGURES

0 1 2 3< UG > / uG

0

1

2

3

< f >

/ u G

fglyc,don, β=50

fglyc,acc, β=50

fglyc,don, β=0

fglyc,acc, β=0

0 1 2 3 4<UG> / uG

0

10

20

30

40

<fAT

P> / u

G

donor, β=50acceptor, β=50donor, β=0acceptor, β=0

0 1 2 3< UG > / uG

-1

0

1

2

3

< f >

/ u G

fLDH, β=50fox, β=50fLDH, β=0fox, β=0

0 1 2 3< UG > / uG

-1

0

1

< f >

/ u G

fLDH, β=50fox, β=50fLDH, β=0fox, β=0

FIG. S1. Glucose intake, ATP production, and oxidative and fermentative fluxes in a donor-acceptor symmetric system. (a, top left)Glucose intakes for two coupled symmetric cells as a function of the total glucose available to the donor-acceptor pair. (b, top right) ATPproduced by the donor and the acceptor cells as a function of the total glucose available to the donor-acceptor pair. (c, bottom left)–(d, bottomright) Average flux through LDH (circles) and PDHm (crosses) as a function of the average glucose supplied to the two-cell system. Curvesdescribe the behaviour obtained for two coupled symmetric HCCN cells with an ATP-maximizing donor (black lines, β = 50) or for anunbiased sampling of the two-cell solution space (blue lines, β = 0). Error bars, which represents s.e.m., are smaller than symbol sizes.

FIG. S2. Lactate shuttling for alternative objective function maximizations. In blue we plot the flux that is maximized in the donor cell:palmitate (top left), proline (top right), cysteine (bottom left), and serine (bottom right) as a function of the total glucose supply, for a systemformed by a lactate donor and a lactate acceptor. The lactate fluxes of donor and acceptor cells are depicted in green and red, respectively. Anegative flux correspond to lactate influx.

19

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FIG. S3. Pearson correlation coefficients among all fluxes of lactate donor and lactate acceptor for maximum glucose uptake UG = 0.3.The fluxes of the acceptor cell are identified by the trailing letters “acc”.

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CIT

RATE

_SHUTT

LE,a

ccVOLU

ME_

CONST

RAIN

T,ac

cATP

_con

sum

ptio

n,ac

c

ACYPDPGMENOFBA

GAPDPGIPGK

G6PDH2rPDHm

AKGDmFUM

ICDHxmICDHy

SUCD1mACONT

CYOR_u10mNADH2_u10m

CO2_FluxO2_Flux

H2O_FluxH_Flux

ATPS4mLDH_L

Ammonium_out✁uxGlucose_Flux1

Glutamine_FluxPhenylalanine_Flux

Methionine_FluxPROLINE_PRODUCTIONSERINE_PRODUCTION

GLYCINE_PRODUCTIONARGININE_PRODUCTIONGlutamate_consumptionAspartate_consumption

Serine_consumptionASPTA

ASNS1_MODALATA_LGLUDxmGLUDymGLUNm

CITRATE_SHUTTLEVOLUME_CONSTRAINT

ATP_consumptionACYP,accDPGM,accENO,accFBA,acc

GAPD,accPGI,accPGK,acc

G6PDH2r,accPDHm,acc

AKGDm,accFUM,acc

ICDHxm,accICDHy,acc

SUCD1m,accACONT,acc

CYOR_u10m,accNADH2_u10m,acc

CO2_Flux,accO2_Flux,acc

H2O_Flux,accH_Flux,acc

ATPS4m,accLDH_L,acc

Ammonium_out✁ux,accGlucose_Flux,acc

Glutamine_Flux,accPhenylalanine_Flux,acc

Methionine_Flux,accPROLINE_PRODUCTION,accSERINE_PRODUCTION,acc

GLYCINE_PRODUCTION,accARGININE_PRODUCTION,accGlutamate_consumption,accAspartate_consumption,acc

Serine_consumption,accASPTA,acc

ASNS1_MOD,accALATA_L,accGLUDxm,accGLUDym,accGLUNm,acc

CITRATE_SHUTTLE,accVOLUME_CONSTRAINT,acc

ATP_consumption,acc

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

FIG. S4. Pearson correlation coefficients among all fluxes of lactate donor and lactate acceptor for maximum glucose uptake UG = 1.The fluxes of the acceptor cell are identified by the trailing letters “acc”.

20

ACYP

DPG

MEN

OFB

AGAPD PG

IPG

KG6P

DH2r

PDHm

AKG

Dm

FUM

ICDHxm

ICDHy

SUCD1m

ACONT

CYO

R_u

10m

NADH2_

u10m

CO2_

Flux

O2_

Flux

H2O

_Flu

xH_F

lux

ATP

S4m

LDH_L

Am

mon

ium

_out

�

uxGlu

cose

_Flu

x1Glu

tam

ine_

Flux

Phen

ylal

anin

e_Fl

uxM

ethi

onin

e_Fl

uxPR

OLI

NE_

PRODUCTI

ON

SERIN

E_PR

ODUCTI

ON

GLY

CIN

E_PR

ODUCTI

ON

ARGIN

INE_

PRODUCTI

ON

Glu

tam

ate_

cons

umpt

ion

Asp

arta

te_c

onsu

mpt

ion

Serine

_con

sum

ptio

nASP

TAASN

S1_M

OD

ALA

TA_L

GLU

Dxm

GLU

Dym

GLU

Nm

CIT

RATE

_SHUTT

LEVOLU

ME_

CONST

RAIN

TATP

_con

sum

ptio

nACYP

,acc

DPG

M,a

ccEN

O,a

ccFB

A,a

ccGAPD

,acc

PGI,a

ccPG

K,ac

cG6P

DH2r

,acc

PDHm

,acc

AKG

Dm

,acc

FUM

,acc

ICDHxm

,acc

ICDHy,

acc

SUCD1m

,acc

ACONT,

acc

CYO

R_u

10m

,acc

NADH2_

u10m

,acc

CO2_

Flux

,acc

O2_

Flux

,acc

H2O

_Flu

x,ac

cH_F

lux,

acc

ATP

S4m

,acc

LDH_L

,acc

Am

mon

ium

_out

�

ux,a

ccGlu

cose

_Flu

x,ac

cGlu

tam

ine_

Flux

,acc

Phen

ylal

anin

e_Fl

ux,a

ccM

ethi

onin

e_Fl

ux,a

ccPR

OLI

NE_

PRODUCTI

ON,a

ccSE

RIN

E_PR

ODUCTI

ON,a

ccGLY

CIN

E_PR

ODUCTI

ON,a

ccARGIN

INE_

PRODUCTI

ON,a

ccGlu

tam

ate_

cons

umpt

ion,

acc

Asp

arta

te_c

onsu

mpt

ion,

acc

Serine

_con

sum

ptio

n,ac

cASP

TA,a

ccASN

S1_M

OD,a

ccALA

TA_L

,acc

GLU

Dxm

,acc

GLU

Dym

,acc

GLU

Nm

,acc

CIT

RATE

_SHUTT

LE,a

ccVOLU

ME_

CONST

RAIN

T,ac

cATP

_con

sum

ptio

n,ac

c

ACYPDPGMENOFBA

GAPDPGIPGK

G6PDH2rPDHm

AKGDmFUM

ICDHxmICDHy

SUCD1mACONT

CYOR_u10mNADH2_u10m

CO2_FluxO2_Flux

H2O_FluxH_Flux

ATPS4mLDH_L

Ammonium_out✁uxGlucose_Flux1

Glutamine_FluxPhenylalanine_Flux

Methionine_FluxPROLINE_PRODUCTIONSERINE_PRODUCTION

GLYCINE_PRODUCTIONARGININE_PRODUCTIONGlutamate_consumptionAspartate_consumption

Serine_consumptionASPTA

ASNS1_MODALATA_LGLUDxmGLUDymGLUNm

CITRATE_SHUTTLEVOLUME_CONSTRAINT

ATP_consumptionACYP,accDPGM,accENO,accFBA,acc

GAPD,accPGI,accPGK,acc

G6PDH2r,accPDHm,acc

AKGDm,accFUM,acc

ICDHxm,accICDHy,acc

SUCD1m,accACONT,acc

CYOR_u10m,accNADH2_u10m,acc

CO2_Flux,accO2_Flux,acc

H2O_Flux,accH_Flux,acc

ATPS4m,accLDH_L,acc

Ammonium_out✁ux,accGlucose_Flux,acc

Glutamine_Flux,accPhenylalanine_Flux,acc

Methionine_Flux,accPROLINE_PRODUCTION,accSERINE_PRODUCTION,acc

GLYCINE_PRODUCTION,accARGININE_PRODUCTION,accGlutamate_consumption,accAspartate_consumption,acc

Serine_consumption,accASPTA,acc

ASNS1_MOD,accALATA_L,accGLUDxm,accGLUDym,accGLUNm,acc

CITRATE_SHUTTLE,accVOLUME_CONSTRAINT,acc

ATP_consumption,acc

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

FIG. S5. Pearson correlation coefficients among all fluxes of lactate donor and lactate acceptor for maximum glucose uptake UG = 3.The fluxes of the acceptor cell are identified by the trailing letters “acc”.

0.1 1 10 100

β

0

0.5

1

f AT

P/f

MA

XA

TP

UG/u

G1=0.5

UG/u

G1=1

UG/u

G1=5

FIG. S6. By increasing β, an HCCN increases the ATP production eventually saturating the capacity at around β ' 50. We displaynormalized ATP production fluxes for maximal glucose supply UMAX equals to 0.5 (black circels), 1 (red squares), and 5 (blu diamonds).

21

SUPPLEMENTARY TABLES

Variable symbol Flux identifiedUG Glucose supplyfglyc, fHEX Glycolysis fluxfox, fPDH Oxidative phosphorylation fluxfLDH Flux through lactate dehydrogenasefATP ATP production

TABLE S1. List of more relevant variables appearing in the manuscript.

22

Metabolite short name Metabolite extended name13 DPG 3-Phospho-D-glyceroyl phosphate23 DPG 2,3-Disphospho-D-glycerate2 PG D-Glycerate 2-phosphate3 PG 3-Phospho-D-glycerate6 PGC 6-Phospho-D-gluconate6 PGL 6-phospho-D-glucono-1,5-lactoneACCoA Acetyl-CoAADP Adenosine diphosphateAKG 2-OxoglutarateATP Adenosine triphosphateCIT CitrateCO2 Carbon dioxydeCoA Coenzyme ADHAP Dihydroxyacetone phosphateE4 P D-Erythrose 4-phosphateF6 P D-Fructose 6-phosphateFAD Flavin adenine dinucleotide oxidizedFADH2 Flavin adenine dinucleotide reducedFDP D-Fructose 1,6-bisphosphateFICYTC Ferricytochrome CFOCYTC Ferrocytochrome CFUM FumarateG3 P Glyceraldehyde 3-phosphateG6 P D-Glucose 6-phosphateGLC D-GlucoseH2O Water moleculeH[M] Hydrogen ion as an electromotive force (mitochondrial)H Hydrogen ion in cytoplasmICIT IsocitrateLAC−L L-LactateMAL−L L-MalateNAD Nicotinamide adenine dinucleotideNADH Nicotinamide adenine dinucleotide - reducedNADP Nicotinamide adenine dinucleotide phosphateNADPH Nicotinamide adenine dinucleotide phosphate - reducedO2 Molecular oxygenO2S Superoxide anionOAA OxaloacetatePEP PhosphoenolpyruvatePi PhosphatePYR PyruvateQ10 Ubiquinone-10Q10 H2 Ubiquinol-10R5 P alpha-D-Ribose 5-phosphateRU5 P−D D-Ribulose 5-phosphateS7 P Sedoheptulose 7-phosphateSUCC SuccinateSUCCoA Succinyl-CoAXU5 P−D D-Xylulose 5-phosphategln L Glutamineglu L GlutamateNH4 AmmoniaCH3 Methil groupasp L Aspartateala L Alanineasn L Asparaginepro L Prolineser L Serinegly Glycinearg L Argininecys L Cysteinemet L Methioninetyr L Tyrosinephe L Phenylalaninehdca Palmitic acid

TABLE S2. List of metabolites appearing in a single HCCN model.

23

Enzyme ReactionACONT CIT←→ ICITACYP 13 DPG + H2O −→ 3 PG + H + PiAKGDm AKG + CoA + NAD −→ CO2 + NADH + SUCCoAATPS4 m ADP + 4 H + Pi −→ ATP + 3 H[M] + H2OCSm ACCoA + H2O + OAA −→ CIT + CoA + H[M]CYOOm3 4 focytC + 7.92H[M] + O2 −→ 4 ficytC + 4 H + 1.96H2O + 0.02O–

2

CYOR u10m 2 ficytC + 2 H[M] + Q10H2 −→ 2 focytC + 4 H + Q10DPGM 13 DPG←→ 23 DPG + HDPGase 23 DPG + H2O −→ 3 PG + PiENO 2 PG←→ H2O + PEPFBA FDP←→ DHAP + G3 PFUM FUM + H2O←→ MAL−LG6PDH2r G6 P + NADP←→ 6 PGL + H + NADPHGAPD G3 P + NAD + Pi←→ 13 DPG + H + NADHGND 6 PGC + NADP −→ CO2 + NADPH + RU5 P−DHEX1 ATP + GLC −→ ADP + G6 P + HICDHxm ICIT + NAD −→ AKG + CO2 + NADHICDHy ICIT + NADP −→ AKG + CO2 + NADPHLDH LAC−L + NAD←→ H + NADH + PYRMDH MAL−L + NAD←→ H + NADH + OAANADH2 u10m 5 H + NADH + Q10 −→ 4 H + NAD + Q10 H2

PDHm CoA + NAD + PYR −→ ACCoA + CO2 + NADHPFK ATP + F6 P −→ ADP + FDP + HPGI G6 P←→ F6 PPGK 3 PG + ATP←− 13 DPG + ADPPGL 6 PGL + H2O −→ 6 PGC + HPGM 2 PG←→ 3 PGPYK ADP + H + PEP −→ ATP + PYRRPE RU5 P−D←→ XU5 P−DRPI R5 P←→ RU5 P−DSUCD1m FAD + SUCC←→ FADH2 + FUMSUCOASm ATP + CoA + SUCC←→ ADP + Pi + SUCCoATALA G3 P + S7 P←→ E4 P + F6 PTKT1 R5 P + XU5 P−D←→ G3 P + S7 PTKT2 E4 P + XU5 P−D←→ F6 P + G3 PTPI DHAP←→ G3 PO2S reduction NADPH + O–

2 + 2 H −→ NADP + 2 H2OFAD regeneration Q10 + FADH2 −→ Q10 H2 + FADATP consumption ATP + H2O −→ ADP + Pi + H

TABLE S3. Internal reactions of an HCCN with the corresponding enzyme that catalyzes it: main catabolic pathways.

24

Enzyme ReactionALATA L AKG + ala L←→ glu L + PYRASPTA AKG + asp L←→ glu L + OAAGLUDxm glu L + H2O + NAD −→ AKG + H + NADH + NH4

GLUDym glu L + H2O + NADP −→ AKG + H + NADPH + NH4

GLUN gln L + H2O −→ glu L + NH4

Tyrosine production phe L + NADH + O2 + h −→ tyr L + NAD + H2OAsparagine production asp L + (2)ATP + gln L + (2)H2O −→ (2)ADP + asn L + glu L + (2)H + (2)PiProline production glu L + ATP + (2)NADPH + 2 H+ −→ pro L + ADP + Pi + H2O + (2)NADPSerine production 3 PG + NAD + H2O + glu L −→ ser L + AKG + NADH + Pi + HGlycine production ser L + H2O + (2)NADP −→ gly + CO2 + (2)NADPH + (2)H

Arginine production asp L + glu L + gln L + (5)ATP + NADPH + 3 H2O + CO2 −→arg L + FUM + AKG + (5)ADP + (5)Pi + 5 H + NADP

Cysteine production met L + ser L + COA + (4)ATP + NAD + CO2 + (4)H2O −→cys L + SUCCOA + (4)ADP + (4)Pi + 4 H + NADH + NH4 + CO2 + CH3

Palmitic acid production (8)ACCOA + (23)ATP + 8 NADH + (6)NADPH + 17 H2O −→hdca + (8)COA + (8)NAD + (6)NADP + (23)ADP + (23)Pi + (10)H

TABLE S4. Internal reactions of an HCCN with the corresponding enzyme that catalyzes it: glutamate metabolism and anaboliceffective reactions.

Description Lower bound Upper bound ReactionFlux of carbon dioxide -1000 1000 CO2←→ outFlux of oxygen -1000 1000 O2←→ outFlux of water -1000 1000 H2O←→ outFlux of ion hydrogen -1000 1000 H←→ outGlucose intake 0 UG in←− GLC

Lactate flux 0-1000

10001000

LAC-L −→ out (donor)LAC-L←→ out (acceptor)

Glutamine intake 0 1 in←− gln LPhenylalanine intake 0 1000 in←− phe LMethionine intake 0 1000 in←− met LAmmonia flux -1000 1000 NH4 ↔ outMethyl group flux -1000 1000 CH3 ↔ out

TABLE S5. Exchange reactions of an HCCN with the corresponding bounds. In this model, we assume that carbon dioxide, water, oxygen,and proton can freely diffuse in and out of the cell. The maximum glucose intake for each cell cannot exceed the total glucose supply. Lactateflux distinguishes donor and acceptor cells as the donor can only secrete lactate, while the acceptor can also intake lactate. In Table S8, theconstraints on glucose and lactate fluxes for the donor-acceptor couple are presented.

25

Enzyme Lower bound Upper bound Enzyme extended nameACONT -1000 1000 AconitaseACYP 0 1000 AcylphosphataseAKGDm 0 1000 2-Oxoglutarate dehydrogenaseATPS4m 0 1000 ATP synthaseCSm 0 1000 Citrate synthaseCYOOm3 0 1000 Cytochrome C oxidase, mitochondrial Complex IVCYOR u10m 0 1000 Ubiquinol-6 cytochrome C reductase, Complex IIIDPGM -1000 1000 DiphosphoglyceromutaseDPGase 0 1000 Diphosphoglycerate phosphataseENO -1000 1000 EnolaseFBA -1000 1000 Fructose-bisphosphate aldolaseFUM -1000 1000 FumaraseG6PDH2r -1000 1000 Glucose 6-phosphate dehydrogenaseGAPD -1000 1000 Glyceraldehyde-3-phosphate dehydrogenaseGND 0 1000 Phosphogluconate dehydrogenaseHEX1 0 1000 HexokinaseICDHxm 0 1000 Isocitrate dehydrogenaseICDHy 0 1000 Isocitrate dehydrogenaseLDH -1000 1000 Lactate dehydrogenaseMDH -1000 1000 Malate dehydrogenaseNADH2 u10m 0 1000 NADH dehydrogenase, mitochondrialPDHm 0 1000 Pyruvate dehydrogenasePFK 0 1000 PhosphofructokinasePGI -1000 1000 Glucose-6-phosphate isomerasePGK -1000 0 Phosphoglycerate kinasePGL 0 1000 6-phosphogluconolactonasePGM -1000 1000 Phosphoglycerate mutasePYK 0 1000 Pyruvate kinaseRPE -1000 1000 Ribulose-5-phosphate 3-epimeraseRPI -1000 1000 Ribose-5-phosphate isomeraseSUCD1m -1000 1000 Succinate dehydrogenaseSUCOASm -1000 1000 Succinate–CoA ligaseTALA -1000 1000 TransaldolaseTKT1 -1000 1000 TransketolaseTKT2 -1000 1000 transketolaseTPI -1000 1000 Triose-phosphate isomeraseLumped reaction 0 1000 Reduction of superoxyde anionLumped reaction 0 1000 FAD regenerationEffective reaction fmin

ATP 1000 ATP hydrolysis

TABLE S6. Internal reactions of an HCCN with the corresponding bounds: catabolic pathways.

Enzyme Lower bound Upper bound Enzyme extended nameALATA L -1000 1000 alanine transaminaseASPTA -1000 1000 aspartate transaminaseGLUDxm -1000 1000 glutamate dehydrogenase (NAD)GLUDym 0 1000 glutamate dehydrogenase (NADP)GLUN 0 1000 glutaminaseLumped reaction 0 1000 Tyrosine productionLumped reaction 0 1000 Asparagine productionLumped reaction 0 1000 Proline productionLumped reaction 0 1000 Serine productionLumped reaction 0 1000 Glycine productionLumped reaction 0 1000 Arginine productionLumped reaction 0 1000 Cysteine productionLumped reaction 0 1000 Palmitic acid production

TABLE S7. Internal reactions of an HCCN with the corresponding bounds: glutamate metabolism and anabolic effective reactions.

26

Variable constrained Lower bound Upper bound Variable expressed by reactionsDonor resource 0 ΦATP aglycfHEX,don + aoxfPDH,don + aLDHfLDH,don

Acceptor resource 0 ΦATP aglycfHEX,acc + aoxfPDH,acc + aLDH|fLDH,acc|Total glucose intake 0 UG UG,don + UG,acc

Total lactate flux 0 1000 ULAC,don + ULAC,acc

TABLE S8. Constraints on maximum resources available for ATP production and on total glucose and lactate fluxes. The first columncontains a description of the variable that is constrained, while the last column the variable expressed as a function of the elementary fluxes ofthe metabolic network. The second and the third column contain the minimum and the maximum value that the variable can take, respectively.ΦATP is set to 0.4 and states that at most 40% of the cellular resources can be devoted to ATP production. The sum of the glucose intaken bydonor and acceptor cannot exceed the total glucose supply UG. The constraint on total lactate flux establishes that there is no external sourceof lactate in the system and that the donor-acceptor couple can only produce lactate.

Metabolite Biomass coefficientH2O -20.651ATP -20.651ADP 20.651H 20.651Pi 20.971glu L -0.38587asp L -0.35261asn L -0.27942ala L -0.50563cys L -0.046571gln L -0.326gly -0.53889ser L -0.39253arg L -0.35926met L -0.15302tyr L -0.15967phe L -0.25947pro L -0.41248hdca -0.112R5 P -0.045G6 P -0.275

TABLE S9. Biomass objective function coefficients adapted from [47].