Downloaded by guest on September 26, 2021 ainratos ne h yia taysaeadgot rate growth and steady-state typical the Under degra- and reactions. flux synthesis dation their given is metabolites the of GEMs through conservation optimiza- constraints mass studying stochiometric linear employs a for that formulates problem method which tion (9), key (FBA) analysis A balance (6–8). study cultures to cell techniques optimization and modeling constraint-based the at diauxie describing formulation a growth. for level. bacterial need proteome a and of thus understand dynamics is to intracellular There milestone the d’ important raison engineer an its better is elucidate physiology fully cell pres- to evolutionary of able of being product the and is sure, mechanism control a such evolved, of regula- the an the of by is network controlled is tion Diauxie occurrence protein its (3). and the behavior, phenomenon of complex first. the description the detailed during a of can dynamics lack depletion models these after diauxie, computational predict consumed optimality-based consumed, only current preferentially is is Although second sugar the sequential one the and sugars: to the corresponds of of behavior growth consumption cellular optimal this allow (2), to culture Hypothesized phenomenon diauxie. plateau—a called a he by separated curves exponential tinct a Salvy Pierre cellular in constraints strategy allocation growth resource optimal under an as diauxie of Emergence PNAS of I diauxie ensure to system control genetic robust growth. that a optimal propose of the We implementation maximiza- cell. as biological of the such objective of networks, evolutionary growth regulatory the specific of the result of the tion the during is behavior lag diauxic cells car- the time that of of associated suggest models the availability Our captures phenotype. the the diauxie it in of and content fluctuations sources, the to bon in response reprogramming cel- in the simulated proteome show The also diauxie. states of lular manifestation a in uptake , lactose preferential of cul- the over cell predicts of successfully of simulations networks, method dynamic produce The metabolic to tures. and scale, cou- expression, genome we the and gene at assumptions, uptakes, consumption modeling kinetic minimal on ple of With based growth. evolution sim- method cellular kinetic to dynamic models the flux iterative ulate enabled thermodynamically improved. robust and be a expression- can diauxie under- developed in the dynamics We proteome particular, the In of behavior. standing intracellular diauxic the explaining underlying of dynamics in short used fall techniques biotechnology computational industrial competi- the Currently, a advantage. bacterium tive its the maximize growth—giving to specific organism instantaneous the allows which bac- strategy evolutionary in 2020) as 23, carbohydrates July of such review for consumption teria (received sequential 2021 the 15, January or approved Diauxie, and Denmark, Copenhagen, Institute, BioInnovation Nielsen, Jens by Edited aoaoyo opttoa ytm Biotechnology, Systems Computational of Laboratory rnhbooitJcusMnd()osre httegrowth the the that cultures, observed bacterial (1) of Monod growth Jacques the biologist on French work pioneering his n eoesaemdl fmtbls GM)combine (GEMs) metabolism of models Genome-scale shrci coli Escherichia 01Vl 1 o e2013836118 8 No. 118 Vol. 2021 | yai FBA dynamic a a enhpteie ob an be to hypothesized been has coli, Escherichia n asl Hatzimanikatis Vassily and namxueo abhdae olwdtodis- two followed carbohydrates of mixture a in | lac eoreallocation resource prnin .coli E. .coli E. utrsgono mixture a on grown cultures lac | prnin operon Emodels ME 4 ) h emergence The 5). (4, a,1 cl oyehiu F Polytechnique Ecole ´ | r the are coli, E. iterative ˆ tei terms in etre ed ´ rl eLuan,C-05Luan,Switzerland Lausanne, CH-1015 Lausanne, de erale ´ inefcs swl smsegrRA(RA n and (mRNA) RNA dilu- messenger describe and as fully well degradation, concentrations. as also studying synthesis, effects, models for enzyme tion ME of ideal level. requirements is proteome the expression that the gene at paradigm models) also diauxie modeling but (ME a allocation expression proteome mechanisms, and include global metabolism 14) using more (13, of toward reactor, push batch models a a uptake In models, in constraint. the with sources allocation predicts carbon proteome (12) a different correctly al. five method of et order crowd- Their Beg molecular lim- with (FBAwMC). by analysis protein to ing balance of demonstrated flux able role of was be formulation the their diauxie may end, cells in this Toward in itation diauxie. models limitation for proteome Therefore, efficient account for 11). most account the 10, cat- that utilize (2, limited that its combination pathways distribute substrate/enzyme is preferentially toward it a will capacity constraint, has allocation cell alytic can proteome the cell it a that call a likely of will amount we Because total which features. house, the on biological constraint other physiological from into come looked not does carbon diauxie of that constraints. contradicts phases achieve stoichiometric suggests sequential this to and distinct, However, sources consumption of (3). carbon observation growth more Monod’s possible or maximum two the of simultane- the consumption predict ous models FBA assumptions, maximization doi:10.1073/pnas.2013836118/-/DCSupplemental at online information supporting contains article This 1 ulse eray1,2021. 18, February Published BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article This interest.y competing no declare authors The wrote V.H. and P.S. and data; con- paper. analyzed P.S. y V.H. the research; and P.S. performed tools; P.S. reagents/analytic research; new designed tributed V.H. and P.S. contributions: Author owo orsodnemyb drse.Eal vassily.hatzimanikatis@epfl.ch.y Email: addressed. be may correspondence whom To prn sacnrlsse oipeetgot optimality growth implement level. to cellular the system at control a (the diauxie is inducing operon) mechanism regulation we Finally, the medium lactose. that growth We and claim glucose a sugar, cell. of on types a behavior two and containing diauxic in results a experimental renewal predict reproducing and con- successfully by under amount model behavior our protein optimal validate the an on growth as cell simply straints in lag explained associated be its combining and can diauxie framework that show expres- a gene to and sion metabolism developed of bacterium models We and the methods dynamic diauxie. disposition, called its ior at are coli Escherichia sugars several When Significance oacutfrdaxebyn tihoercmdln,we modeling, stoichiometric beyond diauxie for account To osmste naseicodrabehav- order—a specific a in them consumes https://doi.org/10.1073/pnas.2013836118 . y raieCmosAttribution-NonCommercial- Commons Creative . y https://www.pnas.org/lookup/suppl/ | f11 of 1 lac

SYSTEMS BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY Since diauxie is also a time-dependent phenomenon, we chose dard mixed integer linear programming (MILP) solvers (19). We to complement ME models with a dynamic modeling approach. herein leverage ETFL for dynamic analysis, in a method called Dynamic flux balance analysis (dFBA) (15) is a generalization of dETFL. It includes a method based on Chebyshev centering to FBA for modeling cell cultures in time-dependent environments. robustly select a representative solution from the feasible space In its original static optimization approach formulation, the time at each time step. The representative solution captures phe- is discretized into time steps, and an FBA problem is solved at notypic and genotypic differences between cells precultured in each step. At each iteration, kinetic laws and the FBA solution different media. The ETFL method, on which dETFL is built, are used to update the boundary fluxes, extracellular concentra- does not need dedicated quadprecision solvers, unlike previous tions, and cell concentration, based on the amount of substrate ME model formulations (13, 14, 16, 17, 20). It also does not use consumed, by-products secreted, and biomass produced by the strict equality coupling between flux rates and enzyme concen- cells. We expected that the combination of a dFBA and ME tration, unlike the previous state-of-the-art ME model methods models would yield a formulation that can describe diauxie at (16, 17). This strict coupling was instrumental in improving the the proteome level. solving performance of these formulations at the cost of reduc- However, we identified three major challenges in the con- ing the predictive capacity of the methods, in particular with ception of dynamic ME models. First, while dFBA studies of respect to the prediction of the lag phase (discussion is in SI metabolic networks can be solved by common linear solvers, Appendix, Note S1). Instead, (d-)ETFL relies on a combination ME models are nonlinear by nature and significantly more com- of scaling methods and MILP formulation, which allows models plex. The species and reactions introduced and considerations to be solved efficiently. As a result, whole-proteome reconfigura- of the interactions between enzyme expression and metabolism tion during sugar consumption can be simulated with reasonable result in nonlinear problems that are often one to two orders solving times, which enabled the modeling of the lag phase in of magnitude bigger in terms of constraints and variables than diauxie. the corresponding linear (d-)FBA problem. The increase in com- Herein, we model the emergence and dynamics of diauxie aris- plexity is compounded when iteratively solving an optimization ing at the proteome level. We first propose a small conceptual problem. As a result, combining ME models and dynamic studies model of a cell, with a limited in proteome, and demonstrate brings along difficulties that arise from the high computational its ability to predict diauxie under a minimal set of assump- cost of solving multiple times, with different conditions, these tions. Using the dETFL method, we subsequently show these large, nonlinear problems. Second, the use of iterative meth- assumptions hold in E. coli and reproduce experimental results ods presents the additional challenge of alternative solutions, of bacterial growth. Finally, we apply the dETFL framework which can span several physiologies. It is thus necessary to find, to the growth of E. coli in a glucose/lactose mixture in a batch for each time step, a suitable representative solution that will reactor and demonstrate that it robustly predicts diauxie. In par- be used to integrate the system. This also poses the problem of ticular, we capture the preferential consumption of glucose over finding a set of initial conditions for the system. Third, the cur- lactose, the emergence of a time delay when the cell changes rent state-of-the-art models present limitations at the proteome substrate, the proteome origin of this delay, and differential level. Lloyd et al. (16) developed an efficient ME model for E. behavior depending on preculture conditions. This dynamic for- coli, and Yang et al. (17) used it to formulate a dynamic anal- mulation of ME models is able simulate at the genome-scale ysis framework (dynamic ME) similar to dFBA. However, the evolution of the proteome in diauxic conditions, including the assumptions introduced to alleviate the computational complex- time lag it induces and the effect of preculture conditions. ity of their model limit some aspects of the modeling capabilities Overall, dETFL offers a method to robustly survey intracellular of their method. To improve the capacity of our models to dynamics of cellular physiology under changing environmental generalize, we sought a formulation relying on less stringent conditions. assumptions, which allowed us to capture more biological fea- tures. We provide in SI Appendix, Note S1 a detailed list of items Results in which dETFL (dynamic Expression and Thermodynamics- Conceptual Model for the Emergence of the Diauxie Phenotype from enabled Flux models) differs from dynamic ME and a list of Proteome Limitation. We designed a simplified conceptual model elements we believe our method addresses better. In particular, to illustrate diauxie from proteome limitations, as described in dynamic ME forces a strict coupling between enzyme concen- Fig. 1A. The model includes both glucose and lactose as sub- trations and fluxes. However, a change in the growth conditions strates, and it is a simplified version of the E. coli metabolism will trigger a change in the proteome allocation to adapt to a based on four considerations. new metabolic state or lag phase. During that time, it is expected Consideration 1: The biomass carbon yield on glucose is slightly that some previously active enzymes will not be able to carry higher than that of lactose (21). flux in the new conditions. Therefore, enzyme flux and con- Consideration 2: Glucose and lactose are taken up and converge centration will decouple, unless the enzyme composition of the to a common intermediate metabolite, glucose 6-phosphate proteome changes at the same rate as the environment. As a (G6P). Glucose is transformed into G6P by a glucokinase. The result, this previous method cannot simulate lag phase during lactose pathway (Leloir pathway) splits the lactose, a disaccha- glucose depletion and proteome reallocation. dETFL does not ride, into its glucose and galactose subunits. The galactose is then use such coupling between enzyme concentrations and reac- converted to G6P by a series of enzymes. tion rates, which allowed the prediction of a lag phase in our Consideration 3: The Leloir pathway requires one enzyme to simulations. split the lactose into glucose and galactose, four enzymes to Both dynamic models and models including gene expres- convert galactose into glucose-1-phosphate (22, 23), and one to sion mechanisms are important components in the development convert glucose-1-phosphate into G6P; this bring the total to six of successful predictive biology (18). We propose a dynamic enzymes needed for the synthesis of two G6P, which is equivalent method that tackles the challenges mentioned above and mod- to three enzymes per G6P. els diauxie at the proteome level. To this effect, we used our Consideration 4: The molecular mass of the each of the enzymes recently published framework for ME models, ETFL (19). The in the lactose pathway is around 60 to 90 kDa (24), which formulation of ETFL permits the inclusion of thermodynamics is heavier than the 33-kDa glucokinase (Uniprot identification constraints in expression models, as well as the ability to describe A7ZPJ8). the growth-dependent allocation of resources. ETFL is faster Based on these considerations, we devised a conceptual model than previous ME model formulations thanks to the use of stan- of glucose and lactose metabolism for E. coli. The model

2 of 11 | PNAS Salvy and Hatzimanikatis https://doi.org/10.1073/pnas.2013836118 Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 loae ato t rtoefrezmsnee o lactose for needed enzymes gradually for system proteome the its and of reduced, part is flux allocates uptake the depleted, of all (Consideration invest ments will (Consideration substrate glucose, system yielding of Assumption highest the metabolism the both the that phe- the is into fact which diauxic resources in the enzyme The switch (limited) to 1C). the a due (Fig. by is time controlled nomenon over of 1D), composition consumption (Fig. preferential proteome lactose the over shows model glucose The system. our allocation proteome (12). for FBAwMC accounting as such approaches that other to similar in is constraint found This independent. not are constraints enzymes corresponding constraints. the variation enzyme of two function the and a concentrations, of as for lactose equation metabolism and the conservation constrain glucose that one inequalities two balance, enzymes, total mass one involves 6. Assumption step. time each at maximum 5. Assumption similar. are G6P ing metabolism 4. lactose Consideration Assumption for on metabolism—based required glucose are for enzymes than more times three and single summarizing 3. a Assumption enzymes by modeling pathways. modeled two two these in step, resulting single metabolism, a tose in Consideration on summarized enzyme—based is G6P diate 2. Assumption Consideration on lactose—based 1. assumptions. make modeling Assumption to five used made then both thus is We which biomass. that substrates, metabolite two intermediate the an of synthesize consumption the for accounts time. over reactor batch the of content and time, over MW nye r sue ob h ae he nye r sue ob eesr opoueteitreit eaoieGPfo ats,adol one only and lactose, from G6P metabolite intermediate the model. produce the to represent necessary to be used to problem assumed Optimization (B) are glucose. enzymes from Three required same. is the enzyme be to assumed are enzymes 1. Fig. mrec fdaxea notmlgot taeyudrrsuc loaincntansi cellular in constraints allocation resource under strategy growth optimal metabolism an as diauxie of Emergence Hatzimanikatis and Salvy h ocpulmdli bet rdc iui eairin behavior diauxic predict to able is model conceptual The activity maximum two the conservation, enzyme total to Due 1B) (Fig. problem the of formulation mathematical The r oeua weights, molecular are ocpulmdlue o h rlmnr nlss hr gc tnsfrguoead“cs tnsfrlcoe h aayi fcece fthe of efficiencies catalytic The lactose. for stands “lcts” and glucose for stands “glc” where analysis, preliminary the for used model Conceptual (A) )adteoewt h eetezm require- enzyme fewest the with one the and 1) dt h aayi ciiiso h w nye synthesiz- enzymes two the of activities catalytic The The h oa nyeaon ntecl slimited. is cell the in amount enzyme total The h aito fezm ocnrtosrahsa reaches concentrations enzyme of variation The steitgainitra.(C interval. integration the is h lcs eaoimlaigt h interme- the to leading metabolism glucose The lcs a lgtyhge abnyedthan yield carbon higher slightly a has Glucose oeua egt fteezmsaetesame, the are enzymes the of weights molecular and 3 CD AB ρ stems rcino h elocpe yteezmsw consider, we enzymes the by occupied cell the of fraction mass the is Assumption 1. 2. h aei oefrtelac- the for done is same The hne nsugar in Changes (D) substrates. mixed a on growing model conceptual the for time over content Enzyme ) ) steguoeis glucose the As 3). 3. and 1 orewl hnet ats fadol if only and if lactose to change will source concentrations at enzymes the respectively, lactose, yields substrate and each glucose on rates for this growth carbon specific for note, a responsible we as parameters If lactose of behavior. to identification switches the and system source the which under ditions state previous cell. the the on of process, depending optimization physiology the new to the memory constraining the a to add adapted They better break conditions. ones con- new to new enzyme synthesize cell and the of enzymes of old change down limitation of catalytic rate the represent the centrations phys- on subsequent unrelated constraints yield and The just iologies. process, will change memory conditions Indeed, as a optimization S1 ). lack However, Fig. methods proteome. its adapt steady-state Appendix, to change time switch needs (SI of cell proteome a rate physiologically, instantaneously the the occur which on without limitations would accurately concentrations, the enzyme is to sub- phase in constraint sugar lag both important the of describe another utilization However, simultaneous the diauxie, strates. predict to to fails leads approach FBA which original the why is straints capacity proteome the only constraint. from still phenomenon can diauxie it describe mechanisms, repression con- catabolite this lacks While system. model experimental ceptual an in to phase lag corresponds observed reallocation the proteome gradual This metabolism. rcino h oa elms i rm e rm fdidcells dried of grams per grams (in mass cell total the of fraction h ocpulmdlas losfrtesuyo h con- the of study the for allows also model conceptual The con- catalytic enzyme and limitation proteome of lack The fteaon faalbeezmsi ersne by represented is enzymes available of amount the If Y v glc r fluxes, are , Y Y lcts glc n h aayi aeconstants rate catalytic the and , E ˙ · max E k Y cat glc r nyeconcentrations, enzyme are glc stemxmlvraino nyeconcentration enzyme of variation maximal the is · v · E biomass glc v glc glc max max E glc < < < , E https://doi.org/10.1073/pnas.2013836118 v Y Y biomass lcts lcts lcts lcts hntepeerdcarbon preferred the then , · · v v k , biomass glc lcts cat max lcts · , E E , 0 v lcts r eeec values, reference are max biomass lcts . PNAS k h carbon the , cat glc , | k sa as ρ, cat lcts f11 of 3 [1] [2] [3] of

SYSTEMS BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY −1 [g.gDW ]), and assuming different molecular weights MW E , To conduct these studies, we used the E. coli model published the proteome limitation constraint will be written by Salvy and Hatzimanikatis (19) that is based on the genome- scale model by Orth et al. (25) (iJO1366) and was assembled

MW Eglc · Eglc + 3MW Elcts · Elcts = ρ. [4] using ETFL. This model is significantly bigger than the concep- tual model studied in the previous section, with 5,295 species, The maximal achievable values for the enzyme concentra- 8,061 reactions, and 578 enzymes. A summary of the model is max max tions will be Eglc = ρ/MW Eglc and Elcts = ρ/ (3MW Elcts ). available in Table 1. Replacing these values in Eq. 3 directly gives the condition For the integration of the dynamic method, it is important to choose a time step that respects the quasisteady-state assump- lcts Y k MW E glc < cat · glc . [5] tions on which the FBA and ETFL frameworks depend (19). We glc used a time step of 0.05 h = 3 min for the numerical integration, Ylcts kcat 3 · MW Elcts as this is around 10 times smaller than a typical doubling time

In our conceptual model, MW Eglc = MW Elcts , and we can for E. coli and efficiently balances the integration approximation simplify Eq. 5: and solving time. lcts Diauxic growth on glucose and acetate. Yglc kcat We compared the accu- 3 · < glc . [6] racy of our computational modeling of diauxie with experimental Ylcts k cat findings. Specifically, we studied the diauxic growth of E. coli This expression identifies the boundary in the parameter space on glucose in batch reactors using experimental data published that separates the preferential use of glucose vs. lactose. in Varma and Palsson (6) and Enjalbert et al. (26). Previously, In this section, we showed that a constraint symbolizing pro- Varma and Palsson (6) used their data to validate a stoichiomet- teome limitation was sufficient to induce a diauxic behavior of ric model of E. coli in quasisteady state, whereas the data from the cell, thus validating the hypothesis that catalytic limitation is Enjalbert et al. (26) were used to validate a population-based at the origin of diauxie (2, 10, 11). These calculations can be gen- approach of dFBA by Succurro et al. (27). eralized for a more realistic model by accounting for the molecu- To reproduce the results of these two batch growth exper- lar weight of the enzymes and setting an adequate proteome frac- iments, we applied constraints to the uptake of glucose and tion allocated to carbon metabolism. In particular, Wang et al. oxygen in the dETFL model (Materials and Methods). The ini- (11) propose an in-depth analysis of the optimal protein alloca- tial uptake rate of glucose is set to 15 mmol·gDW−1·h−1. This tion for a broad range of carbon sources, connected to the central value is characteristic of a typical physiology for E. coli growing carbon metabolism at different levels. In practice, in our model, on glucose with excess oxygen (6, 15, 25, 27). the catalytic efficiencies of the glycolytic enzymes are also higher We also set the initial concentrations of cells, glucose, and than those of the Leloir pathway (Assumption 3 and SI Appendix, acetate to values matching the experimental data. Oxygen trans- Table S1), and the Leloir pathway enzymes are heavier (Consid- fer was considered free (no kinetic law on uptake) in a first eration 4 and Assumption 4), which favors glucose consumption approximation, as done by Succurro et al. (27). even more. Additionally, we did not consider the synthesis cost Our simulations agreed with the published experimental data. of the enzymes used to carry the fluxes in each pathway. Taking The temporal evolution of the glucose and acetate concen- such property into account would also strengthen the pref- trations in the simulated batch reactor agreed with both the erence toward glucose, as fewer enzymes are needed for its Varma and Palsson (6) and the Enjalbert et al. (26) datasets, metabolism.

Diauxie in Genome-Scale ME Models with Thermodynamic Con- Table 1. Properties of the vETFL model generated from iJO1366 straints. Going beyond a conceptual model, we next used dETFL to model diauxie in an ME model of E. coli. dETFL is built Property Value on top of the existing ETFL formulation (summarized in SI Growth upper bound µ 3.5h−1 Appendix, Note S1) using additional rate-of-change constraints No. of bins N 128 (detailed in Materials and Methods) and the robust selection of µ −1 Resolution N 0.027h reference solution at each time step using a method based on No. of constraints 69,323 the Chebyshev center of polytopes (detailed in Materials and No. of variables 50,010 Methods). This method allowed us to study metabolic switches No. of species 5,295 in response to a changing environment, under the aspect for Metabolites 1,806 intracellular enzyme and mRNA concentrations. To do this, we Enzymes 578 studied how ME models can describe diauxie in experiments Peptides 1,433 where E. coli is grown in two different conditions. First, we inves- mRNAs 1,433 tigated the growth of E. coli on glucose. In this experiment, the tRNAs 21 ×2 cells exhibit overflow metabolism, or the secretion of acetate, rRNAs 3 even under aerobic conditions. Experimentally, the bacterium No. of reactions 8,061 reutilizes the secreted acetate after glucose depletion, a form Metabolic 1,840 of diauxic behavior. This type of study was also used as the Transport 733 first proof of concept for dFBA (15). Thus, we first validated Exchange flux 330 the dETFL model by demonstrating its ability to model a first 1,433 diauxic phenotype: overflow metabolism and acetate secretion in Translation 1,433 the presence of excess glucose, followed by acetate reutilization Complexation 578 on glucose depletion. Second, we reproduced ’s Degradation 2,011 experiment of the diauxic growth of E. coli in an oxygenated 0o No. of metabolites ∆f G 1,737 batch reactor (1) with a limited carbon supply made of a mix- 0o No. of reactions ∆r G 1,787 ture of glucose and lactose. We aimed at reproducing the results 0o Metabolites ∆f G ,% 93.9 shown in the conceptual model on a model of a real organ- 0 Reactions ∆ G o,% 69.5 ism and characterizing the intracellular dynamics underlying the r glucose/lactose diauxic behavior. rRNA, ribosomal RNA; tRNA, tranfer RNA.

4 of 11 | PNAS Salvy and Hatzimanikatis https://doi.org/10.1073/pnas.2013836118 Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 mrec fdaxea notmlgot taeyudrrsuc loaincntansi cellular in constraints allocation resource under strategy growth optimal metabolism an as diauxie of over Emergence line) (dashed Hatzimanikatis rate and Salvy growth and line) (solid concentration extracellular Cell (crosses). lines) (D) (solid dataset (crosses). simulated (26) dataset the al. (6) ( of Palsson et crosses]. evolution concentrations. glucose and dataset; Enjalbert Temporal Varma (26) of and (B) al. and concentrations et simulation crosses]. simulation extracellular [Enjalbert time: time: dataset; simulated data over (6) experimental the line) vs. Palsson of lines) (dashed and evolution (solid rate [Varma Temporal acetate (A) data and glucose crosses. experimental of by vs. concentrations represented lines) are (solid data acetate experimental and lines; solid by represented 2. Fig. measurements experimental reproduce to able is models were ME data no which for conditions, preculture available. on compo- biomass depend known. that parameters that are showed are maintenance recently (non–)growth-associated and (28) overexpression) sition al. knock- enzyme et (gene Biselli account differences activities, Finally, can genetic enzyme particular the if in outs, variability respec- ETFL strain their and the by models for rate ME uptake authors. glucose tive reported the in experimen- always the difference the models including in variability the setup, Second, but tal phases. alter two model, same can the the a parameters of show chose these behavior We Changing quantitative agree- studies. datasets. the qualitative good both (27) showed with al. Varma that ment et parameters the of Succurro between set and variability common 50% (6) a Palsson secretion with and maximal reported acetate several the are for and First, rates rates, acetate factors. uptake and several maximal oxygen, experiment the glucose, to including the attributed parameters, and be simulation can simulation points the the data on between phases. based lag discrepancy experimental curves for The the account to of shifting depletion point glucose time of starting time then, the and only simulation dETFL—given cell the of the predictions in the in left are is drop acetate sharp no when a ends observe medium. simulation the the is We and acetate metabolism. cell. rate, residual overflow growth the the the by depleted, to is consumed is due glucose acetate cell, extracellular time, until the that When sustained by During is medium. secreted the which steadily in glucose, depleted on 2 is steadily (Fig. glucose trend grow similar bacteria a the follow also rate C growth specific 2 and Fig. tion in shown as vrl,teerslsso httedTLfaeokfor framework dETFL the that show results these Overall, results The fitting. no with curves simulated these achieved We and oprsno iuae n xeietldt fguoedpeinoe ie[am n aso 6 n nabr ta.(6] iuae aaare data Simulated (26)]. al. et Enjalbert and (6) Palsson and [Varma time over depletion glucose of data experimental and simulated of Comparison .Bt ftesmltospeitafis hs where phase first a predict simulations the of Both D). A and CD AB .coli E. epciey h elconcentra- cell The respectively. B, tan a loacutfrthe for account also can strain, ∗ xeietlvle eei pia est O)adwr ierysae orpeetcell represent to scaled linearly were and (OD) density optical in were values Experimental zto tp erntesmlto codn otemethod the to according simulation the ran before. detailed at we set step, is and physiologically ization concentration the 1 at cell lactose of The and concentrations glucose condi- of initial relevant mixture these uptake a with run on an subsequently tions with is model before, The as present. model physiology the of standard running rate by same glucose the in with preculturing the simulate lac- first glucose shorter the the in with compared involved figure G6P, The steps pathway. form 3. to additional Fig. pathway in multiple tose summarized the are G6P highlights to metabolism the tose of sources. allocation carbon different competitive As of the transport proteome. the capture overall to models proteome the synthesis cost ME to the proteomic result, contribution describe the a their models and ME with enzymes contrast, associated of In term uptake. their sources no carbon of dFBA includes both proteome. of it uptake describe bacterial since simultaneous to the predict models always of ME an will of reorganization is ability this dynamic Therefore, the source. the challenge carbon are to new that system a reconfiguration to ideal shift proteomic the and by phases caused 30). of lag (29, growth the lactose capture diauxic then glucose/lactose, the and of first Modeling medium glucose mixed lactose. consume a preferentially and on glucose substrate that, mixed show on in growth diauxie Diauxic growth investigating and batch for substrates media. the multiple utility on study its communities to suggest or method organisms single modeling of a as dETFL in the framework validate production findings biomass Our growth. and diauxic secretion, glucose–acetate acetate uptake, glucose of oiiilz h oe o h iuaino iui rwh we growth, diauxic of simulation the for model the initialize To lac- and glucose the to related pathways the reference, For 5mmol·gDW 15 −1 ·h C −1 elcnetain(oi ie n growth and line) (solid concentration Cell ) o lcs n olcoeinitially lactose no and glucose for https://doi.org/10.1073/pnas.2013836118 5g 0.05 .coli E. mmol 2 ·L −1 iui experiments Diauxic ihdTLshould dETFL with fe hsinitial- this After . ·L −1 respectively. , PNAS .coli E. | f11 of 5 will

SYSTEMS BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY Fig. 3. Possible uptake routes for glucose (blue) and lactose (orange) toward G6P. The splitting of lactose by LACZ can be done either intracellularly or in the periplasm. Routes toward the main central carbon metabolism are in gray. The figure was made using Escher (31).

The time evolution of the extracellular metabolite concentra- high (Fig. 4C). Relative to glucose, lactose is taken up at lower tions, cellular exchange fluxes, specific growth rate, and total rates (Fig. 4A). In the second phase, the specific growth rate biomass of the culture exhibit four phases (Fig. 4). We observe decreases sharply, while the proteome reallocates its enzymes a first phase similar to the previous experiment, where glucose is for lactose metabolism. We also observe a drop in acetate secre- taken up at a rapid rate until its depletion, with the simultane- tion during the proteome switch and short period of acetate ous production of acetate through overflow metabolism (Fig. 4 reconsumption. This is the lag phase, where acetate is used A and B). During this phase, the growth rate is steady and as a carbon source, while the proteome is reconfigured to

AB

CD

Fig. 4. Diauxic simulation with glucose-only preculture. (A) Temporal evolution of the extracellular concentrations of glucose (blue), lactose (orange), and acetate (green). (B) Exchange rates of the cell. Positive exchange rates mean production, and negative exchange rates mean consumption. (C) Cell concentration (solid line) and growth rate (dashed line) of the culture over time. (D) Mass of enzymes allocated to the transformation of glucose (blue) and lactose (orange) in G6P. The dashed gray line shows the levels of β-galactosidase (LACZ) enzyme (in the Leloir pathway).

6 of 11 | PNAS Salvy and Hatzimanikatis https://doi.org/10.1073/pnas.2013836118 Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 mrec fdaxea notmlgot taeyudrrsuc loaincntansi cellular in constraints allocation resource under strategy growth optimal metabolism an as diauxie (C of Emergence consumption. mean Hatzimanikatis rates and Salvy exchange negative of and levels production, the ( shows mean time. line over rates gray culture dashed exchange the The Positive of G6P. line) in cell. (dashed (orange) the rate lactose growth of and rates line) (solid Exchange concentration (B) (green). acetate and 5. Fig. 5 uptake (Fig. glucose depletion The glucose preculture. until smaller glucose increases significantly gradually the a then of at rate that up taken than is rate glucose reuti- acetate switch Initially, 4) and proteome lization. consumption, lactose 2) 3) consumption, uptake, acetate glucose with preferred 1) phases: concentrations. cell and lactose, glucose, previous of to conditions terms initial in identical experiments with simulation the ran we of wherein rate simulation. preculture uptake the a in included used that the conditions simulation initial a the conducted lactose of We of utilization artifact delayed an the whether was determine to and tion on in and found details glucose respectively. be More the can acetate. of pathways of concentrations Leloir lactose enzyme that that time-dependent to indicates the it preferred lac- threshold, after is low happens a it consumption below Since falls secreted. uptake of acetate instead tose the residual up In the taken scarce, scarce. being becomes becomes is cell lactose lactose and when as uptake phase, decline lactose fourth a in The by peak 4D). enzymes followed a (Fig. of by growth, G6P characterized mass to is total lactose phase the of third conversion in convert the increase that for an enzymes responsible and of reconfigura- mass G6P proteome total into the The glucose of cell. reduction the a shows in tion active trans- operate. in already acetate were to diffusive that because cell mostly possible the is is for port consumption source is acetate depleted carbon acetate Direct been best-available available, next are has enzymes the glucose lactose-metabolizing since few and Indeed, lactose. metabolize vrtecus fti iuain eosre h aefour same the observed we simulation, this of course the Over predic- diauxie our of robustness the assess to sought next We .coli E. eprleouino h xrclua ocnrtoso lcs bu) ats (orange), lactose (blue), glucose of concentrations extracellular the of evolution Temporal (A) preculture. lactose-only with simulation Diauxic oe ntal nyhdacs olcoe ihan with lactose, to access had only initially model mmol·gDW 5 CD AB .coli E. −1 ·h −1 is 2adS3, and S2 Figs. Appendix, SI 3)adivle pathways involves and (32) olwn hspreculture, this Following . β glcoiae(AZ nye(ntelcoepathway). lactose the (in enzyme (LACZ) -galactosidase A aia rwhrt swa sosre miial salag a as empirically new observed a is reach what to is needed con- rate being time substrate growth additional the maximal when This time, changes. generation growth sumed a specific qualitative the to in close strong dip rate, a observe shows we Additionally, which agreement. measurements, diauxic experimental reports work S6 their different two-phase Although a of a growth sugars. with of concentrations extracellular consumption precul- sim- of lactose a showed evolution the (30) al. ilar and et Kremling glucose particular, In the conditions. tured both for data experimental in and found glucose respectively. be the can of pathways on concentrations Leloir reuti- details enzyme acetate More the conditions. time-dependent phase, scarce result, the final under the a again In initiates As lization increases. shifts consumption. rate proteome to uptake lactose the conversion lactose phase, accommodate lactose third to the reuti- for again In acetate needed resynthesized. observe are enzymes also G6P the we second while the depletion, lization, In proteome glucose phase. a after this from to in phase, consumption switch consumption glucose lactose proteome for for optimized to initiating optimized delay for proteome preculture this time, glucose a attribute the doubling with We We cell compared experiment. 5D). the glucose (Fig. of to utilization available close the is delay, glucose transform- a while enzymes observe was of decreases model amount lactose total the that ing the Although to lactose, 5C). similar the in (Fig. is of precultured experiment rate phase previous growth first the the the of of during evolution increases The rate experiment. uptake glucose the and aso nye loae otetasomto fguoe(le and (blue) glucose of transformation the to allocated enzymes of Mass D) hs iuain hwsrn ulttv gemn ihthe with agreement qualitative strong show simulations These h oprsnbtenorsimulated our between comparison the .Cmaaiey h ats paert ty o,while low, stays rate uptake lactose the Comparatively, B). .coli E. tan erpr in report we strain, https://doi.org/10.1073/pnas.2013836118 is 4adS5, and S4 Figs. Appendix, SI .coli E. Fig. Appendix, SI -2adtheir and K-12 PNAS | f11 of 7 Cell )

SYSTEMS BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY phase. Here, we see it can be interpreted as the proteome switch controller to implement robust dynamic control of the optimal time, the result of the cost of rearranging the proteome alloca- growth. In this regard, the catabolite repression through the tion to adapt to new culture conditions. This lag phase is not observed in wild-type E. coli can be considered as predicted by the previous state of the art in dynamic ME mod- a control system that ensures optimal growth of the organism. els, dynamic ME (17). These simulations also demonstrated that Under the selective pressure of evolution, the system might have intracellular LACZ enzyme levels increase when lactose is the evolved the lac operon to preferentially metabolize glucose in sole substrate left and decrease when glucose is consumed—even mixtures of sugars as it guaranteed an evolutionary advantage after a lactose preculture (Figs. 4D and 5D). These agreements (faster growth) compared with substrate coutilization (5). were achieved without adjusting any of the parameters or set- As an approach, dETFL avoids the pitfalls of simplifying tings of the original ME model. However, a key element for the modeling assumptions used in the current state-of-the-art com- consistency between the model simulations and the cellular state putational models of metabolism and . Because is a robust accounting of the intracellular states (mRNA species, of this, dETFL is a dynamic ME model formulation that can enzymes, and fluxes) between consecutive time steps. This has model lag phase and gradual proteome reconfiguration. How- been made possible by the use of the Chebyshev centering of the ever, despite these innovative findings, there are still drawbacks cellular states in the dETFL formulation as detailed in Materials to dynamic constraint-based models that need refinement. For and Methods. example, finding a good representative solution at each time Our results strongly suggest that diauxie in E. coli is an opti- step is extremely important. Here, we used the Chebyshev ball mal growth behavior. Our conceptual study suggests it is the approach, as it is a single linear problem that is computation- consequence of the maximization of the cell-specific growth rate ally simpler than other methods such as variability analysis or under the constraint of a limited proteome. This optimal behav- sampling. While we have reduced the computational burden of ior of privileging glucose consumption over lactose does not ME models enough to efficiently perform iterative solving, there come from the preculturing step but instead, from the optimality are opportunities to further alleviate the computational cost of of the system itself under the constraint of proteome alloca- simulations. Directions to explore include fixing the integer vari- tion for sugar consumption. We performed additional studies ables of subproblems to reduce the nonpolynomial hardness of and demonstrated that this behavior is not due to differences the model and using quadratic programming, for instance, to per- in enzyme catalytic efficiencies between the two pathways, as form an ellipsoid approximation of the enzyme solution space. switching the kcat values does not change the trend (SI Appendix, Additionally, systematically reduced models, where less impor- Fig. S7). Finally, we showed that the lag time observed in exper- tant parts of metabolic machinery are omitted, can also be used iments is determined by the proteome reallocation and quan- to reduce the complexity of the simulations (34, 35). With a titatively predicted changes in the amount of enzyme for each reduced computational cost of simulations, exciting research tar- pathway. gets are also within reach, such as the dynamic effects of gene knockouts or drug-induced changes in cell physiology. Discussion The computational formulations developed herein also offer We devised both a conceptual model and a dynamic ME model opportunities to test other hypotheses that explain diauxie. Suc- that reproduce a diauxic behavior in E. coli, a phenomenon that curro et al. (27) postulated the existence of two subpopulations cannot be captured with current state-of-the-art models. From of E. coli where one obligately consumes glucose, while the other simulation, we determined that the preferential consumption of consumes acetate. Although the study of communities including glucose over lactose in E. coli is a combined effect of its lim- thermodynamics-enabled ME models is, for now, a computational ited proteome size, enzyme properties, and substrate yield. Our challenge, cross-testing the hypothesis we present in this paper model demonstrates, at the proteome level, the mechanisms of with a similar community-based context would certainly yield the proteome switch between conditions and provides a method important insights on the respective role of proteome limitation to resolve the intracellular dynamics of bacterial growth. In and substrate competition in the emergence of diauxic behavior. agreement with experimental observations, our model predicts The inhibitory effect of glucose on certain parts of the a diauxic behavior on a medium of mixed sugars. metabolism is multiple, and includes catabolite repression, tran- In our simulations, we observed lag phases concurrent with sient repression, and inducer exclusion (36). Moreover, more proteome switching. The co-occurrence of the proteome real- complex regulation mechanisms are found in natural environ- location and acetate reutilization suggests secreted acetate can ments. For example, it has been shown that, on its natural marine work as an energy reserve and help the cell adapt to chang- substate, the bacterium Pseudoalteromonas haloplanktis evolved ing environmental conditions. The dETFL model was also able regulation mechanisms allowing simultaneous diauxie and sub- to capture different dynamic trajectories in cell fates that were strate coutilization (37). Such high-order behavior might also dependent on the preculture conditions. have its origin in an optimal growth program, and finding the The preferential consumption of one carbon source vs. the biochemical constraints responsible for it would yield valuable other is the result of an optimal trajectory of the system under insight on the optimal growth of organisms on complex media. the constraints of mass balance, resource allocation, and ther- In general, elucidating the emergence of regulation mechanisms modynamics. These constraints are directly connected to the in the context of evolutionary pressure will considerably increase chemistry of the metabolic pathways in bacteria. Our conceptual our understanding and ability to engineer regulation systems, model suggests that the diauxic phenomenon might be controlled which are ubiquitous in biology from wild-type E. coli to cancer through the engineering of three aspects: 1) the specific activ- cells. dETFL is an important step forward in this direction. Its ity of enzymes (kcat), 2) the molecular weight of the enzymes, use to uncover the optimality principles guiding the emergence and 3) the number of steps involved in the substrate metabolism. of cellular regulatory control systems is key to a better under- The molecular weight and activity of enzymes can be altered standing and ultimately, mastery of metabolic engineering, be through protein engineering, and alternative chemistries from it applied to industrial hosts or the development of cell-based heterologous pathways provide avenues for modifying substrate therapies. metabolism (33). While dETFL does not account for catabolite repression, it Materials and Methods can quantitatively describe the behavior of a cell operating under Rate of Change of Fluxes. One of the important points in the original for- the influence of the lac operon. Our results imply that the genetic mulation of dFBA is that the rate at which intracellular fluxes change is circuits responsible for catabolite repression are evolved as a constrained. In the dFBA formulation, one imposed constraint is

8 of 11 | PNAS Salvy and Hatzimanikatis https://doi.org/10.1073/pnas.2013836118 Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 nerr hc sgvnb h resolution the has by approximation given this However, is balances. which of mass error, linearization the term the an in dilution allows term turn, the bilinear in the in This, of macromolecules. function of piecewise-constant balances a mass the by organism step. time the next of of calculation the for space feasible the the constrain in because will important variable they are explicit concentrations an these dETFL, are In concentrations problem. optimization macromolecule that is ETFL Concentrations. in Macromolecule of Estimation the in Variability concentration, variables: enzyme dETFL of the terms in of these rewrite (synthesis) can We increase respectively. and degradation) and where of change rate programming: linear maximal with the compatible bound is to that fashion possible a hence in is fluxes it ETFL, from ships erdto aecntn ftemcooeue nETFL, In by macromolecule. the of constant rate degradation macromolecule; the of respectively, negative: strictly other rewrite the can we where en taltimes: all at means ETFL: in written is assumption steady-state this from concentration macromolecule a of quantity. estimation the of resolution that follows directly it this, From positive. strictly rates the of all with following the in expressed be can balance, (19): mass way dynamic the concentra- as enzyme well and as flux tion, between relationship the However, points. time where mrec fdaxea notmlgot taeyudrrsuc loaincntansi cellular in constraints allocation resource under strategy growth optimal metabolism an as diauxie of Emergence Hatzimanikatis and Salvy by given is ETFL of resolution and the model the points, of rate growth maximum the h omlto fEF eiso h prxmto ftegot rate growth the of approximation the on relies ETFL of formulation The (dilution decrease the in limitation the represent constraints two These h asblneo macromolecule a of balance mass The rmEq. From µ b = v pη v ˙ syn max with , , v · n h eainhpgvni e.1,w a rewrite can we 17, ref. in given relationship the and 15, deg ∆t and , p sdfie stemxmmcag fflxbtentwo between flux of change maximum the as defined is {0..N ∈ k 0 deg v v ≤ d dil (x dt E [X ˙ E + v −v , 0 max r h ytei,dgaain n iuinrates, dilution and degradation, synthesis, the are }. j t t syn i ] ≤ dE dt µ b ∆t + µ − = = = deg η E + E ˙ ntocmoet,oesrcl oiieand positive strictly one components, two in E E ∈ ˙ ˙ 0, v v E max sterslto fti prxmto,which approximation, this of resolution the is = j t + − max max i syn syn j t +1 −  η i 2 +1 v ) µ b = syn v − − − ≤ ≤v ≤v − − dil ≤ v v ˙ E v ˙ k v [ − max E mu (x X + max syn deg ≤ (x η 2 deg deg  j t ]≤ i , v , , v E . t ≤ ˙ deg µ t = b j deg + ) − − · stegot ae and rate; growth the is max ) [X ≤ [X k − v ≤ k v η E v deg j syn ˙ − η = ] + cat dil k ] dil − max ≤ v = η ˙ 2 − X N cat ftedsrtzto.Given discretization. the of max v v , E · , ˙  + v v dil j dil h ubro discretization of number the max µ µ ∆t k N sn xrsinrelation- expression Using . E tconcentration at . syn syn deg µ , , b ecnesl banthe obtain easily can We .  · · v , [X , , ∆t · − syn ∆t + ], , η 2 µ . . µ sapproximated is e element key A k [X deg ] [dEN under [dEP sthe is [13] [12] [16] [15] [11] [10] [18] [14] [17] [19] [8] [7] [9] µ j j ] ] epc owihteCeyhvcne ilb acltd e sas denote also us Let define calculated. we be will effect, center E this Chebyshev To the constraints. which to of respect set limited inter- a are in we and role particular, which a concentrations, In macromolecule play sphere. for only solution this representative of a inscribing in inequality ested and the considered definition to be solution to the need the in constraints of and variables distance all the not However, maximizes constraints. which size, its ing problem: inequality optimization linear following the with solving defined (39). is polytope b problem a the is as space dETFL, solution opti- in by the case found constraints. if The be the problem space. can is solution center, linear It the Chebyshev single in the a sphere called inscribed mizing sphere, chose maximally this we the of method find The center to approaches. try three sam- to these makes is between models difference dETFL the of of size sheer the impractical. but pling space, of feasible mean the however, the sample This, observation solution. optimal as single use a requires to than of rather is representative analysis, solution variability good first meth- the a One Several finding space. on space. based solution solution all the issues, considered represent these the alleviate not of can do ods extent which simplex), full explored the the accurately of (corners the the results of of extremal realization all each on factors, results. dependent different two yield be might these procedure of will integration change Because values of pro- solutions. rate flux the previous the point, and on time transcriptome, subsequent applied associated each teome, constraints and in the proteomes concentrations, to macromolecule different of due yield addition, will In cost, (piece- fluxes. enzyme pathways a different similar using often a states, optimal with most two example, is For physiologies. for there ent solution point, concen- macromolecule optimal time values, flux unique trations each (including a solutions at of not continuum fact, wise) but In objective variables. the global the unique for a guarantees solu- value only of LP optimum in multiplicity principle the optimality is the solving Indeed, tions. iterative and optimization linear integer) Center. section. Chebyshev previous the in explained as solution, previous step. the of time value next the in problem and where ehdsnew led edt ov hl IPpolmt compute step. to time problem each little at MILP problem is whole dETFL a there the Euler solve to however, explicit to solution the need the case, than already rather our we method since In implicit method an state. using current linearized with a associated meth- the (e.g., cost explicit calculations to contrast, of to set In function) defined equation step. a implicit state time apply an to each us solve at require system to simply drawback ods the us a of require Usually, state they (38). the integration that problems define Euler is stiff (implicit) schemes handle as backward implicit to is a of ability that use its quality to given solution chose scheme We a possible. guarantee robust as to a using good necessary effect, this is To points. scheme time integration two between model the of gration Scheme. Integration Euler to Backward used then is which step. step, time time current the previous constraint the at concentrations molecules i , h e feult osrit fteproblem; the of constraints equality of set the hsi iia oadn omnsakt l nqaiisadmaximiz- and inequalities all to slack common a adding to similar is This form the of inequalities by defined polyhedron a of case the In and sparse give to tend solvers simplex-based that is issue additional An nti otx,w a ert Eqs. rewrite can we context, this In macro- the on uncertainty the represent to expression this use We x E ∈ J j c R (t E stesto nqaiycntansadvrals epciey with respectively, variables, and constraints inequality of set the as + n j . . . i O +1 0 stecnetaino ie nyea h rvostm step time previous the at enzyme given a of concentration the is nigteCeyhvcne fteslto pc mut to amounts space solution the of center Chebyshev the finding , ≤ piiain ob are u.Aohrwywudb to be would way Another out. carried be to optimizations (2n) htcnstsyamxmlgot aewiedsrbn differ- describing while rate growth maximal a satisfy can that ) ) , E v j j deg (t i ) ( − i.S8 Fig. Appendix, SI t 0 n motn su hndaigwt oh(mixed both with dealing when issue important One i +1 E ≤ ujc to subject j maximize ) (t , E v j i +1 j (t deg r ,x i +1 )≤ (t )− i +1  E v j E ) j deg and , (t j a ( tec iese,w prt ninte- an operate we step, time each At r i t i > ) ( i https://doi.org/10.1073/pnas.2013836118 t dEP )≤ savral osrie rudthe around constrained variable a is x i +1 v hw w-iesoa example two-dimensional a shows + j dil v j )+ ka j syn and (t i v i (t k +1 j dil 2 i dEN +1 ) r (t ≤ r aibe ftedETFL the of variables are a )· i i +1 , b j c ∆t nterElrform: Euler their in i i . )  h ethn ieof side left-hand the , · ∆t , PNAS | a f11 of 9 [dEN [dEP i > [20] x I ≤ j j c ] ]

SYSTEMS BIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY the inequality and equality constraints, respectively; and bi, di their right- [Glc]t+1 = [Glc]t + vglc · X · ∆t, [25] hand side, respectively. From there, we can define the modified centering [Ac]t+1 = [Ac]t + vac · X · ∆t, [26] problem: [Lcts]t+1 = [Lcts]t + vlcts · X · ∆t. [27] maximize r r,x We use these linearized equations to update the extracellular medium subject to µ=µ*, after the solution to each time step has been computed. > 1 [21] ai x + k Jc ◦ aik r≤ bi, ∀i ∈ Ic 2 Model. The model used is the vETFL model of iJO1366, presented in the orig- > ai x≤ bi, ∀i ∈/ Ic inal ETFL publication (19). Fifteen additional enzymes were added to the > model to properly account for the protein cost of transporting glucose, lac- c x=dk, ∀k ∈ E, k tose, and galactose from periplasm to the cytoplasm. A simplified metabolic where µ* is the maximal growth rate calculated at this time step, r is the map of the glucose, lactose, and galactose pathways to G6P is shown radius of the Chebyshev ball, x is the column vector of all of the other vari- in Fig. 3. ables of the ETFL problem, 1J has for jth element zero if j ∈ Jc and else c glc one, and ◦ denotes the elementwise product between two vectors. Thus, Kinetic Information. The Michaelis–Menten parameter KM = 0.015 mM for 1 glucose was taken from the original dFBA paper (15). The lcts = 1.3 mM k Jc ◦ aik2 is the norm of the projection of the constraint vector onto Jc. KM We show an example illustration in three dimensions in SI Appendix, Fig. S9. for lactose was obtained from a study by Olsen and Brooker (40) on the For enzymes, for example, it is akin to making the model produce more specificity of lactose permeases. Details on the added enzymes are available glc enzymes than necessary to carry the fluxes while respecting the total pro- in SI Appendix, Table S1. The Michaelis–Menten parameter Vmax = 15 mM teome constraint. By maximizing the radius of the sphere inscribed in the was used similarly to previous work (15). lcts solution space, at maximal growth rate, we are effectively choosing a repre- Because of uncertainty in the values found in the literature, Vmax was sentative solution of the maximal growth rate feasible space. We then use directly computed from the catalytic rate constants of enzymes consum- this solution as a reference point for the next computation step. ing periplasmic lactose (LACZpp, LCTStpp, LCTS3ipp). Since ETFL gives access lcts We chose the Chebyshev center as a representative solution for its advan- to enzyme concentrations, we can rewrite the expression of Vmax using tages in computational complexity, as well as its low-bias physiological j catalytic rate constants kcat: implications. Indeed, the Chebyshev center can be interpreted as the solu- tion that produces a similar amount of excess enzyme for all enzymes. max X j   Vlcts = kcat · Ej , [28] The choice of such a low-bias objective was also important to show the j∈L spontaneous emergence of diauxie under the most limited set of assump- tions. It is also possible to choose a reference solution with more bias, as where L is the set of periplasmic enzymes consuming lactose. Taking this can be the case when qualitative biological knowledge is available. Meth- lcts into account allows us to replace the parameter Vmax by an explicit internal ods that allow this include lexicographic optimization and minimization of variable. adjustment methods, as described in Salvy and Hatzimanikatis (19). Acetate transport is assumed to be mostly diffusive (32); its secretion rate All simulations in this paper perform Chebyshev centering on enzyme was bounded at 5 mmol·gDW−1·h−1 and its uptake to 3 mmol·gDW−1·h−1. variables at each time step. Oxygen is assumed to be nonlimiting and is given a maximal uptake rate of 15 mmol·gDW−1·h−1. These values are of the same order of magnitude as Initial Conditions. Since dETFL is an iterative method, it is necessary to set in previous studies (6, 15, 27). an initial reference point (initial conditions) from which the dynamic anal- ysis will integrate over time. The initial solution is set up as follows: 1) set Implementation. The code has been implemented as a plug-in to pyTFA typical uptake fluxes for carbon sources and oxygen, 2) perform a growth (41), a Python implementation of the thermodynamics-based flux anal- maximization using ETFL, 3) fix the growth to the optimum, and 4) find the ysis method, and ETFL (19), an implementation of ME models account- Chebyshev center of the solution space. ing for expression, resource allocation, and thermodynamics. It uses The solution reported by the latter optimization problem is then used as COBRApy (42) and Optlang (43) as a back end to ensure compat- a starting solution for the dETFL analysis. ibility with several open-source (GLPK, scipy) as well as commercial (CPLEX, Gurobi) solvers. The code is freely available under the APACHE Extracellular Concentrations. At each time step, extracellular concentrations 2.0 license at https://github.com/EPFL-LCSB/etfl and https://gitlab.com/EPFL- are updated following a standard Euler scheme, similarly to what is done in LCSB/etfl under the folder ./work/detfl. Mahadevan et al. (15). The extracellular concentrations of glucose, lactose, and acetate follow a system of ordinary differential equations: Data Availability. Code data have been deposited in GitHub (https://github. com/EPFL-LCSB/etfl). d [Glc] = vglc · X, [22] ACKNOWLEDGMENTS. We thank Dr. Ljubisaˇ Miskoviˇ c´ and Dr. Mar´ıa Masid dt Barcon´ for valuable discussions around this project and Dr. Kaycie But- d [Ac] ler for valuable input on the wording and flow of this manuscript. P.S. = vac · X, [23] dt thanks Kilian Schindler and Prof. Daniel Kuhn for the valuable discussion d [Lcts] around the formulation of the Chebyshev problem. This work has received = v · X. [24] funding from the European Union’s Horizon 2020 Research and Innovation dt lcts Program Marie Skłodowska-Curie Grant 722287, the European Union’s Hori- zon 2020 Research and Innovation Program Grant 686070, and the Ecole We linearize this system into the following forward Euler scheme: Polytechnique Fed´ erale´ de Lausanne.

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10 of 11 | PNAS Salvy and Hatzimanikatis https://doi.org/10.1073/pnas.2013836118 Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 1 .Mi,L ilas .Frni nuneo rnpr nriaino h growth the on energization transport of Influence Ferenci, T. Williams, L. Muir, M. 21. 5 .D Orth D. J. 25. and —structure in metabolism Galactose Reece, J. R. Campbell, pathway. N. Leloir R. Sellick, the A. of a C. Enzymes Kalckar, into 24. M. glucose H. Kurahashi, diphosphate K. Maxwell, uridine S. of E. transformation 23. enzymatic The Leloir, F. L. 22. Yang and L. in integration 20. understanding multi-omics allows Modelling, formulation ETFL biology: The Hatzimanikatis, Predictive V. Salvy, Collins, P. J. 19. J. Lopatkin, Dynamic J. Dynamicme: A. Palsson, 18. O. B. Saunders, A. M. Lloyd, J. C. Ebrahim, A. Yang, L. 17. 7 .Scur,D er,O Ebenh O. Segre, D. Succurro, A. deter- exposure 27. Acetate Letisse, F. Portais, J.-C. Cocaign-Bousquet, M. Enjalbert, B. 26. 5 .Mhdvn .S dad,F .DyeII yai u aac nlsso diauxic of Lloyd analysis balance flux Dynamic J. C. III, Doyle 16. Genome-scale J. F. Palsson, Edwards, O. S. B. J. Mahadevan, Hyduke, R. R. D. 15. Chang, L. R. Lerman, A. J. O’Brien, J. E. 14. Lerman A. J. 13. mrec fdaxea notmlgot taeyudrrsuc loaincntansi cellular in constraints allocation resource under strategy growth optimal metabolism an as diauxie of Emergence Hatzimanikatis and Salvy il of yield Bioinf. .Bacteriol. 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