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It of fold Timothy systems and proteins. to chaperone Rye cell’s S. help Hays a by that reviewed gregate complexes 2016; 16, protein December review are for Chaperones (sent 2017 18, February Dill, A. Ken by Contributed and 11794; NY Brook, Stony a 0 ftecl’ rtisfl usd hs he lse and classes three these outside fall about proteins addition, cell’s In pro- GroEL GroEL. the both I use use of proteins class can III 80% proteins (7): class II and GroEL class DnaK, GroEL, and with need interaction not do of teins (8). and classes more (TF), have three cells into Factor Complex (7). 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BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY the text and Fig. 1, we just give an overview. More complete state) (SI Appendix, Derivation of a Protein’s Dwell Time in the details are given in SI Appendix, Eqs. S2–S6, Figs. S1–S4, and Misfolded State): Tables S1 and S2 (the rate parameters) and SI Appendix, Table kf + kum + kmu S3 (the concentrations of proteins and respective chaperones). τ0 = , [1] The model validation against 20 experimentally determined rate kf kmu curves and the sensitivity analysis of the parameters are given in where kf , kum , and kmu are rates of folding [unfolded (U)→N], SI Appendix, Fig. S5 and Table S4, respectively. misfolding (U→M), and unmisfolding (M→U), respectively. The subscript zero indicates that this is a property of just the Proteostasis Machine Performs Dynamical Sorting protein in the absence of chaperones and aggregation; τ0 reflects This kinetic model shows proteostasis to be a dynamical sorting how long an isolated protein spends in M before it folds to N. machine. The different classes of protein are routed differen- The dwell time is a measure of lability. We also compute τK and tially through the different chaperones (Fig. 2). Class I proteins τG , the dwell times of the protein in M in the presence of KJE fold mainly via TF and KJE. Class II proteins fold mainly through and GroE, respectively (SI Appendix, Eqs. S17 and S18): KJE, and class III proteins use mainly the GroE system (7) kf + kum + kmu + kK (Fig. 2). Under normal (nonstress) conditions, the flux through τK = [2] B is negligible (13). The Ln protease degrades proteins on a kf (kmu + kK ) timescale that is slow enough that the protein has time to attempt 1 to fold or be chaperoned first. The extents of degradation are τG = , [3] ∼1, ∼30, and ∼26% for classes I–III proteins, respectively. The kf populations of native states (N states) are relatively higher and where kK is the rate of KJE cycle. We take the value to be kK = are not shown in Fig. 2. The steady-state yields of N state 1 s−1, which corresponds to the rate-limiting GrpE release step are ∼98, ∼61, and ∼68% for classes I–III, respectively. Below, in the KJE cycle. The other rate parameters used to compute the we explain how this kinetic network encodes these trafficking τ values are given in SI Appendix, Table S1. decisions. Fig. 3A shows the folding rates of different proteins in the pres- ence of different chaperones. Proteins of class I are fast sponta- How Does a Client Protein Reveal Its Sickness? neous folders. They can also be folded by the KJE and GroE We find that a key property that explains dynamical sorting is chaperones at the same speed that they fold spontaneously. In the dwell time (τ0) of the client protein in its misfolded state (M contrast, proteins of classes II and III get stuck in M states over a

Fig. 1. Proteostasis network of E. coli. The six subsystems are folding/misfolding/aggregation (gray), the KJE chaperone system (magenta), the GroE chaperone system (light yellow), the B + KJE disaggregation system (cyan), TF-mediated folding (light green), and degradation by Lon (Ln) protease (light magenta). Protein synthesis (σ) is indicated by arrows. The subscripts T and D refer to the ATP- and ADP-bound states of chaperones, respectively. For the sake of simplicity, some of the intermediate steps have been skipped, and multiple intermediates are merged together in the diagram. The detailed diagrams are shown in SI Appendix, Figs. S1–S4. The model includes cell growth rate (λ), which is not shown in the diagram. A, aggregate; E, GrpE; G, GroEL; J, DnaJ; K, DnaK.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1620646114 Santra et al. Downloaded by guest on October 5, 2021 Downloaded by guest on October 5, 2021 atae al. two et by Santra determined is protein a proteins of III proteins sickness class the I and short, class “frail,” In are sickness: sick. proteins are folding II of class pro- degree “healthy,” the label are their can by We 3). clients (Fig. III tein assistance class kinetically most and the chaperones, also need from is proteins assistance least that U the state of need M proteins conversion stable fast a through U →M. have accessible of rate proteins well), slow with III (deeper but state U Class from M accessible populated kinetically is stable it state. a and U have their proteins to converts II readily Class that diagram) energy free the on chaperones. proteins any of of classes absence three the for protein in the of M UN rates for depths) curves unfolding blue The and 3B stability. given Fig. folding client a in therefore, identical of and N an values and U on same between focus the we having 3B, protein Fig. In 3B diagrams. landscape Fig. GroE quantities. by only physical–chemical fold proteins both III by class folded 3A). whereas be (Fig. can GroE, aggregate; proteins A, and time. II KJE cell-doubling Class 40-min timescale. a slower to much corresponds rate Growth arrows. of thicknesses the by shown are DnaK. K, species GroEL; between G, fluxes and heights, bar 2. Fig. ls rtishv ekyppltdMsae(hlo well (shallow state M populated weakly a have proteins I Class in encoded is sickness of degree protein’s a how is Here h yaia otn ftrecasso rtistruhtedfeetcaeoesses rdce ocnrtoso pce r hw yred by shown are species of concentrations Predicted systems. chaperone different the through proteins of classes three of sorting dynamical The hwtekntc brirhihs n qiira(well equilibria and heights) (barrier kinetics the show lutae sn energy using illustrates .Teeoe ls I class Therefore, →M. ontert rmUt ,lmtn saefo neither in U from an escape slows also is limiting it holdase N, and to A M, U N. to from U to from rate U rate the an from the down is rate down slows foldase the that A up action definitions: speeds following that whole the the action by adopt performed chaper- is We individual action machine? the of of type actions What the components. own chaper- one from its distinct have clients of action can protein of machine zoo of proteostasis type zoo whole the the the of Still, with with 20–24). spectrum ways broader, (7, The complex as in (14–19). interacting seen N) ones now to (accel- U is foldases from mechanisms folding or transition a degradation) the (shielding and erating holdases aggregation either from were distinction protein early This chaperones An protein? study. that client much its of was subject on the chaperone been each has of problem action the is What Client Its Repair Proteins? Machine Proteostasis the Does respectively. How min, 29 and 33, (τ 1, are times I–III dwell classes estimated Our Eqs. pathway. chaperone in sponding (τ given times protein dwell a individual of time protein: dwell any The of quantities physical ntecl sacmiainof combination a is cell the in ) ∆G 1–3 )and (U→M) egtdb h corre- the by weighted NSEryEdition Early PNAS k (U→M). | f8 of 3 for )

BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY AB reducing the barrier of M→U to a value corresponding to a rate of about 1 s−1. KJE does not change the barrier of reverse pro- cess U→M. In contrast, GroE fully destabilizes M, completely unfolding any client protein. Here is a summary from the perspective of the client proteins. Class III proteins bind to the GroE chaperone about five times faster than class II proteins (SI Appendix, Table S1). Therefore, class III proteins preferentially flow through GroE (Fig. 2). Class II proteins preferentially flow through the KJE sys- tem, (i) because KJE tips the equilibrium specifically for class II proteins from M→U (Fig. 3B) and thus, accelerates rate of folding (Fig. 3A) and (ii) because class II proteins are kineti- cally excluded from GroE by GroE’s faster capture of class III proteins. Class I proteins do not misfold much. They fold spontaneously, or they dwell in U (Fig. 3). TF and KJE assist in the U→N tran- sition (Fig. 2). Although this observation summarizes the average trafficking, the model also shows that the process is stochastic. A protein will sometimes enter the “wrong” host chaperone and still be chan- neled toward its N state.

This Sorting Mechanism Is Capable of Handling Any Possible Protein Properties of three classes of proteins with and without chaperones. Fig. 3. A cell must be able to fold any protein, even a protein that it has (A) Rates of protein folding from the M state in the absence of chaper- ones (blue), the presence of excess KJE (orange), and the presence of excess never seen before, because cellular proteins evolve. How is the GroE (green). The rates are defined as k = 109/τ(minutes). The dwell times bacterial chaperone system able to provide this flexibility? This (τ values) are computed using Eqs. 1–3 with the rate parameters given in collection of four chaperone systems can handle any protein. The SI Appendix, Table S1. The black dashed lines represent growth rates cor- model shows that the proteostasis machine sorts clients on the responding to 40-min cell duplication time. (B) Free energy diagrams of basis of two biophysical properties: the stability of its M state (M M U N for three classes of proteins. Blue indicates client protein alone relative to U) and its rate of conversion from U to M relative to in the absence of chaperones. Orange indicates with KJE present. Green its folding rate kf . indicates with GroE present. The rates and barrier heights are computed using rate parameter values given in SI Appendix, Table S1. For class II, the parameter set of DCEA has been used here. G, GroEL; Spont, spontaneous. Dynamical Sorting Is Energy-Efficient for the Cell This dynamical sorting mechanism is energy-efficient. The cell expends more energy folding sick proteins than healthy ones. direction. Additionally, a third type of action, which we call an GroE is the most expensive in ATP use (seven ATPs per cycle). unmisfoldase, speeds up the rate from M to U. We note a matter Also, 70% of GroE activity is on class III proteins (SI Appendix, of terminology. The field has also used the term unfoldase for Fig. S6A). Classes II and I proteins occupy only ∼27 and ∼3% of this process, but our terminology is more in keeping with stan- GroE, respectively, consistent with the data of Kerner et al. (7). dard enzyme terminology in that “-ase” refers to the process that The extents of GroE-mediated folding are in accordance with it involves and does not refer to the product. their GroE enrichments (SI Appendix, Fig. S6B). A key measure Fig. 2 shows two conclusions from the model: (i) the proteosta- of how a protein uses the chaperone systems is the number of sis machine is a foldase for class I proteins, and (ii) it is an un- times that it cycles through different chaperones before folding. misfoldase for classes II and III proteins. For those proteins, A class I protein visits TF, KJE, and GroE on averages of 0.42, GroE and KJE act on the M states of proteins and convert them 0.42, and 0.1 cycles, respectively, before it folds. For class II pro- to U states, from which they can then proceed to N. teins, these values are ∼5, ∼16, and ∼6 cycles, respectively. For class III proteins, they are 0.24, ∼13, and ∼37 cycles, respectively Sorting Is Dictated by the Client Protein, Not the Chaperone (SI Appendix, Fig. S6C). Therefore, the sickest proteins (class III) Proteins of class 0 fold spontaneously and rapidly enough to use the most energy-intensive chaperone (GroE). Frail proteins mostly evade the GroE and KJE chaperone systems altogether. (class II) mostly use the next most energy-intensive chaperone Class I proteins use both TF and KJE to fold but not GroE. KJE (KJE), whereas the healthiest proteins are the cheapest to assist mostly folds class II proteins, and GroE mostly folds class III (SI Appendix, Fig. S6D). proteins. How is this decision made? Is it by the chaperone or the client protein? Fig. 3B shows that the GroE system is not Different Cellular Growth Rates Impose Different Demands selective (green lines in Fig. 3B). GroE acts the same way on all on Proteostasis classes of proteins, namely as an unmisfoldase, converting mis- Chaperones Are More Filled Up in Faster-Growing Cells. How filled folded to unfolded conformations. However, SI Appendix, Table up is an average chaperone with its client proteins? It depends S1 shows that class III proteins bind to GroE rapidly, whereas on the growth rate. At fast growth, a cell produces new proteins class II proteins bind to GroE much slower. Therefore, the class rapidly, and therefore, chaperones are relatively full; Fig. 4 shows III “patient chooses the doctor” (GroE), relegating the class II the model predictions for fixed chaperone concentration. GroEL proteins to the KJE system. is ∼100% full with client proteins under the fast-growth condi- What is the difference between the chaperone systems? The tions of a 40-min duplication time (red dashed vertical line in Fig. model shows that KJE and GroE destabilize the M state of 4). TF and KJE are also near their saturation limit. At slower- all three classes of proteins. Therefore, KJE and GroE are un- growth rates, chaperones are less filled up by client proteins. It misfoldases operating on M, not holdases operating on U. Even follows that fast-growing cells will have reduced capacity to han- so, KJE and GroE act differently. KJE partly destabilizes M by dle additional stresses on protein folding that would increase

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Fig. time. doubling 40-min a to con- of corresponding the in fraction indicate GroEL 5 the Fig. of indicates in centration asterisks 5 indicates white Fig. con- 5 The in proteins. Fig. indicate degraded brown in 5 red and Fig. nonnative; states, in is aggregated than “cliffs” proteome more brown the is where and ditions “sea” proteome red blue a The the which folded. which 90% in in situations diagrams, represents “sea–cliff” region shows concentra- 5 GroEL a and Fig. as rate tion. growth folding cell proteome combined of the extent of function the chaperone shows its 5 control can Fig. cell concentrations. concen- a realistically, chaperone More the fixed. that is tration assumption simplified the under Different ing Require Rates Concentrations. Growth Chaperone Different Down: Up/Shields Shields than stress showed heat which handle yeast, to able in cells. less slower-growing (26) exper- are cells the coworkers faster-growing with and that consistent Botstein is of prediction iment This misfolding. client in given are proteins S3. Table client of and time chaperones doubling 40-min of a Concentrations to corresponding growth cell indicates line 4. Fig. atae al. et Santra h edfrGoLa ifrn elgot ae oke h rtoefle.Ntv rti rcinwt ayn ocnrto fGoLand GroEL of concentration varying with fraction protein Native folded. proteome the keep to rates growth cell different at GroEL for need The rE lsu ihcin rtisa atrgot ae.Tered The rates. faster-growth at proteins client with up fills GroEL .coli E. bv,w osdrdcaeoefill- chaperone considered we Above, xettert fuflig(k unfolding of rate the except , S2 and S1 Tables Appendix , SI ne odto fagot rate growth a of condition under IAppendix, SI .coli. E. ltl,irsetv fGo ocnrto rgot ae Fig. rate. growth or concentration 5, GroE of irrespective 5, pletely, Fig. stability. ing etwt xeietldt ntewr yVlee ta.(25) al. et consis- Valgepea is by prediction work the This in cell. data the experimental protect with tent to sufficient shields therefore, is and aggregation, down avoid to enough slow and tion folding and binding protein 6, faster chaper- (Fig. by of degradation Up-regulation prevents rates. one low-growth pro- at Hsp104) stress-response and up-regulated Hsp82, that are Hsp78, Hsp42, show Hsp30, data Hsp26, (Hsp12, Those teins slow- (28). prediction in yeast action This shields-up growing proteins. showing experiments synthesizing with of consistent up Chaper- is cost shields cell’s important. Therefore, the degradation. becomes reduces with degradation competes protein folding oned of rate the to proteins inability synthesized kinetic newly not enough. of cell’s fast numbers does a large the stress causes fold it and growth-rate capture stability; contrast, constants) protein’s In (rate a proteins. change properties client folding the protein changes of of thus, thermodynamics the and heat stres- affects stability from a shock different is Heat is stress growth stress. fast-growth shock fast However, high short, cell. the In a of (27). sor maintains expression and protein of misfolding level prevents factor the regulating pro- by transcription with chaperones to consistent tend of overexpression is needed they that explanation observation are and This chaperones aggregation. rapidly, against more model. expressed tect therefore, the are from aggregate; proteins explanation to the growth, At is fast growth. Here pro- 6). In slow and its or 5 (shields protect (Figs. chaperone growth less down) to needs fast cell the up) conditions: rates, The intermediate-growth (shields two down. under chaperones up/shields teome more shields needs terminology cell movie and gaming (not similar are limitations 5). KJE Fig. Fig. time. but in doubling degradation, shown limitations, 40-min folding, a GroE of of shows rate edge 5 growth a cliff’s at the aggregation on and balanced are tively, Center i.5sosasacifpo o rti aigmria fold- marginal having protein a for plot sea–cliff a shows 5 Fig. nemdaegot ae r ateog oaoddegrada- avoid to enough fast are rates Intermediate-growth growth, slow In cell. the of stressor a also is growth Slow of terms in expressed be can needs chaperoning cell’s The oe Left). Lower and Right u ,wihi 3 is which ), hw htcassI n I rtis respec- proteins, III and II classes that shows Left × hw htcasIpoen odcom- fold proteins I class that shows 10 −5 s −1 h ocnrto fcochaperone of concentration The . NSEryEdition Early PNAS | f8 of 5 σ 32

BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY Fig. 6 The flow of client protein at different growth rates and the effect of GroEL overexpression on its folding. Schematic representation of flow of client protein at different growth. U, N, and GU represent nonnative, native, and GroEL-bound protein complexes, respectively; φ indicates degradation. The fluxes in steady state are shown by arrows. The magnitude of flux is proportional to the width of the arrow. The most populated state is indicated by a bold letter. A, aggregate; G, GroEL.

(yellow triangles in Fig. 5, Right). In short, in going from low ducing too many chaperones means that most chaperones will be growth to medium growth to fast growth, the cell needs shields- empty, meaning that the cell has wasted energy and biomass. On up→shields-down→shields-up chaperone concentrations. We do the other hand, producing too few chaperones means that chap- not explicitly consider oxidative damage here, but that would add erones will be near maximum capacity most of the time, meaning even more stress to the cell at slow-growth rates (29). The result that the cell could not handle additional stresses. Fig. 7 is a sea– in Fig. 5 at slow growth looks contradictory to the result shown in cliff diagram that shows the degree of proteome folding at dif- Fig. 4. Fig. 4 suggests that, at slow growth, the majority of chaper- ferent GroE and KJE concentrations at 37◦C (at a growth rate ones are free, whereas Fig. 5 indicates the need of excess chaper- of a 40-min doubling time). The white asterisks in Fig. 7 indicate one. The schematic diagram in Fig. 6, Left can explain this appar- standard chaperone concentrations in E. coli (SI Appendix, Table ently contradictory observations. According to the diagram, there S3). Fig. 7 shows that the normal chaperone concentrations are is continuous flow of protein from N to U state, and the U state at a balance point: they keep most of the proteome folded but is degraded by protease irreversibly, which reduces the overall not all. Fig. 7 also shows (i) that class I proteins can fold inde- population of nonnative protein, causing a decrease in chaperone pendent of the GroE and KJE chaperones, (ii) that class II can occupancy. The need of excess chaperone under this condition as use either GroE or KJE chaperones to fold, and (iii) that GroE shown in Fig. 5 is to prevent this degradation by rapidly binding is irreplaceable for folding class III proteins. those unfolded proteins by excess chaperone.

Cell Achieves an Economical Balance; Just Enough Protein Expression Is Near the Aggregation Tipping Point Chaperones to Fold the Proteome Fig. 8 shows how aggregation depends on protein expression lev- In principle, evolution fixes a cell’s average chaperone concentra- els. It supports the proposal of “life at the edge” (30), which is tions at some optimal level for cell fitness. On the one hand, pro- the idea that proteins are expressed at levels just below their

Fig. 7. Sea–cliff plots showing that KJE and GroE concentrations are just sufficient to fold all three classes of proteins. They show that KJE can trade off for GroE for class II proteins but not for class III proteins at growth rates corresponding to a 40-min cell-doubling time. The asterisks indicate cellular concentrations of DnaK and GroEL under normal conditions (SI Appendix, Table S3). The concentrations of cochaperones DnaJ and GrpE are changed accordingly, maintaining stoichiometry of DnaK:DnaJ:GrpE (30:1:15). Similarly, GroES is varied, keeping GroEL:GroES (1:0.83) fixed. Concentrations of all other chaperones and proteins are kept constant (SI Appendix, Table S3).

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Downloaded by guest on October 5, 2021 Downloaded by guest on October 5, 2021 atae al. et Santra h ouino hs ifrnileutosgvstm-eedn concen- time-dependent gives equations synthe- differential for these equations of solution of in The shown set is representative folding/misfolding/aggregation A and sis species. each in of con- are time-dependent centrations involving (details equations differential 1 time-dependent Fig. of set in described model The Methods their and Materials overflow folded. fully nearly barely is proteome proteins the and client chaperones, Under degrees. conditions, different at most to growing chaperones fast-growth the the Cells up use fill GroE). proteins rates is different sickest which protein the client chaperone, collec- (only any energy-expensive This handle energy-efficient to GroE. is capacity by the and proteins has III chaperones class by proteins of encoded of tion sickest largely speed is the capture sorting high and This a GroE. system, through KJE flow II) III) the (class (class proteins through do Sicker mostly proteins chaperones. the flow healthiest with through The much III kinetics. engage and not and II equilibria pro- types M U un- of the as their proteins act and to client decisions seems network affecting The machine misfoldase, proteostasis whole protein). The kinetic chaper- properties. client folding the (the tein’s (the in hospital patient encoded a a are and of system) those the one resemble Metaphorically, (7). decisions measurements machine rate extensive on draws the modeled have We on depend Conclusions should it folders, slow activity. chaperone for and whereas rate stability the aggregation state, with N correlated be to their to pro- likely expected of of are is levels fast expression alone fold The that factors cell. teins chap- the these in of and levels one expression rate, predict no aggregation Hence, rate, activity. fold- misfolding erone of interplay stability, complex rate, a from ing model the or in aggre- arise aging, These points small aggregation. gation shock, drive that heat could is protein) damage, proteins of overexpression oxidative III and as How- II (such points. classes perturbations aggregation for their class implication below is the well ever, exception are maximal The which or 8). proteins, 7) I (Fig. (Fig. cells chaperone levels that minimal expression implies of client–protein It points chaperones. their of near presence predicted are are the proteins in proteins those which aggregate GroEL-interacting at to of points the classes to three close quite of are 8) red Fig. by in (shown arrows concentrations (SI levels physiological cellular The normal limits. their solubility at be to taken are concentrations Chaperone time. doubling 40-min a of rate the at S3). growing Table cells Appendix, in found is that abundance 8. Fig. rti grgto s bnac.Bu niae ocaeoe.Bakidctsta l hprnsaepeet e rosidct h protein the indicate arrows Red present. are chaperones all that indicates Black chaperones. no indicates Blue abundance. vs. aggregation Protein .coli E. rtotssmcie h model The machine. proteostasis IAppendix SI Eqs. Appendix, SI sa is S1–S4) Figs. , S2–S6. rtoso h eann pce r ae ob eo h rwhrate growth The zero. h be 1.04 to be taken to are taken species is remaining (λ) the concentrations assistance in of initial for given trations The are compete model. chaperones proteins the free 252 param- in of rate these systems with with Thus, chaperone proteins proteins DAPA. different III II proteins from as class class II 84 same 63 and class the SYT, DCEA, 63 as eters of same ENO, those the of parameters to those rate identical to parameters identical rate parameters with rate their folding). with their GroEL. in by GroEL degrees from different assistance to folded active are get they (ii not Therefore, do and they GroEL) dif- (SYT; by kDa two assisted 60 are be there (i can class: that they this is (DCEA; in II for proteins for reason class of one The types for and ferent classes. SYT), proteins respective and representative their (DCEA two of II choosing representative class four as for chose (DAPA), two We III (ENO), class. class I its class of each average for that the one assuming of proteins, representative level is proteome proteins the refolding these at vitro of them in use from We substrates data. GroEL of experimental classes three of proteosta- protein sentative whole the three model we these Here, characterize theoretical 32). of model machine (13, Recent sis substrates simulation proteins). GroEL (84 FoldEco of III the classes class pro- three on (42 and into based I proteins), class divided studies (126 GroEL: with been II systems interaction class have of chaperone teins), extent proteins their use These on depending They (7). classes aggregation-prone. folding for highly extensively be to reported Proteostasis. Vivo In in given are values parameter in rate shown S1. estimated Table are the Appendix, data and experimental S5, Mathematica. Fig. corresponding using Appendix, fit and we graphs that for best-fit S1 ) parameters Table (six The Appendix, parameters (SI rate parameters five are remaining SYT) The (31). Dill and Ghosh hs fohrsae r ae ob eo h rti oseicrate nonspecific protein The zero. be to (k taken from taken are are available states parameters in use other noted we as of proteins, chaperones these Those and of U Appendix, each of (SI For concentrations (7) conditions. initial al. diverse et Kerner for by S5) reported Fig. been have DAPA) and SYT, DCEA, Refolding. Vitro In remaining data. the refolding protein, vitro in given using any found For exist- are parameters using S1). unknown Table estimated five be Appendix, S2). Table also (SI Appendix , can theories (SI parameters ing literature protein-specific the the in of available Some and indepen- are type parameters protein rate of the of dent each Most of values. concentrations parameter initial rate and requires species them solve To species. each of trations f ete opt rpriso rtotssb aig4 ls proteins I class 42 taking by proteostasis of properties compute then We repre- each for parameters rate chaperoning and folding derive first We /k u scmue sn h rti hi eghdpnetfruaof formula length-dependent chain protein the using computed is ) .coli E. h xeietlrfligdt ffu rtis(ENO, proteins four of data refolding experimental The n t adigo hs 5 proteins. 252 these of handling its and Among −1 h tblt fteNstate N the of stability The . S2 Table Appendix , SI hc orsod oa4-i cell-doubling 40-min a to corresponds which , ,0 rtisin proteins ∼4,000 h nta concen- initial The S3. Table Appendix, SI rtiso iesalrta 0kDa 60 than smaller size of proteins ) rtiso iebge than bigger size of proteins ) NSEryEdition Early PNAS .coli, E. . S5 Fig. Appendix , SI 5 rtisare proteins ∼250 | f8 of 7 SI SI

BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY time. The protein synthesis fluxes of each class of proteins (σ) are computed fluxes, growth rates, and rate parameters, we ran the numerical differential using the relationship between growth rate, desired concentrations of pro- equation solver in Mathematica. At long times, the system reaches steady- tein in the steady state given in SI Appendix, Table S3, and synthesis flux (SI state concentrations, which are the values reported in this paper. Appendix, Eq. S9). The steady concentrations of each protein are obtained ACKNOWLEDGMENTS. We thank Evan T. Powers and Lila M. Gierasch, who from the PaxDB database (33) by taking the geometric average over pro- introduced us to this problem and created the FoldEco model. We also thank teins belonging to a given class. They are as follows: class I: 6 µM (total Dr. Adam M. R. de Graff, Dr. Kingshuk Ghosh, Dr. Timothy O. Street, and 252 µM); class II: 0.6 µM (total 75.6 µM); and class III: 0.35 µM (total Dr. Rakesh S. Singh for insightful discussions and suggestions, and Dr. Sarina 29.4 µM) (SI Appendix, Table S3). After setting up the proteostasis machine Bromberg for help with graphics. We were supported by the Laufer Center at time t = 0 by assigning initial concentrations of each species, synthesis and National Science Foundation Grant 1205881.

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