Iterative Annealing Mechanism Explains the Functions of the Groel and RNA Chaperones

Iterative Annealing Mechanism Explains the Functions of the GroEL and RNA Chaperones

D. Thirumalai1, George H. Lorimer2, and Changbong Hyeon3 1 Department of Chemistry, The University of Texas at Austin, Texas, 78712 2 Biophysics Program, Institute For Physical Science and Technology, University of Maryland, College Park, MD 20742 3 Korea Institute for Advanced Study, Seoul 02455, Korea

(Dated: September 18, 2019) arXiv:1909.07508v1 [q-bio.BM] 16 Sep 2019

1 Abstract

Molecular chaperones are ATP-consuming biological machines, which facilitate the folding of proteins and RNA molecules that are kinetically trapped in misfolded states for long times. Unassisted folding occurs by the kinetic partitioning mechanism according to which folding to the native state, with low probability as well as misfolding to one of the many metastable states, with high probability, occur rapidly on similar time scales. GroEL is an all-purpose stochastic machine that assists misfolded substrate proteins (SPs) to fold. The RNA chaperones (CYT-19) help the folding of ribozymes that readily misfold. GroEL does not interact with the folded proteins but CYT-19 disrupts both the folded and misfolded ribozymes. Despite this major difference, the Iterative Annealing Mechanism (IAM) quantitatively explains all the available experimental data for assisted folding of proteins and ribozymes.

Driven by ATP binding and hydrolysis and GroES binding, GroEL undergoes a catalytic cycle during which it samples three allosteric states, referred to as T (apo), R (ATP bound), and R00 (ADP bound).

In accord with the IAM predictions, analyses of the experimental data shows that the efﬁciency of the

GroEL-GroES machinery and mutants is determined by the resetting rate , which is largest for kR00→T the wild type GroEL. Generalized IAM accurately predicts the folding kinetics of Tetrahymena ribozyme and its variants. Chaperones maximize the product of the folding rate and the steady state native state fold by driving the substrates out of equilibrium. Neither the absolute yield nor the folding rate is optimized.

2 INTRODUCTION

Molecular chaperones have evolved to facilitate the folding of proteins that cannot do so spontaneously under crowded cellular conditions1–3. This important task is accomplished without chaperones imparting any additional information beyond what is contained in the amino acid sequence. Furthermore, chaperones assist the folding of proteins whose folded structures bear no relationship to one another. In other words, chaperones are “blind" to the architecture of the folded proteins. Most of the protein chaperones belong to the family of heat shock proteins (HSPs) that are over expressed when the cells are under stress. Among the many classes of chaperones, the bacterial chaperonin, GroEL, has been most extensively investigated, possibly because it was the ﬁrst one to be discovered4–6. Although less appreciated, RNA chaperones have also evolved to enable the folding of ribozymes7,8, which are readily kinetically trapped implying that in vitro only a very small fraction folds to the functionally competent state on a biologically relevant time scale9. Both GroEL and RNA chaperones (CYT-19), which we will collectively refer to as molecular chaperones from now on, are not unlike molecular motors, such as kinesin, myosin, and dynein. There are many similarities between motors and chaperones. (i) Both motors and chaperones are enzymes that undergo a catalytic cycle, which involves binding and hydrolysis of ATP. Molecular motors hydrolyze one ATP per step, thus converting chemical energy to mechanical work in order to walk on the linear cytoskeletal ﬁlaments (actin or microtubule). Both GroEL and RNA chaperones consume copious amounts of ATP (see below). They couple the hydrolysis of ATP to perform work by partially unfolding the misfolded RNA or proteins. Indeed, helicase activity is attributed to RNA chaperones, such as CYT-19. Helicases are biological machine that separate double stranded DNA or RNA and translocate on single stranded nucleic acids. (ii) During the catalytic cycle, the enzymes (motor head in the case of motors and the subunits in the GroEL particle) undergo spectacular conformational changes, which are transmitted allosterically (action at a distance) throughout the complex (see10 for a recent review). Indeed, it is impossible to rationalize the functions of motors or chaperones without allosteric signaling, which we illustrate more fully for GroEL in this article. (iii) Some of the rates in the catalytic cycles of molecular motors also depend on the presence of actin or microtubule. Similarly, ATPase functions of GroEL are stimulated in the presence of substrate proteins (to be referred to as SPs from now on). For these reasons, a quantitative understanding of the functions of

3 molecular chaperones mandates that they be treated as molecular machines.

GroEL-GroES machine

The complete chaperonin system consists of GroEL, the co-chaperonin GroES, which together form both a 2:111 and 1:112 complex depending on whether SPs are present or absent. For it to function, which means assist in the folding of a vast number of SPs that otherwise could aggregate, it requires MgATP as well. The availability of a number of structures that GroEL visits during the catalytic cycle11–13 and theoretical developments3,14 have made it possible to obtain insights into the function of GroEL-GroES system. GroEL, assembled from seven identical subunits, is a homo oligomer with two rings that are stacked back-to-back, which confers it in an unusual rare seven fold symmetry in the resting (T or taut) state. Major changes in the structures take place between the allosteric (T , R, and R00) states in response to ATP and GroES binding (Figs.1 and 2a). The dynamics of allosteric changes in GroEL has been reviewed recently10,15–17. The ATP binding sites are localized in the base of GroEL corresponding to the equatorial (E) domain, connecting the two rings (Fig. 2a). The E carries bulk (roughly two thirds) of the inertial mass of a single subunit. Binding sites for the co-chaperonin GroES are localized in the apical (A) domain, which also coincides with the region of interaction of the SPs with GroEL in the T state. We present a schematic in Fig. 3b of the reaction cycle in a single ring. We ought to emphasize, right at the outset, that recent advances show that when challenged with SPs the functioning state is the symmetric 14-mer GroEL-GroEs complex, resembling a football18,19 (see Fig.2b), and not the asymmetric bullet structure as had been thought for a long time. The parts list of this complex machine is GroEL, GroES, MgATP, and the SPs, that require assistance to reach the folded structures. A few words about the SPs are in order. It has been shown long ago that GroEL is a promiscuous machine that interacts with a vast majority of E. Coli. proteins as long as they are presented in the misfolded states20. This observation and the subsequent demonstration that most of the SPs used to study GroEL assisted folding are ones not found in E. Coli. further buttresses this point. For discussion purposes, we distinguish between permissive and non-permissive conditions. Under permissive conditions,

folding to the native state occurs readily in vitro on a biologically relevant time scale (τB),

4 which is between 20–40 minutes for E. coli proteins at 37◦C. Under non-permissive conditions, spontaneous folding does not occur in vitro with sufﬁcient yield of the folded protein on the time scale, τB. The SPs satisfying this criterion are deemed to be stringent substrates for GroEL. Several in vitro experiments show that in most cases the SP folding rates in the wild type GroEL are enhanced only modestly, which is fully explained using theoretical studies21–23. In fact, the folding rate could even decrease although this has been shown experimentally24 using only a mutant form of GroEL, referred to as SR1, from which GroES does not easily dissociate. In contrast, it is the native yield over a period of time that is maximized14,25 by the GroEL machinery.

RNA Chaperones

The tendency of RNA enzymes (ribozymes) to misfold in vitro is well established26–30. Sev- eral in vitro experiments have ﬁrmly established that self-splicing ribozymes, such as Tetrahy- mena ribozyme, fold to functionally competent state extremely slowly. Only a very small fraction of the initially unfolded ensemble reaches the folded state rapidly ( one second)9,31. The ∼ reason for RNAs to be kinetically trapped in metastable states is due to the high stability of the base-paired nucleotides. In addition, RNA molecules have considerable homopolymers-like characteristics, which do not fully discriminate between a large number of relatively low free energy structures. These factors render the folding landscapes of RNAs more rugged than proteins32. We showed that only 54 % of RNA secondary structures is made of helices33 ∼ with intact Watson-Crick base pairs, which implies that substantial number of nucleotides are engaged in non-canonical base pairs, bulges, and internal multi loops. It is estimated that the life time of a helix made of 6 bps, which gives rise to a free energy barrier of δG‡ 10 15 ≈ − kcal/mol, can be as large as 105 sec ( 1 month). Thus, if a helix forms incorrectly during ∼ ∼ the folding process, spontaneous unfolding of the helix would not occur on a reasonable time scale. Typical time scales for escape from low free energy some kinetically trapped metastable states can easily exceed hundred minutes (see for example Eq. (2) in9). The arguments given above suggest that the folding landscape of most RNA molecules ought to consist of multiple metastable minima with similar stability that are separated from the native state by large (compared to kBT where kB is the Boltzmann constant and T is the temperature, which unfortunately is the same notation used for the T state in GroEL) free

5 energy barriers34. The structures in the metastable states often share many features that are common with the folded state. Such rugged landscapes govern the functions of many RNA molecules, such as riboswitches that are involved in transcription and translation. These biological processes are associated with switching between at least two alternative structures. In riboswitches the switch between the states is modulated by metabolites or metal ions. Of relevance here is the in vitro folding of Tetrahymena ribozyme, which is a self-splicing intron. For this enzyme it is found that only a fraction (Φ = 0.08) of the initial population of unfolded molecules directly folds to the functionally competent native state in about 1s, and the rest (1 Φ = 0.92) of the molecules are kinetically trapped in competing basins of attraction9,35,36 − for arbitrarily long times. For Tetrahymena ribozyme to function, it is essential that several key native tertiary contacts form. Incorrect formation of these tertiary contacts leads results in functionally incompetent ribozyme. For example, without the formation of the pseudo knot (P3 helix) the two domains (P5-P4-P6 and P7-P3-P8) cannot be stabilized (see Fig.4a). Formation of alternative helix (Alt-P3) and other misfolded structures impair the function of ribozymes. An introduction of a single point mutation (U273A) stabilizing the P3 pseudoknot helix was shown to increase Φ as high to 0.837. DEAD-box protein CYT-19, which belongs to a general class of RNA chaperones7,8,38–43, comprises of a core helix domain and arginine rich C-terminal tail. Cyt-19 recognizes surface- exposed RNA helices (duplexes) and unwinds them, like helicases belonging to the SF2 family (see Fig.4b for the yeast analogue of CYT-1942), into single stranded RNA by expending free energy due to ATP hydrolysis. It is likely that ATP triggered conformational changes promotes local unwinding of RNA helices. Because of the helicase activity of CYT-19, the microscopic mechanism does involve local unfolding of the accessible helices. Thus, both GroEL and CYT-19 perform work on the misfolded structures by forcibly unfolding them, at least partially. This is another common theme linking the functions of CYT-19 and GroEL.

Our goals in this article are the following: (1) We present a uniﬁed theoretical perspective on the functions of GroEL and RNA chaperones. The essence of the assisted folding mechanism of the SPs is illustrated using the well investigated GroEL-GroES system. Although the two enzymes, exhibiting machine-like activity, are quite different we show that the theory based on the Iterative Annealing Mechanism (IAM) quantitatively explains a vast amount of experimental data in chaperone-assisted folding of proteins as well as ribozymes. A major conclusion of the

6 theory is that these ATP-consuming chaperones are stochastic machines that drive the SPs SS or ribozymes out of equilibrium. This implies that in the steady state, PN , (long time limit) EQ the yield of the folded protein does not correspond to that expected at equilibrium PN , which would be exp( β∆GNU ) where β = 1/kBT , and ∆GNU is the free energy of stability of ∝ − the native N state with respect to the unfolded state (U). In other words, P SS = P EQ. (2) N 6 N The differences between GroEL-GroES system and RNA chaperones naturally arises from the IAM predictions, and highlights the likely inefﬁciency (large consumption of ATP relative to the production of the folded state) of RNA chaperones. (3) Because the GroEL structures in different nucleotide states are known, we illustrate the conformational changes that occur during the allosteric transitions in the GroEL in response to ATP and SP binding and link these changes to the folding of the SPs. (4) Finally, we outline recent developments, which provide incontrovertible evidence for the quantitative validity of the IAM, which establishes that GroEL-GroES system is a parallel processing stochastic machine that simultaneously anneals two misfolded molecules by sequestering one each in the two chambers of the symmetric complex. Remarkably, the symmetric complex forms only when the GroEL-GroES system is subject to load, i.e., challenged with SPs that require assistance to reach the native state.

ITERATIVE ANNEALING MECHANISM (IAM) FOR GROEL-GROES

In this section we systematically develop the physical basis for the IAM by dissecting the fate of the SPs in the absence of the chaperonin machinery. We begin by considering how SPs, which do not recruit GroEL-GroES, fold spontaneously. This is followed by a brief description of the dynamics of allosteric transitions that GroEL undergoes in response to ATP and GroES binding and hydrolysis of ATP. Lastly, the physical picture of the link between allosteric transitions and SP folding is described, which vividly reveals the machine-like characteristics of the GroEL-GroES system. The applications of the theory of the IAM to experimental data cement the quantitative validity of the active role GroEL plays in assisted folding.

E. Coli does not have enough GroEL to process the entire proteome

Over twenty years ago Lorimer44 showed, using data for E. Coli B/r growing in minimal ◦ glucose medium at 37 C with a cell doubling time (τD of 40 minutes, that the number of ∼ 7 GroEL particles can only process between (5 -10)% of the proteome. The crux of the that argument can be summarized as follows. The rate of protein synthesis is kS = NP /τD where

NP is the number of polypeptide chains in a cell. Assuming that the total mass of proteins −13 4 6 per cell is 1.56 10 g and the average mass is 4 10 g/mol then NP 2.35 10 , ≈ × ≈ × ≈ × 4 4 which implies that kS = 6 10 chains/min. Given there are about 3.5 10 ribosomes, it × × follows that this strain of E. Coli synthesizes about one polypeptide chain every 35 seconds. Needless to say, most if not all of the proteins have to reach the the folded state in the crowded environment without the assistance of GroEL. The average cell contains about NGroEL =1,580

GroEL14 particles, and about nearly twice as many GroES molecules. The typical measured −1 values of the rates of assisted folding in vitro, kF s are in range (1 2) min . Thus, the − available GroEL particles can assist in the folding of N kF 3160 polypeptide chains/min, GroEL ≈ which is clearly far less than the synthesis rate of 6 104 chains/min. Thus, only about (3 5) × − % of the proteome can recruit GroEL-GroES in order to fold. Nevertheless, removal of the GroE gene is lethal to the organism, attesting to its importance in E. Coli growth. These estimates raise the following two important questions: (a) What are the potential SPs that fold with the assistance of the GroEL-GroES system? (b) How do the vast majority of proteins ( 95%) fold without the chaperonin machinery? We will answer the second question here ≈ and refer the readers to relevant papers45–49, except to note that GroEL does not discriminate between proteins based on their folded structures because the very SP residues that interact with GroEL are buried in the folded state47.

Stringent SPs and ribozymes fold by the Kinetic Partitioning Mechanism (KPM)

Spontaneous folding of small proteins or those with relatively simple native topology is well understood. Proteins, such as SH3 domain or Chymotrypsin Inhibitor 2, fold in an ostensibly two state manner although when examined using high spatial and temporal resolution it is found they too fold by multiple routes to the native state. For these proteins the yield of the native state is sufﬁciently large that their folding does not require the assistance of chaperones. However, from the perspective of assisted folding, it is more instructive to consider the folding of SPs whose folding landscapes are rugged containing many free energy minima (Fig. 3(a)) separated by sufﬁciently large barriers (several kBT s) that they cannot be overcome readily. Although the structures in the low energy minima could have considerable overlap with the

8 folded state they are misfolded because they are likely to contain incorrect tertiary contacts and/or secondary structures. These are targets for recognition by molecular chaperones. After an initial rapid compaction of the SPs or the ribozyme many of the molecules are trapped in one of several low free energy minima.

The KPM, which explains the folding mechanisms of proteins succinctly, follows immediately from the rugged folding landscape in Fig.3(a) (see also Fig.5). According to KPM50,51, a fraction of molecules Φ folds rapidly without being trapped in one of the low free energy minima. These are sometimes referred to as the fast track molecules for which, following an initial “speciﬁc" collapse, folding to the native state is rapid52. Explicit simulations using lat- tice models53 have shown that the folding characteristics (dynamics of compaction and the increase in the fraction of native contacts as a function of time) of the fast track molecules are identical to sequences for which the folding landscape is simple with one dominant minimum. The remaining fraction (1 Φ) of molecules are trapped in an ensemble of low free energy − structures because their initial collapse produce structures containing interactions that are not present in the native state. The resulting misfolded structures have to overcome activation barriers in order to reach the folded state. Thus, after the ensemble of unfolded molecules

undergoes rapid collapse they partition to the native state at a rate kIN or transition to the

misfolded ensemble at a rate kIM . The fraction of fast track molecules, referred to as the partition factor is associated with the rates in the 3-state cyclic model for chaperone-assisting folding depicted in Fig.5 as, Φ = kIN . It is the value of Φ, which depends on a number of kIN +kIM extrinsic factors such as ionic strength, pH, and temperature that governs the need of a SP or the ribozyme for the chaperone machinery (see below).

We classify the misfolded structures into slow folders and no folders, depending on the magnitude of the activation free barriers separating them from the native state. The time scale for slow folders to reach the native state could range from milliseconds to several minutes whereas for no folders the transition to the native state could occur on time scales that exceed biologically relevant times. The effects of external conditions might be appreciated by noting that ribulose bisphosphate carboxylase oxygenase (RUBISCO) behaves as a no folder at low ionic strength but becomes a slow folder at high ionic strength54. Similarly, by increasing the temperature from about (10 20) ◦C to physiological temperature (37 ◦C) both malate − dehydrogenase and aspartate transaminase transition from being a slow to no folders54,55. The no folders, with low Φ, are prime candidates, which can fold with the aid of the complete

9 chaperonin machinery.

EXPERIMENTAL EVIDENCE FOR KPM

The KPM has been validated in a number of experiments. The value of Φ has been measured in ensemble and single molecule experiments. (i) For example, Kiefhaber56 showed, using interrupted folding (final folding condition is 0.6 M GdmCl, pH = 5.2 and T = 20◦C), that Φ = 0.15 for hen egg white lysozyme (HEWL). The fast track molecules fold in about 50 ms. The remaining fraction are slow folders, which reach the native state on a time scale about 400 ms. By varying pH the value of Φ was found to increase to about 25 %57 while the time constants for the fast track molecules were roughly identical to the earlier study56. Because of the time for reaching the folded state by the molecules in the slow track is relatively small compared to biological times, it might be correctly concluded that folding of HEWL would not require the assistance of chaperones. (ii) The yield of the folded RUBISCO obtained in the direct folding, under non-permissive conditions, inferred from chaperonin-mediated folding (see below) is extremely small and is only order of (2-5)%. For both Tetrahymena and RU- BISCO most of the molecules ought to be classified as no folders, which imply their folding requires molecular chaperones. (iii) An indirect estimate of Φ was first made using theory and experiments for Tetrahymena ribozyme. It was found9 that Φ 0.1, which was subsequently ≈ confirmed in smFRET experiments36. These values were obtained at sufficiently high Mg2+ concentration. At cellular Mg2+ the value is expected to be much less. Introduction of a single point mutation (U273A) stabilizing the P3 pseudoknot helix was shown to increase Φ as high to 0.837, which shows that both sequence and external conditions determine the value of Φ. For both Tetrahymena and RUBISCO most of the molecules ought to be classified as no folders, which imply their folding requires molecular chaperones. In addition to the above mentioned experiments single molecule pulling experiments on several proteins (Tenacin, Fi- bronectin, T4Lysozyme, Calmodulin) using both Atomic Force Spectroscopy58,59, and optimal tweezer techniques have established the validity of the KPM.

10 Size and kinetic Constraints

Two constraints must be satisﬁed for GroEL-GroES assisted folding. First, pertains to the size of the SPs. The radius of gyration, Rg, of folded states of globular proteins is fairly 1/3 60 accurately given by Rg = 3N Å . Small Angle X-Ray Scattering experiments on a few proteins have shown that the typical sizes of misfolded SPs is about (5-10)% larger than the folded states. This implies that the size of the RUBISCO monomer, with N = 491, in the misfolded state is 32Å. The size of the expanded cavity, when both ATP and GroES are ≈ bound to GroEL, is 185,000 Å3. If the cavity is approximated as a sphere the apparent ≈ radius would be 35 Å, which implies that if RUBISCO is fully encapsulated in the expanded cavity there would be room for about one layer of water molecules. Thus, GroEL can process

SPs that contain . 500 residues by fully encapsulating them. The second and a more important constraint is kinetic in nature. As argued before only a small fraction, Φ of the SPs, reaches the folded state rapidly without being kinetically trapped in one of the many metastable states. If the average rate for molecules that fold by the slow track is ks then in order to prevent aggregation the pseudo first order binding rate, kB, of the misfolded SP to bind to GroEL must greatly exceed kA where kA is a pseudo first order rate for SP aggregation. The kinetic constraint shows clearly that the efficacy of assisted folding depends on the concentrations of both the SPs and GroEL.

Allosteric Transitions in GroEL

Because the equilibrium and non-equilibrium aspects in the spectacular allosteric transitions in GroEL have been recently reviewed10,15,16,61, we describe only brieﬂy the key events that impact the nature of assisted folding. Although the functional state of GroEL-GroES in the presence of SPs is the symmetric structure with the co-chaperonin bound to both the rings18,19,62,63, let us consider for illustration purposes only the hemicycle, thus allowing us to describe events in one ring. The T , R, and R00 are the three major allosteric states (Fig.1). The misfolded SPs, with exposed hydrophobic residues, preferentially interact with the T state, which has almost a continuous hydrophobic region lining the mouth of the GroEL cavity. The presence of the hydrophobic region is due to the alignment of seven subunits that join several large hydrophobic residues in the two helices (H and I) in the apical domain of

11 each subunit. The T R transition, resisted by the SP, is triggered by ATP binding to the → seven sites in the equatorial domain. The rates of the reversible T R transition were ﬁrst ↔ measured in pioneering studies by Yifrach and Horovitz64,65 who also established an inverse relation, predicted using computations21, between the extent of co-operativity in this transition and the folding rates of slow folding SPs66. Binding of GroES, which predominantly occurs only after ATP binds, drives GroEL to the so-called R0 state, which is followed by an irreversible non-equilibrium transition to the R00 state after ATP hydrolysis. It is suspected there is little structural difference between the R0, with ATP-bound, and the R00, containing ADP and inorganic phosphate, states. In both these transitions strain due to ATP binding and hydrolysis at the catalytic site propagates through a network of inter-residue contacts67, thus inducing large scale conformational changes. That such changes must occur during the reaction cycle of GroEL is already evident by comparing the static crystal structures in different allosteric states, such as the T and R00 states11. Release of ADP and the inorganic phosphate from the R00 state resets the machine back to the taut state from which a new cycle can begin. The allosteric transitions that GroEL undergoes during the catalytic cycle is intimately related to its function (see below). As we discuss later, it is not sufﬁcient to deal with the catalytic cycle in a single ring because under load it is the symmetric football-like structure that is the functional state.

Iterative Annealing Mechanism integrates GroEL Allostery and assisted SP folding

The importance of GroEL allostery in assisted folding can be appreciated by understanding the interaction of the SP with the GroEL-GroES system in different allosteric states. The changes in the SP-GroEL interaction occur in three stages corresponding to the allosteric transitions between the three major allosteric states (see Fig.1). (i) The continuous lining of the hydrophobic residues in the T state ensnares a misfolded SP with exposed hydrophobic residues. At this stage in the catalytic cycle the SP is predominantly in a hydrophobic environment, resulting in the formation of a SP-GroEL complex that is stable but not hyper stable so that the SP can be dislodged in to the cavity upon GroES binding68,69. (ii)The dynamics of the T R transition, upon ATP binding, reveals that there is a downward tilt in two helices ↔ near the E domain that closes off the ATP binding sites, and which is followed by multiple salt

12 bridge disruption (within a subunit) and formation of new ones across the adjacent subunits70. As these events unfold cooperatively, the stability of the initial SP-GroES complex decreases. More importantly, the adjacent subunits start to move apart, which imparts a moderate force that is large enough to at least partially unfold the SPs35,71. (iii) Both GroES and SP bind to the same sites, which are located in the crevices of helices H and I in the apical domain. Thus, when GroES binds, displacing the SP into the expanded central cavity, there are major structural changes in the GroEL cavity with profound consequences for the annealing mechanism. Only 3–4 of the 7 SP binding sites are needed to capture the SP, leaving 3–4 sites available for binding of the mobile loops of GroES. This ensures that the subsequent displacement of the SP occurs vectorially into the central cavity of GroEL. First, there is a signiﬁcant conformational change in the A domain, which undergoes a rotation and twist motion. Each subunit results in the two helices (K and L) in each subunit undergo an outside-in movement (Fig.1). As a result, polar and charged residues, which are solvent exposed in the T state, line the inside of the GroEL cavity. This in turn creates a polar microenvironment for the SP (Fig.1). Second, these large scale conformational changes are facilitated by the formation of several inter subunit salt bridges and disruption of intra subunit salt bridges70.

From the perspective of SP, there are major consequences that occur as a result of the allosteric transitions in GroEL. First, by breaking a number of salt bridges the volume of the central cavity increases two fold (85,000 Å3 185,000 Å3). In such a large central cavity, → enough to fully accommodate a compact protein with 500 residues, folding to the native ≈ state could occur if given sufﬁcient time as is the case in the SR1 mutant. But in the wild type the residence time of the encapsulated SP is very short (see below). Second, and most importantly, the SP-GroEL interaction changes drastically during the catalytic cycle. In the T state, SP-GroEL complex is (marginally) stabilized predominantly by hydrophobic interactions. However, during the subsequent ATP-consuming and irreversible step R R00 transition the → microenvironment of the SP is largely polar (see the discussion in the previous paragraph). Thus, during a single catalytic cycle, that is replicated in both the rings, the microenvironment of the SP changes from being hydrophobic to polar. We note parenthetically that even during the T R transition there is a change in the SP-GroEL interactions, which explains the ob- → servations that GroEL can assist of the folding of certain SPs (non stringent substrates) even in the absence of GroES. The annealing capacity of GroEL is intimately related to the changes in the SP-GroEL interactions that occur during each catalytic cycle. Hence, the function of the

13 GroEL-GroES system cannot be understood without considering the complex allosteric transitions that occur due to ATP and GroES binding. As a result of these transitions, the SP is placed stochastically from one region in the folding landscape, in which the misfolded SP is trapped, to another region from which it could undergo kinetic partitioning with small probability to the folded state or be trapped in another misfolded state. The cycle of hydrophobic to polar change is repeated in each catalytic cycle, and hence the GroEL-GroES system iteratively anneals the misfolded SP enabling it to fold to the native state. Because this process is purely stochastic, GroEL plays no role in guiding the protein to the folded state nor does it sense the architecture or any characteristics of the folded state. In other words, the information for protein self-assembly is fully encoded in the amino acid sequence as articulated by Anﬁnsen72. GroEL merely alters the conformation of the SP stochastically as it undergoes the reaction cycle, enabling the SP to explore different regions of the folding landscape. In this sense the action of GroEL is analogous to simulated annealing used in optimization problems73 although the latter is a more recent realization of an evolutionary event that took place millions of years ago.

Theory underlying the IAM

The physical picture of the IAM described above can be formulated mathematically to quantitatively describe the kinetics of chaperonin-assisted folding of stringent in vitro substrate proteins74. According to theory underlying IAM (see Fig. 3), in each cycle the SP folds by the KPM, as the microenvironment for the SP changes as GroEL undergoes the reaction cycle. Thus, with each round of folding the fraction of folded molecules is Φ, and the remaining fraction gets trapped in one of the many misfolded structures. After n such cycles (or iterations) the yield of the native state is, n Ψ = Λss [1 (1 Φ) ] (1) − − where Λss is the steady state yield. The mathematical model accounts for all the available experimental data, and shows that for for RUBISCO the partition factor Φ 0.02, which ≈ means that only about 2% of the SP reaches the folded state in each cycle. From this ﬁnding we could surmise that the GroEL chaperonin is an inefﬁcient machine, which consumes ATP lavishly and yet the yield of the folded protein per cycle is small. A prediction of the IAM is that GroEL should reset to the starting T state as rapidly as possible in the presence of SPs.

14 By rapidly resetting to the T state the number of interactions can be maximized, which would maximize the yield of the folded state for a speciﬁed amount of time75. Indeed, this is the case, which we delve into detail below.

Rate of R00 T transition is a maximum for the wild-type (WT) GroEL →

A clear implication of the IAM is that rapid turnover of the catalytic cycle would produce the maximum yield of the native state in a given time. Examination of the reaction cycle shows that the rate determining step (resetting of the machine) should correspond to release of ADP and the inorganic phosphate. In other words, maximization of the rate, kR00→T returns GroEL to the acceptor state for processing a new SP. In order to illustrate that this is indeed the case, we first extracted the rates of the allosteric transitions by fitting the solutions of the kinetic equations74 by simultaneously fitting the experimental data for assisted folding at various GroEL concentrations. For this purpose, we used the data for RUBISCO for which the yield of the folded state as a function of eight values of the GroEL concentration are available14. The excellent fits at various GroEL concentrations (Fig.6), with a fixed initial concentration of Rubisco, were used to extract Φ. We find that Φ 0.02, which means that only about 2% of ≈ the SP reaches the folded state in each catalytic cycle. Armed with the rates that describe the allosteric transitions, we used the IAM theory based to analyze experimental data on the folding of other SPs. Because the reversible transition ATP-induced T R transition occurs at equilibrium even in the absence of SP65 it is rea- ↔ sonable to assume that they are relatively insensitive to the nature of the SP. Indeed, the extracted values of the T R rates using the RUBISCO data (see Table 1 in74) are very sim- ↔ 65 ilar to measurements made in the absence of SP . This leaves the rate kR00→T that results in the resetting the machine after ATP hydrolysis to the taut (T ) state as the most important factor

in determining the efficiency of GroEL or its mutants. Thus, maximizing kR00→T should result in optimizing the native state yield at a fixed time. This most significant prediction of IAM can be quantitatively demonstrated by analyzing the data reported by Lund and coworkers76. They measured the activity, which we assume is proportional to the yield of the folded state, as a function of time for GroEL and five mutants including SR1. The two SPs used in these studies were mitochondrial Malate Dehydrogenase (mtMDH) and citrate synthase (CS). The results,

15 SS reproduced in Fig.7 shows that PN and indeed the yield at any time is largest for the WT and is least for the SR1 mutant from which GroES disassociates in 300 minutes. The curves in ≈

Fig.7 were calculated by adjusting just one parameter, the rate kR00→T while keeping the rates for other allosteric rates ﬁxed at the values extracted by analyzing the RUBISCO data. The IAM predictions are in quantitative agreement with experiments for both the proteins and for −1 GroEL and its mutants. The value of kR →T 60 s (one second) is largest the WT GroEL. 00 ≈ This implies, as predicted by IAM, that GroEL catalytic cycle is greatly accelerated when SP is present, a point that requires further elaboration.

GENERALIZED IAM FOR RNA CHAPERONES

Compared to GroEL-GroES chaperonin, details of the catalytic cycle of CYT-19 are not known. Consequently, it is not possible to link the structural transitions that occur during the CYT-19 assisted folding of the misfolded ribozymes, as we did for the GroEL-GroES machinery. We should note that the structures and biophysical studies of the DEAD-box protein Mss116p, Saccharomyces cerevisiae analogue of CYT-19, showed the expected helicase activity, resulting in the disruption of the structure of the misfolded ribozyme. These studies and the still undetermined ATPase cycle could be used in the future to provide a molecular basis of the IAM for CYT-19 assisted folding. Nevertheless, the mathematical formulation of the IAM theory could be adopted to investigate the interesting experimental findings by Bhaskaran and Russell40. The most significant experimental findings of CYT-19 assisted folding of ribozymes are: (i) When incubated in CYT-19 under somewhat destabilizing conditions ([Mg2+]< 2 mM), ribozymes show a low cleavage activity. (ii) Regardless of the initial population, the native and the misfolded ribozyme reach a steady state value for the folded ribozyme fraction, which is not unity (P SS = 1). (iii) The deactivation of the ribozyme function was observed at longer N 6 pre-incubation times in CYT-19. Deactivation of native ribozyme was also observed at higher CYT-19 concentration. Taken together, these observations imply that CYT-19 destabilizes the native as well as the misfolded ribozyme. The finding that CYT-19 interacts with the fully folded ribozyme is in stark contrast with GroEL, which does not interact with the folded states of proteins. In light of the experimental observations the IAM theory has to be generalized

16 (see Fig.8). The results in this study inspired us to generalize the IAM theory using the master equation77. More recently, we proposed a simpler version that describes the functions of GroEL and RNA chaperones on equal footing75. The resulting theory, which gives rise a complicated expression for the folded state of P5a variant of the Tetrahymena sketched below, provided a quantitative agreement (Fig.9) of the experimental data40. The KPM description of ribozyme folding9,51 shows that upon increasing the Mg2+ concentration a fraction of the initial unfolded population, Φ, folds to the native state and the remaining

fraction, M1 = 1 Φ, collapses to one of many misfolded states. Consider the fate of the mis- − folded states, with population, M1, as they interact with CYT-19. In the presence of the RNA

chaperone, a fraction Φ of M1 reaches the native state (Φ(1 Φ)) and 1 Φ of M1 to one of − − the misfolded states ((1 Φ)2). Because CYT-19 also acts on the native state we also have to − consider the fate of the folded ribozyme, as it interacts with CYT-19 (see Fig.8). Let a fraction κ denote the fraction of the initially folded ribozyme reach the misfolded state (bottom right circle in Fig.8) while the 1 κ remain in the native state (top right circle in Fig.8). In the subsequent − round, Out of κΦ, κΦ2 of them goes to native and κΦ(1 Φ) reaches the misfolded state. − 2 Therefore, M2 = κΦ(1 Φ) + (1 Φ) is the total of the misfolded ribozyme in the second − − round of IAM, which accounts for accumulation from both the folded and misfolded states in the ﬁrst round. In order to obtain an expression for the yields of both the folded and misfolded states of the ribozyme the branching process from both the accumulated folded and misfolded states of the ribozyme in the previous round has to be taken into account. A recursion relation for this iterative process may be written down, such that the amount of misfolded state at the

n-th round is the sum of Mn−1 (1 Φ) from the misfolded ensemble, and κ(1 Mn−1)(1 Φ) × − − − from the native ensemble. In short, Mn = Mn−1(1 Φ) + κ(1 Mn−1)(1 Φ) (see Fig.8). As − − − a result, the total yield of native state in the N-th round of annealing process (ΨN = 1 MN ) − can be calculated in order to obtain yield of the native ribozyme from the generalized version of IAM, 1 (1 κ)N (1 Φ)N ΨN = Φ − − − , (2) κ + (1 κ)Φ − and the steady state solution (N ) is → ∞ Φ Ψ = . (3) ∞ κ + (1 κ)Φ − For κ = 0, corresponding to the situation that the RNA chaperone does not recognize the native state , the yield in the N-th round is identical to the conventional IAM expression. For

17 κ = 1 in which RNA chaperone recognizes the native state equally as well as misfolded states, there would be no gain in the native yield by the action of RNA chaperone. The action of chaperones on substrate RNA can be mapped onto 3-state kinetic model of RNA with transitions between the native (N), misfolded (M), and intermediate states (I).

When the partition factor Φ in terms of the rate constants, Φ = kIN /(kIN + kIM ), is plugged

into Eq.3, Ψ∞ is

kIN Ψ∞ = . (4) κkIM + kIN

SS In addition, the expression for the steady state value of fraction native (PN ), which is equiva-

lent to Ψ∞, can be obtained using 3-state kinetic model (Fig.5) under the following conditions:

(i) kNM , kMN kIN , kIM , kNI , kMI , and (ii) kNI kIN . With these assumptions we ﬁnd that,

SS kIN PN . (5) ≈ kNI kIM + kIN kMI

Therefore, comparison between Eq.4 and Eq.5 gives

k ([C], [T ]) κ = NI (6) kMI ([C], [T ]) where the dependence of unfolding rates kNI and kMI on chaperone and ATP concentration is made explicit. It turns out that κ, deﬁned as the unfolding efﬁciency of chaperone for the native state with reference to the misfolded ensemble, is effectively the ratio between chaperone- induced unfolding rate from the native and misfolded state. A sketch of the native state as a function of κ, which depends both on the chaperone and ATP concentration, is given in Fig.10.

DISCUSSION

What do chaperones optimize?

The question of what quantity a biological machines optimize subject to the constant of available free energy does not have a general answer. However, in the rare case of chaperones a plausible answer has been recently proposed, which we illustrate here75. It is note- worthy that despite the critical difference between CYT-19 and GroEL, with the former that

18 disrupts both the folded and misfolded states of ribozymes whereas the latter does not interact with the folded proteins, the mechanisms of their functions are in accord with the predictions of IAM. Both GroEL and RNA chaperones function by driving the SPs and ribozymes out of equilibrium75. Remarkably, we showed by analyzing experimental data on ribozymes and MDH that the quantity that is optimized by GroEL and RNA chaperones is,

SS ∆NE = kF PN (7)

SS where kF is the folding rate and PN is the steady state yield (see Fig.11). Thus, neither the folding rate nor the steady state yield is maximized but it is the product of the two that is optimized by the molecular chaperones. It follows from Fig.11 that, for a given SP and external conditions, which would ﬁx kF , the steady state yield would have the largest value for the wild type GroEL than any other mutant. That this is indeed the case is vividly illustrated in SS Fig. 6. In the case of GroEL, the value of PN (or PN (t) at any t) is critically dependent only on kR00→T , which has the largest value for the WT GroEL. The optimality condition given in

Eq.7 is determined by the value of kR00→T , which in turn depends on the dynamics of allosteric transitions as well the presence of SP.Thus, the function of GroEL, and most likely CYT-19 and related RNA chaperones, cannot be understood without considering the details of the reaction cycle and how they are directly related to SP folding. The IAM theory, which accounts for all the complexities of the reaction cycle, explains the available experimental data quantitatively

(see for example Fig. 6) using a single parameter (kR00→T ).

When it does SP folding occurs in the expanded GroEL cavity

Does SP folding occur in the expanded cavity or in solution after ejection? This question has unnecessarily plagued the discussion of GroEL-assisted folding, causing substantial con- fusion largely because of insistence by some that GroEL merely encapsulates the SP in the cavity until it reaches the native state with unit probability78. Such an inference that GroEL is a passive Anﬁnsen cage has been made principally using experiments based on a single ring mutant (SR1) from which discharge of GroES and the SP occurs on a time scale of 300 minutes is erroneous. For starters, the life time of the encapsulated SP in the wild type (WT) cycling GroEL is about 2 seconds63 that is four orders of magnitude shorter than the SR1 lifetime! Furthermore, neither the passive or active cage model can explain how the commu-

19 nication to discharge the ligands (ADP and the inorganic phosphate), GroES, and the folded SP takes place.

Does folding to the folded state occur within the cavity in the WT GroEL? We answer this question in the affirmative by using the following argument. Assisted folding requires that the kinetic constraint, kF < kB be satisfied where kB pseudo first order binding rate of SP to

GroEL. In the opposite limit (kF > kB), which is relevant at low GroEL concentrations, folding is sufﬁciently rapid compared to diffusion controlled binding that the chaperonin machinery would not be needed. Thus, assuming that the kinetic constraint (kB kA) is always satisﬁed for stringent substrates under non-permissive conditions then the SP upon ejection from the GroEL cavity, roughly every two seconds, rebinds (presumably to the same GroEL molecule) rapidly. If the ejected SP is in the folded state then it would not be recognized by GroEL because the hydrophobic recognition motifs would no longer be solvent exposed. Thus, the fate of SP, which occurs by the KPM, is decided entirely within the cavity during the lifetime of its residence. Both folding and partitioning to the ensemble of misfolded states occur rapidly while the SP is encapsulated for a brief period in either chamber.

We provide evidence to substantiate the physical arguments given above. The theory underlying IAM was used to obtain the parameters for the rates in the catalytic cycle and the intrinsic rates for assisted folding of RUBISCO. The time for RUBISCO molecules to reach

−1 74 the folded state by the fast track, τF = kF = 0.6 s (Table 1 in ), which is less than the encapsulation time of about 2 seconds. This implies only the fast track RUBISCO molecules −1 fold in the cavity because time for slow track Rubisco molecules τS(= kS ) to fold is about 333 minutes (Table 1 in74). The slow track molecule would rapidly rebind upon exiting the cavity, and the process is iterated multiple times till the majority of unfolded SPs reach the native state. One can use the same argument for reconstituting Citrate Synthase (CS) using

76 GroEL. The ﬁts to the experimental data in Fig. 6 yields τF = 0.6 s whereas τS = 100 minutes74, which again shows that KPM resulting in folded and misfolded states occurs while CS is encapsulated in the cavity. Thus, we can conclude that when SP folding occurs it occurs in the expanded cavity. It is worth emphasizing that because the IAM theory takes into account the coupling between the events in the reaction cycle of GroEL and SP folding it naturally explains the allosteric communication needed for discharge of the SP, whether it is folded or not, and other ligands. However, only a very small fraction reaches the folded state in each cycle, and hence the need to perform the iterations as rapidly as possible. Remarkably, GroEL

20 has evolved to do just that by functioning as a parallel processing machine in the symmetric complex when challenged with SP18,19.

Symmetric Complex is the Functioning Unit of the GroEL-GroES machine

The IAM predicts that the yield of the folded SP increases with each iteration. It, therefore, follows that for highly efﬁcacious function it would be optimal if GroEL-GroES functions as a parallel processing machine with one SP in each chamber. This would necessarily involve

formation of a symmetric complex GroEL14-GroES14, which was shown as the functioning unit only recently13,19,63. In particular, using a FRET-based system Ye and Lorimer63 have established unequivocally that the response of the GroEL-GroES machinery is dramatically different with and without the presence of SP. In order to unveil the differences they had to

follow the fate of ADP and Pi release in real time. These experiments showed that in the

absence of the SP the rate determining step involves release of Pi before ADP release from the trans ring of the dominant asymmetric complex (GroES bound to the cis ring). In sharp contrast, when challenged with the SP, ADP is released before Pi. The symmetric particle, with GroES bound to both the rings (Fig.2b), is the predominant species in the presence of SP. In principle, the symmetric particle can simultaneously facilitate the folding of two SPs one in each chamber. Thus, it is likely the case that the functional form in vivo is the symmetric particle , which is activated when there is a job to do, namely, help SPs fold. There was one other major ﬁnding in the Ye-Lorimer study63. They discovered that the ATP hydrolysis rate ( 0.5−1) is the same in the presence and absence of the SP. In the presence ∼ of SP, hydrolysis of ATP is rate limiting, which in the language used to describe motility of motors means that GroEL is ATP-gated. In other words, symmetry breaking (or inter ring communication) events that determine the ring from which GroEL disassociates depends on extent of ATP hydrolysis in each ring. Remarkably, the release of ADP from the trans ring is accelerated roughly 100 fold in the presence of SP. We note parenthetically that release of ADP from the nucleotide binding pocket of conventional kinesin is accelerated by nearly 1000 fold in the presence of microtubules80, hinting at the possibility that there is a uniﬁed molecular basis for nucleotide chemistry in biological machines. By greatly enhancing ADP release in the presence of SP, resetting to the initial SP accepting state occurs rapidly (kR00→T is maximized in the WT GroEL), which allows GroEL to process as many SP molecules as

21 fast as possible. Clearly, these ﬁndings are in complete accord with the IAM predictions and debunk the Anﬁnsen cage model78,81.

CONCLUSIONS

In this perspective, we have shown that, despite profound differences, the functions of GroEL-GroES machine and RNA chaperones are quantitatively described by the theory underlying the Iterative Annealing Mechanism. We are unaware any experiment of assisted folding of the SPs or ribozymes that cannot be explained by the theory. We conclude with the following additional comments.

1. It is sometimes stated that the mechanism of how GroEL functions is controversial because of the proposal that the cavity in the GroEL could act as an Anﬁnsen cage in which folding can be completed unhindered by aggregation. Such a conclusion was reached based mostly on experiments on the SR1 mutant (an asymmetric GroEL complex) from which GroES disassociates in 300 minutes. Although experiments using SR1 (with ADP

and Pi locked into the equatorial domain) provide insights into effects of confinement on SP folding they are irrelevant in understanding of WT GroEL function. Finally, in the Anfinsen cage model there is no necessity for invoking allosteric transitions and how they are linked to assisted folding. In the SR1 mutant, ATP binding and hydrolysis occurs once, which means that the SP is trapped in a hydrophilic cavity for extremely long times, and hence lessons from the SR1 mutant neither inform us about the intact WT GroEL nor are they biologically relevant. On the other hand, the stochastic WT GroEL comes alive when presented with SPs, undergoes a series of allosteric transitions by binding, hydrolyzing ATP, and releasing the products, which permits the SPs multiple chances to fold in the most optimal fashion (see Eq.7). The quantitative success of the IAM should put to rest the inadequacy and the erroneous Anfinsen cage model78 for describing the function of the WT GroEL. For instance, the results in Fig. 7 cannot be understood within the Anfinsen cage model.

2. The machine-like non-equilibrium characteristics of chaperones are most evident by the SS demonstration that the steady state yield, P = PEQ where the equilibrium yield of the N 6 22 folded state, PEQ, is given by the Boltzmann distribution,

EQ 1 PN = , (8) (1 + e−∆GNM /kB T )

where ∆GNM is the free energy of the folded state with respect to the manifold of mis- SS folded states. The values of PN for the two SPs and Tetrahymena ribozyme and its variants depend both the chaperone and ATP concentrations75, which itself is evidence

of departure from equilibrium. In addition, the measured value of ∆GNM for the WT EQ ribozyme is 10 kBT , which implies that P 0.99 according to Eq. 8. However, the ∼ N ≈ measured value in the presence of ATP is far less, which shows that P SS = P EQ. The N 6 N SS ﬁnding that PN values of RUBISCO and MDH are dependent on GroEL concentration also implies that in the presence of GroEL Eq. 8 is not valid. Taken together they imply that in the process of assisted folding both GroEL and CYT-19 drive the misfolded SPs and ribozymes out of equilibrium (see also79).

3. GroEL and RNA chaperones burn copious amount of ATP because in each round only a small fraction (Φ 1) of misfolded molecules reach the native state. Consider RU- BISCO folding for which Φ = 0.0274. The yield of the native state at t = 20 min with the concentration of GroEL roughly equal to the initial unfolded RUBISCO (both at 50 nm) is about 0.7 (see Fig.6). The value of P SS 0.8 from which we obtain n 100 using Eq.1. N ≈ ≈ In each catalytic cycle between (3 4) ATP molecules are consumed, which implies that − in order to fold roughly 88% of RUBISCO in the steady state between (300 400) ATP − molecules are hydrolyzed. As pointed out elsewhere14, this is but a very small fraction of energy required to synthesize RUBISCO, a protein with 491 residues. Thus, the beneﬁts of GroEL assisted folding far outweighs the cost of protein synthesis. However, from a thermodynamic perspective it can be argued that GroEL is less efﬁcient than Myosin V,

which consumes one ATP molecule (available energy is about EAT P (20 25) kBT ) ≈ − per step (s 36 nm) while walking on actin ﬁlament, resisting forces on the order of ≈ sfs about fs 2 pN. Thus, a rough estimate of Myosin V efﬁciency is is very high. ≈ EAT P

ACKNOWLEDGMENTS

We are grateful to Bernard Brooks, Shaon Chakrabarti, Eda Koculi, George Stan, Riina Tehver, and Scott Ye for collaboration on various aspects of the works described here. DT

23 acknowledges useful conversations with Alan Lambowitz on RNA chaperones. This work was supported in part by a grant from National Science Foundation (CHE 19-00093) and the Collie-Welch Chair (F-0019) administered through the Welch Foundation.

[1] Goloubinoff P, Gatenby AA, Lorimer GH (1989) GroEL heat-shock proteins promote assembly

of foreign prokaryotic ribulose bisphosphate carboxylase oligomers in Escherichia-Coli. Nature

337:44–47.

[2] Sigler PB, Xu Z, Rye HS, Burston SG, Fenton WA, Horwich AL (1998) Structure and function in

groel-mediated protein folding. Annu. Rev. Biochem. 67:581–608.

[3] Thirumalai D, Lorimer GH (2001) Chaperonin-mediated protein folding. Ann. Rev. Biophys.

Biomol. Struct. 30:245–269.

[4] Ellis J (1987) Proteins as molecular chaperones. Nature 328:378–379.

[5] Lorimer GH (2001) A personal account of chaperonin history. Plant physiology 125:38–41.

[6] Hemmingsen SM, Woolford C, van der Vies SM, Tilly K, Dennis DT, Georgopoulos CP, Hendrix

RW, Ellis RJ (1988) Homologous plant and bacterial proteins chaperone oligomeric protein as-

sembly. Nature 333:330.

[7] Mohr S, Stryker JM, Lambowitz AM (2002) A dead-box protein functions as an atp-dependent rna

chaperone in group i intron splicing. Cell 109:769–779.

[8] Woodson SA (2010) Taming free energy landscapes with rna chaperones. RNA biology 7:677–

686.

[9] Pan J, Thirumalai D, Woodson SA (1997) Folding of rna involves parallel pathways. J. Mol. Biol.

273:7–13.

[10] Thirumalai D, Hyeon C, Zhuravlev PI, Lorimer GH (2019) Symmetry, rigidity, and allosteric signal-

ing: From monomeric proteins to molecular machines. Chem. Rev. 119:6788–6821.

[11] Xu Z, Horwich AL, Sigler, PB (1997) The crystal structure of the asymmetric GroEL-GroES-(ADP)7 chaperonin complex. Nature 388:741.

[12] Fei X, Yang D, LaRonade-LeBlanc N, Lorimer GH (2013) Crystal structure of a groel-adp complex

in the relaxed allosteric state at 2.7 Å resolution. Proc. Natl. Acad. Sci. U. S. A. 110:E2958–E2966.

[13] Fei X, Ye X, LaRonade NA, Lorimer GH (2014) Formation and structures of groel:groes2 chaper-

onin footballs, the protein-folding functioal form. Proc. Natl. Acad. Sci. U. S. A. 111:12776–12780.

24 [14] Todd MJ, Lorimer GH, Thirumalai D (1996) Chaperonin-facilitated protein folding: Optimization of

rate and yield by an iterative annealing mechanism. Proc. Natl. Acad. Sci. U. S. A. 93:4030–4035.

[15] Gruber R, Horovitz A (2016) Allosteric mechanisms in chaperonin machines. Chem. Rev.

116:6588–6606.

[16] Thirumalai D, Hyeon C (2018) Signalling networks and dynamics of allosteric transitions in bacte-

rial chaperonin GroEL: implications for iterative annealing of misfolded proteins. Phil. Trans. Royal

Soc. B: Biol. Sci. 373:20170182.

[17] Gruber R, Horovitz A (2018) Unpicking allosteric mechanisms of homo-oligomeric proteins by

determining their successive ligand binding constants. Phil. Trans. R. Soc. B 373:20170176.

[18] Yang D, Ye X, Lorimer GH (2013) Symmetric GroEL:GroES2 complexes are the protein-folding

functional form of the chaperonin nanomachine. Proc. Natl. Acad. Sci. U. S. A. 110:E4298–E4305.

[19] Takei Y, Iizuka R, Ueno T, Funatsu T (2012) Single-molecule observation of protein folding in

symmetric GroEL-(GroES) 2 complexes. J. Biol. Chem. 287:41118–41125.

[20] Viitanen PV, Gatenby AA, Lorimer GH (1992) Puriﬁed chaperonin 60 (groel) interacts with the

nonnative states of a multitude of escherichia coli proteins. Protein Science 1:363–369.

[21] Betancourt MR, Thirumalai D (1999) Exploring the kinetic requirements for enhancement of pro-

tein folding rates in the GroEL cavity. J. Mol. Biol. 287:627–644.

[22] Baumketner A, Jewett A, Shea J (2003) Effects of conﬁnement in chaperonin assisted protein

folding: rate enhancement by decreasing the roughness of the folding energy landscape. Journal

of molecular biology 332:701–713.

[23] Cheung MS, Thirumalai D (2006) Nanopore–protein interactions dramatically alter stability and

yield of the native state in restricted spaces. J. Mol. Biol. 357:632–643.

[24] Hofmann H, Hillger F, Pfeil SH, Hoffmann A, Streich D, Haenni D, Nettels D, Lipman EA, Schuler

B (2010) Single-molecule spectroscopy of protein folding in a chaperonin cage. Proceedings of

the National Academy of Sciences 107:11793–11798.

[25] Fedorov AN, Baldwin TO (1997) GroE modulates kinetic partitioning of folding intermediates be-

tween alternative states to maximize the yield of biologically active protein. J. Mol. Biol. 268:712–

723.

[26] Treiber DK, Rook M, Zarrinkar PR, Williamson, JR (1998) Kinetic Intermediates Trapped by Native

Interactions in RNA Folding. Science 279:1943–1946.

25 [27] Treiber DK, Williamson JR (1999) Exposing the kinetic traps in RNA folding. Curr. Opin. Struct.

Biol. 9:339–345.

[28] Thirumalai D, Woodson SA (1996) Kinetics of Folding of Proteins and RNA. Acc. Chem. Res.

29:433–439.

[29] Russell R, Herschlag D (2001) Probing the folding landscape of the tetrahymena ribozyme: Com-

mitment to form the native conformation is late in the folding pathway. J. Mol. Biol. 308:839–851.

[30] Woodson SA (2005) Structure and assembly of group i introns. Curr. Opin. Struct. Biol. 15:324–

330.

[31] Russell R, Herschlag D (1999) New pathways in folding of the Tetrahymena group I RNA enzyme.

J. Mol. Biol. 291:1155–1167.

[32] Thirumalai D, Hyeon C (2005) RNA and Protein folding: Common Themes and Variations. Bio-

chemistry 44:4957–4970.

[33] Hyeon C, Denesyuk, NA, Thirumalai D (2014) Development and Applications of Coarse-Grained

Models for RNA. Israel J. Chem. 54:1358–1373.

[34] Solomatin SV, Greenfeld M, Chu S, Herschlag D (2010) Multiple native states reveal persistent

ruggedness of an RNA folding landscape. Nature 463:681–684.

[35] Thirumalai D, Lee N, Woodson SA, Klimov DK (2001) Early Events in RNA Folding. Annu. Rev.

Phys. Chem. 52:751–762.

[36] Zhuang X, Bartley L, Babcock A, Russell R, Ha T, Hershlag D, Chu S (2000) A single-molecule

study of RNA catalysis and folding. Science 288:2048–2051.

[37] Pan J, Deras ML, Woodson SA (2000) Fast Folding of a Ribozyme by Stabilization Core Interac-

tions: Evidence for Multiple Folding Pathways in RNA. J. Mol. Biol. 296:133–144.

[38] Lorsch JR (2002) Rna chaperones exist and dead box proteins get a life. Cell 109:797–800.

[39] Tijerina P, Bhaskaran H, Russell R (2006) Nonspeciﬁc binding to structured RNA and preferential

unwinding of an exposed helix by the CYT-19 protein, a DEAD-box RNA chaperone. Proc. Natl.

Acad. Sci. U. S. A. 103:16698–16703.

[40] Bhaskaran H, Russell R (2007) Kinetic redistribution of native and misfolded rnas by dead-box

chaperone. Nature 449:1014–1018.

[41] Grohman JK, Del Campo M, Bhaskaran H, Tijerina P, Lambowitz AM, Russell R (2007) Probing

the mechanisms of DEAD-box proteins as general RNA chaperones: the C-terminal domain of

CYT-19 mediates general recognition of RNA. Biochemistry 46:3013–3022.

26 [42] Mallam AL, Jarmoskaite I, Tijerina P, Del Campo M, Seifert S, Guo L, Russell R, Lambowitz AM.

(2011) Solution structures of dead-box rna chaperones reveal conformational changes and nucleic

acid tethering by a basic tail. Proc. Natl. Acad. Sci. U. S. A. 108:12254–12259.

[43] Russell R, Jarmoskaite I, Lambowitz AM (2013) Toward a molecular understanding of RNA re-

modeling by DEAD-box proteins. RNA Biology 10:44–55.

[44] Lorimer GH (1996) A quantitative assessment of the role of the chaperonin proteins in protein

folding in vivo. FASEB J. 10:5.

[45] Stan G, Brooks BR, Lorimer GH, Thirumalai D (2005) Identifying natural substrates for chaper-

onins using a sequence-based approach. Prot. Sci. 14:193–201.

[46] Chaudhuri TK, Gupta P (2005) Factors governing the substrate recognition by groel chaperone:

a sequence correlation approach. Cell stress & chaperones 10:24.

[47] Stan G, Brooks BR, Lorimer GH, Thirumalai D (2006) Residues in substrate proteins that interact

with groel in the capture process are buried in the native state. Proc. Natl. Acad. Sci. U. S. A.

103:4433–4438.

[48] Noivirt-Brik O, Unger R, Horovitz A (2007) Low folding propensity and high translation efﬁ-

ciency distinguish in vivo substrates of GroEL from other Escherichia coli proteins. Bioinformatics

23:3276–3279.

[49] Endo A, Kurusu Y (2007) Identiﬁcation of in vivo substrates of the chaperonin GroEL from Bacillus

subtilis. Biosci. Biotech. Biochem. 71:1073–1077.

[50] Guo Z, Thirumalai D (1995) Kinetics of Protein Folding: Nucleation Mechanism, Time Scales, and

Pathways. Biopolymers 36:83–102.

[51] Thirumalai D, Klimov DK, Woodson SA (1997) Kinetic partitioning mechanism as a unifying theme

in the folding of biomolecules. Theor. Chem. Acc. 96:14–22.

[52] Thirumalai D (1995) From Minimal Models to Real Proteins: Time Scales for Protein Folding

Kinetics. J. Phys. I (Fr.) 5:1457–1467.

[53] Ziv G, Thirumalai D, Haran G (2009) Collapse transition in proteins. Phys. Chem. Chem. Phys.

11:83–93.

[54] Schmidt M, Buchner J, Todd MJ, Lorimer GH, Viitanen PV (1994) On the role of groes in the

chaperonin-assisted folding reaction. three case studies. J. Biol. Chem. 269:10304–10311.

[55] Mattingly JR, Iriarte A, Martinez-Carrion M (1995) Homologous proteins with different afﬁnities for

groel the refolding of the aspartate aminotransferase isozymes at varying temperatures. J. Biol.

27 Chem. 270:1138–1148.

[56] Kiefhaber T (1995) Kinetic traps in lysozyme folding. Proc. Natl. Acad. Sci. U. S. A. 92:9029–9033.

[57] Matagne A, Chung EW, Ball LJ, Radford SE, Robinson CV, Dobson CM (1998) The origin of the

α-domain intermediate in the folding of hen lysozyme. J. Mol. Biol. 277:997–1005.

[58] Peng Q, Li H (2008) Atomic force microscopy reveals parallel mechanical unfolding pathways

of t4 lysozyme: evidence for a kinetic partitioning mechanism. Proc. Natl. Acad. Sci. U. S. A.

105:1885–1890.

[59] Stigler J, Ziegler F, Gieseke A, Gebhardt J, Rief M (2011) The Complex Folding Network of Single

Calmodulin Molecules. Science 334:512–516.

[60] Dima RI, Thirumalai D (2004) Asymmetry in the shapes of folded and denatured states of proteins.

J. Phys. Chem. B 108:6564–6570.

[61] Lorimer GH, Horovitz A, McLeish T (2018) Allostery and molecular machines. Philos Trans R Soc

Lond B Biol Sci. 373:20170173.

[62] Sameshima T, Iizuka R, Ueno T, Funatsu T (2010) Denatured proteins facilitate the formation of

the football-shaped GroEL–(GroES) 2 complex. Biochem. J. 427:247–254.

[63] Ye X, Lorimer GH (2013) Substrate protein switches groe chaperonins from asymmetric to sym-

metric cycling by catalyzing nucleotide exchange. Proc. Natl. Acad. Sci. U. S. A. 110:E4289–

E4297.

[64] Horovitz A, Fridmann Y, Kafri G, Yifrach O (2001) Review:allostery in chaperonins. J. Struct. Biol.

135:104–114.

[65] Yifrach O, Horovitz A (1995) Nested cooperativity in the ARPase activity of the oligomeric chap-

eronin GroEL. Biochemistry 34:5303–5308.

[66] Yifrach O, Horovitz A (2000) Coupling between protein folding and allostery in the groe chaperonin

system. Proc. Natl. Acad. Sci. U. S. A. 97:1521–1524.

[67] Tehver R, Chen J, Thirumalai D (2009) Allostery wiring diagrams in the transitions that drive the

groel reaction cycle. J. Mol. Biol. 387:390–406.

[68] Thirumalai D (1994) Statistical Mechanics, Protein Structure, and Protein - Substrate Interactions,

ed. Doniach, S. (Plenum, New York), pp. 115 – 134.

[69] Orland H, Thirumalai D. (1997) A kinetic model for chaperonin assisted folding of proteins. J.

Phys. I France 7:553–560.

28 [70] Hyeon C, Lorimer GH, Thirumalai D (2006) Dynamics of allosteric transition in GroEL. Proc. Natl.

Acad. Sci. U. S. A. 103:18939–18944.

[71] Corsepius NC, Lorimer GH (2013) Measuring how much work the chaperone groel can do. Proc.

Natl. Acad. Sci. U. S. A. 110:E2451–E2459.

[72] Anﬁnsen CB (1973) Principles that govern the folding of protein chain. Science 181:223–230.

[73] Kirkpatrick S, Gelatt D, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–

680.

[74] Tehver R, Thirumalai D (2008) Kinetic Model for the Coupling between Allosteric Transitions in

GroEL and Substrate Protein Folding and Aggregation. J. Mol. Biol. 377:1279–1295.

[75] Chakrabarti S, Hyeon C, Ye X, Lorimer G, Thirumalai D (2017) Molecular Chaperones Maximize

the Native State Yield on Biological Times by Driving Substrates out of Equilibrium. Proc. Natl.

Acad. Sci. U. S. A. 114:E10919–E10927.

[76] Sun Z, Scott DJ, Lund PA (2003) Isolation and characterisation of mutants of groel that are fully

functional as single rings. J. Mol. Biol. 332:715–728.

[77] Hyeon C, Thirumalai D (2013) Generalized iterative annealing model for the action of rna chaper-

ones. J. Chem. Phys. 139:121924.

[78] Horwich AL, Apetri AC, Fenton WA (2009) The groel/groes cis cavity as a passive anti-aggregation

device. FEBS letters 583:2654–2662.

[79] Goloubinoff P, Sassi AS, Fauvet B, Barducci A and De los Rios P (2018) Chaperones convert

the energy from ATP into the nonequilibrium stabilization of native proteins. Nat. Chem. Biol.

14:388–394.

[80] Hackney DD (1988) Kinesin atpase: rate-limiting adp release. Proceedings of the National

Academy of Sciences 85:6314–6318.

[81] Brinker A, Pfeifer G, Kerner MJ, Naylor DJ, Hartl FU, Hayer-Hartl M (2001) Dual Function of

Protein Conﬁnement in Chaperonin-Assisted Protein Folding. Cell 107:223–233.

29 AAAB3nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TyAuSEGcnvcmQ2QczvUIIuXoR8aLgJ/kL/o2TxyWJBQNFVQ3d1X6ipCHX/XUyW9s7u3vZ/dzB4dHxSf70rGHiVAusi1jFuuVzg0pGWCdJCluJRh76Cpv+6HHmN19QGxlHNRon2A35IJKBFJysVK318gW36M7BNom3JAVYotLL/3T6sUhDjEgobkzbcxPqTrgmKRROc53UYMLFiA9wMl9vyq6s1GdBrO2LiM3VlRwPjRmHvk2GnIZm3ZuJ/3ntlIL77kRGSUoYicWgIFWMYjbryvpSoyA1toQLLe2GTAy55oLsRXK2urdedJM0SkXvpliq3hbKD8sjZOECLuEaPLiDMjxBBeogAOENPuHLeXZenXfnYxHNOMs/57AC5/sPp2qIhw==

AAAB73icbVDLTgIxFL2DL8TXqEs3jcTEFZlBEl0S3bhEI48EkHTKHWjoPGw7JGTCd7gxxo0m/om/4N9YYDaAJ2lzcs5pes/1YsGVdpxfK7exubW9k98t7O0fHB7ZxycNFSWSYZ1FIpItjyoUPMS65lpgK5ZIA09g0xvdzfzmGKXiUfikJzF2AzoIuc8Z1Ubq2fbjc9qJJQ9wcU97dtEpOXOQdeJmpAgZaj37p9OPWBJgqJmgSrVdJ9bdlErNmcBpoZMojCkb0QGm83mn5MJIfeJH0pxQk7m6lKOBUpPAM8mA6qFa9Wbif1470f5NN+VhnGgM2eIjPxFER2RWnvS5RKbFxBDKJDcTEjakkjJtVlQw1d3VouukUS65V6XyQ6VYvc2WkIczOIdLcOEaqnAPNagDgzG8wSd8WS/Wq/VufSyiOSt7cwpLsL7/ALvrj5Q= T RAAAB3nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TMQ9IQpyd9CZDZh/M9Aoh5OpFxIuCn+Qv+DdOHpckFgwUVTV0V/uJkoZc99fJbGxube9kd3N7+weHR/njk7qJUy2wJmIV66bPDSoZYY0kKWwmGnnoK2z4w/up33hBbWQcPdEowU7I+5EMpOBkpepjN19wi+4MbJ14C1KABSrd/E+7F4s0xIiE4sa0PDehzphrkkLhJNdODSZcDHkfx7P1JuzCSj0WxNq+iNhMXcrx0JhR6NtkyGlgVr2p+J/XSim47YxllKSEkZgPClLFKGbTrqwnNQpSI0u40NJuyMSAay7IXiRnq3urRddJvVT0roql6nWhfLc4QhbO4BwuwYMbKMMDVKAGAhDe4BO+nGfn1Xl3PubRjLP4cwpLcL7/AKR2iIU= R00

ATP GroES

SP SP SP

ATP ATP ATP ATP

85,000 Å3 185,000 Å3 SP micro- Less Hydrophobic Hydrophilic environment Hydrophobic

FIG. 1. Allosteric states in GroEL. The T R transition is driven by ATP, and subsequent → binding of GroES and ATP hydrolysis results in the R R00. As a result of transition from T to → R00, the volume of cavity expands from 85,000 to 185,000 Å3, and the SP experiences the change in microenvironment from hydrophobic in the T state to hydrophilic in the R00 state.

30 a

AAAB3nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TyAuSEGcnvcmQ2QczvUIIuXoR8aLgJ/kL/o2TxyWJBQNFVQ3d1X6ipCHX/XUyW9s7u3vZ/dzB4dHxSf70rGHiVAusi1jFuuVzg0pGWCdJCluJRh76Cpv+6HHmN19QGxlHNRon2A35IJKBFJysVK318gW36M7BNom3JAVYotLL/3T6sUhDjEgobkzbcxPqTrgmKRROc53UYMLFiA9wMl9vyq6s1GdBrO2LiM3VlRwPjRmHvk2GnIZm3ZuJ/3ntlIL77kRGSUoYicWgIFWMYjbryvpSoyA1toQLLe2GTAy55oLsRXK2urdedJM0SkXvpliq3hbKD8sjZOECLuEaPLiDMjxBBeogAOENPuHLeXZenXfnYxHNOMs/57AC5/sPp2qIhw==

AAAB3nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TMQ9IQpyd9CZDZh/M9Aoh5OpFxIuCn+Qv+DdOHpckFgwUVTV0V/uJkoZc99fJbGxube9kd3N7+weHR/njk7qJUy2wJmIV66bPDSoZYY0kKWwmGnnoK2z4w/up33hBbWQcPdEowU7I+5EMpOBkpepjN19wi+4MbJ14C1KABSrd/E+7F4s0xIiE4sa0PDehzphrkkLhJNdODSZcDHkfx7P1JuzCSj0WxNq+iNhMXcrx0JhR6NtkyGlgVr2p+J/XSim47YxllKSEkZgPClLFKGbTrqwnNQpSI0u40NJuyMSAay7IXiRnq3urRddJvVT0roql6nWhfLc4QhbO4BwuwYMbKMMDVKAGAhDe4BO+nGfn1Xl3PubRjLP4cwpLcL7/AKR2iIU= T R RAAAB/HicbVDLSsNAFJ34rPUVdSlCsAiCUJIq6LIoosv66AOaWibTSTt0kgkzN2IJceOvuBFxo+A3+Av+jdM2m7YemOFwzhnm3uNFnCmw7V9jbn5hcWk5t5JfXVvf2DS3tmtKxJLQKhFcyIaHFeUspFVgwGkjkhQHHqd1r38x9OuPVComwnsYRLQV4G7IfEYwaKlt7t0+JG4kWUDHd3rkAn2C5EqKy7u0bRbsoj2CNUucjBRQhkrb/HE7gsQBDYFwrFTTsSNoJVgCI5ymeTdWNMKkj7s0GQ2fWgda6li+kPqEYI3UiRwOlBoEnk4GGHpq2huK/3nNGPyzVsLCKAYakvFHfswtENawCavDJCXAB5pgIpme0CI9LDEB3Vder+5MLzpLaqWic1ws3ZwUyudZCTm0i/bRIXLQKSqja1RBVUTQC3pDn+jLeDZejXfjYxydM7I3O2gCxvcfejKVGg== 00 + GroES b }

FIG. 2. Structures of the chaperonin GroEL-GroES molecular. a. Structures of GroEL in T , R, and R00 states (PDB codes: 1OEL, 2C7E, 1AON). Apical, intermediate, and equatorial domains are colored in red, green, and blue, respectively. In R00 state, GroES is bound on top of the apical domain of GroEL ring structure. b. Football like structure of GroEL GroES complex (PDB code : − 4PKN) is the functional state that is populated in the presence of substrate proteins. A view from the bottom highlights the structure with 7-fold symmetry.

31 (b) (a)

Unfolded

ΦAAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cI9gPaUDbbTbN0dxN2N0IJ/QtePCji1T/kzX/jps1BWx8MPN6bYWZemHKmjet+O5W19Y3Nrep2bWd3b/+gfnjU0UmmCG2ThCeqF2JNOZO0bZjhtJcqikXIaTec3BV+94kqzRL5aKYpDQQeSxYxgk0hDfyYDesNt+nOgVaJV5IGlPCH9a/BKCGZoNIQjrXue25qghwrwwins9og0zTFZILHtG+pxILqIJ/fOkNnVhmhKFG2pEFz9fdEjoXWUxHaToFNrJe9QvzP62cmuglyJtPMUEkWi6KMI5Og4nE0YooSw6eWYKKYvRWRGCtMjI2nZkPwll9eJZ2LpnfZdB+uGq3bMo4qnMApnIMH19CCe/ChDQRieIZXeHOE8+K8Ox+L1opTzhzDHzifP+NjjiE=

1AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgxbKrgh6LXjxWsB/QLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8LK6tr6RnGztLW9s7tX3j9oGpVqyhpUCaXbITFMcMkallvB2olmJA4Fa4Wj26nfemLacCUf7DhhQUwGkkecEuukpn/WrQ95r1zxqt4MeJn4OalAjnqv/NXtK5rGTFoqiDEd30tskBFtORVsUuqmhiWEjsiAdRyVJGYmyGbXTvCJU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6DrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1AJReCv/jyMmmeV/2Lqnd/Wand5HEU4QiO4RR8uIIa3EEdGkDhEZ7hFd6QQi/oHX3MWwsonzmEP0CfP71pjpM= Φ −

∆AAAB+nicdVDJSgNBFHwTtxi3qMdcGoPgKUyioMcQBT1GyAaZOPT09Eya9Cx0vxFCzMFf8SLiRcGP8Bf8GyeLYFwKHhRV9ejX5cRSaDTNDyOztLyyupZdz21sbm3v5Hf3WjpKFONNFslIdRyquRQhb6JAyTux4jRwJG87g/OJ377lSosobOAw5r2A+qHwBKOYSna+YF1wiZRc3ows16W+z9XY8n0ysGsNO18sl8wpiPmLfFlFmKNu598tN2JJwENkkmrdLZsx9kZUoWCSj3NWonlM2YD6fDQ9fUwOU8klXqTSCZFM1YUcDbQeBk6aDCj29U9vIv7ldRP0znojEcYJ8pDNHvISSTAikx6IKxRnKIcpoUyJ9ELC+lRRhmlbue9f/5+0KqXycalyfVKs1uYlZKEAB3AEZTiFKlxBHZrA4B4e4QVejTvjwXgynmfRjDHf2YcFGG+fbn2TQA== G‡ k T ! B } (c) Competing Basins of Attraction

Native Basin of Attraction

generalized tertiary capture model for protein-assisted (CBP2) RNA folding. This scheme closely

FIG. 3. Kinetic partitioning mechanism (KPM) on rugged energy landscapes and chaperones. (a) Rugged folding landscape illustrating the native (NBA) and competing basins of attraction (CBA).

In the absence of chaperones, a fraction Φ of the unfolded state ensemble folds into the NBA and 1 Φ of the ensemble collapses to CBA. (b) IAM for GroEL-GroES showing the coupling between − allosteric transitions and SP folding. The ﬁgure clearly illustrates that partitioning to native state, with probability Φ, and mifolding to a metastable state, with probability (1 Φ) occurs rapidly − within the cavity during the two second life time of the encapsulated state. Although not shown explicitly, the functioning state is the symmetric complex (see Fig. 2). (c) IAM for RNA chaperone (CYT-18/CYT-19) acting on RNA. Both the processes shown in (b) and (c) are energy consuming processes associated with ATP hydrolysis.

32 a b

FIG. 4. a. Folding of T. ribozyme from its secondary structure to three dimensional native state. b. Structure of yeast analogue of CYT-19.

33 (M)

k MI

kNI

FIG.(I) 1: Model for chaperone assisted folding of Tetrahymena ribozyme. The ribozyme in(N) the intermediate (I, brown), native (N, red), misfolded (M, blue) states and RNP complexes are illustrated in the scheme. The CYT-19 is represented in green. Shaon, why don’t we add (N), (M), and (I) next FIG. 5. The modelto the for Native, a uniﬁed Misfolded, description and Intermediate. of chaperone-assisted folding. Tetrahymena ribozyme and CYT-19 (green) are employed for illustration purposes. The model shows the ribozyme in the I (brown), N (red), and M (blue) states and ribozyme-CYT-19. The GroEL-associated folding can similarly be accounted for by replacing CYT-19 with the chaperonin machinery. The ﬁgure was adapted from Ref.75.

34 GroEL Allosteric Transitions and SP Folding and Aggregation 1283 different intrinsic folding timescales and can also affect the GroEL allosteric rates (see below). Never- theless, many of the qualitative results obtained here also provide predictions or illustrations of trends and dependencies of the GroEL efficiency on allosteric transitions or spontaneous folding rates of SP. The data at eight values of GroEL concentration in Fig. 5 of Ref. 19 allow us to demonstrate that our model can reproduce the observed time-dependent Folded Rubisco (nM) yield of the native state of Rubisco and its dependence on the concentration of GroEL. In the process, we can extract the nine kinetic parameters [Eq. (12)]. Fortunately, the number of measurements reported Time (min) in Ref. 19 is large enough that an unbiased fit to the equations can be carried out. In the first testFIG. of Eq. 6. Native-state yield of Rubisco as a function of time at different GroEL concentrations. The (12), we varied, without prejudice, the nine parameters to fit all eight curves at different [EL]chaperonin con- concentrations for the curves from bottom to top are 1 nM, 2 nM, 5 nM, 10 nM, 20 centrations. Further tests of the robustnessnM, of 30our nM, 50 nM, and 100 nM. The lines are the fits to the experimental data from Ref.14 using model are described below. In assessing the validity the kinetic model. The figure was adapted from Ref.74. of the model, it is important to not only check the quality of the fits but also ensure that the values of the parameters are physically reasonable. Our kinetic model fits the experimental data extremely well (Fig. 2). We should emphasize that all the data points were globally fit using only one set of parameters, describing the kinetic reaction cycle of GroEL and intrinsic Rubisco rates. By fitting the experimental data using our kinetic model, we ob- Fig. 2. (a) Native-state yield of Rubisco as a function of tained physically reasonable reaction rates for time at different chaperonin concentrations. The points are experimental data from Ref. 19. The lines are the fits to the Rubisco (see Table 1). The intrinsic timescales data using the kinetic model. The set of parameters that associated with Rubisco are in accord with theore- provide a global fit to all eight data sets is given in Table 1. tical expectations. For example, the timescale for the The initial concentration of Rubisco is 40 nM. The − 1 fast process τF ≈kF is in the range of time expected chaperonin concentrations for the curves from bottom to 38 for a 491-residue protein. The value kS is consistent top are as follows: [EL0]=1 nM, [EL0]=2 nM, [EL0]=5 nM, with measurements that monitored spontaneous [EL0]=10 nM, [EL0]=20 nM, [EL0]=30 nM, [EL0]=50 nM, 39 folding of Rubisco in the presence of chloride ion. and [EL0]=100 nM, corresponding exactly to Ref. 19. (b) 4 Results of sensitivity analysis of the kinetic model [Eq. The ratio kF/kS ≈ 10 ≫1 is in accord with the (12)] using Rubisco parameters (Appendix B). The time- predictions of KPM. Recall that kS is an average rate for the transition from the ensemble of mis- dependent native-state yield [N](t), continuous curves, for three chaperonin concentrations, [EL0]=1 nM, [EL0]= folded structures to the native state. It is likely a 10 nM, and [EL ]=100 nM, is compared with the robust- small fraction of misfolded states that can fold more 0 4 ness measure Δ[N](t), dashed curves [Eq. (23)]. The small rapidly (i.e., 1≪kF/kS b10 ). That, under experi- values of δ[N](t)/[N](t) for all t and at various [EL] mental conditions, Rubisco folding requires the concentrations suggest that35 the allosteric model is robust chaperonin machinery implies that the aggregation to changes in the kinetic parameters around the values must compete with transition from misfolded that provide the best fit in (a). structures, especially considering that the partitioning factor Φ=0.02 is small. Indeed, the pseudo-first- order process for aggregation kA ≈λA[SP]NkS when 19 [SP]0 ≈40 nM, as in the experiments. Efficient allosteric cycle in the presence of saturating SP. Our − 1 − 1 function of GroEL also requires that the binary rate fits yield kR–Rʺ =12 min (Table 1) or kR–Rʺ =5 s, constant for [ELSP] formation exceeds the low-order which is consistent with the GroE hemicycle time. − 1 aggregation rate. Our fits show that λB =0.6 nM The load-dependent GroEL allosteric cycle time τ1 is − 1 40 min ≫λA, and hence GroEL can effectively rescue usually estimated to be 10–15 s. The value of τ1 can stringent substrates. be as small as 6 s, depending on the nature of the ligand that is bound to the trans ring (J. Gresham, J. Extracted kinetic parameters are in accord with Grason, and G.H. Lorimer, unpublished data). The chaperonin binding rate has been measured to be in experiments 7 8 − 1 − 1 9 the range λB ≈ 10 –10 M s . The binding 7 − 1 − 1 We further calibrate our model by comparing constant λB =10 s M obtained from our fits the extracted parameters for GroEL allostery with (Table 1)fallswithinthisrange.TheTT↔ TR rates that have been obtained by independent mea- transitions have been characterized extensively by surements. In particular, the transition R→Rʺ is Yifrach and Horovitz,41 who found that, for various − 1 believed to be the rate-limiting step in the GroEL GroEL mutants, kT–R ∼2000–9000 min . The value 1286 GroEL Allosteric Transitions and SP Folding and Aggregation 1286 GroEL Allosteric Transitions and SP Folding and Aggregation than that of the WT GroEL. The efficiency of SR1 thanmutants that is of linked the WT to a GroEL. decrease The in efficiencykRʺ–T (see below). of SR1 (a) mutants is linked to a decrease in kRʺ–T (see below). Efficiency of SR1 and its variants in folding EfficiencymtMDH and of CSSR1 depends and its variants on intrinsic in folding GroEL mtMDHallosteric and rates CS depends on intrinsic GroEL allosteric rates In the quest to find single-ring constructs that can substituteIn the quest for to GroEL find single-ringin vivo, Sun constructset al. used that genetic can substituteexperiments for to GroEL screenin for vivo SR1, Sun mutantset al. used from genetic which Folded mtMDH (nM) experimentsGroES can to disassociate. screen for18 SR1Two mutants of the from three which SR1 GroESmutants, can namely, disassociate. SR-D115N18 Two and of SR-A399T, the three were SR1 mutants,judged to namely, be nearly SR-D115N as efficient and as SR-A399T, GroEL18 in were that Time (min) judgedthey facilitated to be nearly folding as of efficient mtMDH as and GroEL CS18 almostin that as theywell facilitated as did GroEL. folding They of mtMDHcould also and function CS almost well asin \ wellvivo asat did 37 °C.GroEL. The They third could one, SR-T522I, also function while well con-in (b) vivosiderablyat 37 °C. more The efficient third one, than SR-T522I, SR1, was while unable con- to siderablysustain growth more efficient of E. coli than. The SR1, experiments was unable were to sustainperformed growth under of non-permissiveE. coli. The experiments conditions were such performedthat spontaneous under non-permissive folding of mtMDH conditions and CS such was thatnegligible spontaneous (see Figs. folding 5 and of 6 mtMDH of Ref. 18 and). The CS time- was negligibledependent (see changes Figs. in 5 the and activity 6 of Ref. of mtMDH,18). The which time- dependentpresumably changes mirrors in the the yield activity of theof mtMDH, folded state, which in Folded CS (nM) presumablythe presence mirrors of GroEL, the yield SR1, SR-D115N,of the folded SR-A399T, state, in theand presence SR-T522I of are GroEL, reported SR1, in SR-D115N, Fig. 5 of Ref. SR-A399T,18. and SR-T522I are reported in Fig. 5 of Ref. 18. Time (min) mtMDH mtMDH We fitted the experimental data using our kinetic model,We fittedwith the the experimental same GroEL parameters data usingFIG.(varying our7. Yield kinetic only of SPs as a function of time. (a) The data points are taken from the experiment for model,kRʺ–T forwith different the same GroEL mutants; parameters see below)(varying as only we 74 kobtainedRʺ–T for in different the analysis mutants; of Rubisco seefolding below) (Table of as 1 mtMDH.) and we The lines are the fits using the kinetic model developed in . The black line obtainedtreating in mtMDH the analysis folding of withinRubisco the (Table single-ring 1) and treatingcavity using mtMDH Eq. (8). folding In particular, within weis the for analyzed single-ring spontaneous the folding. Assisted folding in the presence of GroES and SR1 (purple), SR-T522I cavitydata in using Fig. 5 Eq. of (8).Ref. In18 particular,in two steps. we First, analyzed we used the (blue), SR-A399T (red), and SR- D115N (dark red) was used to assess the efficiencies of these three datathe data in Fig. of spontaneous5 of Ref. 18 in folding two steps. and GroEL-assisted First, we used thefolding data of from spontaneous the figure folding to determine andsingle-ring GroEL-assisted the intrinsic chaperonins relative to GroEL (green). (b) The same as (a) except the SP is CS. foldingrates of from mtMDH the figure folding. to determineWhile GroEL the allosteric intrinsic ratesrates of are mtMDH dependent folding. on the While SP and GroEL on the allosteric specific ratesexperimental are dependent conditions, on the we SP used and the on rates theλ specificB, kR–Rʺ, Fig. 3. Yield of SPs as a function of time. (a) The points k , k , and k from Table 1 for the WT GroEL. 18 experimentalT–R R–T conditions,Rʺ–T we used the rates λB, kR–Rʺ, areFig. taken 3. Yield from of the SPs experimentalas a function of measurements time. (a) The pointsfor kThus,k for mtMDH,k the intrinsic folding rates are as folding mtMDH. The lines are fits to the data using18 the T–R, R–T, and Rʺ–T from− 1 Table 1 for the− WT1 GroEL. are taken from the experimental measurements for Thus,follows: fork mtMDH,F =100 min the, intrinsickS =0.0012 folding min rates, Φ=0.002, are as foldingkinetic model. mtMDH. The The black lines line are is fits for to spontaneous the data using folding. the −−11 − 1 − 1 Assisted folding in the presence of GroES and SR1 follows:and λA =0.021kF =100 nM min min, kS =0.0012. Consistent min , withΦ=0.002, these kinetic model. The black line is for spontaneous folding. − 1 − 1 (purple), SR-T522I (blue), SR-A399T (red), and SR- rates, the values for kN and kM [kN =Φk̄R –T and kM = Assisted folding in the presence of GroES and SR1 and λA =0.021 nM min . Consistentʺ with these D115N (dark red) was used to assess the efficiencies of (1−Φ̄)kRʺ–T] for GroEL are as follows:̄ Φ̄=0.002 (purple), SR-T522I (blue), SR-A399T (red), and SR- rates, the values for−k1N and kM [kN =ΦkRʺ–T and kM = these three single-ring chaperonins relative to GroEL and k – =60 min . For SR1, we used the same D115N (dark red) was used to assess the efficiencies of (1−Φ̄)kRRʺʺ–TT] for GroEL are as follows: Φ̄=0.002 (green). (b) The same as (a) except that the folding of CS λ , k – , k – , and− 1k – values; however, k – =1/ these three single-ring chaperonins relative to GroEL andB kRRʺR–Tʺ =60T R min .R ForT SR1, we used theRʺ sameT instead of mtMDH is analyzed. The only GroEL allosteric 300 min− 1. The corresponding data and curves are (green). (b) The same as (a) except that the folding of CS λB, kR–Rʺ, kT–R, and kR–T values; however, kRʺ–T =1/ insteadrate that of is mtMDH varied in is obtaining analyzed. the The results only GroEL in (a) and allosteric (b) is shown in− 1 Fig. 3. 300 min . The corresponding data and curves are ratekRʺ–T that, while is varied all others in obtaining were taken the results from Table in (a) 1 and. (c) (b) Pre- is Second, we fitted the yield data for mtMDH for dictions for the time-dependent yield of Rubisco for shown in Fig. 3. kRʺ–T, while all others were taken from Table 1. (c) Pre- theSecond, three weSR1 fitted mutants the (SR-D115N,yield data for SR-A399T, mtMDH and for dictionsGroEL (green) for the and time-dependent the single-ring yield mutants of Rubisco SR-T522I for theSR-T522I), three SR1using mutants only one (SR-D115N, free parameter SR-A399T,each, namely, and GroEL(blue), SR-A399T (green) and (red), the SR-D115N single-ring (dark mutants red), SR-T522Iand SR1 (purple) based on chaperonin allosteric rates. The values the Rʺ→T transition rate, kRʺ–T, which reflects the (blue), SR-A399T (red), SR-D115N (dark red), and SR1 SR-T522I), using only one free parameter each, namely, of k were taken by analyzing the mtMDH data in (a). In kinetic efficiency of inter-ring communication. For (purple)Rʺ–T based on chaperonin36 allosteric rates. The values the Rʺ→T transition rate, kRʺ–T, which reflects the (a) and (b), superstoichiometric concentrations of chaper- the mutants SR-T522I, SR-A399T, and SR-D115N, of kRʺ–T were taken by analyzing the mtMDH data in (a). In kinetic efficiency of inter-ring communication.− 1 For− 1 onins were used. the fits yielded kRʺ–T =2.5 min , kRʺ–T =11 min , (a) and (b), superstoichiometric concentrations of chaper- the mutants SR-T522I,− 1 SR-A399T, and SR-D115N, and kRʺ–T =14 min , respectively.− 1 The fits are shown− 1 onins were used. the fits yielded kRʺ–T =2.5 min , kRʺ–T =11 min , in Fig. 3. While it− is1 likely that other SR1 kinetic rates and kRʺ–T =14 min , respectively. The fits are shown inwereFig. also 3. While altered it is as likely a result that of other these SR1 mutations, kinetic rates we mutants (Table 5 of Ref. 18) indicate an increased werechose also the alteredkRʺ–T transition as a result rate of these because mutations, it is most we mutantsdissociation (Table rate 5 forof Ref. the18 mutants) indicate compared an increased with closely related to the GroES dissociation rate (Table 5 SR1. A larger GroES dissociation rate would corres- chose the kRʺ–T transition rate because it is most dissociation rate for the mutants compared with closelyof Ref. related18). The to the GroES GroESKd dissociationmeasurements rate for (Table these 5 SR1.pond A to larger speeding GroES ofdissociation the Rʺ→ rateT transition would corres- rate of Ref. 18). The GroES Kd measurements for these pond to speeding of the Rʺ→T transition rate (1 ) 1  + (1 ) ….. +  2  YAAAB8HicdVDLSgMxFM3UV62vqks3wSK4ccjU1rYLoejGjVDB1pZ2GDJppg3NPEgyQhn6FW5cKOLWz3Hn35hpR1DRAxcO59zLvfe4EWdSIfRh5JaWV1bX8uuFjc2t7Z3i7l5HhrEgtE1CHoquiyXlLKBtxRSn3UhQ7Luc3rmTy9S/u6dCsjC4VdOI2j4eBcxjBCst9XpO+dw6uXbKTrGEzEa92qhUITLRHCmpokrtDFqZUgIZWk7xfTAMSezTQBGOpexbKFJ2goVihNNZYRBLGmEywSPa1zTAPpV2Mj94Bo+0MoReKHQFCs7V7xMJ9qWc+q7u9LEay99eKv7l9WPl1e2EBVGsaEAWi7yYQxXC9Hs4ZIISxaeaYCKYvhWSMRaYKJ1RQYfw9Sn8n3TKpnVqoptKqXmRxZEHB+AQHAML1EATXIEWaAMCfPAAnsCzIYxH48V4XbTmjGxmH/yA8fYJWwuPdw== =1 M 2 2

(1 ) 1 + ….. (1 )2 M1 M2

Mn = Mn 1(1 )+(1 Mn 1)(1 )

FIG. 8. Schematic of the generalized IAM of chaperone-assisted substrate folding. Depicted are the logical steps in a branching process that leads to the recursion relation for the total yield of the misfolded state after n-th annealing process (Mn), given at the bottom. Yi(= 1 Mi) and Mi − are respectively the yield of native and misfolded states from the ith iteration. Φ is the kinetic partitioning factor.

37 hkaat tal. et 50). Chakrabarti (49, well as structures ATP-inde- RNA between drive conversions Chaperones proteins pendent DEAD-box RNA certain for rearrangements, Remarks yield. state RNA Additional N native final the to in CYT-19 shifts of significant in binding result and can recognition in changes ( parameters other the with pared constant binding example, the and Interestingly, domains—for rates chaperone hydrolysis the ATP the in changed be mutations conceivably making could that by parameters ( other ribozyme the WT of (Eq. the some parameter of yield the values state of fit N best perturbations long-term to the due how changes be analyzed could we parameters CYT-19, P5a the of more report we Since fit, used. robustly are WT the S1 on compared ribozyme, concentrations native we the CYT-19 of and kinetics unfolding ATP chaperone-induced varying the of effect the assess term exponential the ing by governed (see primarily is model state three-state the of Information porting behavior kinetic overall the ( value with WT the with compared stable less showing is P5a results of experimental state with N accord the rough that in is value This ference, of values fitted “ eigenvalue trend in the second fits only the the of from dependence obtained CYT-19 ters the is curve | The and 30). native ref. of CYT-of mixture inactivate a to introduced for ( is obtained K 19. proteinase is when green ribozymes WT in misfolded curve The CYT- (pink). of 1 CYT-19 concentrations (brown), in various CYT-19 for (given no concentration ATP WT 19: mM the 2 in for ribozyme parameters WT plots best are the curves mine the while data, of experimental represent Circles ribozyme. 4. Fig. t1m T sn h aaeesfrtePavratin variant P5a the for parameters the using 80 ATP to mM 40 1 roughly at is states, ribozyme M the and in N the both unwinds chaperone data. experimental the large the in given scatter satisfactory especially is faithfully agreement of The model ribozyme. kinetic folding/unfolding our CYT-19–mediated that describes indicate experimen- corresponding values their measured with tally parameters fit data best experimental the the and with curves these of agreement reasonable 5 Fig. on data with AB 2 G ial,t etteiprac fteNsaercgiinby recognition state N the of importance the test to Finally, eigen- dominant one ribozyme, the of variants three all For h ratio The Eq. eotie ulttvl iia eut fprmtr from parameters if results similar qualitatively obtained We . | | B NM bandfo u oe (see model our from obtained WT eedneof Dependence ) S2 C 2 | | ⇠ h v eso aain data of sets five The . oprbet h xeietlyosre rate observed experimentally the to comparable o 5)adAPcnetain(i.5 (Fig. concentration ATP and P5a) for nlsso Y-9mdae odn fteWT the of folding CYT-19–mediated of Analysis 2 10 | G ⇡ (see then if: hand, other the On satisfied. is the- inequality: Our the if model. [C] our of by that explained predicts fully ory is behavior contrasting E10922 because reduce ( of ATP definition intuitive the with 7 accord in is which chaperonin-induced with associated constants unfolding: rate two of ratio P limit while the taking and yield long-time framework, the of factor, mapping discrimination approximate Y an the making of by dent meaning physical The tt.I ses oso httentgi ntefato fN of fraction the in gain net the n that show N to after same state easy the is in fraction It unperturbed A state. remains states. population M native and original N the between redis- erone and RNA or protein where of the states on M act the tributes to to addition allowed in is state chaperone N con- The not (42). was previously which CYT-19, sidered chaperone, RNA the by recognition cycles reaction reach such theoretically of function yield a The as Y again. state over N and over the form, repeated expanded of is more process the whole to the it the and reverting of likelihood all regions in hydrophobic GroEL, molecules fraction, exposed remaining chaperone The the the protein. to misfolded state, bind N kineti- and the these recognize to rescue proteins rest, To trapped states. The cally metastable spontaneously. long-lived state in N trapped the reach fraction ble small a only Factor. Recognition erones, State N the and IAM Generalized increased. is concentration GroEL as Rubisco of Eq. yield native in inequality the MDH satisfies from and parameters fit best the Similarly, Rubisco Eq. of in inequality Folding of Mediated Folding Mediated edto: lead + Y k k ti ot oigthat noting worth is It . NM B N N N k k ⇤ =1 k usiuigteprmtr from parameters the Substituting h eeaie A 3)alw o h osblt fNstate N of possibility the for allows (35) IAM generalized The MI P | ( obs obs T obs ( ( ( N ((1 n n n acltdfrom calculated , ( k uprigInformation Supporting / P 2 0 , N , IM ) )=1 1,48). (11, P ftemse qainfruaindescribes formulation equation master the of ) | k !1 nrae s[]icess smeaningful. is increases” [C] as increases 2 [ .  1 ( | safnto fCT1 Fg 4 (Fig. CYT-19 of function a as T 5 Fg )i therefore is 3) (Fig.  NI ,... N cmue rmteprmtr in parameters the from (computed ( , .Tu,tetm vlto ftefato fN of fraction the of evolution time the Thus, ). k  (0 k N www.pnas.org/cgi/doi/10.1073/pnas.1712962114 )= ] and 1 NI yincreasing by , hc unie o niciiaeythe indiscriminately how quantifies which , ocnrtos hc ugssta ti osbeto possible is it that suggests which concentrations, ) obs n 1 .Tettlyedo h tt after state N the of yield total The ). , < ([ ) µ k 1 )(1 ) 5 gi into again P trtosi ie by given is iterations oteeuvln xrsini u master-equation our in expression equivalent the to MI fW ioyeo Y-9(aafo gr 1d figure from (data CYT-19 on ribozyme WT of ) ( C Y-9(e) 2 (red), CYT-19 M  µ k (1 sms estv ocagsin changes to sensitive most is Y ) k ( ilb oooial eraigfnto f[C] of function decreasing monotonically a be will NI ] ocnrto ag fCT1 and CYT-19 of range concentration M < N / IN , N A with / P [ k , k 6 ie h ag xeietluncertainty, experimental large the Given . T 1) (  NI MI 1 n ( is k k Y k side aifidby satisfied indeed is )) k k N ]) IN A ) cat )= cat ATP stedge fdsrmnto ytechap- the by discrimination of degree the is cat ATP cat ) ATP ) k ATP N o nytePavariant. P5a the only for n k uprigInformation Supporting 0 , IM ysubstituting By . n snegligible. is , aebe tsmlaeul odeter- to simultaneously fit been have , MN , 1  , ( M . N M As . N n [ + 10 n  1 C Tetrahymena i.S2 Fig. ⇡ ) n h odtosof conditions the and , k )  !1 > , k IN < ] samntnclyicesn function increasing monotonically a is ! 2 NM eed ntecaeoe( chaperone the on depends k n k k or Y Therefore, . i.S2 Fig. o details). for NM MI NI ttsand states N 1 (1 N eoe ag,tentv il can yield native the large, becomes M N 0  M N 5 eobtain we , . [ ([ ([ ( T µ . and hsepann h nraein increase the explaining thus , +(1 fteoiia noddensem- unfolded original the of ,teeyidctn that indicating thereby ), ⌧ 12 n C C k k Y-9(le,ad3 and (blue), CYT-19 M ] IM )= . IM vdnl,frGroEL, for Evidently, . ]  ] h reeeg dif- energy free The . 1 , al S1 Table , ((1 eie ATP-driven Besides ) .W loperturbed also We ). [ and , [ + + k T n k T k al S2 Table IM (1 MN IM k ]) ]) al S1 Table k Ribozymes  + cat ATP cat ATP , ) Tetrahymena  D , , k = M ie 2.6 gives  M (1 sapoiaeythe approximately is .( ). NI )(1 B (1 N ,w e htthe that see we ), Y o 5) The P5a). for ,wt parame- with ), , e k ) Tetrahymena (1 IN n N M hwta Rubisco that show o Tand WT for ⌧ A rudthe around Tetrahymena k + ieisof Kinetics ) ( IN (1 | al S1 Table 1 k k 1 ( . )  IM ihu chap- Without (see  2 IN i.S2 Fig. < sasse by assisted is , )(1 k )) and | )= N n t ie nEq. in given n obs 1 n mak- , ) noEq. into  Table eue to reduces sgvnby given is k com- ribozyme. and iterations ) Sup-  states, M B  CYT-19– To . [ 1 remain , κ varying yield, of effect The 10. FIG. was figure The CYT-19. inactivate Ref. to from added adapted is K proteinase when ribozymes variant P5a variants. misfolded P5a (triangles) CYT- misfolded different 0.5 for or folding are (circles) variant native CYT-19s P5a were of conditions Kinetics Starting (b) (black). concentrations. ATP 19 mM 2 and (pink), ATP mM 1 100 (brown), ATP no concentrations: of variant (1 P5a 19 the of folding of Analysis 9. FIG. sevi- is , C GroEL- µ k  T (where IM 3 0 = M k fthe of ). ] )+  IN ) ) and )  . + ! < k [6] [5] [8] [7] . Y 05 IN µ ... a N 0 1 )idcdkntc ftentv 5 ain ioyei MMg mM 5 in ribozyme variant P5a native the of kinetics M)-induced 7 , n onn ifrn ocnrtoso rE using GroEL of concentrations different correspond- the curves, nine of we time-evolution to addition theory, nine ing upon our the Using time fit GroES). with simultaneously and increased the (GroEL state states), system N M chaperonin the from trapped Starting of kinetically (17). yield (in reported a Rubisco was In concentration func- acid-denatured folding. a GroEL ensure as machinery of Rubisco to full of tion folding required the GroEL-assisted is the that study, GroES sense previous the and in ATP GroEL including for MDH. and substrate Rubisco gent of Folding GroEL-Mediated maintained. be still will captured model action three-state chaperone the the by of nature nonequilibrium the chap- but RNA the of on In details acting molecular erone hydrolysis. the ATP CYT-18, being of presence force nature, the in driving nonequilibrium thermodynamic is the mech- chaperone with detailed the the of of action regardless This the that anisms, CYT-19. show of to us differences concentrations allowed the different analysis maxi- explaining by on with driven been refolding refold has in not focus may our therefore efficiency, mal and CYT-19 for substrate pathways ATP-dependent the hence and dominate. There- slower, much present, predicts. doubt are no model they are pathways equi- our ATP-independent the to as while fore, out lead exactly pathways are concentrations, ATP-independent that librium The concentrations equilibrium. substrate ATP-dependent of the to only lead that demonstrate rearrangements (49) al. et Yang rear- tion, structural [protein] ATP-driven when the only than rangements rates and lower inefficient much highly at is occurs process independent ATP the However, 4 frf 0,rsetvl.TelnsaeCT1 rAPdpnec fthe of in dependence fits ATP the or from CYT-19 figure are from (data lines concentration eigenvalue The ATP second respectively. and 30), 30) ref. ref. of to of S4c S3 introduced figure is from (data K 19 proteinase when ( ribozymes CYT-19. inactivate variant P5a misfolded and 0.5 are concentrations 1 Cyt-19 pri- and (blue) variants. or P5a (circles) (triangles) native misfolded primarily marily were conditions Starting concentrations. ( 19 (black). ATP mM 2 and 100 Mg mM 5 in ribozyme ant of S1 plots are curves B the while data, mena 5. Fig. CD AB ( (green), n eefi iutnosyt eemn etfi aaees(ie in (given parameters fit best determine to simultaneously fit were .( ). ) ial,atog the although Finally, eu to h vial aa r ossetwt h direct the with consistent are data, available the of in fit given neous parameters, fit best the extracted S1 we Table dataset, the for 4 5 (Fig. (Fig. WT variant the for data of case sets (N) for active varied of cases was production for tion the varied on was probe ATP CYT-19 To and state, K. CYT-19 proteinase of of effects addition the con- by ( inactivated initial and chaperone ribozymes, different 19 (M) misfolded under primarily time, from starting of function ( a ditions: 4 as ribozyme (Figs. native probed mutant of fraction was P5a the and experiments, the WT In respectively). the the- on our and data of CYT-19 tal validity varying the using by establish by probed ory We were (30). ribozyme concentrations the ATP of WT ( the P5abc-deleted and mutant [P5a of a on trap kinetic (46). the scale resolve time viable autonomously biologically to unlikely highly be is to estimated is stacks O base of disruption neous is barrier timescale energy The free the RNA, secondary in a helix, ubiquitous conformations base-paired motif misfolded six a structure the disrupt to stabilizes example, For case pairs (46). the base In incorrect 13). of (11, inactive (at the remain 10% of to ribozymes 6% of only majority is CYT-19 the of absence the in value the ity 1), (Fig. KPM the with of accord prin- In general folding. the RNA reveal of to ciples used workhorse the been intron has ribozyme I group thermophila the in Tetrahymena activity enzymatic self-splicing of discovery of Folding CYT-19–Mediated and state: N the large Eq. ing 3. Fig. be to estimated in out experiments fraction pointed the the are var- and the than affect rates could ious differences CYT-19) These theory. of our using analyzed absence in Mg or (temperature, cited conditions concentration, results different experimental under the performed of some (see that estimates and measurements experimental safnto fnme fcycles of number of function a as µ G A T bu) 200 (blue), ATP M (10 h vrl rnsi h aaees xrce ysimulta- by extracted parameters, the in trends overall The folding the facilitate to CYT-19 of ability the analyze first We ioye h ice n netdtinlsrpeetexperimental represent triangles inverted and circles The ribozyme. Tetrahymena Y-9(1 CYT-19 ) 7 ‡   ,a ucino ubro cycles of number of function a as ), κ µ nlsso Y-9mdae odn ftePavratof variant P5a the of folding CYT-19–mediated of Analysis = 75 fteWT the of n nteyedo h tt.Soni h lto h yield, the of plot the is Shown state. N the of yield the on = 5 bu)ad1 and (blue) M 0 = ) . hrfr eed on depends therefore Tetrahymena 0t 5ka/o = stacks (=5 kcal/mol 15 to 10 eeaie A o rtisadRA hwn h feto vary- of effect the showing RNA, and proteins for IAM Generalized 1 .Tu,oc rpe noamsardcnomto,it conformation, mispaired a into trapped once Thus, s. . µ . (black). 0 i . trigfo opeeyfle N ioye,( ribozymes, (N) folded completely from starting ) rd.Tecrei re sotie o itr fnative of mixture a for obtained is green in curve The (red). M 3  A bon,and (brown), = Eq. and (red), 0 Tetrahymena µ | ⌧ frbzmsta oddrcl oteNsaewas state N the to directly fold that ribozymes of C A Tetrahymena ioye iersle ieiso w variants two of kinetics Time-resolved ribozyme. )idcdkntc trigfo h aiePavari- P5a native the from starting kinetics M)-induced ⇠ 0 2 and and S2 κ | B . PNAS 09 ⌧ bandfo u oe,wt aaeesobtained parameters with model, our from obtained µ . nteyedo h tt..Soni h lto h native the of plot the is Shown . state. N the of yield the on B o ioye ti upce htteformation the that suspected is it ribozyme, T rd,400 (red), ATP M µ oqatttvl ta ra fexperimen- of array an fit quantitatively to ieiso 5 ain odn o ifrn CYT- different for folding variant P5a of Kinetics )  e 2 D 3) while (30), B rd.Tecrei re sframxueo aieand native of mixture a for is green in curve The (red). M + = eedneof Dependence ) µ G ioye yacutn quantitatively accounting By ribozyme. ) κ tvrosAPcnetain:n T (brown), ATP no concentrations: ATP various at Tetrahymena i T bu) 200 (blue), ATP M 0 ‡ | nttl eue u hoyt tfive fit to theory our used we total, In . .  / 1 = 1(blue), 01 k ioye(34) the (43–45), ribozyme h fcec fcaeoercgiinof recognition chaperone of efficiency the , ulse nieDcme ,2017 7, December online Published B ioyeta tan aayi activ- catalytic attains that ribozyme Tetrahymena T . ioyewl otlkl ealtered, be likely most will ribozyme 0 n 38 A with bak.Tefiuewsaatdfo Ref. from adapted was figure The (black). T n 1st fdt o h P5a the for data of sets 11 and ) o varying for Eq. ioyei h rsneo Y-9 a CYT- (a) CYT-19. of presence the in ribozyme . al S1 Table b n  h aiefato ntelmtof limit the in fraction native The . i ⇥ 5b)rbzm]a elas well as ribozyme] P5abc) (Eq. S2 ⌧ = and µ sbtae 4,5) naddi- In 50). (49, [substrate] o o3ka/o e stack). per kcal/mol 3 to 2 0 T gen,1m T (pink), ATP mM 1 (green), ATP M h 1st fdt in data of sets 11 The . ⇡ . k ioyei o natural a not is ribozyme 5(green), 05 obs ii 1 1 acltdfo our from calculated ) n T concentra- ATP and , o h 5 mutant, P5a the For . µ κ Ribozymes. µ T rd,400 (red), ATP M ftePavrato CYT- on variant P5a the of values: 4)frasponta- a for (47) s A uic sastrin- a is Rubisco al S1 Table n 5 and al S1 Table  hkaat tal. et Chakrabarti O 25 κ Tetrahymena = (10 0 = 0 ic the Since . A iii C (brown), 3 Y 1 ,while ), N CYT- ) .Note ). ) (red), and 2+ ( . n were (see ) 2+ ⌧ | Eq. ion µ Tetrahy- ii B . T (green), ATP M tvrosATP various at κ E10923 ) A , Table 0 = and S2 µ M . 01 . 75 (blue), .

BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY hkaat tal. et Chakrabarti ohpoenadRAa hw nFg 7 Fig. in shown as for RNA concentration yield and chaperone steady-state protein of values of both low amount at and saturation rate reaches folding between balance be must concentrations in chaperone under the regulated. large that arbitrarily that and be conditions suggests vivo cannot also concentration This CYT-19 maximized. 7 nonequi- the is (Fig. the yield nor low rate steady-state con- very folding librium become the CYT-19 neither would that larger ribozyme indicating strongly of much yield at native CYT- the saturates of rate ( function centration folding increasing the an around is that 5 even rate concentration, Fig. folding 19 and the that ribozyme show WT (Fig. CYT-19 ribozyme the increasing the for of since yield maximized, native 7 not the is decreases yield concentration the and of Rubisco value for completely yield native in mena the results of concentration behavior opposite chaperone the increasing chaper- steady-state protein ques- mena the and S2, this and RNA yield S1 to Tables both native in answer of parameters fold- general the scenario the Using unified a ones. or the states suggests in N theory tion the Our of rate? yield and Concentration. ing absolute Annealing Chaperone the either Iterative of maximize by Regulation Yield Vivo Finite-Time in the of Maximization nonequilibrium very of a role makes the it with dynamics. that compared show contribution final Therefore, calculations the minor observed. establishing our in was yields, role some steady-state aggregation plays MDH likely visible and aggregation accord while (17) no Rubisco in where of also (36), observations are experimental scales the calculations time with the different These on in the concentrations. N for resulting and GroEL observed aggregation, M, concentrations of I, steady-state rate of different cumulative states of the three will states than the current three faster probability within the steady-state up within a set result, dynamics be a present, the As be than substrate. clearly slower the could This much aggregation nM. is though 30 it even is concentration that the GroEL demonstrates the in when be smaller magnitude will 6 of Rubisco Fig. order will nM an in rate is 50 aggregation rate than the aggregation true smaller of the the Clearly, all and state. not M limit, since upper smaller an be is is we rate), rate this aggregation aggregation that the the for that limit con- find Rubisco upper misfolded an the (setting for elaborate nM centration an 50 by using and obtained (27) aggregation framework Rubisco for constant rate of estimate nM the 0.77, 30 min using 0.39, 9.29 and are and theory 20, 6.64, our 3.58, 10, 1.86, by predicted 5, as 2, concentrations of GroEL 1, values corresponding the of The For substoichiometric. concentrations are analysis). GroEL this 6 so for Fig. used, neglected in ( be shown circulation can experiments into and back smaller Rubisco much misfolded in brings results marily the From aggregation? 6 S2 substrate Table Fig. chaper- of see consequence of a values; concentrations be different substoichiometric at (especially steady- substrate different one of the yields Could and parameters. one state least additional at many introduces which likely aggregation, of possibility any Aggregation. of Effect B neetnl,teproduct the Interestingly, .Tefligrt ( rate folding The ). ioyei i.7 Fig. in ribozyme ioye hsimdaeysget htteabsolute the that suggests immediately This ribozyme. ecluae h hprn-rvnrate chaperone-driven the calculated we , A 1n) hl ti vrtoodr fmagnitude of orders two over is it while nM), (1 ⇠ P k ( MI N 80 , µ 1 o h mletGoLcnetainused concentration GroEL smallest the for M ) .A uhhg ocnrtos however, concentrations, high such At ). oeta u he-tt oe neglects model three-state our that Note 0 spotdfrbt uic and Rubisco both for plotted is . k 001 obs A sntmxmzdete—i.4 either—Fig. maximized not is ) nM A and NE 1 0n eaue uic was Rubisco denatured nM 50 , epciey nteohrhand, other the On respectively. , 4 = 1 0 B C . | min 05 h gr ihihsthat highlights figure The . 6 2 o h 5 ain both variant P5a the for µ | min P M 1 ( N u hoypredicts theory Our . o h second-order the for , C 1 1 oeoc again once Note . and ) ochaperones Do uniyn the quantifying k D k MI MI . A o these for htpri- that Tetrahy- Tetrahy- NE and k MN sa is B gr a dpe rmRef. from adapted was figure time ( unit per Yield panels) ( (Bottom Rubisco concentration. folded the of yield annealing. iterative Steady-state by panels) yield finite-time the of Maximization 11. FIG. C b is B ), ,a hprn ocnrto.Fralo h uvs T ocnrto a e o1m.The mM. 1 to set was concentration ATP curves, the of all For concentration. chaperone a as ), ) auainrgo fFg 7 Fig. 37 of of volume is region yeast I Mss116p saturation average of group chaperone an concentration mitochondrial the Given and RNA yeast (55), (54). CYT-19 introns of yeast to II splicing similar and the efficient structurally is the of which catalyzes molecules 54), of (53, maximization 10,300 Mss116p the are suggesting results There our support also times. relevant biologically few short a in of yields range 7 the (Fig. in in maximized shown be is may to is concentrations regulated chaperone MDH be that for well shows plot result same intriguing The This protein. the and concentra- at chaperone values saturation the reaching tion, of function increasing monotonically ATP. of concentrations other for in change given not ribozyme, do results P5a qualitative mutant the for parameters S1 fit best the using in given system, Rubisco in curves ( protein ( concentration. chaperone of tions B 7. Fig. h ieycnrsigeprmna eut npoenadRNA and protein on results experimental contrasting widely the increases times. long concentration at chaperone yield native the the pro- of increasing folding GroEL-mediated where to teins, contrast however, of stark in yield Surprisingly, states, final (30). N the the decreases minutes concentration of CYT-19 matter the fold- increasing the a accelerate to experi- can CYT-19 (11), process of vitro ing addition in the days that of show scales ments time large a over Though trapped hours. kinetically few ( a ribozyme of the order of the fraction on be should for state N scale the time There- viable (56). cells the hours, eukaryotic fore, 2 free-living multiplying about fastest of the of time doubling a With vivo in Remarks under Concluding scales time the relevant maximize that to biologically idea possible conditions. in is the it yield to equilibrium native support of 7 out additional functioning Fig. provide by from results seen two 5.2 be These of can concentration As a 7-mer). the being 1 of volume 5.2 about 1 CD AB

taysaeyedo h tt fRbso( Rubisco of state N the of yield Steady-state ) SS SS

o l ftecre,teAPcnetain[]wsstt M The mM. 1 to set was [T] concentration ATP the curves, the of all For . k P (min ) AAACBHicbVBLSwMxGMzWV62vVY9egqVQD5bdKuixKIgnqdQ+oI8lm2bb0OyD5FuxLHv14l/xIuJFwbN/wX/jtt1LWwcCw8yEZMYOBFdgGL9aZmV1bX0ju5nb2t7Z3dP3DxrKDyVldeoLX7ZsopjgHqsDB8FagWTEtQVr2qPrid98ZFJx33uAccC6Lhl43OGUQCJZemFk3VStu15Uq8W4A+wJIlyMZ8TlXtyLTs34xNLzRsmYAi8TMyV5lKJq6T+dvk9Dl3lABVGqbRoBdCMigVPB4lwnVCwgdEQGLJqWiHEhkfrY8WVyPMBTdS5HXKXGrp0kXQJDtehNxP+8dgjOZTfiXhAC8+jsIScUGHw8WQT3uWQUxDghhEqe/BDTIZGEQrJbLqluLhZdJo1yyTwrle/P85WrdIQsOkLHqIhMdIEq6BZVUR1R9ILe0Cf60p61V+1d+5hFM1p65xDNQfv+AyOkl5A= P F AAAB5XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LLoxpVUah/Q1pJJM21oJjMkd4Qy9BPciLhR8Hf8Bf/GtJ1NWw8EDueccO+5fiyFQdf9ddbWNza3tnM7+d29/YPDwtFxw0SJZrzOIhnplk8Nl0LxOgqUvBVrTkNf8qY/upv6zReujYjUE45j3g3pQIlAMIpWalV7D89prTbpFYpuyZ2BrBIvI0XIUO0Vfjr9iCUhV8gkNabtuTF2U6pRMMkn+U5ieEzZiA54OttyQs6t1CdBpO1TSGbqQo6GxoxD3yZDikOz7E3F/7x2gsFNNxUqTpArNh8UJJJgRKaVSV9ozlCOLaFMC7shYUOqKUN7mLyt7i0XXSWNcsm7LJUfr4qV2+wIOTiFM7gAD66hAvdQhTowkPAGn/DlDJxX5935mEfXnOzPCSzA+f4Dm0mLcg== nti ok ehv eeoe hoeia oe ostudy to model theoretical a developed have we work, this In ial,ruhetmtso hprn ocnrtosi vivo in concentrations chaperone of estimates rough Finally, N N aiiaino h nt-ieyedb trtv neln.( annealing. iterative by yield finite-time the of Maximization C A n ioye( ribozyme and ) and µ [GroEL]( [GroEL]( AAAB9XicbVDLSsNAFJ34rPUVFdy4GSxC3ZSkCrosiuhCoYJ9QBPKZDpph04yYeZGLbGf4kbEjYK/4S/4N6ZtNm09MHA45wz33uNFgmuwrF9jYXFpeWU1t5Zf39jc2jZ3dutaxoqyGpVCqqZHNBM8ZDXgIFgzUowEnmANr3858huPTGkuwwcYRMwNSDfkPqcEUqlt7rccYM+QXCt5dTt0cdEJYnx33DYLVskaA88TOyMFlKHaNn+cjqRxwEKggmjdsq0I3IQo4FSwYd6JNYsI7ZMuS8ZbD/FRKnWwL1X6QsBjdSpHAq0HgZcmAwI9PeuNxP+8Vgz+uZvwMIqBhXQyyI8FBolHFeAOV4yCGKSEUMXTDTHtEUUopEXl09Pt2UPnSb1csk9K5fvTQuUiKyGHDtAhKiIbnaEKukFVVEMUvaA39Im+jCfj1Xg3PibRBSP7s4emYHz/AU5fkNw= AAAB9XicbVDLSsNAFJ34rPUVFdy4GSxC3ZSkCrosiuhCoYJ9QBPKZDpph04yYeZGLbGf4kbEjYK/4S/4N6ZtNm09MHA45wz33uNFgmuwrF9jYXFpeWU1t5Zf39jc2jZ3dutaxoqyGpVCqqZHNBM8ZDXgIFgzUowEnmANr3858huPTGkuwwcYRMwNSDfkPqcEUqlt7rccYM+QXCt5dTt0cdEJYnx33DYLVskaA88TOyMFlKHaNn+cjqRxwEKggmjdsq0I3IQo4FSwYd6JNYsI7ZMuS8ZbD/FRKnWwL1X6QsBjdSpHAq0HgZcmAwI9PeuNxP+8Vgz+uZvwMIqBhXQyyI8FBolHFeAOV4yCGKSEUMXTDTHtEUUopEXl09Pt2UPnSb1csk9K5fvTQuUiKyGHDtAhKiIbnaEKukFVVEMUvaA39Im+jCfj1Xg3PibRBSP7s4emYHz/AU5fkNw= M C µ eeotie sn h etfi aaeesfrteGroEL– the for parameters fit best the using obtained were m teeae1501-e rE oeue na in molecules GroEL 14-mer 1,580 are (there 75 a 3 . in PNAS µM µM C al S2 Table shrci coli Escherichia D and µ ,a ucin fcaeoecnetain The concentration. chaperone of functions as ), ) ) | M 1 D ulse nieDcme ,2017 7, December online Published h uvsin curves The . D ,teeyalwn o ihrnative higher for allowing thereby ), si h auainrgo swell. as region saturation the in is rE ocnrto nvv is vivo in concentration GroEL . 39 C Tetrahymena and with a n ioye( ribozyme and ) ⇠

∆ SS 1 SS

D k P (min ) AAACBHicbVBLSwMxGMzWV62vVY9egqVQD5bdKuixKIgnqdQ+oI8lm2bb0OyD5FuxLHv14l/xIuJFwbN/wX/jtt1LWwcCw8yEZMYOBFdgGL9aZmV1bX0ju5nb2t7Z3dP3DxrKDyVldeoLX7ZsopjgHqsDB8FagWTEtQVr2qPrid98ZFJx33uAccC6Lhl43OGUQCJZemFk3VStu15Uq8W4A+wJIlyMZ8TlXtyLTs34xNLzRsmYAi8TMyV5lKJq6T+dvk9Dl3lABVGqbRoBdCMigVPB4lwnVCwgdEQGLJqWiHEhkfrY8WVyPMBTdS5HXKXGrp0kXQJDtehNxP+8dgjOZTfiXhAC8+jsIScUGHw8WQT3uWQUxDghhEqe/BDTIZGEQrJbLqluLhZdJo1yyTwrle/P85WrdIQsOkLHqIhMdIEq6BZVUR1R9ILe0Cf60p61V+1d+5hFM1p65xDNQfv+AyOkl5A= P NE

AAAB5XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LLoxpVUah/Q1pJJM21oJjMkd4Qy9BPciLhR8Hf8Bf/GtJ1NWw8EDueccO+5fiyFQdf9ddbWNza3tnM7+d29/YPDwtFxw0SJZrzOIhnplk8Nl0LxOgqUvBVrTkNf8qY/upv6zReujYjUE45j3g3pQIlAMIpWalV7D89prTbpFYpuyZ2BrBIvI0XIUO0Vfjr9iCUhV8gkNabtuTF2U6pRMMkn+U5ieEzZiA54OttyQs6t1CdBpO1TSGbqQo6GxoxD3yZDikOz7E3F/7x2gsFNNxUqTpArNh8UJJJgRKaVSV9ozlCOLaFMC7shYUOqKUN7mLyt7i0XXSWNcsm7LJUfr4qV2+wIOTiFM7gAD66hAvdQhTowkPAGn/DlDJxX5935mEfXnOzPCSzA+f4Dm0mLcg== 0 F N N lt of Plots ) . 5 ihtefntoa unit functional the with , ⇠ = ⇡ A B k 0 1 n ioye( ribozyme and ) Tetrahymena and 0 F . µ . 5 P 1 M N µ ioyefligto folding ribozyme SS [Cyt-19]( [Cyt-19]( iflsadstays and misfolds ) AAAB9nicbVDLSsNAFJ3UV62vqAsXbgaLUBeWpArqrtiNG6GCfUATymQ6aYdOHszcSEPIr7gRcaPgZ/gL/o1pm01bDwwczjnDvfc4oeAKDONXK6ytb2xuFbdLO7t7+wf64VFbBZGkrEUDEciuQxQT3Gct4CBYN5SMeI5gHWfcmPqdFyYVD/xniENme2Toc5dTApnU1096FrAJJI0YLs271MYVy4vw40VfLxtVYwa8SsyclFGOZl//sQYBjTzmAxVEqZ5phGAnRAKngqUlK1IsJHRMhiyZrZ3i80waYDeQ2fMBz9SFHPGUij0nS3oERmrZm4r/eb0I3Fs74X4YAfPpfJAbCQwBnnaAB1wyCiLOCKGSZxtiOiKSUMiaKmWnm8uHrpJ2rWpeVWtP1+X6fV5CEZ2iM1RBJrpBdfSAmqiFKErRG/pEX9pEe9XetY95tKDlf47RArTvP402kPQ= AAAB9nicbVDLSsNAFJ3UV62vqAsXbgaLUBeWpArqrtiNG6GCfUATymQ6aYdOHszcSEPIr7gRcaPgZ/gL/o1pm01bDwwczjnDvfc4oeAKDONXK6ytb2xuFbdLO7t7+wf64VFbBZGkrEUDEciuQxQT3Gct4CBYN5SMeI5gHWfcmPqdFyYVD/xniENme2Toc5dTApnU1096FrAJJI0YLs271MYVy4vw40VfLxtVYwa8SsyclFGOZl//sQYBjTzmAxVEqZ5phGAnRAKngqUlK1IsJHRMhiyZrZ3i80waYDeQ2fMBz9SFHPGUij0nS3oERmrZm4r/eb0I3Fs74X4YAfPpfJAbCQwBnnaAB1wyCiLOCKGSZxtiOiKSUMiaKmWnm8uHrpJ2rWpeVWtP1+X6fV5CEZ2iM1RBJrpBdfSAmqiFKErRG/pEX9pEe9XetY95tKDlf47RArTvP402kPQ= D µ M b uhthat such M o ohteRNA the both for o uic ( Rubisco for aebe produced been have NE ,a ucino chaperone of function a as ), a hc si the in is which , C rE and GroEL = b | and 2 µM µM | P r some are B ( ,a func- as ), N i.S2 Fig. i.S1 Fig. | a , ) ) n ribozyme and ) b 1 E10925 A µ Table y-9 (Top Cyt-19. for ) NE and m NE 3 . , .

BIOPHYSICS AND PNAS PLUS COMPUTATIONAL BIOLOGY