Cochaperones enable Hsp70 to fold proteins like a Maxwell’s demon

The heat shock protein 70 (Hsp70) chaperones, vital to the proper folding of proteins inside cells, consume ATP and require cochaperones in assisting protein folding. It is unclear whether Hsp70 can utilize the free energy from ATP hydrolysis to fold a protein into a native state that is thermodynamically unstable in the chaperone-free equilibrium. Here we present a model of Hsp70-mediated protein folding, which predicts that Hsp70, as a result of differential stimulation of ATP hydrolysis by its Hsp40 cochaperone, dissociates faster from a substrate in fold-competent conformations than from one in misfolding-prone conformations, thus elevating the native concentration above and suppressing the misfolded concentration below their respective equilibrium values. Previous models would not make or imply these predictions, which are experimentally testable. Our model quantitatively reproduces experimental refolding kinetics, predicts how modulations of the Hsp70/Hsp40 chaperone system affect protein folding, and suggests new approaches to regulating cellular protein quality.


Introduction
The discovery of chaperones and their roles in assisting protein folding amended the long-held view that proteins spontaneously fold into their native structures [1][2][3] . Large, multi-domain proteins may take many hours to fold, or fail to fold properly altogether on their own 2,4 . ATPconsuming chaperones-including Hsp70s-provide critical assistance in the in vivo folding and the biological functions of broad sets of substrate proteins 3 . Extensive experimental studies have firmly established that the Hsp70 chaperones can greatly accelerate the folding of their substrate proteins 5,6 . Despite tremendous progress in the mechanistic studies of the Hsp70 chaperones 2,7 , including the development of theoretical models [8][9][10] , it remains unclear why ATP consumption is indispensable to these chaperones-enzymes do not need to consume free energy in catalyzing chemical reactions-and whether these chaperones can utilize the free energy from ATP hydrolysis to drive a protein to fold into a native state that is thermodynamically less stable than other conformational states or an aggregated state. In addition, Hsp70s require Hsp40-also known as J proteins 11 -cochaperones in assisting protein folding. It is yet unexplained why cochaperones are absolutely necessary.
The Hsp70 chaperones, such as the bacterial DnaK, adopt an open conformation in the ATPbound state (ATP-state), which allows the substrate to bind and unbind at high rates, whereas in from ATP hydrolysis to drive its substrate toward the native state such that fN / fM > fN,eq / fM,eq, where fS is the fraction of the substrate in state S at the steady state of Hsp70-mediated folding, and fS,eq is the corresponding fraction at the folding equilibrium in the absence of the chaperone.
Previous models 9,28 mostly considered the chaperone as an unfoldase/holdase-which need not consume free energy-that pulls the substrate out of the misfolded state and holds it in an unfolded state. It was proposed that the free energy from ATP hydrolysis was used to achieve ultra-affinity in substrate binding 8 . As an unfoldase/holdase, Hsp70 would also pull the substrate out of the native state into the unfolded state; unless Hsp70 has a higher affinity for the native substrate than for the misfolded substrate, these models would predict fN / fM ≤ fN,eq / fM,eq.
Here we propose a model of Hsp70-mediated protein folding, in which Hsp70 and Hsp40 together constitute a molecular machine that uses the free energy from ATP hydrolysis to actively drive a protein toward its native state, so that fN / fM > fN,eq / fM,eq. It suggests that without Hsp40, Hsp70 alone cannot change the ratio fN / fM from the equilibrium value fN,eq / fM,eq. Our model thus answers the question why Hsp70 requires both the Hsp40 cochaperones and ATP consumption in assisting protein folding. Our model explains the puzzling non-monotonic dependency of folding efficiency on the chaperone and cochaperone concentrations. It makes quantitative predictions on how protein folding is affected by modulations of the chaperone environment, including changes in the ATPase activity or the nucleotide exchange rate of Hsp70. These predictions may be readily tested by experiments, and inform rational approaches to manipulating chaperone-mediated protein folding.

Results
In our model (detailed in Methods), we consider two additional conformational states-besides the M and N states-of a protein: the unfolded and aggregation-prone state, U, and the foldcompetent state, F. A protein in the F state is unfolded but poised to fold into the native state ( Fig. 1a, b). Such intermediate states of folding have been observed experimentally 4 .
Conformational transitions can occur between M and U, between U and F, and between F and N ( Fig. 1b). We assume that a protein in the U state has more exposed hydrophobic sites than in the F state-which is consistent with the experimental observations 4 -and a protein in the M and N states has nearly zero such sites, as both folding and aggregation (including oligomerization) bury the protein's hydrophobic sites. Key to our model is the assumption that Hsp40, like Hsp70, has different affinities for the substrate in different conformations 19 , favoring conformations with more exposed hydrophobic sites. Thus Hsp70 and Hsp40 can bind to a substrate molecule in the U and F states-with higher affinities for the U state than for the F state-but not to one in the M and N states (Fig.   1b). As a result, an Hsp70 molecule bound to a substrate molecule in the U state will on average have substantially higher ATP hydrolysis rate-because of the more probable cis stimulation by an Hsp40 molecule bound to the same substrate molecule-than if it is bound to a substrate molecule in the F state. If the nucleotide exchange rate is between these two hydrolysis rates, an Hsp70 bound to a substrate in the U state will be driven toward the ADP-state, where it slowly dissociates from the substrate, while an Hsp70 bound to a substrate in the F state will be driven toward the ATP-state, where it rapidly dissociates from the substrate. Acting like a Maxwell's demon 29 , Hsp70 releases the fold-competent substrate but clasps the aggregation-prone substrate, driving the folding along the reaction path of M → U → U · C · ATP → U · C · ADP → F · C · ADP → F · C · ATP → F → N (S · C · X represents the complex between a substrate in conformation S and the chaperone C bound to nucleotide X = ATP, ADP) (Fig. 1b). One ATP molecule is consumed in this reaction path and the free energy is used to compel the substrate into the native state.
The extent to which Hsp70 biases protein folding can be quantified by the excess free energy: ΔΔG ≡ R T (ln(fN / fM) − ln(fN,eq / fM,eq)), where R is the gas constant and T the temperature. A positive excess free energy requires not that more chaperones bind to the substrate in the U state than to the substrate in the F state, which is true and reflected in previous models, but that an individual chaperone molecule, when bound to a substrate, resides longer on it if the substrate is in the U state than if it is in the F state. For this, Hsp70 needs both ATP consumption and a cochaperone: it can be shown algebraically (see Methods) and numerically (Fig. 1c) that without cochaperones, ΔΔG = 0. We applied our model to the analysis of DnaK/DnaJ/GrpE-mediated refolding of luciferase 5 and its variant LucDHis6 28 . Most of the relevant kinetic parameters for this bacterial Hsp70 system have been carefully determined experimentally 30 (Table 1). Our model quantitatively reproduces the experimentally observed refolding kinetics under various conditions, capturing the slow spontaneous refolding and denaturation of luciferase, the acceleration of refolding with chaperone assistance, and the necessity of GrpE (Fig. 2a, b). The refolding speed and yield reach a maximum at the DnaK concentration of 1M, which is captured by our model (Fig. 2b, c) 28 , our model suggests that the Hsp70-accelerated refolding proceeds in two steps: 1) rapid unfolding of the misfolded substrate, stabilized by the ADP-bound DnaK, followed by 2) slow conversion to native state (Fig. 3a).
At the steady state, the reactive flux along the ATP-driven cycle U → U · C · ATP → U · C · ADP → F · C · ADP → F · C · ATP → F (→ U) (Fig. 3b) keeps the protein folding out of equilibrium, elevating the native population above and suppressing the misfolded population below their respective equilibrium values (Fig. 2c). The excess free energy at the steady state always increases with increasing DnaK concentrations, but the native population reaches a maximum and then decreases (Fig. 2c), because at high DnaK concentrations, the substrate is trapped in the DnaK-bound state and thus prevented from folding into the native state. We used our model to estimate the ATP consumption in the DnaK/DnaJ/GrpE-mediated folding of LucDHis6 (Fig. 4). In the initial minutes of refolding, approximately 150 ATP molecules are consumed to refold one LucDHis6 (Fig. 4a), which is reasonably close to the experimental result of ~50 ATP molecules consumed per refolded LucDHis6 when the stoichiometry of DnaK:LucDHis6 is 1:1, significantly higher than the experimental number of ~5 when LucDHis6 is in excess of DnaK, and significantly lower than the estimates of >1000 for many other substrates in other experiments 28,[31][32][33] . The discrepancy between the model and the experimental results may be partially attributable to the approximations in our model and the inaccuracies in the input kinetic parameters. ATP hydrolysis continues at the steady state and the free energy is utilized to promote the native state and suppress the misfolded state (Fig. 4b, c).
As [DnaK] exceeds 1 µM, the ATP consumption rate increases rapidly without commensurate increase in the excess free energy. Our analysis thus suggests that DnaK may be most free energy efficient at maintaining protein folding out-of-equilibrium when its concentration is in the sub-micromolar range, a prediction that may be tested experimentally.
Our model suggests that Hsp70 can keep a protein folded even if it thermodynamically favors aggregation. The chaperone is thus able to play a critical role in maintaining protein conformations, not just in the folding of nascent chains 34 . Higher DnaK concentrations are required to suppress aggregation at increasing substrate concentrations ( Fig. 5a) or at decreasing substrate stabilities (Fig. 5b). This may explain how cells that overexpress DnaK can tolerate higher numbers of mutations in the chaperone's substrates 35 . Because the excess free energy plateaus at high chaperone concentrations (Fig. 2c), our results imply a limit on the chaperones' capacity to prevent aggregation, in that there exists a threshold of aggregation tendency ( Fig. 5a, b, the black arrows) above which the chaperone can no longer maintain high levels of native concentrations and prevent aggregation at the same time.
Our model suggests that Hsp70 only drives the folding of proteins with sufficiently slow conversion between U and F states (Fig. 5c, d, e), implying that Hsp70 substrates tend to be slow folding proteins (Fig. 5d). If the conversion between U and F is too fast, the chaperone diminishes, rather than increases, the native fraction in comparison to the chaperone-free equilibrium. As the conversion slows, the chaperone drives the steady state native fraction higher, but at the price of longer refolding time (Fig. 5e), a trade-off reminiscent of that between speed and specificity in the kinetic proofreading mechanism 36,37 , where the expenditure of free energy (such as from ATP or GTP consumptions) is utilized to increase the specificity of chemical reactions.
Our model explains the observation that folding is less efficient at both low and high DnaJ concentrations 14 (Fig. 6a). At low DnaJ concentrations, ATP hydrolysis is slow, and nucleotide exchange drives DnaK toward the ATP-state, in which it dissociates from the substrate rapidly and thus unable to prevent aggregation. At high DnaJ concentrations, a large fraction of the substrate in the U state is bound to DnaJ. These DnaJ-bound substrate molecules are trapped in the U state, unable to progress toward the F state, resulting in diminished folding.
Our model also explains the observation that folding decreases at both low and high GrpE concentrations 26 (Fig. 6b). For the chaperone to effectively assist folding, nucleotide exchange should be much slower than ATP hydrolysis when the chaperone binds to a substrate in the U state, but much faster than ATP hydrolysis when it binds to a substrate in the F state, so that the chaperone is driven toward the ADP-state in the former case, and toward the ATP-state in the latter case ( Fig. 1b). At low GrpE concentrations, nucleotide exchange is slow, leaving DnaK bound to the substrate in the F state predominantly in the ADP-state-as reflected by the low population of F · C · ATP (Fig. 6b), slowing its dissociation from the substrate and thus preventing the latter from folding to the native state. At high GrpE concentrations, nucleotide exchange is fast, and DnaK is driven into the ATP-state and does not stay bound to the substrate in the U state long enough-as reflected by the decreasing population of U · C · ADP (Fig. 6b)-to prevent the substrate from aggregation. To maximize substrate folding, higher nucleotide exchange rate should accompany higher stimulated ATP hydrolysis rate (Fig. 6c, d, e).
Our model predicts that Hsp70 chaperones with higher Hsp40-stimulated ATP hydrolysis rates can drive substrate folding to higher native fractions (Fig. 6e), at the cost of higher free energy expenditure (Fig. 6f). This result explains a previous experimental observation that a small molecule that enhances ATP hydrolysis by Hsp40-bound Hsp70 can induce higher yields of substrate folding 38 . Modulation of the ATP hydrolysis or the nucleotide exchange rates by small molecules may represent a therapeutic opportunity in the treatment of misfolding-or proteostasis-related diseases 39 .

Discussion
Our model makes two distinct predictions that subject it to future experimental tests and possible falsification. First, it predicts that some thermodynamically unstable substrates depend on continuous Hsp70 assistance to maintain their native structures, and such a substrate in the steady state of Hsp70-mediated folding will gradually lose its native structure if ATP is depleted, for example, by the addition of apyrase. This prediction should be readily tested experimentally.
Second, it predicts that an Hsp70 molecule bound to a substrate molecule will dissociate faster if subsequently the substrate molecule folds into the native state than if the substrate molecule misfolds, and such a difference will vanish in the absence of Hsp40. This prediction may be tested by single molecule experiments 22,40 , if, for instance, separate fluorescence signals are available to detect Hsp70-substrate binding and substrate folding.

Methods
Model of Hsp70-mediated protein folding. We denote Hsp70 as C, Hsp40 (J protein) as J, and the NEF as E.
[Y] denotes the solution concentration of the molecular species Y. There are four types of reactions explicitly considered in our model (Fig. 1b): 1) Hsp70 binding to the substrate.
2) Conformational transitions of the substrate. An Hsp70-free substrate can adopt any of the four conformational states The chaperone-bound substrate can only be in and transition between the U and F states The details of the kinetic rates of the above reactions are described below. Conformational transitions of the substrate. The transition rates between conformations S and S' are different between a chaperone-free substrate ( → ′ ) and a chaperone-bound substrate 1b). The condition of thermodynamic cycle closure dictates that We take the rate of aggregation to be proportional to the substrate concentration: The rates → †

Author Contributions
HX designed and performed the research and wrote the paper.     Table 2, and the conditions of the experiments considered in this paper are summarized in Table 3.  Here, the DnaK concentration is 0.5 M, and other kinetic parameters are given in Table 1 and     b. Substrate-free DnaK has a basal ATP hydrolysis rate of 0.001 s -1 , but the substrate can further accelerate ATP hydrolysis by up to 9-fold 14,46 . We take the ATP hydrolysis rate of substrate-bound DnaK to be 10-fold higher than the basal rate. The predictions of our model are insensitive to this parameter.

Figures and Tables
c. This is the experimental rate for the temperature T = 25 °C. For the higher temperature T = 30 °C, we used an arbitrary but reasonable 5.5-fold higher value of 10 s -1 because we could not find any reported experimental value for this temperature.
d. The kinetic rates for GrpE binding to DnaK were not determined, but the steady state ratio in the two-step dissociation reaction, • + ⇌ • • ⇌ • + , ′ = ( , • + , )/ , • = 20 μM was determined. We chose an arbitrary diffusion limited association rate for GrpE binding to DnaK in our calculations.
e. We note that (0) and appear too large to be physically meaningful; they should instead be taken simply as numerical parameters that yield an excellent fit of the Arrhenius equation to the experimental data.  a. The predictions of our model depend on the equilibrium constant ⇌ = → / → , but are degenerate with respect to the individual rates. We thus fix an arbitrary but plausible value for → and fit → to the experimental refolding data.
b. For the similar reason as above, we fix → and fit → to the refolding data.
c. There should be more accessible DnaK binding sites in the U state than in the F state.
Here, we arbitrarily set the ratio between the two. Although the values of the other fitting parameters will change accordingly, the quality of the fit and the predictions of our model are insensitive to this ratio (at least for values between 1 and 100).