Efficient conversion of chemical energy into mechanical work by Hsp70 chaperones
Salvatore Assenza, Alberto S. Sassi, Ruth Kellner, Ben Schuler, Paolo De Los Rios, Alessandro Barducci
EE ffi cient conversion of chemical energy into mechanicalwork by Hsp70 chaperones S. Assenza a,b,i , A. S. Sassi c,d,i , R. Kellner e , B. Schuler e,f , P. De Los Rios c,g , A.Barducci h a Laboratory of Food and Soft Materials, ETH Z¨urich, CH-8092 Z¨urich, Switzerland b Departamento de F´ısica Te´orica de la Materia Condensada, Universidad Aut´onoma de Madrid, E-28049Madrid, Spain c Institute of Physics, School of Basic Sciences, Ecole Polytechnique F´ed´erale de Lausanne (EPFL),CH-1015 Lausanne, Switzerland d IBM T. J. Watson Research Center, Yorktown Heights, New York, United States of America e Department of Biochemistry, University of Zurich, CH-8057 Zurich, Switzerland f Department of Physics, University of Zurich, CH-8057 Zurich, Switzerland g Institute of Bioengineering, School of Life Sciences, Ecole Polytechnique F´ed´erale de Lausanne (EPFL),CH-1015 Lausanne, Switzerland h Centre de Biochimie Structurale (CBS), INSERM, CNRS, Universit´e de Montpellier, Montpellier, France i These two authors contributed equally
Abstract
Hsp70 molecular chaperones are abundant ATP-dependent nanomachines that activelyreshape non-native, misfolded proteins and assist a wide variety of essential cellularprocesses. Here we combine complementary computational / theoretical approaches toelucidate the structural and thermodynamic details of the chaperone-induced expansionof a substrate protein, with a particular emphasis on the critical role played by ATPhydrolysis. We first determine the conformational free-energy cost of the substrateexpansion due to the binding of multiple chaperones using coarse-grained molecularsimulations. We then exploit this result to implement a non-equilibrium rate modelwhich estimates the degree of expansion as a function of the free energy provided byATP hydrolysis. Our results are in quantitative agreement with recent single-moleculeFRET experiments and highlight the stark non-equilibrium nature of the process, show-ing that Hsp70s are optimized to convert e ff ectively chemical energy into mechanicalwork close to physiological conditions. Keywords: molecular chaperones, Hsp70, protein folding, non equilibriumthermodynamics
1. Introduction
Even though in vitro most proteins can reach their native structure spontaneously[1],this is not always the case in cellular conditions and proteins can populate misfolded
Email address: [email protected] (A. Barducci) a r X i v : . [ q - b i o . B M ] M a r tates which can form cytotoxic aggregates[2]. In order to counteract misfolding andaggregation, cells employ specialized proteins, called molecular chaperones , which acton non-native protein substrates by processes that stringently depend on ATP hydroly-sis for most chaperone families[4, 5]. Among them, the ubiquitous 70 kDa heat shockproteins (Hsp70s) play a special role because they assist a plethora of fundamentalcellular processes beyond prevention of aggregation.Hsp70s consist of two domains [4, 8]. The substrate binding domain (SBD) inter-acts with disparate substrate proteins, whereas the nucleotide binding domain (NBD) isresponsible for the binding and hydrolysis of ATP. The two domains are allostericallycoupled, and the nature of the nucleotide bound to the NBD a ff ects the structure of theSBD and as a consequence the a ffi nity for the substrate and its association / dissociationrates. More precisely, the chaperone in the ATP-bound state is characterized by bindingand unbinding rates that are orders of magnitude larger than those measured when ADPis bound [9]. Furthermore the coupling is bidirectional: the substrate, together with aco-localized J-domain protein (JDP) that serves as cochaperone, greatly accelerates thehydrolysis of ATP. Substrate binding thus benefits from the fast association rate of theATP-bound state and the slow dissociation rate of the ADP-bound state, resulting in anon-equilibrium a ffi nity ( ultra-a ffi nity ) that can be enhanced beyond the limits imposedby thermodynamic equilibrium [10].Several lines of evidence suggest that the binding of Hsp70s to a polypeptide in-duces its expansion. Nuclear Magnetic Resonance (NMR) measurements have shownthat Hsp70s destabilize the tertiary structure of several di ff erent substrates [11, 12].Biochemical assays revealed that binding of Hsp70 increases the sensitivity of mis-folded Luciferase to proteolysis and decreases its propensity to bind Thioflavin-T,strongly suggesting a loss of compactness [13]. Moreover, a single-molecule studybased on F¨orster resonance energy transfer (FRET) spectroscopy quantified the con-siderable expansion of unfolded rhodanese in native conditions upon binding of mul-tiple Hsp70 chaperones [16]. In particular, this study revealed that the expansion isstringently ATP-dependent, because upon ATP exhaustion the system relaxes to theexpansion values observed in the absence of chaperones [16]. Building on this re-sult, we elucidate here the coupling between the expansion of the substrate and theexternal energy source provided by ATP hydrolysis. To this aim, we first explore thestructural and energetic features of Hsp70-bound rhodanese using Molecular Dynam-ics (MD) simulations. We next integrate this molecular information into a rate modelthat explicitly includes the Hsp70-rhodanese interactions and the chaperone ATPasecycle, thus clarifying the role played by energy consumption in the expansion of thesubstrate.
2. Results
To characterize the main features of chaperone-induced expansion, we performedMD simulations of the Hsp70 / rhodanese complexes. We relied on a one-bead-per-residue Coarse Grained (CG) force field [15], which has been tailored to match ex-perimental FRET data of intrinsically-disordered proteins and satisfactorily reproduce2 o t en t i a l E ne r g y ( kc a l / m o l ) P o t en t i a l E ne r g y ( kc a l / m o l ) P o t en t i a l E ne r g y ( kc a l / m o l ) Figure 1: Probability density maps of substrate potential energy and radius of gyration for representativeHsp70 / rhodanese complexes with one (left), three (center) and six (right) bound chaperones. The di ff erentHsp70 chaperones have been represented with di ff erent colors to ease their discernibility. the compactness of unfolded rhodanese in native conditions without any further tun-ing (see SI). In particular, we focused on hydrophobic and excluded volume interac-tions while neglecting the electrostatic contribution which is negligible in rhodaneseand plays only a minor role in Hsp70 / rhodanese complexes according to FRET ex-periments (see SI). Hsp70 chaperones were modeled with a structure-based potentialbuilt upon the ADP-bound conformation, and restrained on binding sites on the sub-strate. We identified six binding sites on the rhodanese sequence using two distinctbioinformatic algorithms[17, 18]. Considering that each binding site could be eitherfree or bound to a Hsp70 protein, we thus took into account a total of 2 =
64 distinctchaperone / substrate complexes, which were exhaustively simulated. In Fig.1 we reportthe distributions of the substrate potential energy and of the radius of gyration ( R g )for three representative complexes with one (left), three (center) and six (right) boundchaperones. As previously noticed[16], chaperone binding leads to larger radii of gy-ration and higher potential energies, implying that the excluded volume interactionsdue to the large Hsp70s progressively expand the complex and disrupt the attractiveintra-chain interactions in rhodanese.We then calculated the conformational free energy of all the possible chaperone / rhodanesecomplexes to obtain a quantitative picture of the energy landscape governing the chaperone-induced expansion. To this aim, we performed extensive sets of non-equilibrium steer-ing MD trajectories for each complex, and measured the work needed to steer it to acompletely extended reference structure ( R gyr > ff ected by chaperone binding. Equilibrium free-energy di ff erences withrespect to this reference state were then estimated from non-equilibrium work distribu-tions via the Jarzynski equality[36], thus allowing the determination of the conforma-tional free-energy ∆ G of each distinct chaperone / substrate complex.In Fig.2 (main) we report ∆ G for each complex as a function of its mean radius ofgyration using di ff erent colors for di ff erent stoichiometries. The conformational freeenergy increased with the swelling of the substrate due to the progressive binding ofthe chaperones. The increase in substrate potential energy due to the loss of intra-chaininteractions upon Hsp70 binding is therefore only marginally compensated by the gainin conformational entropy. Notably, the conformational free-energy is not uniquely3etermined by the stoichiometry, and is significantly a ff ected by the specific bindingpattern. The conformational free-energy cost ∆∆ G of adding a single chaperone (insetin Fig.2) is positive for all complexes, but it varies from 2 kcal / mol up to 7 kcal / moldepending on the stoichiometry of the complex and on the particular choice of thebinding sites. The increase of ∆ G as a function of R g is quantitatively captured bySanchez theory ([31] and SI) for the coil-to-globule collapse transition in polymers (seeFig.2). Remarkably, the excellent agreement is not the outcome of a fitting proceduresince all the parameters were extracted from experiments (see Methods). This resultfurther reinforces the reliability of our simulations as well as the general applicabilityof the present setup beyond the particular system considered in this work.
20 40 60 80 100 R g ( Å )05101520 G ( k c a l / m o l ) n=0n=1n=2n=3n=4n=5n=6 2 3 4 5 6 7 G ( kcal / mol )010203040 Figure 2: Free energy ∆ G as a function of the radius of gyration R g for all the 64 possible binding config-urations. Di ff erent colors represent di ff erent numbers of chaperones bound. The black curve was obtainedusing the model in [31] (see SI). (inset) Histogram of the free energy cost ∆∆ G of the binding of an additionalchaperone. The structural and thermodynamic characterization obtained by molecular simula-tions can be profitably complemented by a kinetic model encompassing relevant bio-chemical processes in order to determine the probability of each chaperone / substratecomplex as a function of the chemical conditions. Notably, a model of the Hsp70biochemical cycle based on experimental rates was previously used to illustrate how4TP-hydrolysis may result into non-equilibrium ultra-a ffi nity for peptide substrates[10]. Here we extend this result to the more complex case of Hsp70-induced expansionby taking into account multiple chaperone binding events and their consequences onthe conformational free energy of the substrate. k off ATP k off ATP k off ADP k on ADP ∙ [Hsp70 ∙ADP ] k on ATP ∙ [Hsp70 ∙ATP ] k on ATP ∙ [Hsp70 ∙ATP ] k + k h s ex,TD eff k + k s s ex,DT eff Figure 3: Portion of the biochemical cycle. Each binding site of rhodanese (filled in black) can be eitherfree or occupied by an Hsp70 (yellow), which in turn can be either ADP- or ATP-bound. The rate of eachreaction is highlighted, as detailed in the SI. The binding constant, k ATPon and k ADPon , are computed accordingto (1).
In our model each state corresponds to a single configuration of the chaperone / substratecomplex, which is defined by the occupation state of the six Hsp70 binding sites on rho-danese. Each site can be either free or occupied by an ADP- or ATP-bound chaperonefor a total of 3 =
729 di ff erent states. All the relevant molecular processes corre-sponding to transitions between these states are explicitly modeled, including chaper-one binding / unbinding, nucleotide exchange and ATP hydrolysis (see Fig.3). We tookadvantage of available biochemical data for determining the rate constants associatedto all the relevant reactions (see methods and SI). Importantly, kinetic rates for Hsp70binding were modulated by the conformational free-energies determined by CG MDsimulations. Indeed, the unbinding rates of Hsp70 from large-sized protein substrateswere observed to be similar to the ones from small peptides, whereas the binding ratescan be up to two orders of magnitude smaller[16, 9, 26]. This evidence was further cor-roborated by a recent NMR study[12] suggesting a conformational selection scenariowhere the energetic cost due to substrate expansion mostly a ff ects the Hsp70 / rhodanesebinding rate. Following the experiments, we thus considered a substrate-independentunbinding rate constant k o f f , while we expressed the binding rate constant as k on , i j = k on exp[ − β ∆∆ G i j ] , (1)where β = / k b T , k on is the binding rate measured for a peptide substrate, and ∆∆ G i j isthe conformational free-energy cost of Hsp70 binding, which depends on the specificinitial and final binding patterns i and j in the rhodanese / chaperone complex (see Fig.2,5nset). The interactions with JDP cochaperones were not explicitly modeled but thecochaperones were assumed to be colocalized with the substrate, so that their e ff ect wasimplicitly taken into account in the choice of the rate constants for the ATP hydrolysis[34, 14].The analytical solution of the model provides the steady-state probability of eachbinding configuration and allows the exploration of their dependence on the biochemi-cal parameters. It is particularly instructive to investigate the system behavior as a func-tion of the ratio between the concentration of ATP and ADP, which is intimately con-nected to the energy released by ATP hydrolysis. At thermodynamic equilibrium, the[ AT P ] / [ ADP ] ratio is greatly tilted in favor of ADP ([
AT P ] eq / [ ADP ] eq ’ − − − ,[21]) whereas in the cell ATP is maintained in excess over ADP by energy-consumingchemostats ([ AT P ] / [ ADP ] >
1, [22]). The [
AT P ] / [ ADP ] ratio hence determines howfar the system is from equilibrium, thus representing a natural control parameter forthe non-equilibrium biochemical cycle. We thus report in Fig.4 the compound prob-abilities for complexes with the same stoichiometry n as a function of this nucleotideratio. In conditions close to equilibrium (very low values of [ AT P ] / [ ADP ]), the vastmajority of the substrate proteins are free and only about 10% of them are bound to asingle chaperone. The population of equimolar complexes increases for [
AT P ] / [ ADP ]between 10 − and 10 − and gives way to larger complexes with multiple chaperonesfor higher values of the nucleotide ratio.For [ AT P ] / [ ADP ] >
1, most substrates are bound to at least 4 chaperones, with anaverage stoichiometry h n i ∼ .
9. Further increase of the nucleotide ratio does not sig-nificantly change this scenario indicating an almost constant behaviour in large excessof ATP ([
AT P ] / [ ADP ] > Combining the steady-state probabilities derived from the rate model with the re-sults of the MD simulations, we can now exhaustively characterize the structural prop-erties of the system. This provides the opportunity to directly compare our model withthe results from FRET experiments both in equilibrium and non-equilibrium condi-tions. To this aim, we first focused on the average radius of gyration of the system atthermodynamic equilibrium ([
AT P ] (cid:28) [ ADP ]) or in non-equilibrium conditions withATP in large excess over ADP ([
AT P ] / [ ADP ] > ∆ G i . The results are reported as histograms in Fig.5 (top) andthey suggest that at equilibrium the average radius of gyration is extremely close towhat would be measured in the case of free substrate (dashed line). This is in agree-ment with the experimental observation that Hsp70 cannot significantly associate torhodanese in these conditions[16]. Conversely, in large excess of ATP we observe asubstantial swelling of the substrate (75 < R g < ffi ne binding ofHsp70s. This finding is fully compatible with the size of DnaK / DnaJ / rhodanese com-plexes determined by sm-FRET experiments in excess of ATP [16]. In this regime,the limited e ff ects of cochaperone binding on substrate conformations, which are notexplicitly included in the model, play a minor role in determining the global expansionof the complex. 6 [ ATP ] / [
ADP ]0.00.20.40.60.81.0 p n=0n=1n=2n=3n=4n=5n=6 n Figure 4: Probabilities of the state with n chaperones bound as a function of [ATP] / [ADP], for di ff erentstoichiometries n . The dashed line indicates the mean value h n i . A more quantitative comparison between the model and the FRET results can beachieved by back-calculating the transfer e ffi ciencies that were experimentally mea-sured for five distinct pairs of fluorescent dyes[16]. In equilibrium conditions, namelywhen [ATP] / [ADP] (cid:28)
1, the calculated FRET e ffi ciency is ’ ff erence is observed in excess of ATP(red circles), where the expansion of the substrate leads to a significant decrease of thecalculated e ffi ciency, in excellent agreement with the experimental values measured insimilar conditions (black circles)[16]. Remarkably, the results correctly captured thenon-monotonic behaviour of FRET e ffi ciency as a function of the sequence separationbetween the dyes, which was not reproduced in previous calculations[16]. This agree-ment corroborates the prediction of the DnaK binding sites on the rhodanese sequenceand the overall reliability of our model. ffi ciency Molecular chaperones consume energy via ATP hydrolysis in order to expand rho-danese. It is hence important to determine how e ff ective they are as molecular ma-chines, as well as to assess how favourable the physiological conditions are to performtheir biological task.To this aim, we calculated the global increase in the overall conformational freeenergy of the substrate with respect to equilibrium conditions, ∆ G S well (Fig.6, top). This7 R g ( Å ) c o un t s
50 100 150sequence separation0.00.20.40.60.81.0 t r a n s f e r e ff i c i e n c y Figure 5: Histogram of the radius of gyration for equilibrium (blue) and non equilibrium (red) values of[ATP]. The black dashed line indicates the average radius of unbound rhodaneses. (inset) FRET transfer e ffi -ciencies as a function of the sequence separation between the fluorescent dyes. The black circles correspondto the experimental values [16]. Calculated e ffi ciencies taking into account uncertainties are reported as blue(equilibrium conditions) and red circles (ATP excess) quantity measures the excess probability of each complex with respect to equilibriumconditions weighted by its corresponding conformational free-energy ∆ G i . ∆ G S well = X i " p i [ AT P ][ ADP ] ! − p eqi ∆ G i , (2)where p i (cid:16) [ ATP ][ ADP ] (cid:17) is the probability of complex i for a given value of [ AT P ] / [ ADP ] and p eqi is the same quantity computed at equilibrium conditions. In order to investigatethe conversion of chemical energy into mechanical work it is instructive to focus onthe ratio between ∆ G S well and the free energy of hydrolysis of ATP ∆ G h , which reportson the e ff ectiveness of the transduction process. We plot in Fig.6 (top) this quantityas a function of the [ AT P ] / [ ADP ] ratio considering the estimated inaccuracies of themodel as previously done for the gyration radius. Not surprisingly, all these curves ex-hibit a maximum because the probabilities of the di ff erent states, and thus also ∆ G swell ,attain plateaus for [ AT P ] (cid:29) [ ADP ], whereas ∆ G h increases monotonically with thenucleotide ratio (see Methods). The maximal transduction regime intriguingly corre-sponds to values of [ AT P ] / [ ADP ] that are typical of cellular conditions (grey area).8e highlight that in our model Hsp70 functioning encompasses two distinct yetintertwined processes: the ATP-dependent binding of the chaperones to the substrate,and its consequent expansion. Energy transduction occurs then through two steps, andthe amount of energy available for the mechanical expansion is limited by that providedby chaperone binding. To corroborate this picture and to gain a more complete insighton the action of Hsp70s, we analyzed the energetic balance of chaperone binding to asingle site. To this aim, we focused on a simplified reaction cycle, which essentiallycorresponds to a single triangle within the overall scheme in Fig.3 and does not takeinto account the conformational free-energy of the substrate. We report in Fig.6 (lowerpanel, black) the non-equilibrium dissociation constant, K neqd , normalized with respectto its equilibrium value K eqd , as a function of [ATP] / [ADP]. When the ratio between theconcentrations of ATP and ADP approaches the physiological regime, the dissociationconstant drops significantly until it settles at a value that is two order of magnitudelower than its equilibrium counterpart (this result is the core of ultra-a ffi nity). Wecan convert the dissociation constant into a binding free energy excess with respect toequilibrium ∆ G b = − k B T ln K neqd K eqd (3)that we can compare to the free-energy of ATP hydrolysis, ∆ G h , as previously donein the case of ∆ G S well . Interestingly, also in this case the energy ratio is maximalin cellular conditions, suggesting that the optimality of the overall expansion process(Fig.6, top panel) does not depend on specific features of the substrate but it is a directconsequence of the intrinsic kinetic parameters of Hsp70 binding.
3. Discussion
Integrating molecular simulations, polymer theory, single-molecule experimentaldata and non-equilibrium rate models, we have developed a comprehensive frameworkthat provides a quantitative picture of Hsp70-induced expansion of substrate proteinsand o ff ers a broad insight into the cellular functioning of this versatile chaperone ma-chine.We relied on molecular simulations for characterizing the structural and thermo-dynamic features of the complexes formed by the bacterial chaperone DnaK and itsunfolded substrate rhodanese. Notably, we investigated a large variety of possiblechaperone-substrate complexes for determining their conformational free-energy as afunction of stoichiometry and chaperone binding patterns. This computational strategybased on an enhanced-sampling protocol confirmed that excluded volume interactionsupon chaperone binding can greatly perturb the conformational ensemble of the un-folded substrate leading to its expansion. Remarkably, simulation results were foundto be in excellent agreement with the predictions of Sanchez theory for globule to coiltransition, thus providing another example of how polymer theory can be successfullyused to decipher the behaviour of disordered proteins [23, 24, 25]. We then combinedconformational free-energies with available biochemical data to develop an analyticalrate model of the chaperone / substrate reaction cycle, which included both chaperonebinding / unbinding and nucleotide hydrolysis / exchange processes.9 igure 6: (top) Ratio between the conformational free energy and the free energy of ATP hydrolysis, asa function of [ATP] / [ADP]. Light green curves represent single realizations, whereas the dark green curveis the average. (bottom) E ff ective dissociation constant in the case of a single binding site normalized withrespect to the corresponding value in equilibrium, as a function of [ATP] / [ADP] (black). Ratio between thebinding free energy and the free energy of ATP, as a function of [ATP] / [ADP] (purple). The gray regionindicates the interval of the physiological conditions. ff ects due to ATP hydrolysisand represents a natural extension of the ultra-a ffi nity framework originally developedfor peptide substrates with a single Hsp70 binding site[10]. We could thus investigatethe population of each complex and the average structural properties of the system as afunction of the ATP / ADP nucleotide ratio, which measures how far the system is fromthermodynamic equilibrium. The reliability of the model was corroborated by quan-titative comparison with recent sm-FRET data, indicating that our non-equilibriumframework accurately captures the salient features of the ATP-dependent expansion.We then used this unprecedented access to the thermodynamics details of this complexmolecular process to compare the free-energy cost associated with substrate swellingwith the chemical energy released by ATP-hydrolysis. Remarkably, this analysis re-vealed that energy transduction is maximally e ffi cient for ATP / ADP values in cellularconditions. This result hints at the possibility that Hsp70 chaperones have been tunedby evolution to optimize the conversion of chemical energy into mechanical work forsubstrate expansion. Further analysis indicated that this optimality is likely inheritedfrom the intrinsic properties of Hsp70 chaperones, which can convert up to 20% ofthe ATP chemical energy into non-equilibrium, excess binding energy at physiologicalconditions (Fig.6, lower panel).Hsp70s are highly versatile machines that play a fundamental role in a variety of di-verse cellular functions beyond the unfolding of non-native substrates, such as proteintranslocation, protein translation, and disassembly of protein complexes. Nevertheless,all these processes share basic analogies from the mechanistic point of view. Indeed,in all these cases Hsp70 binding to flexible substrates in constrained environments re-quires the energy of ATP hydrolysis (ultra-a ffi nity) and results in the generation ofe ff ective forces due to excluded volume e ff ects (entropic pulling), which ultimatelydrive protein translocation into mitochondria [19, 35], clathrin cage disassembly [41]and / or prevention of ribosome stalling [40]. Here by detailing how energy flows fromATP hydrolysis to mechanical work due to entropic pulling, we have elucidated a gen-eral force-generating mechanism of Hsp70 chaperones. This mechanism does not relyon any power-stroke conformational change but it rather depends on the e ffi cient con-version of ATP chemical energy into ultra-a ffi nity. The non-equilibrium nature of thisprocess allows further spatial and temporal regulation by cofactors, thus paving theway for performing complicated molecular functions such as the ATP-dependent sta-bilization of native proteins [42].
4. Materials and methods
In our MD simulations both rhodanese and Hsp70 were coarse grained at the single-residue level. To this aim, the molecules were represented as collections of beads cen-tered on the C α atom of each amino acid. Rhodanese was modeled according to theforce field from Ref. [15] (see SI for details). We modeled Hsp70 starting from theknown crystal structure of ADP-bound Hsp70 (PDB:2KHO [37]). The influence ofHsp70 in the ATP state on substrate conformation was assumed to give similar results.Following [35], we considered both the NBD and the SBD to be rigid bodies inter-acting only via excluded-volume interactions, while we modeled the flexible linker in11he same way as rhodanese. The residues of the binding site moved rigidly with thecorresponding SBD, thus ensuring that each chaperone was irreversibly bound to thesubstrate. All the simulations were performed with a version of LAMMPS [38] patchedwith PLUMED [39]. The temperature T =
293 K was controlled through a Langevinthermostat with damping parameter 16 ns − . The time step was set equal to 1 fs, andeach residue had a mass equal to 1 Da to speed up equilibration. In equilibrium sim-ulations, the system was equilibrated for 10 time steps starting from a rod-like con-formation, and subsequently sampled for other 10 time steps. At least 10 independentsimulations were performed for each configuration. Statistical errors on the computedquantities were estimated as standard errors of the mean. In the pulling simulations,starting from an equilibrium conformation the substrate was pulled by an external forceacting on its radius of gyration R g at a constant pulling speed v = − Å / fs, until arod-like conformation was reached, which we defined to be at R g =
260 Å. For this setof simulations, 100 independent realizations for each configuration were performed.The indetermination of the computed free energies were estimated according to thebootstrap method. For both equilibrium and pulling simulations, the statistical errorsare smaller than the size of symbols reported in the figures.
For the rate model we considered all possible binding configurations for chaperonesin the ATP or ADP state to the six identified binding sites, resulting in a total of 3 =
729 states. The equation for the average concentration of each binding configuration c i has the form dc i dt = X j k ji c j − X j k i j c i , (4)where k i j is the transition rate from configuration i to j . The transition rates betweendi ff erent states depend on the intrinsic rates of the underlying molecular processes,namely binding and unbinding of a chaperone from a given binding side, the hydrolysisand synthesis of the nucleotides, and the nucleotide exchange, and have been takenaccording to biochemistry experiments [16, 34, 43] (see SI for the rate values and fordetails about the thermodynamic constraints between the rates).We considered the steady state solution, obtained by imposing dc i dt = , (5)for every i . The concentration of chaperones was fixed at [Hsp70] = µ M , and theconcentration of nucleotides was equal to 1 mM as in the experiments in [16]. The freeenergy provided by the hydrolysis of ATP is given by the following formula: ∆ G h = k b T " ln [ AT P ][ ADP ] ! − ln [ AT P ] eq [ ADP ] eq ! , (6)where [ AT P ] eq and [ ADP ] eq are the concentrations of ATP and ADP in equilibrium.The mathematical and physical details of the model, as well as the values of the rates,are described in the Supplementary Information.12 .3. Molecular graphics Molecular graphics in Figs.1 and 6 have been generated with UCSF Chimera, de-veloped by the Resource for Biocomputing, Visualization, and Informatics at the Uni-versity of California, San Francisco, with support from NIH P41-GM103311 [44].
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UCSF Chimera - a visualization system for exploratory re-search and analysis , J. Comput. Chem., 25, 1605-12 (2004).16 upplementary Information for “Efficient conversion of chemicalenergy into mechanical work by the Hsp70 chaperones”
S. Assenza , , A. S. Sassi , R. Kellner , B. Schuler , , P. De Los Rios , and A. Barducci Laboratory of Food and Soft Materials, ETH Z¨urich, CH-8092 Z¨urich, Switzerland Departamento de F´ısica Te´orica de la Materia Condensada, Universidad Aut´onoma de Madrid,E-28049 Madrid, Spain Institute of Physics, School of Basic Sciences, Ecole Polytechnique F´ed´erale de Lausanne (EPFL),CH-1015 Lausanne, Switzerland IBM T. J. Watson Research Center, Yorktown Heights, New York, United States of America Department of Biochemistry, University of Zurich, CH-8057 Zurich, Switzerland Department of Physics, University of Zurich, CH-8057 Zurich, Switzerland Institute of Bioengineering, School of Life Sciences, Ecole Polytechnique F´ed´erale de Lausanne(EPFL), CH-1015 Lausanne, Switzerland Inserm, U1054, Montpellier, France
Details of MD Simulations
Force field.
Following [1], two- and three-body bonded interactions along the substratewere included via harmonic potentials, namely V bond = k l P b ( r b − l ) / V bend = k θ P α ( θ α − θ ) ,respectively. In the previous formulas, r b denotes bond lengths; θ α the bend angles; l =3 . k l /k B T ) − / = 0 .
046 ˚A; θ = 2 .
12 rad; ( k θ /k B T ) − = 0 .
26; and k B T is the ther-mal energy. Four-body bonded interactions were implemented as Fourier terms, V dihed = k B T P d P s =1 [ A s cos ( sφ d ) + B s sin ( sφ d )], where φ d is the torsion angle and A = 0 . A = − . A = − . A = 0 . B = − . B = − . B = 0 . B = 0 . V W CA = P ij V r , where V r = k B T (cid:20)(cid:16) σr ij (cid:17) − (cid:16) σr ij (cid:17) (cid:21) + k B T if r ij ≤ σ otherwise . (S1)In the previous formula, r ij is the distance between beads i and j , while σ = 4 . V hydro = (cid:15) h P ij V h , where V h = (cid:15) ij (cid:20)(cid:16) σr ij (cid:17) − (cid:16) σr ij (cid:17) (cid:21) if r ij ≥ σ − (cid:15) ij otherwise . (S2)In the previous formula, (cid:15) h = 0 . k B T sets the overall strength of the hydrophobic interac-tions, while (cid:15) ij depends on the residues i and j involved in the interaction, and is defined as the1 a r X i v : . [ q - b i o . B M ] M a r eometric mean of their hydrophobicities, (cid:15) ij ≡ √ (cid:15) i (cid:15) j . The values of the hydrophobicities con-sidered are based on a shifted and normalized Monera hydrophobicity scale[1]. Without furthertuning, this force field gives a radius of gyration of unbound rhodanese equal to R g = (23 . ± . R g = (20 . ± .
8) ˚A [2].
Binding sites.
DnaK binding sites were selected by applying the algorithms by van Durmeet al. [3] and R¨udiger et al. [4]. Only the sequences were chosen for which at least partialconsensus between the two approaches was present. In this way, six binding sites were identifiedcentered at the following amino acids along rhodanese: 10, 118, 131, 162, 188, 260.
Experimental assessment of electrostatic interactions.
The choice of neglectingelectrostatic interactions in the CG simulation of DnaK/rhodanese complexes was motivatedby the results of two control experiments, which were performed exactly at the same conditionsexcept for the salt concentration, namely 5 mM and 100 mM KCl. A double-cysteine variant ofrhodanese (K135C/K174C), which was produced by site-directed mutagenesis and prepared asdescribed before for wild-type rhodanese [5], was labeled with Alexa Fluor 488 C5 maleimideand Alexa Fluor 594 C5 maleimide (Invitrogen, Molecular Probes) [6]. A twofold molar excessof the dyes was added to the protein and incubated for 1h at room temperature. Unreacted dyewas removed by gel filtration followed by anion exchange chromatography to reduce the amountof incorrectly labelled protein using a MonoQ 5/50 GL column (GE Healthcare) installed onan ¨AKTA purifier FPLC system equilibrated in 50 mM TrisHCl, pH 7.0, and eluted with agradient from 0 to 500 mM sodium chloride over 60 mL (12 column volumes). The chaperoneproteins DnaK and DnaJ (stock solution concentration 100 µ M in 50 mM Tris HCl, pH 7.7,100 mM NaCl) were gifts from H.-J. Sch¨onfeld (Hoffmann-La Roche Ltd., Basel). Labelledrhodanese was denatured in 4 M guanidinium chloride in buffer (50 mM Tris HCl, 10 mMMgCl , 200 mM β -mercaptoethanol, and 0.001% Tween 20) with either 5 or 100 mM KCladded. The denatured rhodanese was diluted 100x into buffer containing 10 µ M DnaK, 0.5 µ M DnaJ, 1 mM ATP and 5 or 100 mM KCl to form chaperone rhodanese complexes at afinal concentration of 50 pM rhodanese. Single-molecule F¨orster Resonance Energy Transfer(FRET) measurements were started immediately after dilution and data recorded at 22 ◦ C for30 min to construct the FRET efficiency histograms. Data were recorded with a MicroTime 200confocal microscope (PicoQuant) and on a custom-built confocal microscope. All measurementswere obtained with pulsed interleaved excitation [7]. The instrument set up and data reductionwere the same as described before [2].The collected FRET histograms are plotted in Fig.S1. The average FRET efficiencies were0 . ± .
01 with 5 mM KCl and 0 . ± .
01 with 100 mM KCl (errors indicate uncertaintiesestimated from the standard deviation of the larger data set collected at 5 mM salt). The verysimilar values obtained at different salt concentrations suggest that electrostatic interactions donot play a significant role in determining the conformational properties of rhodanese/chaperonecomplexes.
Control simulations on electrostatic interactions.
To further test the role of elec-trostatic for the results of the present work, a direct comparison was performed for the case2f DnaK-free rhodanese by considering a set of simulations including also electrostatics in-teractions following Smith et al.[1]. Similar values of the radius of gyration were found forsimulations with and without electrostatics, namely R g = (22 . ± .
1) ˚A and R g = (23 . ± . µ M DnaK, 500 nM DnaJ, 1 mM ATP andeither 5 mM (grey curves) or 100 mM (red curves) of KCl.
Pulling simulations.
The dependence of conformational free energy of rhodanese onthe set of bound chaperones was assessed via steered Molecular Dynamics simulations. Forany given chaperone configuration, the system was started at a conformation compatible withequilibrium and pulled by adding a harmonic potential acting on the radius of gyration R g of rhodanese. The center of the harmonic trap was increased at a constant pulling speed v = 10 − ˚A / fs from the equilibrium value up to an almost fully stretched conformation, whichwas set at R g = R fin g ≡
290 ˚A (Fig.S3).Free-energy differences between the various configurations of bound chaperones and the caseof simple rhodanese (i.e., no bound Hsp70s) were obtained via the Jarzynski equality [8] in thefollowing way. For each combination of bound chaperones, 100 independent pulling realizationswere performed, starting from uncorrelated initial snapshots extracted from the equilibriumdistribution. For each realization, the work W performed by the bias potential during thepulling process was measured (black curves in Fig. S4). The Jarzynski equality then reads [8] e − δGkBT = D e − WkBT E , (S3)where δG is the free-energy difference between the equilibrium starting point and the statecorresponding to the chosen value of R g , while h . . . i denotes statistical average. Equation (S3)3igure S2: Average FRET efficiency < E > obtained from simulations in the a bsence ofchaperones for the various couples of dyes considered in Ref.[2]. Black triangles were obtainedconsidering the whole force field (excluded volume + hydrophobic + electrostatic non-boundinteracions), while for red circles electrostatic interactions were neglected.Figure S3: Example of “fully-stretched” conformation with three bound chaperones.4nables the computation of δG ( R g ) from the knowledge of the pulling work, leading to the redcurves reported in Fig.S4. Errors were estimated according to the bootstrap method.Due to the large intermolecular distances, the effect of chaperones on the conformational prop-erties of fully-stretched rhodanese is negligible (Fig.S3). Hence, this state can be taken as areference for free-energy computations. The Jarzynski averages were consequently shifted sothat δG ( R fin g ) = 0 for all the chaperone combinations. As an example, in Fig.S5 the averagesconsidered in Fig.S4 are shifted according to this prescription. After such adjustment, thevertical distance between the starting point of each pulling curve and the case with no boundchaperones (arrows in Fig.S5) gives the estimation for the free-energy difference ∆ G consideredin the main text. In order to enhance the robustness of the results, the final values reported inthe main text were obtained as a further average over the values of R fin g within the range 260˚A ≤ R fin g ≤ anchez theory In his theory on polymer coil-to-globule transition, Sanchez [9] considers a Freely-JointedChain (FJC) made of n monomers characterized by attractive interactions of average magni-tude (cid:15) . Let α ≡ R g /R g, be the expansion parameter, where R g is the radius of gyration ofthe polymer, while R g, is the value of R g in the unperturbed case ( (cid:15) = 0 and no excludedvolume present). Assuming a Flory-Fisk distribution for the unperturbed case, the probabilitydistribution of the expansion parameter P ( α ) was shown to be [9] P ( α ) = 1 Z α e − α + nq ( (cid:15),α ) , (S4)where Z is the partition function and q ( (cid:15), α ) = 12 (cid:15) φ α − (cid:18) α φ − (cid:19) ln (cid:18) − φ α (cid:19) , with φ = p / (27 n ). The free energy can then be straightforwardly computed as ∆ G ( α ) = − k B T ln P ( α ), i.e., making the dependence on R g explicit,∆ G ( R g ) = − k B T ln P ( R g /R g, ) . (S5)In the present case, rhodanese is constituted by 293 amino acids, i.e. there are N b = 292bonds. The bond length is b l = 3 . C α atomsbelonging to consecutive residues [10]. In order to apply the Sanchez theory, one needs toconsider the FJC equivalent to rhodanese. To this aim, the Kuhn length b K can be estimatedas b K = 2 l p = 8 ˚A, where l p = 4 ˚A is the persistence length of a protein [10]. The numberof monomers n in the equivalent FJC can then be computed by imposing the total contourlength of rhodanese: n = N b b l /b K . From there, both φ and the unperturbed radius of gyration R g, = p nb K / (cid:15) as the only unknown quantity in Eq. (S4). Thelatter can be fixed by imposing that ∆ G has a minimum at the experimental value R ∗ g = 20 . (cid:15) ’ .
07 kcal/mol.Plugging the values of the parameters derived above into Eq.(S5) gives the black line re-ported in Fig.2 in the main text, in excellent agreement with the simulation results. We stressthat neither the force field nor the MD results were employed in the derivation of the parametersused in the theoretical formula.
Computation of Efficiency
The FRET efficiency E for a given couple of dyes was computed starting from the distance r separating the corresponding amino acids as [2] E = 11 + (cid:16) rr (cid:17) , (S6)6here r = 54˚A. For each realization, the time average of E was computed. The final valuesemployed to compute the results reported in the inset of Fig.5 in the main text were obtained asthe mean between independent realizations. The corresponding indeterminacy was estimatedconsidering the standard error of the mean and computing the error propagation to the finalvalues, and is smaller than the size of symbols. Rate model
For the kinetic model we consider a system in which each binding site can either be occupiedby a chaperone in the ATP or ADP state, or it can be free, so that in total there are 3 = 729possible configurations. The concentration c i of each state evolves in time according to a systemof rate equations dc i dt = X j k ji c j − X j k ij c i (S7)where k ij is the transition rate from state i to state j . The first term in the right hand side( r.h.s. ) of (S7) represents the total flux of molecules from the other states toward state i , whilethe second term in the r.h.s. of (S7) accounts for the flux of molecules from state i to any otherstate. We focused on the steady-state, when the concentrations of the various states do notchange over time, which is defined by dc i dt = 0 (S8)Here we provide a pedagogical example of state-encoding, and of the associated transitionsand equations. For example, in the configuration (0 , T, , D, , • binding/unbinding ( ... ... ) k adpon e − β ∆∆G −−−−−−−− *) −−−−−−− k adpoff ( ...D... )( ... ... ) k atpon e − β ∆∆G −−−−−−− *) −−−−−−− k atpoff ( ...T... ) • hydrolysis/synthesis ( ...D... ) k ss −− *) −− k sh ( ...T... )7 nucleotide exchange ( ...D... ) k effex , DT −−−− *) −−−− k effex , TD ( ...T... ) . We further provide, as an example, the equation for a precise configuration, say (0 , T, , D, , ddt (0 , T, , D, ,
0) = − (0 , T, , D, , ∗ ( k effex,DT + k ss + k effex,T D + k sh + k atpoff + k adpoff )+ (S9)+ (0 , , , D, , · ATP] k atpon e − β ∆∆ G ++ (0 , T, , , , · ADP] k adpon e − β ∆∆ G ++ ( T, T, , D, , k atpoff ++ ( D, T, , D, , k adpoff ++ (0 , T, T, D, , k atpoff ++ (0 , T, D, D, , k adpoff ++ (0 , T, , D, T, k atpoff ++ (0 , T, , D, D, k adpoff ++ (0 , T, , D, , T ) k atpoff ++ (0 , T, , D, , D ) k adpoff . Here below we further detail the rates of our model.It is possible to move from an ATP-state to an ADP-state either via hydrolysis/synthesisor via nucleotide exchange. In the case of exchange, effective constants are used, which takeinto account the unbinding of one nucleotide species and the binding of the different one. Theeffective exchange rates are thus a function of the ratio [ATP]/[ADP] (see also [11]): k effex,DT = k − D k + T [ AT P ][ ADP ] k + D + k + T [ AT P ][ ADP ] (S10) k effex,T D = k − T k + D k + D + k + T [ AT P ][ ADP ] , (S11)where k + D , k + T , k − D and k − T are the binding and unbinding rates for ADP and ATP respec-tively. 8he rates of binding between the chaperone and single peptides have been previously de-termined experimentally [12], and they were corrected in order to take into account the confor-mational change of the full polypeptide substrate upon binding, as we illustrated in the maintext.Substrate binding enhances the chaperone ATPase activity. Furthermore, the stimulationof the hydrolysis of ATP always takes place in cooperation with the co-chaperone JDP. In ourmodel we did not consider it explicitly but its contribution was implicitly included through thechoice of the rate constants.If we call k h the hydrolysis rate in the absence of the substrate and k sh the rate in thepresence of the substrate, we have k h (cid:28) k sh . We assumed that the ratio between the rate ofhydrolysis k h and the rate of synthesis k s is not altered by the substrate: k h k s = k sh k ss . (S12)The substrate binding/unbinding rates, the rates of nucleotide exchange and the hydrolysisand synthesis rates are collectively constrained by thermodynamic relations. Indeed, when theratio between the concentrations of ATP and ADP is equal its equilibrium value (when thespontaneous hydrolysis and synthesis reaction are at steady state and compensate each other),detailed balance must be satisfied [13]. As a consequence, for every closed cycle in the reactionnetwork the product of the rates in one direction must be equal to the product of the rates inthe opposite direction. Therefore, if k atpon , k adpon , k atpoff and k adpoff are the rate of substrate bindingand unbinding from a chaperone in the ATP and ADP states, we must have k atpon k sh k adpoff k s k adpon k ss k atpoff k h = k atpon k adpoff k adpon k atpoff = 1 . (S13). Remarkably, taking the rates as provided in [2, 12, 14], this relation is not satisfied, andwe had thus to modify them. We just calculated the product in the formula above and thencorrected the rates in the following way: k atpon k adpoff k adpon k atpoff = r (S14) k atpon , k adpoff → k atpon /r / , k adpoff /r / (S15) k adpon , k atpoff → k adpon ∗ r / , k atpoff ∗ r / . (S16)The concentration of free chaperones in the ATP and in the ADP states was obtained, atthe leading order, by solving a three-state system whose reactions have the formHsp70 + ADP −− *) −− Hsp70 · ADP −− *) −− Hsp70 · ATP −− *) −− Hsp70 + ATP . (S17)9ince we worked in the assumption of excess of chaperones in the system, once these concen-trations were obtained, they remained fixed once for all, without being considered as a variableof the biochemical network.We report in the following table the rates used in the model. Parameters of the model [2, 12, 14] k atpoff . s − k adpoff ∗ − s − k − T . ∗ − s − k − D . s − k atpon . ∗ M − s − k adpon M − s − k + T . ∗ M − s − k + D . ∗ M − s − k h ∗ − s − k sh . s − To test the robustness of the model for the radius of gyration, the average FRET efficiencyand the free energy ∆ G swell , 100 realizations were implemented, taking each time the values∆ G i from a Gaussian distribution with σ = 0 . 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