Atomic-level observation of macromolecular crowding effects: Escape of a from the GroEL cage

Adrian H. Elcock*

Department of , University of Iowa, Iowa City, IA 52242

Edited by Peter G. Wolynes, University of California at San Diego, La Jolla, CA, and approved January 10, 2003 (received for review August 21, 2002) Experimental work has demonstrated that the efficient operation the repeated cycles of folding undergone by the protein were of the GroEL-GroES chaperonin machinery is sensitive to the forced to occur in the same GroEL cage initially encountered by presence of macromolecular crowding agents. Here, I describe the unfolded protein. Here, I show that this effect can be atomically detailed computer simulations that provide a micro- captured by atomically detailed simulations that explicitly com- scopic view of how crowding effects are exerted. Simulations were pute the complicated, interdependent interactions of protein and performed to compute the free energy required to extract the crowding molecules. Specifically, I use Brownian dynamics sim- protein rhodanese from the central cavity of GroEL into ulations (7, 8) to calculate the free energy profile for removal of containing a range of crowder concentrations. The computed a protein molecule from inside the GroEL cage to the energetics allow the total yield of folded protein to be predicted; environment, in the presence of different concentrations of the calculated yields show a nonlinear dependence on the concen- crowding agent. The calculated (unfavorable) free energy tration of crowding agent identical to that observed experimen- change for this process allows the total yield of folded protein to tally. The close correspondence between simulation and experi- be predicted in the different solution conditions. The predicted ment prompts the use of the former in a truly predictive setting: yields are shown to scale identically with the experimental yields, simulations are used to suggest that more effective crowding indicating that the essential physical aspects of the system are agents might be designed by exploiting an ‘‘agoraphobic effect.’’ faithfully reproduced by the simulations. Because the ultimate goal of simulation methods is not simply to recapitulate exper- he potential role played by the high protein concentrations imental results but to make testable predictions (9), I consider, Tof the cellular environment in affecting macromolecular in a second series of simulations, the possibility of designing more thermodynamics and kinetics has attracted considerable atten- effective macromolecular crowding agents. I suggest that an tion over the years (for recent reviews, see refs. 1 and 2). A analogy between crowding and hydrophobic effects might be thermodynamic consequence of the ‘‘crowding’’ that occurs at exploited in the form of an ‘‘agoraphobic effect’’ that drives the high concentrations is an increase in the activity of all solute association of solute through a desire to mini- species; the fact that the resulting increases are expected to be mize disruption of favorable interactions between crowding greater for compact protein complexes than for their separated molecules. components has long been noted as a potential means of increasing the stability of protein–protein interactions (1). The Methods kinetic consequences of crowding are best illustrated by noting Analysis of Experimental Results. Previous work reported by Martin the significant decreases in protein diffusion coefficients that and Hartl (6) has quantified the amount of protein that success- occur at high concentrations (3). This latter phenomenon has fully refolds in the presence of an excess of a trapping-form of taken on more interest recently with the development of fluo- GroEL (trap-GroEL), as a function of the concentration of the rescence techniques capable of monitoring macromolecular crowding agents Ficoll 70 and . To establish a relation- diffusion in vivo (4). ship between these experimental results and observables that can A full understanding of the effects of macromolecular crowd- be computed through molecular simulation, we conduct the ing requires the construction of a practical theoretical frame- following analysis. work consistent with experimental observations. This develop- Chaperonin-mediated refolding is an iterative process, with ment is by no means straightforward to achieve. On the one each round of folding being accompanied by ATP hydrolysis. At hand, it is certainly possible to derive analytical equations for the end of each ATP cycle, we assume that the protein is in one simplified cases [such as when are approximated as hard of three states: (i) fully folded, (ii) partly folded and escaped spheres (5)], and these models can be very informative of the from the GroEL (such that it is then assumed to be captured by qualitative behavior to be expected in highly crowded solutions trap-GroEL), or (iii) partly folded and still bound to the original (1). On the other hand, it is impossible to derive analytical GroEL cage. We denote the probabilities of each case by pfold, equations for arbitrarily shaped molecules that might interact, in pescape, and Pre-bind, respectively; together, they must sum to 1. If a highly anisotropic manner, via any number of physical effects we further assume that pfold is (i) independent of how ‘‘folded’’ such as electrostatic interactions, hydrophobic effects, hydrogen the protein is at the beginning of the cycle, and (ii) independent bonds, etc. For such cases, which represent many of the most of the presence of crowding agent [a reasonable assumption interesting situations, a perhaps more meaningful way of pro- given that folding occurs primarily within the confines of the ceeding is to look to numerically based methods to explicitly GroEL-GroES cage complex (10)], we can proceed as follows. simulate the macromolecular interactions at a realistic level of The total fractional yield of folded protein will be the sum of detail. Here, I describe a set of atomically detailed molecular the amounts that successfully fold during each cycle. For the first simulations aimed in this direction. cycle, this amount will be simply pfold. For the second cycle, this The specific system under study here is one investigated amount will be the product of pfold and the amount of protein experimentally with the aim of examining the effects of macro- that re-bound to the GroEL cage after the first cycle, i.e., pfold ϫ molecular crowding on the efficiency of protein refolding me- pre-bind (the very small amount of protein that escapes after the diated by the chaperonin GroEL͞GroES (6). Briefly, it was shown that increasing concentrations of crowding agents, such as Ficoll 70 and dextran, hindered the escape of partly folded This paper was submitted directly (Track II) to the PNAS office. rhodanese molecules from the GroEL cage, in such a way that *E-mail: [email protected].

2340–2344 ͉ PNAS ͉ March 4, 2003 ͉ vol. 100 ͉ no. 5 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0535055100 Downloaded by guest on October 2, 2021 Fig. 1. Illustration of the simulation system for GroEL and rhodanese immersed in a 30% (wt͞vol) crowder solution. The figures, which are of a slice through the center of the simulation box, show GroEL in blue, rhodanese in red, and model crowder molecules in green. Rhodanese is positioned in its initial location (Left) and in its final position (Right) (a separation distance of 150 Å). This figure was prepared by using PYMOL (2002) (www.pymol.org).

ϭ first cycle only to bind again to a WT GroEL molecule makes a crowder solution (6). In such a situation, pescape 0, and because ϭ ϭ negligible contribution). In general, the fraction that successfully pfold 0.083 (see above), pre-bind 0.917. By using our computed ϫ iϪ1 ‡ folds in the ith cycle will therefore be pfold pre-bind . Martin values of ⌬⌬G for the 30% crowder solution, together with the o and Hartl (6), who conducted an analysis along similar lines, have fact that we know both pre-bind (0.917) and pre-bind (0.667), we can estimated pescape for rhodanese in the absence of crowder to be obtain A⅐⌬t by rearranging Eq. 2. We are now in a position to use equal to 0.25. Given this value, together with their reported 25% our computed values of ⌬⌬G‡ in other crowder solutions to total yield of folded protein obtained in the absence of crowder, calculate their values of pre-bind; this calculation then allows us (at

ϭ we obtain a value of pfold 0.083. This number is in reasonable last) to predict the total yield of folded protein in each crowder agreement with the value of Ϸ0.05 previously estimated by solution. Martin & Hartl (6); the slight discrepancy is probably due to subtle differences in theoretical schemes. Brownian Dynamics Simulations. The underlying principles behind To proceed further, we assume that escape of the protein is a the simulation methodology used in the present application have first-order process characterized by a rate constant kescape. This been discussed in recent work (11). Briefly, the method is a ϭ Ϫ ⅐ Ϫ assumption allows us to write pre-bind (1 pfold) exp( kescape development of Brownian dynamics techniques pioneered by ⌬t), where ⌬t is the average length of time it takes for a GroES Gabdoulline and Wade (12) and earlier, Northrup and Erickson molecule to bind to GroEL (at which point the protein is (13), for the purposes of simulating the association dynamics of assumed to begin another round of folding; ref. 10) and the a pair of macromolecules treated in atomic detail. Recent work Ϫ preexponential factor, (1 pfold), ensures that the sum of pre-bind has extended the methodology to handle arbitrary numbers of and pfold cannot exceed 1. We now make use of the fact that diffusing and mutually interacting molecules; a first application kescape can be expressed in terms of an activation free energy of the methodology showed that competition between small ⌬ ‡ ϭ ⅐ Ϫ⌬ ‡͞ ( G ) for escape by using kescape A exp( G RT), where A molecule substrates and inhibitors for occupation of an ’s is a proportionality constant, R is the gas constant, and T is active site could be accurately described (11). temperature. Combining these two equations allows us to ex- The molecular system studied here is illustrated in Fig. 1. I ⌬ ‡ press pre-bind in terms of G thus: consider the effects of increasing concentrations of crowding agents on the ease of escape of the protein rhodanese from the ϭ ͑ Ϫ ͒⅐ ͓Ϫ ⅐⌬ ⅐ ͑Ϫ⌬ ‡͞ ͔͒ pre-bind 1 pfold exp A t exp G RT [1] GroEL cage. The conformation used for GroEL is that found in the GroEL–ES complex solved by Sigler and coworkers (14); Finally, if we denote the value of p in the absence of re-bind although GroES is omitted from my simulations, this confor- crowder as p o, we can express the fractional increase of re-bind mation is the one most likely to be encountered by an escaping re-binding due to the presence of crowder as: protein. The structure used for rhodanese is the crystal structure ͞ o ϭ ͓Ϫ ⅐⌬ ⅐ ͑Ϫ⌬⌬ ‡͞ ͔͒ of the fully folded form of the protein (15); I have made no pre-bind pre-bind exp A t exp G RT [2] attempt to model partly folded structures because the simulation where ⌬⌬G‡ is the activation free energy in the presence of methodology is not currently capable of allowing internal flex- crowder minus the activation free energy in the absence of ibility in the macromolecules. Because crowding agents are crowder. Eq. 2 provides the desired connection between an believed to exert greater effects on more extended conforma- experimentally derivable quantity (pre-bind) and a quantity that tions than compact conformations (5, 16), the results obtained can be computed by molecular simulation (⌬⌬G‡; see below). here by using a fully folded model of the protein may, if anything, The one issue left to deal with is how to assign a value to A⅐⌬t. represent an underestimate of the real effects. This value can be assigned by noting that, from experiments, the The crowding agent Ficoll 70 is less straightforward to model total yield of folded protein reaches 100% in a 30% (wt͞vol) because there is no detailed structure available; it has been

Elcock PNAS ͉ March 4, 2003 ͉ vol. 100 ͉ no. 5 ͉ 2341 Downloaded by guest on October 2, 2021 modeled here as a spherical of radius 30 Å,a dimension approximately consistent with that of a globular protein of similar molecular mass (70 kDa; ref. 17). Although it is known that Ficoll 70 solutions are somewhat heterogeneous in terms of molecular weight (18), we have for simplicity opted to consider only homogeneous samples, i.e., all molecules are identical. The surfaces of the Ficoll 70 model molecules are covered with 288 evenly spaced atoms; interactions between these surface atoms and the GroEL and rhodanese are modeled by the repulsive 1͞r12-dependent component of the Lennard– Jones 12-6 potential. A second series of simulations was per- formed with a modified Ficoll model, in which four charge patches were placed on the surface (see Results and Fig. 5 Inset). In these latter simulations, forces on each crowder molecule again consisted of the repulsive Lennard–Jones component, but this time supplemented with electrostatic forces calculated by using the ‘‘effective charge’’ approach (19), as a product of the atomic charges of the crowder and the Poisson-Boltzmann electrostatic fields generated by other molecules (7). With 200 Ficoll 70 molecules, and a single copy each of GroEL and rhodanese, the total number of atoms in each simulated system was therefore 64,029. All simulations were conducted under Fig. 2. Calculated free energy change due to interaction of rhodanese with periodic boundary conditions (20), with the dimensions of the crowder molecules in a 30% (wt͞vol) crowder solution, as a function of periodic adjusted so that the concentration of crowding separation distance. Results from 64 separate, independent simulations are agent matched the solution conditions studied experimentally. In shown in gray; arithmetic and Boltzmann-weighted means of these results are all simulations, the initial positions of the crowder molecules shown in black. were assigned in a step-by-step process (11). Each successive molecule added to the system was placed in 10 random (and nonoverlapping) positions, and the position with the most fa- ducted far faster than would occur in reality, so to obtain vorable energy was selected. Diffusion coefficients for the meaningful estimates of the free energy change for the process, crowder molecules were set to values similar to, or higher than, I made use of an approach developed by Jarzynski (23, 24), who those of proteins of similar molecular weight: translational and has shown that Boltzmann-weighting of the results of multiple rotational diffusion coefficients were set to 0.01 Å2⅐psϪ1 and short simulations (which can be very far from equilibrium) can 0.0001 rad2⅐psϪ1, respectively. Motion of the crowders was yield surprisingly accurate estimates of equilibrium-free energy simulated by using the Ermak–McCammon algorithm (21), with changes. In the present case, between 55 and 85 simulations were a maximum permitted timestep of 10 ps; steps for which any of performed for each concentration of crowding agent; error the interaction energies between molecules increased by more estimates for the final free energy changes were obtained from than 10 kT were reversed and repeated with the timestep halved. the standard deviation of a bootstrap analysis, involving resam- pling of the calculated free energy profiles a total of 1000 times. Calculation of ⌬⌬G‡. The free energy change for removing rho- danese from the GroEL cavity was calculated by using free Results energy perturbation methodology (22). The protein was moved Molecular simulations were performed to calculate the free in steps of 0.5 Å from its initial position in the center of the energy change resulting from progressive displacement of a cavity, to a final position 150 Å further out along the direction rhodanese molecule from the center of the GroEL cage to a of the long axis of the GroEL molecule (Fig. 1). The free energy position fully exposed to solutions containing a range of Ficoll change accompanying displacement of the protein was obtained 70 concentrations. As an illustrative example, the simulated free by using: energy change for a 30% (wt͞vol) Ficoll 70 solution (relative to ⌬ ϭ Ϫ ͗ Ϫ⌬ ͞ ͘ the corresponding free energy change for a solution containing G RT exp E RT x [3] no crowder) is shown in Fig. 2. One clear consequence of the use where ⌬E denotes the difference between the system energies of multiple short simulations to calculate the free energy profile ϩ ͗͘ (see Methods) is that a relatively large spread of values is when the protein is at positions x and x 0.5 Å and x denotes an ensemble average computed with system configurations obtained. A second probable consequence is the tendency of the generated with the protein at position x. In computing the energy calculated free energy changes to continue increasing even at the of each configuration, we omit all contributions from direct furthest distances from the cage (Figs. 2 and 3); although this interactions between the rhodanese and the GroEL. This omis- result might indicate an entropic penalty for Ficoll molecules to sion ensures that the ⌬G we calculate is actually the difference intervene between the rhodanese and GroEL (even when there between the ⌬G in the presence of crowder and the ⌬G in the is more than enough space for them to occupy such positions), absence of crowder, i.e., it is the ⌬⌬G required in Eq. 2. Framed a more likely reason is that the crowder molecules simply do not in this way, the free energy computation is effectively identical have enough time in the simulations to respond to the change. to a potential of mean force calculation (20). Visual inspection of simulation trajectories supports this inter- After an initial equilibration period of 25 ns, the GroEL- pretation (see http:͞͞dadiddly.biochem.uiowa.edu͞GroEL). rhodanese-crowder system was simulated at each x position for Despite the variability in results indicated in Fig. 2, the Boltz- a total of 250 ps, with the average in Eq. 3 being accumulated mann-weighting suggested by Jarzynski (23) gives results that are only in the last 150 ps. Because translating the protein 150 Å is sufficiently precise that the differences between the calculated conducted in a total of 300 steps, the total length of a single free energy values for different crowder concentrations are simulation was 100 ns (25 ns equilibration ϩ 75 ns potential of almost all outside of the estimated error and so are sufficiently mean force calculation). The removal process is clearly con- accurate to draw meaningful conclusions (Fig. 3).

2342 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0535055100 Elcock Downloaded by guest on October 2, 2021 Fig. 3. Calculated Boltzmann-weighted mean free energy changes for Fig. 5. Calculated relative free energies of activation (⌬⌬G‡) for removal of removal of rhodanese in crowder solutions of various concentrations. Error rhodanese as a function of crowder concentration for purely steric and bars indicate the standard deviations obtained from bootstrap analysis of the charge-interacting crowder molecules (latter shown in Inset). Calculated val- calculated free energies. ues are for a separation distance of 75 Å; identical qualitative effects are obtained when values from other separation distances are considered.

In Methods, I outline a way in which the computed free energies can be used to calculate the expected total yield of at a distance of 75 Å away from the initial position. It should be folded protein; the goal for the simulations is to reproduce the remembered that, in our theoretical scheme, the calculated and experimentally observed trends. To make this comparison, we experimental numbers are, by definition, in exact agreement in must first identify a separation distance that corresponds to the the 0% and 30% crowder solutions (see Methods). However, transition state for escape. As it turns out, the exact choice of this there is no such constraint on the agreement for all intervening distance does not seem to be important: the simulations fit the concentrations; the fact that the simulations are successful at reproducing the very nonlinear relationship between crowder experimental results very well over a wide range of candidate BIOPHYSICS distances. Fig. 4 shows the comparison between calculated and concentration and total yield of folded protein is therefore notable (Fig. 4). experimental total yields of folded protein obtained when we This close agreement suggests that the essential physics of the assume that the transition state for escape of the protein occurs molecular system are captured correctly by the simulations, but to turn a simulation approach into a proper research tool requires that it be useable in a predictive setting. In the present case, we attempt to demonstrate this utility by considering the following idea. As noted in the introduction, crowding effects are usually interpreted in terms of excluded volume contributions (1, 2); from such a viewpoint, interactions between crowder mole- cules are generally considered to be entirely steric in nature. But simulations of the type described above can just as easily be performed with additional favorable interactions operating be- tween the crowders. I hypothesize that, if crowder molecules can be made to interact favorably with each other (and not with solute proteins), then dissociation of the solute proteins should be further discouraged because of the necessity of limiting disrup- tion of crowder–crowder interactions. Clearly, this idea implies an analogy with the hydrophobic effect (25), in which the association of nonpolar molecules in water is driven by the need to restrict, as much as possible, unfavorable disruption of water–water interactions. Here, I have investigated whether this analogy might be exploited to design crowding agents more powerful than those that operate predominantly via volume exclusion effects (1, 2). To this end, I devised modified crowding molecules containing two positive charge patches and two negative charge patches placed on their surface in a tetrahedral Fig. 4. Correlation of calculated and experimental total yields of folded arrangement (Fig. 5 Inset); this arrangement is intended to mimic rhodanese in crowder solutions of various concentrations. Calculated values are obtained by using free energy values from Fig. 3 at a separation distance that of the protons and electron lone pairs in a water molecule. of 75 Å; a snapshot of the system at this separation distance is shown in Inset. Conducting simulations in exactly the same way as above, we find Experimental yields are taken from table 4A of ref. 6; the point indicated by that these crowders cause a substantial increase in the free an asterisk was interpolated from this figure and is therefore only an esti- energy required to remove rhodanese from the GroEL cage, mated value. relative to the value obtained with the purely steric crowders

Elcock PNAS ͉ March 4, 2003 ͉ vol. 100 ͉ no. 5 ͉ 2343 Downloaded by guest on October 2, 2021 (Fig. 5), but only when the highest concentration of crowder is for this idea is an analogy with the hydrophobic effect. If one used. At all lower crowder concentrations, the favorably inter- views our model crowders as high molecular weight analogues of acting crowders actually cause a slight decrease in ⌬⌬G‡,an water molecules (Fig. 5), it might be legitimate to term their effect that seems to result from a tendency (although slight) for predicted effect on the association of solute proteins as an the crowders to form small ‘‘clumps’’ that exclude less volume agoraphobic effect. than separated, noninteracting crowders. If an attempt is made to put this design principle into practice there will be a number of aspects to keep in mind. Although Discussion globular proteins may not be the ideal starting point in a search There are two main results reported here. First, atomically for better crowding agents (because of their propensity to detailed computer simulations have been shown to provide very aggregate, for example; D. M. Hatters, personal communica- good reproduction of the experimental effects of a crowding tion), they nevertheless represent attractive vehicles for pursuing agent on the operation of a complex macromolecular machine. experimental design strategies for the following reasons. First, Although some aspects of these results might also be attainable central to any truly rational approach, 3D structures of many by application of simpler analytical theories (5), as a proof-of- proteins are available (www.rcsb.org); any protein that forms an principle, the present study opens the way to studying a host of approximately tetrahedral shape might be considered a prom- molecular phenomena, dependent on atomic-level interactions ising ‘‘lead’’ candidate. Second, mutations (which can that cannot be described analytically. This potential is demon- usually be produced straightforwardly), would need to be made strated here by our second main result: that a macromolecule only at surface positions, where substitutions are, in any case, capable of forming favorable interactions with itself may be a most easily tolerated. There are of course several potential more effective crowding agent than one that relies solely on problems. For example, if electrostatic interactions are chosen as steric interactions, but only when added in sufficiently high the means to promote crowder–crowder interactions (as in the concentrations. current simulations), care would need to be taken to ensure that This latter result may be of significance given that one of the excessive repulsion of the like-charged residues comprising the hopes for macromolecular crowding agents has been that their surface patch does not occur. This, however, is a problem that addition to a solution might enhance the formation of otherwise has already been solved by the many proteins that generate weak protein–protein complexes. This is a goal of considerable strong electrostatic potentials to accelerate catalysis or binding importance given that many biologically important complexes to other molecules (7). Another problem that is easy to note but seem to be stable only in the high macromolecular concentra- difficult to predict is the potential for phase separation: protein tions encountered in vivo (e.g., ref. 26). To this end, several cases solutions are known to exhibit complicated phase behavior [even have already been reported in which macromolecular crowding including the coexistence of liquid–liquid phases (29)], and these agents have been used to stabilize homo-oligomeric assemblies aspects will probably come into play given that the agoraphobic (e.g., ref. 27; other examples in ref. 28). The simulations respon- effect appears only at high concentrations of crowder (Fig. 5). sible for the results shown in Fig. 5 should be seen in this light Despite these caveats, it should be remembered that remarkable as a preliminary attempt to guide the design of more effective feats of protein engineering, such as the generation of caged macromolecular crowding agents. Clearly, the design is predi- assemblies (30), have already been achieved. Rational design of cated on the idea that the best way of improving such agents is more effective macromolecular crowding agents might therefore to extend consideration beyond purely excluded volume inter- ultimately prove to be an achievable goal. actions and to incorporate additional physical interactions. Note Added in Proof. There is a variety of ways in which this might be achieved. The For an alternative theoretical treatment of macromolecular crowding effects (applied to ), see present work has examined the incorporation of complementary ref. 31. charged patches on the crowders, but any approach that strength- ens crowder–crowder interactions without also strengthening I thank Dr. Chris Jarzynski for helping me to understand some basic solute–crowder interactions might be appropriate. Whatever physics. I am thankful for financial support from the Irene Wells Medical approach is adopted, it should be clear that the real inspiration Research Fund administered through the University of Iowa.

1. Minton, A. P. (2001) J. Biol. Chem. 276, 10577–10580. 17. Creighton, T. E. (1993) Proteins (Freeman, New York). 2. Ellis, R. J. (2001) Trends Biochem. Sci. 26, 597–604. 18. Wenner, J. R. & Bloomfield, V. A. (1999) Biophys. J. 77, 3234–3241. 3. Muramatsu, N. & Minton, A. P. (1988) Proc. Natl. Acad. Sci. USA 85, 19. Gabdoulline, R. R. & Wade, R. C. (1996) J. Phys. Chem. 100, 3868–3878. 2984–2988. 20. McCammon, J. A. & Harvey, S. C. (1987) Dynamics of Proteins and Nucleic 4. Verkman, A. S. (2002) Trends Biochem. Sci. 27, 27–33. Acids (Cambridge Univ. Press, Cambridge, U.K.). 5. Minton, A. P. (2000) Biophys. J. 78, 101–109. 21. Ermak, D. L. & McCammon, J. A. (1978) J. Chem. Phys. 69, 1352–1360. 6. Martin, J. & Hartl, F.-U. (1997) Proc. Natl. Acad. Sci. USA 94, 1107–1112. 22. Kollman, P. (1993) Chem. Rev. 93, 2395–2417. 7. Elcock, A. H., Sept, D. & McCammon, J. A. (2001) J. Phys. Chem. B 105, 23. Jarzynski, C. (1997) Phys. Rev. Lett. 78, 2690–2693. 1504–1518. 24. Egolf, D. A. (2002) Science 296, 1813–1815. 8. Gabdoulline, R. R. & Wade, R. C. (2002) Curr. Opin. Struct. Biol. 12, 204–213. 25. Kauzmann, W. (1959) Adv. Protein Chem. 16, 1–63. 9. Elcock, A. H. (2002) Curr. Opin. Struct. Biol. 12, 154–160. 26. Srere, P. A. (1987) Annu. Rev. Biochem. 56, 89–124. 10. Hartl, F.-U. & Hayer-Hartl, M. (2002) Science 295, 1852–1858. 27. Rivas, G., Fernandez, J. A. & Minton, A. P. (2001) Proc. Natl. Acad. Sci. USA 11. Elcock, A. H. (2002) Biophys. J. 82, 2326–2332. 98, 3150–3155. 12. Gabdoulline, R. R. & Wade, R. C. (1997) Biophys. J. 72, 1917–1929. 28. Zimmerman, S. B. & Minton, A. P. (1993) Annu. Rev. Biophys. Biomol. 22, 13. Northrup, S. H. & Erickson, H. P. (1992) Proc. Natl. Acad. Sci. USA 89, 27–65. 3338–3342. 29. Galkin, O., Chen, K., Nagel, R. L., Elison Hirsch, R. & Vekilov, P. G. (2002) 14. Xu, Z. H., Horwich, A. L. & Sigler, P. B. (1997) Nature 388, 741–750. Proc. Natl. Acad. Sci. USA 99, 8479–8483. 15. Gliubich, F., Gazerro, M., Zanotti, G., Delbono, S., Bombieri, G. & Berni, R. 30. Padilla, J. E., Colovos, C. & Yeates, T. O. (2001) Proc. Natl. Acad. Sci. USA (1996) J. Biol. Chem. 271, 21054–21061. 98, 2217–2221. 16. Zhou, H.-X. & Dill, K. A. (2001) Biochemistry 40, 11289–11293. 31. Kinjo, A. R. & Takada, S. (2002) Phys. Rev. E 66, 031911.

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