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-Type PotentialDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts
Correspondence: Address reprint requests to Eugene Shakhnovich, E-mail: eugene{at}belok.harvard.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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-type potential to study the mechanical unfolding pathway of ubiquitin. A detailed, robust, well-defined pathway is found, confirming existing results in this vein though using a different model. Additionally, we identify the protein's fundamental stabilizing secondary structure interactions in the presence of a stretching force and show that this fundamental stabilizing role does not persist in the absence of mechanical stress. The apparent success of simulation methods in studying ubiquitin's mechanical unfolding pathway indicates their potential usefulness for future study of the stretching of other proteins and the relationship between protein structure and the response to mechanical deformation. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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) and unfolding rates (2), to the larger body of work on the protein folding problem.
Ubiquitin was selected for study in this context as a result of its small size and the substantial body of experimental work that has focused on it. This 76-residue protein (see Fig. 1 and Table 1) is primarily involved in marking other proteins that have been targeted for degradation within a cell. Its thermal, chemical, and pressure-induced unfolding have been investigated in detail; in particular, it has been shown that ubiquitin unfolds thermally at a temperature of 83°C (3,4). Ubiquitin's folding transition state ensemble has been described and has been shown to contain conformations with a common native-like core (5). Additionally, a "native-like intermediate" has been observed during the molecule's low-temperature refolding, though the structure of the intermediate was not ascertained (6).
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–14). Although the results thereby produced are valuable in providing insight into the mechanical properties of polypeptides, they present difficulties at the interface with simulation. In the case of Monte Carlo simulation, the notion of an unfolding velocity as a rate of change with respect to time is not well-defined. Thus, because the aforementioned experiments were performed at constant unfolding velocity with variable stretching force, the specification of a potential that will generate the appropriate behavior is not straightforward.
Fortunately, advances in experimental methodology have in recent years allowed the execution of single-molecule mechanical unfolding under constant force. These techniques were of course first applied to proteins such as titin that had already been the focus of study (15). However, experimental work focused on the constant-force mechanical unfolding of ubiquitin has also emerged (16,17). These experiments measure the end-to-end distance of single chains of ubiquitin domains subjected to constant stretching forces. Although the minimum force at which ubiquitin unfolds has not been precisely determined, it has been shown to lie between 
50 pN and 200 pN (17). When the polypeptide is subjected to forces within this range, the unfolding event for a single ubiquitin domain is generally marked by a sharp two-state step in end-to-end distance of 20.3 ± 0.9 nm. However, for 5% of 821 observed events, three-state unfolding occurred, with steps of size 8.1 ± 0.7 nm and 12.4 ± 1.0 nm (17).
Such experimental work has provided a basis for the development of models and accompanying simulations of the mechanical unfolding process. Molecular dynamics has been applied in this context using models and yielding results having varying degrees of detail (8,18–21). To our knowledge, molecular dynamics has not yet been used to study the mechanical unfolding pathway of ubiquitin. Monte Carlo simulation has also been employed in the investigation of mechanical unfolding, though in general to test relatively coarse kinetic models (2,17,22). In particular, Szymczak and Cieplak have recently used a coarse G-like model to examine the statistics of the mechanical unfolding times of ubiquitin and integrin (23). Furthermore, Paci et al. have made a number of contributions to the investigation of the mechanical unfolding of a number of different proteins using molecular dynamics, G-type models, and other methods, in general to the end of examining molecules' energetic and mechanical properties such as resistance to force (24–28). More in the vein of this article, recent work by Irbäck et al. has examined the unfolding of ubiquitin under a stretching force using all-atom Monte Carlo simulation with a sequence-based potential (29). That work predicts an unfolding pathway for the protein, and so our results may be considered complementary.
We have used all-atom Monte Carlo simulation with a G-type potential to elucidate in detail the mechanical unfolding pathway of ubiquitin. The computationally tractable nature of this simulation method has allowed us to obtain robust results that hold at different forces across a large number of simulations. Our model gives good qualitative agreement with experimental results for a number of measurable aspects of the process and provides an explanation of the underlying manner in which it may occur. Additionally, our proposed unfolding pathway agrees with that proposed by Irbäck et al. (29) despite our use of a fundamentally different potential, thereby further confirming the robustness of these results. In contrast to such previous studies of the ubiquitin unfolding pathway, we have used a G-type model that guarantees that the ground state of the model is in fact the protein's native state, thus rendering our simulations a truly equilibrium study of the adiabatic unfolding pathway.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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,31) in constructing our simulations. All nonhydrogen atoms in the residue chain were represented as hard spheres having radii proportional to their van der Waals radii; a proportionality factor of 0.75 was used. No hydrogen atoms or solvent (e.g., water) molecules were explicitly included in the model. At each Monte Carlo step, one backbone and ten side-chain moves were performed. Only local backbone moves and torsional side-chain rotations were permitted, and moves were accepted or rejected based on the Metropolis rule (through which the simulation "temperature" T entered). Each backbone move consisted of a small random rotation of the 
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angles of up to three nearby residues. Thus, although these moves were not strictly enforced to be wholly local, the majority of them resulted in largely local changes. Please see Shimada et al. (30) for more details regarding the structure of the Monte Carlo simulations. Because we only studied unfolding, all simulations were started with ubiquitin in its native state as specified by the structure given by Protein Data Bank code 1UBQ (32).
The potential used has the form
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is a G-type (33) square well contribution, EH is a backbone hydrogen bonding contribution, and EF is the force term. For any atom a, let #(a) denote the index of the residue of which a is a member. Thus, with k 
0 a constant,
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ab and interatom distance r,
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(i, j) = –1 if i, j are in contact (i.e., ab 
r < 
ab) in the native state and (i, j) = 1 otherwise. The hard core distance ab is simply the sum of the aforementioned hard sphere radii for atoms a and b, and the value is a constant. We use = 1.8 in accordance with the findings of Shimada et al. (30). The backbone hydrogen-bonding contribution to the potential was included primarily to allow tuning of secondary structure stability and has the form
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is a vector extending from the N-terminal nitrogen atom to the C-terminal carbon atom of the backbone. The vector 
is the Monte Carlo stretching force; in our simulations, remains constant. Note that these definitions imply that the stretching force is applied at the terminal amino acids of the protein, in accordance with experimental methodology (16,17).
To complete the energy model's specification, it is necessary to calibrate it to the observed thermodynamics of ubiquitin (in the absence of a stretching force) by selecting values for the constants k and h such that the appropriate relative stabilities of the entire protein and its constituent secondary structures across the temperature range of interest are achieved. Let Tc be the Monte Carlo temperature at which ubiquitin transitions thermally from its folded to its unfolded state. Thus, Tc corresponds to a physical temperature of 83°C = 356 K (3). To allow comparison with experimental results, we performed our simulations of mechanical unfolding at the Monte Carlo temperature To corresponding to a physical temperature of 21°C = 294 K. Note that To is therefore determined by Tc as To = (294/356)Tc, and we must ensure that our model exhibits the appropriate thermodynamic behavior for T To.
As predicted by AGADIR (34), 
1 (see Table 1 for definitions of secondary structure labels) in isolation (i.e., a free helix with the residue sequence of 1) has helical propensity <1.2% at all temperatures between 273 and 373 K (and pH = 7). Although 2 is predicted to have no helical propensity at any of these temperatures, we will not explicitly consider this fact because the extremely small size of 2 causes it to always have stability significantly less than that of 1. To ensure that an isolated 1 is not stable at To, simulations of length 50 x 106 steps were run for a variety of values of k, h, and T to screen across values of T and determine Tc and 
, the location of the thermal unfolding transition of the isolated 1 helix. Selected results of these screens are shown in Table 2.
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1 is vastly overstabilized when k = 2; this relatively low value of k allows the plethora of local interactions in the helix to dominate its energetics. As expected, this effect is significantly mitigated, though not eliminated, when k is increased to 3. Additionally, increasing k from 2 to 3 does not significantly destabilize the remainder of the protein, because the existence and contribution of local (i.e., i – (i + 2) in this case) interactions outside of the helices in the native state are minimal. However, even with k = 3 (and h = 0), 
. As a result, we introduce h > 0 to further destabilize 1 relative to the remainder of the protein by lessening the stabilizing energetic contribution of the i – (i + 4) hydrogen bonds of the -helix. Although h = 1.5 yields a significant decrease in helix stability, inspection of the resulting thermal unfolding simulation data indicates that this relatively high value of h causes the remainder of the molecule to become excessively unstable. It is worth noting that a similar problem was observed for k 4, even with h = 0. Because h = 0.5 yields 
and increasing h to 0.75 or 1.0 does not give decreased helix stability, we use h = 0.5. As a result, the parameter values selected are
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applied in the direction of the vector 
extending from the N-terminal nitrogen atom to the C-terminal carbon atom of the polypeptide's backbone in the native state. Given this directionality, the stretching force in any simulation can now be specified merely by giving its magnitude 
.
To ascertain the force magnitude Fc at which ubiquitin transitions from its folded to its unfolded state at temperature To, simulations were run on the molecule for 50 x 106 steps with stretching force magnitudes throughout a range of values. The results of this screen across force magnitudes are shown in Fig. 3 and indicate that 13 < Fc < 14; although Fc has not yet been precisely determined experimentally in physical units, it has been shown approximately that 50 pN < Fc < 200 pN (17). Therefore, to examine the mechanical unfolding process, large collections of independent simulations were run at force magnitudes of F = 16 and F = 26. Each simulation was allowed to run until the polypeptide unfolded and subsequently for an additional 5 x 106 steps.
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RESULTS |
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), with the transition from the folded to the unfolded state appearing as a sharp 20 nm step. However, closer inspection (see Fig. 4 b) reveals a subtle intermediate plateau in the value of 
that occurs during the primary unfolding event. Its existence suggests the possibility of an unstable intermediate on the mechanical unfolding pathway. Although the majority of experimentally derived mechanical unfolding trajectories are similar to that seen in Fig. 4 a, three-state unfolding as suggested by Fig. 4 b is observed for 5% of trajectories (17).
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1). More significantly, the unfolding of secondary structure involving the ß-strands is consistently characterized by sharp transitions from the presence of a finite number of native-state contacts to the absence of such contacts, with few fluctuations thereafter. As a result, we define the step in a simulation during which a given secondary structure interaction (e.g., 1–ß1) unfolds to be the step after which the fraction of native-state contacts intact within that interaction remains beneath 0.1 for the remainder of the simulation. Unfolding trajectories for 1 and 2 are not included in Fig. 6 because these structures are consistently the final secondary structure elements to unfold. Their unraveling occurs more gradually in comparison to the aforementioned sharper transitions characterizing interactions involving the ß-strands. In general, when 
nm for ubiquitin (as in Fig. 4), the entire molecule with the exception of the -helices has unfolded; a subsequent more gradual increase of from 20 to 25 nm consists of the subsequent unfolding of 1 and 2.
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2–2 unfolding events are simply exchanged):
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2 unfolds before 1 in all but 4/113 simulations. Furthermore, the final unfolding event of block 1 is ß3–ß5 in 112/113 simulations. Thus, the vast majority of the variability in the sequence of unfolding events arises in the remainder of block 1, as seen in Table 3. Therefore, the mechanical unfolding pathway of ubiquitin is in general as follows, with only minor deviations:
1, ß5, and each other. ß5 may separate from 1 either concurrently or immediately thereafter, while remaining in contact with ß3. The end of this step in the pathway marks the beginning of the plateau seen in Fig. 4 (point I); the resulting structure is given by structure I in Fig. 5. 1 and only subsequently separate from each other. The completion of this step occurs at point III in Fig. 4, with the resulting structure shown in Fig. 5. 1 and 2 unfold. The completion of this final step yields the complete unfolding of the protein and is indicated by becoming greater than 25 nm.
These results and the form of ubiquitin's mechanical unfolding trajectory suggest that structure I is an unstable intermediate in the stretching process; we also expect that structure II is similarly unstable. To validate these expectations, we created two structures corresponding to structures I and II by eliminating secondary structure elements from the original molecule. The first, which we name UBQiA, corresponds to structure I, consists of ubiquitin's residues 21–76, and was formed by removing ß1 and ß2 from the original structure of ubiquitin. The second, UBQiB, corresponds to structure II and was formed by also eliminating ß5, leaving only residues 21–63 of the original ubiquitin structure. Simulations with parameters identical to those of the simulations discussed above (i.e., T = To = 2.1) were performed throughout a range of forces for each structure; the results are shown in Fig. 7. As expected, both structures are unstable at low forces (F < 4 for UBQiA and F < 2 for UBQiB). Furthermore, UBQiB is less stable under a stretching force than UBQiA, as expected from the roles of structures I and II in ubiquitin's mechanical unfolding pathway: the formation of structure I marks the beginning of the plateau in the value of , and the formation of structure II marks its end.
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DISCUSSION |
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1 or the remainder of the ß-sheet. It follows that the interaction between these ß-strands and the other secondary structure of the protein provides a fundamental stabilizing contribution in the presence of a stretching force. This conclusion is further supported by our examination of the structures UBQiA and UBQiB, which are drastically destabilized under a stretching force as a result of their lack of ß1, ß2, and ß5 (in the case of UBQiB). Interestingly, the seemingly fundamental stabilizing energetic contribution of these ß-strands to the protein is only essential to its stability in the presence of a stretching force. Simulations of UBQiA and UBQiB at various Monte Carlo temperatures in the absence of a stretching force indicate that both structures remain stable at temperatures up to 2.2 (see Fig. 8). Although this value is less than Tc = 2.6, the difference between these transition temperatures is sufficiently small to allow us to conclude that the removed secondary structure elements and their interactions with the remainder of the protein are not fundamental to the stability of ubiquitin in the absence of a stretching force. Rather, the small amount of observed destabilization is presumably largely the result of a decrease in the number of bonds (or contacts, in our case) that must be broken to unfold the structure thermally.
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-helices unfold gradually in comparison to the remainder of ubiquitin's secondary structure is also of interest. This observation is at first glance somewhat surprising given that, as seen above, the helices have negligible helical propensity in isolation within the temperature range of interest in the absence of a stretching force. However, the timescales and simulation lengths characteristic of thermally induced unfolding are significantly greater than those for force-induced unfolding. Furthermore, the structure of the helices and the interactions within them differ greatly from the structure and interactions of the ß-sheet. The vast majority of the interactions within the ß-sheet and between ß-strands and 1 are nonlocal. Additionally, the number of contacts per atom is on average less for atoms in the ß-sheet than for atoms in the -helices. The helices are in contrast characterized by dense local interactions. These physical considerations provide some insight into the causes of the more gradual unfolding observed for the helices even after the remainder of the protein has unfolded. It is also possible that the helices are to some extent shielded from the applied force by the presence of a significant portion of the amino acid chain between their ends and the terminal residues of the protein. Finally, it must be noted that the unfolding behavior of the helices could simply be an artifact of the model and potential chosen, with the helices resultantly overstabilized.
As seen above, our computationally derived mechanical unfolding trajectories for ubiquitin yield good qualitative agreement with those that are observed experimentally. The unfolding transition is in both cases generally characterized by a sudden 20 nm step. However, upon examining the simulated transition at higher resolution—which is not currently possible experimentally—we observe a plateau in the value of when 
nm. Analagously, three-state unfolding is seen in 5% of experimentally derived trajectories (17). In these cases, a marked plateau in the value of occurs when 
; this value is an average over events observed in a large collection of experiments. Our results in this vein are thus within the realm of quantitative agreement with experiment, thereby providing a potential explanation of the empirically observed three-state unfolding: this phenomenon is observed in the relatively few instances in which an unstable intermediate consisting of structure I transitioning to structure II persists for a detectably long period of time.
In addition to being consistent with experimental observations, our results are complementary to and provide confirmation of existing computational results regarding the mechanical unfolding of ubiquitin. Also using all-atom Monte Carlo simulation but with a sequence-based potential, Irbäck et al. (29) have studied this process. Our work differs fundamentally in that we use a G-type rather than a sequence-based potential, thus ensuring that our model's ground state is the protein's native state; as a result, our simulations do in fact constitute an equilibrium study of the unfolding pathway. We have nevertheless obtained similar results regarding the qualitative properties of unfolding trajectories, the location of the plateau in the value of , and the form of the mechanical unfolding pathway. Therefore, we have strikingly demonstrated the robustness of these findings.
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CONCLUSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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-type potential, we have elucidated a detailed and robust mechanical unfolding pathway for ubiquitin. Our results also identify the protein's essential stabilizing interactions under a stretching force and suggest the structure of an unstable intermediate on the pathway. In addition to obtaining good agreement with and a possible explanation for experimental observations, we have confirmed a number of existing results in this vein, though using a fundamentally different energy model.
As applications of all-atom simulation methods such as those presented here to the detailed study of the mechanical unfolding of proteins are only now emerging, rich opportunities for future work exist. In particular, it will be interesting to examine other proteins such as titin, which, unlike ubiquitin, has a role in natural processes that involves mechanical deformation. Consideration of other polypeptides will allow a more thorough assessment of the structural properties that determine the behavior of a protein under a stretching force. Furthermore, the simulations presented here can be extended to probe other interesting features of the process, including the effects of applying the stretching force at nonterminal amino acids and the results of allowing a protein to refold under low force after it has been unfolded at a higher force. These variants of the mechanical unfolding process have already been examined experimentally (8,16) and have bearing on the mechanical unfolding of proteins as it occurs in biological systems.
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ACKNOWLEDGEMENTS |
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Submitted on January 17, 2006; accepted for publication November 3, 2006.
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= 3.847 nm.
