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Proc Natl Acad Sci U S A. Sep 20, 2005; 102(38): 13427–13432.
Published online Sep 7, 2005. doi:  10.1073/pnas.0501581102
PMCID: PMC1224613

Dissecting the mechanical unfolding of ubiquitin


The unfolding behavior of ubiquitin under the influence of a stretching force recently was investigated experimentally by single-molecule constant-force methods. Many observed unfolding traces had a simple two-state character, whereas others showed clear evidence of intermediate states. Here, we use Monte Carlo simulations to investigate the force-induced unfolding of ubiquitin at the atomic level. In agreement with experimental data, we find that the unfolding process can occur either in a single step or through intermediate states. In addition to this randomness, we find that many quantities, such as the frequency of occurrence of intermediates, show a clear systematic dependence on the strength of the applied force. Despite this diversity, one common feature can be identified in the simulated unfolding events, which is the order in which the secondary-structure elements break. This order is the same in two- and three-state events and at the different forces studied. The observed order remains to be verified experimentally but appears physically reasonable.

Keywords: all-atom model, force-induced unfolding, Monte Carlo simulation

The 76-residue protein ubiquitin fulfills many important regulatory functions in eukaryotic cells through its covalent attachment to other proteins (1, 2). In many cases, the ubiquitin tag consists of a chain of ubiquitin domains (polyubiquitin), which is formed by linkages between an exposed lysine side chain (Lys-11, -29, -48, or -63) of the last ubiquitin of a growing chain and the C terminus of a new ubiquitin. The fate of a polyubiquitin-tagged protein depends on the linkage. For example, Lys-48-C-linked polyubiquitin marks the protein substrate for proteasomal degradation (3).

Recently, Fernandez and coworkers (4-7) and Chyan et al. (8) investigated the mechanical properties of polyubiquitin by single-molecule force spectroscopy. It was shown that Lys-48-C-linked as well as end-to-end (N-C)-linked polyubiquitin can withstand a stretching force; the average unfolding force was 85 pN for Lys-48-C linkage and ≈200 pN for N-C linkage (4). In these experiments, the polyubiquitin chains were pulled with a constant velocity. In another experiment on N-C-linked polyubiquitin, the stretching force was kept constant (6). At constant force, the fraction of unfolded ubiquitin domains was found to show an approximately single-exponential time dependence, as expected if the unfolding of individual domains is a simple Markovian two-state process. Nevertheless, the unfolding of individual domains sometimes occurred through intermediate states. The precise nature of these different unfolding pathways, including the structure of the intermediate states, remains to be determined. We stress that these intermediates are states along forced unfolding trajectories. To what extent there are significant folding intermediates for small proteins is a debated (9, 10), but different, issue. Here, we use Monte Carlo (MC) simulations to examine the unfolding of ubiquitin under a constant stretching force in atomic detail. Our calculations are based on a simplified force field, which was developed through folding studies of several α-helical and β-sheet peptides with ≈20 residues (11-13). The same model also was used to study the oligomerization properties of a fibril-forming fragment of the Alzheimer's Aβ peptide, with very promising results (14). The model is computationally efficient and allows for the collection of large amounts of unfolding events for ubiquitin, which is important because of the existence of multiple unfolding pathways.

It turns out that the model is able to reproduce key features observed in the above-mentioned experiments. In particular, we find that both one-step unfolding and unfolding through intermediate states occur in our simulations. Having verified this property, we turn to more detailed measurements aimed, in particular, at characterizing the typical intermediate state. This goal is a nontrivial challenge because this state is nonobligatory and is located far away from the native state, so that a few representative unfolding events would be inadequate to characterize it. In a previous study, Li and Makarov (15) used the CHARMM force field and a continuum solvation model to calculate the force required to initiate the unfolding of ubiquitin. The intermediate appears at a later stage of the unfolding process, in a region not explored in their study. An intermediate was, by contrast, observed in simulations of force-induced unfolding of the I27 Ig domain from titin (16, 17). This domain was found to unfold through an obligatory intermediate located close to the native state. Atomic-level simulations of force-induced unfolding also have been performed for other proteins (18, 19), and several groups have used simplified protein representations to study the mechanisms of force-induced unfolding (20-25).

Model and Definitions

The model we use (11-13) contains all atoms of the protein chain, including hydrogen atoms, but no explicit water molecules. It assumes fixed bond lengths, bond angles, and peptide torsion angles (180°), so that each amino acid only has the Ramachandran torsion angles ϕ, ψ and a number of side-chain torsion angles as its degrees of freedom. In absence of the stretching force, the interaction potential

equation M1

is composed of four terms. The term Eloc is local in sequence and represents an electrostatic interaction between adjacent peptide units along the chain. The other three terms are nonlocal in sequence. The excluded volume term Eev is a 1/r12 repulsion between pairs of atoms. Ehb represents two kinds of hydrogen bonds: backbone-backbone bonds and bonds between charged side chains and the backbone. The last term Ehp represents an effective hydrophobic attraction between nonpolar side chains. It is a simple pairwise additive potential based on the degree of contact between two nonpolar side chains. The precise form of the different interaction terms and the numerical values of all of the geometry parameters held constant can be found elsewhere (11, 13). All our simulations were carried out at a fixed temperature of 288 K.

This potential does not require prior knowledge of the native structure, making it a sequence-based rather than Gō-type (26) potential. Computationally, it is almost as fast as a Gō potential, because the most expensive term, Eev, is essentially the same. The terms Ehb and Ehp contain nonnative interactions that are ignored in a simple Gō-type potential. Such interactions may play a role even in force-induced unfolding; in fact, nonnative hydrogen bonds do form, and break, in our ubiquitin simulations (see below). It has been shown (13) that the potential in Eq. 1, despite its simplicity, is able to provide a good description of the structure and folding thermodynamics of several peptides with different native geometries, for one and the same choice of parameters. Ubiquitin is significantly larger than these peptides. However, our present work focuses on unfolding, which is easier to simulate than folding to a unique native state.

Specifically, here we investigate the response of ubiquitin to constant stretching forces equation M2 and equation M3 acting on the C and N termini, respectively. In the presence of these forces, the energy function becomes

equation M4

where E0 is given by Eq. 1 and equation M5 denotes the vector from the N to the C terminus.

Fig. 1 shows the NMR-derived (28) native structure for ubiquitin, which contains a five-stranded β-sheet, an α-helix (residues 23-34), and two short 310-helices (residues 38-40 and 57-59). The organization of the β-sheet is illustrated to the right.

Fig. 1.
Schematic illustrations of the native structure of ubiquitin with our labels for secondary-structure elements, A-E. (Left) A 3D model (Protein Data Bank ID code 1d3z, first model) drawn with rasmol (27). (Right) The organization of the β-sheet. ...

Despite its limited degrees of freedom, the model offers an accurate representation of the experimental structure. A model approximation of this structure was derived by using the auxiliary energy function Ê = E0 + κΔ2, where κ is a parameter and Δ denotes the rms deviation from the NMR structure (calculated over all nonhydrogen atoms). By simulated-annealing-based minimization of Ê, a structure with Δ < 0.5 Å was found. This optimized model structure served as the starting point for all our unfolding simulations.

To get a precise picture of the unfolding process, native backbone hydrogen bonds were identified and monitored in the simulations. A hydrogen bond is considered formed if the energy is lower than a cutoff (13).

To identify unfolding intermediates, we constructed a histogram of equation M6 for each unfolding event, based on measurements taken at regular time intervals. During an unfolding event, r increases essentially monotonically, and significant peaks in the histogram of r are candidates for intermediate states. To reduce noise, a moving average smoothing procedure (29) was used, in which data in each bin were replaced by the average over the three nearest bins. After this smoothing, a cutoff in the height and area of the peaks was used to filter out all but the most significant peaks. Pairs of peaks close in r for which the level between the two maxima does not fall below half the average height of the peaks were combined into single peaks. In most events, all of the bins at intermediate r were so sparsely populated that the above procedure left no peak, which would count as an intermediate, corresponding to two-state unfolding. However, some events were qualitatively different with one or two very prominent peaks at intermediate r.

Our simulations were performed by using MC methods. To avoid large unphysical deformations of the chain, a semilocal update was used for the backbone degrees of freedom. This update, Biased Gaussian Steps (30), works with up to eight backbone torsion angles that are turned in a coordinated manner. Side-chain torsion angles were updated one by one. In addition to these updates of internal coordinates, we also included small rigid-body rotations of the whole molecule. The fractions of attempted backbone moves, side-chain moves, and rigid-body rotations were 24%, 75%, and 1%, respectively.

The system was restarted from the native state as soon as an unfolding event had occurred. The maximum length of a run was 108 elementary MC steps. If this limit was reached, the run was stopped even if the chain was still folded, and a new run was started. The fraction of runs in which the system remained folded after 108 MC steps varied from 65% at 100 pN to 1% at 200 pN.


We study the unfolding of ubiquitin for three different strengths of the applied force, namely 100, 140, and 200 pN, a choice that enables us to directly compare with the experiments at constant force by Schlierf et al. (6). For each force, a set of >500 MC runs was performed, all of which were started from the native structure but with different random number seeds.

Let us first briefly describe the different phases encountered in the simulations. Fig. 2 shows the time evolution of the end-to-end distance equation M7 in two typical runs. All runs start with a rapid relaxation of r from its native value of 3.9 nm to a value rf, which is rf = 4.6 nm for 100 pN and rf = 4.7 nm for 200 pN. Here, the chain ends get stretched out, whereas the rest of the chain remains largely unaffected. This initial adjustment of r is followed by an extended “waiting time,” a phase where r is confined to a small range around rf. In this phase, structural fluctuations occur, but the protein remains native-like with essentially intact secondary structure. Actually, in some runs, the β-sheet temporarily increases in length along the applied force, through the formation of nonnative hydrogen bonds. This phase is terminated by a sudden jump in r. In many runs, the unfolding from the native-like state occurs in a single step, as in run R1 in Fig. 2. However, in agreement with the experiments (6), we also observe several examples of unfolding through intermediate states, as in run R2 in Fig. 2. In all our runs, r shows an essentially monotonic increase with time; no transition from an intermediate state back to the native state was observed. In the final phase of the runs, r fluctuates around a force-dependent mean ru, which increases from ru = 25.2 nm to ru = 25.9 nm as the force increases from 100 to 200 pN. The unfolded value ru may be compared with the value 76 × 0.36 = 27.4 nm for a fully extended 76-residue protein and to the average value 23.0 nm for a worm-like chain (31) with contour length 27.4 nm and persistence length 4 Å (32), at 100 pN and 288 K.

Fig. 2.
MC evolution of the end-to-end distance r in two representative runs. The protein chain unfolds in a single step in run R1 and by means of an intermediate state in run R2. The pulling force is 100 pN in both runs.

For the total step size Δrtot = ru - rf, a value of Δrtot = 20.3 ± 0.9 nm was reported from the experiments (6), which was an average over data at different forces. In our model, we find a weak but steady increase in Δrtot with force: Δrtot = 20.6 nm for 100 pN, Δrtot = 21.0 nm for 140 pN, and Δrtot = 21.2 nm for 200 pN. These results are consistent with the experimental value. However, further experimental data are required to verify the force dependence of Δrtot.

In the experiments, the average unfolding curve showed an approximately single-exponential time dependence, with a rate constant that increased exponentially with force (6). Fig. 3 shows the average of r against time from our simulations. A single exponential provides a reasonable description of the data at 200 pN, whereas higher statistics would be required to study the functional form at 100 and 140 pN. Our fitted time constant at 200 pN is τ = 1.0 × 107 MC steps. At 100 pN, the best way to estimate τ is from the folded population after 108 MC steps (65%), which, assuming a single exponential, gives τ = 2.3 × 108 MC steps. The experiments obtained τ = 0.05 s at 200 N and τ = 2.77 s at 100 pN (6), so unfolding was 55 times faster at 200 pN than at 100 pN. This ratio is a factor of 2 smaller in our simulations.

Fig. 3.
End-to-end distance r against MC time for the three different forces studied. Each data point represents an average over all the runs for a fixed force. The curves are fitted single exponentials.

The time behavior of the experimental data is consistent with a simple two-state picture. Nevertheless, unfolding intermediates were observed in the experiments. Schlierf et al. (6) saw intermediates in ≈5% of their 800 unfolding events, recorded at different forces. The most common distance between the initial and intermediate states was Δr = 8.1 ± 0.7 nm.

Although the steps are not as sharp as in the experiments, unfolding proceeds in a clear step-wise fashion in the simulations as well (see Fig. 2), and a useful operational definition of intermediate states can be easily devised (see Model and Definitions). To avoid the initial and final states, we restrict our analysis of intermediate states to the range 6.5 nm ≤ r ≤ 18.5 nm. Here the upper limit is rather conservatively chosen. The reason for this choice is that the last part of the unfolding process is somewhat slow in the simulations (see Fig. 2), which prevents us from unambiguously identifying intermediates with r > 18.5 nm.

All our unfolding events contain either zero, one, or two intermediate states. The relative frequencies of two-, three-, and four-state events are shown in Table 1. For all three force magnitudes, two-state events are more common than three-state events, which in turn are more common than four-state events. The fraction of events with intermediate states is comparable with the above-mentioned value of 5%, which was an average over experiments at different forces. From Table 1 we also note that intermediates occur more frequently as the force gets lower. This force dependence was not investigated experimentally.

Table 1.
The total number of observed unfolding events and the respective fractions of two-, three-, and four-state events for the three different forces studied

Fig. 4 shows the distribution of r for the intermediate states we observed at 100 pN. The distribution is sharply peaked around a typical unfolding step of Δr ≈ 7 nm, which is slightly lower than the most common step size in the experiments (8.1 ± 0.7 nm). This deviation could indicate that the stability of the intermediate state is somewhat low in the model, so that we lose this state before it reaches its optimal orientation. However, the deviation is small, and one should keep in mind that the experimental value represents an average over experiments under varying conditions. Our step-size distribution at 140 pN (data not shown) is similar to that in Fig. 4 but noisier because of fewer events. For scarcity of events, the step-size analysis is not meaningful for 200 pN. Neither is it possible to draw any statistical conclusions specifically about four-state events. We note, however, that the intermediate state with Δr ≈ 7 nm occurs in our four-state events as well (see Fig. 4). Further, this intermediate state has the longest lifetime among the observed intermediates (data not shown).

Fig. 4.
Histogram of r for intermediate states at 100 pN. For this force, we observed 43 three-state events and 4 four-state events. The light gray and dark gray parts of the bars correspond to intermediates from three- and four-state events, respectively.

To check whether the character of the unfolding event depends on the waiting time, we divided the 183 unfolding trajectories for 100 pN into two groups of 93 and 90 events. Those in the first group unfolded in the first third of the maximal simulation time. The second group had waiting times greater than that. Both the frequency of occurrence of intermediate states and their typical location in r are very similar between these groups, indicating that the unfolding behavior is largely independent of the waiting time.

We now turn to a more detailed description of the unfolding process. To delineate the unfolding process, we follow five key elements of the native structure, labeled A-E and indicated in Fig. 1. The structure A is the α-helix, and B, C, D, and E are the four different pairs of adjacent strands in the β-sheet. The experiments mainly focused on the end-to-end distance and therefore provide only limited information about the structure of the intermediate states. However, Schlierf et al. (6) proposed a three-state scenario, based on the observation that the ubiquitin sequence can be split into two halves that correspond to well-defined clusters packing against each other in the native structure. The first cluster includes the structures A and B, and the second cluster includes the structures D and E (C has one strand in each cluster). The unraveling of the second cluster would give an intermediate unfolding step of ≈8 nm, which agrees with the most common step size in the experiments (6). In this scenario, the intermediate is composed of the structures A and B.

To examine the order in which the structures A-E break, we study the presence of native backbone hydrogen bonds as a function of r. To this end, simulated conformations were divided into different intervals in r. For each interval, the frequency of occurrence of the different hydrogen bonds was computed. Fig. 5 shows the result of this calculation for all of the native backbone hydrogen bonds in the structures A-E at 100 pN. From this figure, it is immediately clear that the structures do not break in a random order but instead in a statistically preferred order, namely CBDEA. That C breaks first is expected, because the rest of the structure is shielded from the action of the pulling force as long as C is in place (see Fig. 1). The structures C and B tend to break below the typical r for intermediate states, rI ≈ 12 nm (see Fig. 4), whereas D, E, and A tend to break above rI. The data in Fig. 5 thus suggest that the typical intermediate is composed of A, D, and E rather than A and B. An analogous analysis using native contacts also was performed, with very similar results.

Fig. 5.
The frequency of occurrence as a function of the end-to-end distance r for all native backbone hydrogen bonds in the respective structures A-E at 100 pN. Each curve represents one hydrogen bond. The numbers of native backbone hydrogen bonds are 8, 4, ...

Fig. 6 shows the results obtained when performing the same analysis as in Fig. 5 for two- and three-state events separately. For clarity, Fig. 6 does not show data for individual bonds but only averages for the different structures. Overall, the curves for two- and three-state events are similar in shape, which in particular suggests that the structures A-E break in the same order in both cases. The biggest difference we see is for structure D, which tends to break just above rI.At rI, D is essentially intact in three-state events, whereas one or two of its hydrogen bonds typically are missing in two-state events. This difference strongly indicates that structure D plays a crucial stabilizing role in the typical intermediate state. A small difference between two- and three-state events also can be seen in the results for structure B; remnants of B are present near rI in two-state events but not in three-state events. The results for the three structures A, C, and E show, by contrast, no significant differences between two- and three-state events.

Fig. 6.
The fraction of formed hydrogen bonds as a function of r for the structures A-E for two-state (solid lines) and three-state (dashed lines) events, at 100 pN. Each curve represents an average over all of the native backbone hydrogen bonds of a given structure. ...

The analysis of Figs. Figs.55 and and66 does not tell how strong the statistical preference is for the unfolding order CBDEA. To quantify this preference, we directly analyzed the time of breaking of structures A-E and determined a path (a permutation of ABCDE) for each individual event. For this purpose, we regard a structure to be intact if one-third or more of its native backbone hydrogen bonds are present. To filter out short-term thermal fluctuations and thereby focus on genuine long-term changes due to unfolding, we define the time of breaking of a structure as the last point in time after which the structure is never found intact.

In the 100-pN case, we find that 61% of the events follow the path CBDEA unambiguously. Another 23% of the events have the order of B and D apparently interchanged, the path being CDBEA. In these events, B does unfold before D but then partially reforms after D is gone, so that the definition above assigns a later breaking time for B. This partial refolding can actually be seen in Fig. 5; the β-hairpin B is almost completely dissolved at rI, but two of its four hydrogen bonds (those closest to the turn) occur again around r = 14 nm with a nonnegligible frequency. The partial refolding of B takes place just after D breaks and is possible because a large chain segment is released as D breaks. As this segment gets stretched out, B dissolves again. The reformation of B is not a step back toward the native state, because when B reforms, D is gone, and the system has a larger end-to-end distance. Hence, this class of events can be regarded as a minor variation of the path CBDEA, which means that 84% of our unfolding events follow the same basic pathway. All other of the 120 possible paths have probabilities of <5%. Given the uncertainties due to the intrinsically somewhat arbitrary definition of a path, these low-probability paths were not analyzed any further. As expected from Fig. 6, separate analyses of two- and three-state events showed that the fraction of events following the main pathway is similar in both cases.

To check the force dependence of our findings, the analysis of Figs. Figs.55 and and66 as well as the path analysis of individual events were repeated for 140 and 200 pN, with results similar to those at 100 pN. Although intermediate states occur more frequently at low force, we thus find that the order in which the structures break is the same at the different forces.

That the α-helix A survives up to very large r hints at the possibility that its intrinsic stability could be very high. Therefore, we performed equilibrium simulations of the α-helix segment in isolation. We found that this excised segment makes an α-helix in the model, which unfolds at ≈15 pN. This force is much smaller than those studied above, which implies that the effective force felt by the α-helix in the ubiquitin simulations must be small compared with the applied force.

Finally, Fig. 7 shows a snapshot of the typical intermediate state, taken from run R2 in Fig. 2. The structures A, D, and E are still present, whereas B and C are missing. Structure B, which together with A forms the intermediate in the above-mentioned three-state unfolding scenario of Schlierf et al. (6), breaks early in our simulations.

Fig. 7.
The native structure and a snapshot illustrating the shape of the typical intermediate state in our simulations. The chain segment that makes structure B, a β-hairpin, is marked in black.

Summary and Discussion

We have performed a MC study to elucidate at the atomic level the unfolding behavior of ubiquitin under a constant stretching force. The study consists of two parts. First, we investigated the behavior of the end-to-end distance, as was done in the experiments. Properties such as the size of the unfolding step, the frequency of occurrence of intermediate states, and the position of the typical intermediate state all were found to be in reasonable agreement with experimental data. For the range of forces studied here, we further found that the size of the unfolding step increased with force, whereas intermediate states occurred more frequently at lower force. These dependencies on the strength of the applied force remain to be tested experimentally.

In the second part of our study, we analyzed the order of breaking of five major secondary-structure elements. This analysis revealed that these structures, A-E (see Fig. 1), tend to break in a definite order. Disregarding interchanges of B and D due to the partial refolding of the β-hairpin B, we found that >80% of the events followed the same unfolding path, CBDEA. The order was the same in events with and without intermediate states, and it did not change between the three different forces studied.

To what extent this predicted unfolding order is correct is not obvious, given the lack of detailed experimental data to examine it. In fact, as mentioned in Results, a markedly different unfolding scenario has been proposed (6). Therefore, it should be stressed that several aspects of the calculated unfolding behavior can be understood in terms of topology and pulling geometry. That C breaks first is inevitable; the other parts cannot sense the force until C is broken (see Fig. 1). The native state is mechanically resistant because C is pulled longitudinally, so that several bonds must break simultaneously. Once C is gone, nothing keeps the β-hairpin B from unzipping, one bond at a time; unzipping requires less force than separation by longitudinal pulling (33, 34). Similarly, for the three structures A, D, and E, which form the typical intermediate state, it is clear that the β-hairpin E is protected by D (see Fig. 7), because the two strands of D are on different sides of E along the sequence. Visual inspection of snapshots from the simulations suggests that D is pulled in a direction neither parallel nor perpendicular to its β-strands, making it semiresistant, which is consistent with our conclusion that D is important for the stability of the intermediate (see Fig. 6). When D is gone, E is free to unzip. Compared with unwinding the α-helix A, the unzipping of E requires less force, which makes E break before A.

This picture of the unfolding process highlights the importance of topology and pulling geometry. These two factors make the four β-sheet structures B-E behave very differently. C and D play important stabilizing roles in the native and typical intermediate states, respectively. B and E are, by contrast, easy to pull apart and survive only as long as they are protected; they do not cause any traps on the unfolding pathway. Previous studies (4, 34) have shown the importance of pulling geometry, in relation with native topology, as a determinant of a protein's mechanical stability. We extend that picture and show that geometrical and topological factors play important stabilizing roles along the entire unfolding pathway.

Our study was based on nonequilibrium simulations but contains a sufficient number of unfolding events to speculate, nevertheless, on the underlying free-energy landscape. Of particular interest is the typical intermediate state with Δr ≈ 7 nm. Our step-size and secondary-structure analyses strongly suggest that this unfolding intermediate corresponds to a local free-energy minimum with a rather well-defined 3D structure. It is worth stressing that in the limit of zero force, this minimum might very well be irrelevant, because the smallest force studied, 100 pN, is still large enough to strongly tilt the energy landscape (100 pN × 7 nm ≈ 100 kcal/mol).

The folding and unfolding of ubiquitin at zero force has been studied extensively by using both chemical denaturants (35) and temperature denaturation (36, 37), and the flexibility of the native state also has been examined (38). In a study of thermal unfolding based on IR spectroscopy, Chung et al. (37) found that the β-strands I and II are more stable than the β-strands III and V, which contrasts sharply with our results. It thus appears that the pathways followed in thermal and force-induced unfolding of ubiquitin indeed are different.

The physical forces governing the force-induced unfolding of proteins are the same as those governing protein folding and aggregation, which makes it tempting to search for a unified computational approach to these phenomena. The model used in the present calculations has been used previously to study folding (13) as well as aggregation (14). All of the parameters of the model could be kept unchanged between these three different studies. This fact does not, of course, imply that the model is perfect, but it is nevertheless encouraging. In particular, it strongly suggests that computational studies of force-induced unfolding could complement detailed experimental studies to provide a better physical picture of the mechanical unfolding of different proteins. Such studies also could provide useful information for refining the force field, which in turn would improve our understanding of protein folding and aggregation.


We thank Michael Schlierf for useful discussions. This work was supported in part by the Swedish Research Council. The computer simulations were performed at the LUNARC facility at Lund University.


This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: MC, Monte Carlo.


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