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A Directed Essential Dynamics Simulation of Peptide Folding

Changjun Chen ^*, Yi Xiao ^* and Linsen Zhang 

^* Biomolecular Physics and Modeling Group, Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China; and Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, Hubei, China

Correspondence: Address reprint requests to Yi Xiao, E-mail: yxiao{at}mail.hust.edu.cn.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
We present a directed essential dynamics (DED) method for peptide and protein folding. DED is a molecular dynamics method based on the essential dynamics sampling and the principal component analysis. The main idea of DED is to use principal component analysis to determine the direction of the most active collective motion of peptides at short intervals of time (20 fs) during the folding process and then add an additional force along it to adjust the folding direction. This method can make the peptides avoid being trapped in the local minima for a long time and enhance the sampling efficiency in conformational space during the simulation. An S-peptide with 15 amino acids is used to demonstrate the DED method. The results show that DED can lead the S-peptide to fold quickly into the native state, whereas traditional molecular dynamics needs more time to do this.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Protein folding is one of the most important and unresolved problems in life science. It has been studied for many years, and many methods for investigating it have been developed, from lattice to all-atom ones (Boczko and Brooks, 1995; Gutin et al., 1995; Irback et al., 2000; Kaya and Chan, 2000; Shakhnovich and Gutin, 1993; Takada et al., 1999). After decades of extensive research, some comparably sophisticated theories have been proposed. Among them, the folding funnel theory is widely accepted (Bryngelson et al., 1995; Leopold et al., 1992; Mirny and Shakhnovich, 2001). According to this theory, proteins go down along the energy landscape during the folding process and quickly approach their native states.

Many folding/unfolding simulations have been carried out on proteins at the atomic level (Gsponer and Caflisch, 2002; Jaenicke, 2000; Knapp-Mohammady et al., 1999; Luque et al., 1996; Majeux et al., 2001; Galzitskaya et al., 2000; Pande and Rokhsar, 1999; Zagrovic et al., 2001; Zhang et al., 2001). These studies enable a thorough understanding of protein folding/unfolding. One of the important results is that both folding and unfolding processes of proteins go through a few conformational clusters (so called local minima) before they arrive at their native states. In the past, most of the simulations of protein folding were done under traditional molecular dynamics (MD). It was found that traditional MD is not very efficient for exploring conformational space because it spends too much time on staying in local minima and is hard to find an outdoor to get out (Pande and Rokhsar, 1999). So to get a full folding path of a protein, simulations must be carried out from different initial structures to enhance the probability of getting out of local minima and reaching the native state, or simply increase the simulation time. For example, Duan and Kollman have made a successful 1-µs simulation at 300 K for a 36-residue peptide, and the final structure of the peptide was very close to the native state (Duan and Kollman, 1998). This encourages using simulation to study protein folding.

In the last 10 years, many methods have been proposed to improve sampling efficiency on energy landscape. A lot of them are based on the normal mode analysis (NMA; Brooks and Karplus, 1983; Go et al., 1983; Levitt et al., 1983) or the principal component analysis (PCA; Amadei et al., 1993; Balsera et al., 1996; Kitao and Go, 1999; Teeter and Case, 1999; Zhang et al., 2003). In NMA, the potential energy is approximated as harmonic terms and normal modes are obtained by diagonalization of the Hessian matrix of a structure, which stays in a local minimum. In other words, NMA describes the atomic motions in terms of single frequency modes within a harmonic energy minimum, and so it can give important information on protein structure, such as flexibility and stability. However, since NMA assumes that the system is harmonic, when dealing with anharmonic systems, like peptides and proteins at ordinary temperatures, it is not really useful. Furthermore, the Hessian matrix is calculated from the second derivation of the potential energy described by an all-atom force field, and so NMA costs too much CPU time.

PCA provides a more useful tool to study protein folding. It does not use any harmonic assumption and simply provides those collective degrees of freedom, in general not at all associated to a single frequency, which decompose the whole conformational fluctuation into the largest-smallest components. It is known that protein structure is always represented by three-dimensional Cartesian coordinates, and so the degrees of freedom of a protein are very large (i.e., 3N – 6, where N is the number of atoms). It is difficult to explore the energy landscape in this multidimensional space by traditional MD. PCA tries to solve the problem by generating new 3N – 6 vectors from those Cartesians coordinates. Then we can pick out the main collective degrees of freedom necessary to describe dynamics of protein folding. These special collective degrees of freedom span an essential subspace, and all the conformations in the MD trajectories can be projected into it and form a visual image of the folding process. A lot of researches have reported that this subspace can reflect global motion of proteins with a high approximation. Using some special sampling techniques in this space, it is possible to explore the energy landscape more extensively and get a general view of protein folding. In the last 10 years, there were some successful applications of PCA in protein simulations (Amadei et al., 1993; De Groot et al., 2001; Elmaci and Berry, 1999; Garcia, 1992; Hayward et al., 1995; Ota and Agard, 2001; Van Aalten et al., 1995), which exhibit that PCA is perspective and can help us to investigate the protein folding problem in depth.

Essential dynamics sampling (EDS; Amadei et al., 1993, 1996) is one of the simulation methods based on PCA. In EDS, a simulation is done at first (typically lasting for nanoseconds), and the principal components or collective degrees are derived from it. Then these principal components can be used to lead protein folding in subsequent normal simulation. Recently Daidone et al. (2003) studied protein folding in explicit water with EDS and obtained a full folding path for a real protein (cytochrome c) successfully. EDS is also robust in analyzing relative motion, for example, hinge-bending between domains in protein (Cregut et al., 1998; Van Aalten et al., 1998). But it must be noted that the principal components may vary significantly during protein folding, i.e., when the conformations undergo large transformations. So, when applying EDS to protein folding, we should pay more attention to conformation change.

Recently some improvements to EDS have been reported for protein folding. For example, Zhang et al. have proposed a useful technique named amplified collective motion (ACM; Zhang et al., 2003) to supplement EDS. They analyzed the recorded trajectory and obtained normal modes at short intervals. Then they coupled the normal modes in different subspaces with different thermal baths. The normal modes in the essential subspace are coupled to a higher temperature bath, and the rest are coupled to a normal temperature bath. This method is effective in practice and enables short peptides to fold into the native state successfully, but there is one thing to be considered: in this method all the normal modes are obtained from the anisotropic network model (Atilgan et al., 2001). The anisotropic network model is an elastic network model and can be seen as a reduced model of NMA. It treats protein structures as many beads connected by harmonic springs. The potential energy of this system is very different from that of an all-atom force field and is related with cumulated harmonic energies of residue pairs. So the normal modes derived from this model may not be consistent with all-atom models, especially with protein structures far away from the native state.

In this study, we present a new method for peptides and protein folding, i.e., a directed essential dynamics (DED) analysis, which is motivated by EDS and ACM. The main idea of DED is as follows: we hope to make the peptide move away from the local minima quickly and try to use the most active collective degrees of freedom to do this. EDS tells us that the dynamics of the molecule can be described by some essential collective degrees of freedom, whereas PCA provides a method to obtain these essential degrees. In practice, we use PCA to capture the essential motions from the most recent structures in the trajectories at short intervals (20 fs) during protein folding. Then, instead of coupling with different temperatures as in ACM, we add a weak force on the peptide along the resultant direction of these essential motions to direct the folding of the peptide. The main difference between EDS and DED is the simulation time used to obtain collective motions. In EDS, nanosecond simulations are typically used, whereas in DED only 20 fs are utilized. This is because EDS and DED have different aims. EDS wants to get the collective degrees of freedom of peptide in its native state accurately to use them as acceptance conditions in the subsequent folding or unfolding simulation to guide the molecule moving toward or away from the native state and thus enhance the sampling efficiency. The aim of DED is to use the most active collective degrees of freedom of peptide in the unfolded state to guide the molecule to jump out of local minima or across barriers. So it does not need the information of the collective motions in the native state but the most flexible collective degrees of current conformation at intervals. The intervals should not be too large because the structures in the intervals would vary too much and the calculated collective degrees would not be suitable for current conformation. We have applied DED to the folding simulation of S-peptide and analyzed typical physical characteristics, such as folding path, hydrogen bond formation, and so on. The results show that DED is efficient in exploring conformation space and provides a novel tool to study protein folding.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Now let's have a brief review on PCA. It works as follows: When we have done a folding simulation of a peptide, we can get a covariance matrix C from the trajectory X(t). The element of the matrix C is determined by

(1)

where donates the average over all the structures sampled in the trajectory and is the ith Cartesian coordinate of the conformation at time t. So the covariance matrix C represents the correlation between atomic motions in Cartesian coordinate space. Then the collective degrees of freedom can be derived from the eigenvectors of the diagonalized matrix C.

Since C is symmetric, it can be diagonalized by using an orthogonal transformation matrix T:

(2)

where is a diagonal matrix and every diagonal element in it is an eigenvalue of the covariance matrix C, and its corresponding eigenvector is the ith column in the matrix T. Each eigenvalue and eigenvector pair represents one collective motion. The eigenvectors corresponding to larger eigenvalues represent more flexible directions in the trajectory. So the eigenvectors with the largest eigenvalues will hold most of the characters of the motions or, in other words, all the complex motions of protein can be reduced to only approximately a few collective motions.

Based on this idea, DED determines the collective motions at short intervals during the simulation. We analyze the trajectory every 20 fs and construct a covariance matrix for all the nonhydrogen atoms immediately. This means that we only consider the collective motions of nonhydrogen atoms. This is because these "heavy atoms" can describe the conformation of the molecule adequately and furthermore can reduce the calculation time. We choose 20 fs because unfolded states are not stable and the conformations of S-peptide change quickly. So we select this short interval to make a reliable evaluation on the collective motions of the current conformation.

Then we pick out the six eigenvectors (as collective coordinates) corresponding to the six largest eigenvalues in the covariance matrix and build a new vector by the linear combination of these eigenvectors with their eigenvalues as the coefficients

(3)

where c is

(4)

is considered as the principal component of the motions in the local state, and the peptide is directed to move along this direction by adding a weak force:

(5)

where is a coefficient. Here we select only the first six eigenvectors because we found that the sum of the first six eigenvalues is always more than 90% of the sum of the total eigenvalues. So we assume that the first six eigenvectors are enough to describe the peptide motion. Furthermore, after many tests, we find that, when equals 5.0 kcal/mol Å, the force is strong enough to pull the molecule and, at the same time, is weak enough to avoid distorting the molecule. Since the force breaks contacts quickly, it can increase the probability of the molecule to jump out of the local minima. Thus the simulation based on DED may overcome the difficulty met by traditional MD.

It must be pointed out that, to increase the sampling efficiency, the particular choice of the number of eigenvectors used and the linear combination in Eq. 3 are not unique. Our simulation results show that any choice of the number of eigenvectors and the linear combination can give a similar effect, i.e., any steering that breaks and forms contacts quickly can lead the peptide fold correctly and rapidly. However, we suggest using the eigenvectors corresponding to the first largest eigenvalues to investigate the folding pathway of the molecule. This is because the first modes are the most flexible motions and adding forces to them can avoid disturbing the original folding pathway of the molecule too much. If we apply forces to the less flexible direction, we disturb the original motion of the molecule significantly although we still can enhance the sampling efficiency.

The model system we used in this study is an S-peptide, which forms a helix in proteins (Søgaard et al., 1999; Tirado-Rives and Jorgensen, 1991). It is selected from 15 N-terminal amino acids of ribonuclease A and capped with the acetyl and N-methyl groups: KETAAAKFERQHMDS.

In this study, we use a GA/SA model (Qiu et al., 1997; Still et al., 1990) as an implicit solvent model to simulate the aqueous environment. GB/SA is a reduced model from the continuum model, which treats the water as a continuous medium, and there are usually three terms included in the free energy of solvation:

(6)

where is a solvent-solvent cavity term corresponding to the free energy of creating a cavity of solute in the solvent continuum; is the free energy term representing the interactions between the solute and solvent; and denotes electrostatic interactions between the solute and solvent.

The advantage of this model is that it need not treat solvent molecules explicitly and costs much less time, although some accuracy may be lost. The MD software we used is Tinker (see http://dasher.wustl.edu/tinker/) with Charmm27 force field (MacKerell et al., 1998). All the simulations were done under 298 K and 1 atm.

To test DED, we carried out "native simulations" and "folding simulations" by traditional MD and DED, respectively. The native simulations and folding simulations refer to the simulations with the native and unfolded structures as their initial structures, respectively. The time for all the simulations is 20 ns. We carried out four DED and four traditional MD native simulations and eight DED and four traditional MD folding simulations. We found that the corresponding results of each kind of simulation are similar. So we only present the most typical trajectory in the following analysis.

The initial structure of the S-peptide in native simulation is obtained from its crystal structure directly (Fig. 1 a). And for folding simulations, we set its initial structure as a ß-strand (Fig. 1 b) to eliminate the correlations between the initial and native structures. Some parameters, such as radius of gyration and hydrogen bonds, are calculated from all the backbone atoms (including two caps). RMSD is calculated from backbone atoms too, but only between residues 3 and 13 to avoid the end effect. This is because the native structure (Fig. 1 a) is cut from the whole protein structure, and so its two ends are connected with other parts of the protein. But in our simulation, the two ends of the S-peptide are free and interact with water molecules directly.

View larger version (15K): [in this window] [in a new window]   FIGURE 1  Initial structures for the S-peptide in (a) native simulations and (b) folding simulations.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Figs. 2 and 3 show the results of native simulations under traditional MD and DED methods, respectively. From Fig. 2 we can see that when the S-peptide is in its native state, it shows high stability under the traditional MD simulation. The root mean-square difference (RMSD) of its conformations from its native state is 1.0 Å all the time (Fig. 2 b), and its radius of gyration and hydrogen bonds (Fig. 2, c and d) also conform to this. But in the DED simulation, we find that the S-peptide is more active. Fig. 3 b shows that the RMSD has some large fluctuations and even exceeds 4.0 Å once in a while, although it maintains 1.0 Å on the whole. This is because DED finds the collective motions in every short period (20 fs), and immediately directs the S-peptide to move along its directions in the next period. So during the DED simulation, the S-peptide is always trying to move in the most flexible direction. However, it does not go away from the native state for a long time and comes back to it quickly. It is just one of the advantages of DED to prevent peptides from staying in the local energy minima for a long time.

View larger version (26K): [in this window] [in a new window]   FIGURE 2  Four parameters via time for native simulations by traditional MD: (a) potential energy, (b) RMSD, (c) radius of gyration, and (d) H-bond.

View larger version (33K): [in this window] [in a new window]   FIGURE 3  Four parameters via time for native simulations by DED: (a) potential energy, (b) RMSD, (c) radius of gyration, and (d) H-bond.

Fig. 4 describes the folding of the S-peptide beginning from a ß-strand under traditional MD simulation. The RMSD curve in Fig. 4 b shows that the S-peptide cannot fold to its native state in 20 ns and the RMSD is up to 2.0 Å all along and without large fluctuations, so it may have been trapped into a local stable state. To check our guess, we selected five structures every 2 ns from 2 ns to 10 ns and found that these five structures overlapped with each other on the whole (Fig. 5 a). We found that in most of these structures LYS2-ALA6, GLU3-ALA7, and PHE9-GLN12 form stable backbone hydrogen bonds, and the side chains of THR4, PHE9, and MET14 are almost always packed together tightly (Fig. 5 b). Fig. 6 gives a detailed description for this. It shows that, although the distances between the three pairs (THR4-PHE9, THR4-MET14, and PHE9-MET14) vary with time, they maintain at 5 Å for most of the simulation time and only move away from each other once in a while. So they form a core for most of the time. Obviously, this is a stable conformation cluster different from the native state. When the S-peptide goes into this cluster, it can hardly jump out. This is the notable problem encountered by traditional MD. Of course, if the simulation time is long enough, it may overcome all the local minima eventually, just as the work of Duan and Kollman on a 36-residue peptide found (Duan and Kollman, 1998).

View larger version (26K): [in this window] [in a new window]   FIGURE 4  Four parameters via time for folding simulations by traditional MD: (a) potential energy, (b) RMSD, (c) radius of gyration, and (d) H-bond.

View larger version (34K): [in this window] [in a new window]   FIGURE 5  (a) Overlapped five structures selected every 2 ns from 2 ns to 10 ns during traditional MD folding simulation. (b) Highlight presentation of THR4, PHE9, and MET14, which packed together tightly. The structure is picked up at 6 ns.

View larger version (33K): [in this window] [in a new window]   FIGURE 6  Distances between the centers of the side chains of three residues via time: (a) THR4-PHE9, (b) THR4-MET14, and (c) PHE9-MET14.

View larger version (33K): [in this window] [in a new window]   FIGURE 7  Four parameters via time for folding simulations by DED: (a) potential energy, (b) RMSD, (c) radius of gyration, and (d) H-bond.

View larger version (37K): [in this window] [in a new window]   FIGURE 8  Smoothed energy landscape for the S-peptide constructed from the RMSD and radius of gyration. The "start point" donates the initial position in the simulation. Similarly, the "end point" donates the final position in the simulation. From this picture, we can clearly see that the energy landscape for the S-peptide is full of frustrations, but at the end point, the energy is much lower than other places.

View larger version (23K): [in this window] [in a new window]   FIGURE 9  Average potential energies for different RMSDs in the DED folding simulation. It shows that the potential energies for the structures far away from the native state fluctuate very much, and some of them are even close to that of the native state.

View larger version (99K): [in this window] [in a new window]   FIGURE 10  The residue's conformations via time. The darker the color is, the closer it is to the standard helical conformation.

View larger version (29K): [in this window] [in a new window]   FIGURE 11  PCA on the DED folding simulation. From this we can see that all the conformations in the trajectory aggregate to a few clusters. The time used to transfer between the clusters is very short.

View larger version (23K): [in this window] [in a new window]   FIGURE 12  Projections of all the conformations on the first three principal components via time on the first (a), second (b), and third (c) principal components. It is noted that only the first principal component varies greatly and the other two principal components only fluctuate randomly around zero.

Finally, it is noted that DED can be considered as a kind of biased sampling method. Fig. 13 shows that the samplings in energy space in both traditional MD and DED methods have the similar Gaussian distributions. The exponential histogram is calculated by using the formula

(7)

Here is the exponential histogram between and , is the number of conformations in this energy gap, and is the total number of conformations. E_i is the total energy of the system, including the free energy term due to the implicit solvent. The curves are those of the fitted Gaussian functions. The main difference is that the distribution for DED is wider than that for traditional MD, i.e., the sampling space of the former is larger than the later. As mentioned above, the force (5.0 kcal/mol Å) is strong enough to pull the molecule but, at the same time, is too weak to distort the molecule. Furthermore, the steering direction changes rapidly in a very short time. So DED is not as the standard steered molecular dynamics and will not introduce significant distortions or drift in the steering direction. This can be clearly seen from the DED unfolding simulation of CI2 (Fig. 14). The motions of the structure are similar to those of high temperature unfolding simulation (Day et al., 2002). In the work of Day et al., an unfolding simulation of CI2 is performed. At the first stage of the simulation (0–0.26 ns), where most of structures are near the native conformation, the motion of CI2 is mainly present at its loop, whereas in the two relatively rigid groups, one consists of three ß-sheets and the other consists of one helix, is almost maintained. It is noted that the behavior of CI2 seems different from that of the S-peptide, which fluctuates around the native state as shown in Fig. 3 b. This is due to the short simulation time (only 0.23 ns), which strongly restricts the sampling in the conformation space of CI2. The whole conformation space for CI2 is much larger than S-peptide. So even though the same simulation time as for S-peptide is used for CI2, the sampling is still not sufficient. Here, the short time simulation of CI2 is only used to show that the DED will not introduce significant distortions or drift in the steering direction as the standard steered molecular dynamics.

View larger version (12K): [in this window] [in a new window]   FIGURE 13  The sampling distributions in energy space for traditional MD (a) and DED (b) during the folding simulation, respectively. The data are fitted by Gaussian functions (real line).

View larger version (18K): [in this window] [in a new window]   FIGURE 14  A 0.23 ns unfolding simulation for CI2 starting from its native state.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

The DED simulation results of the S-peptide are also in agreement with the current viewpoint on protein folding. First, the energies of most unfolded structures are very high, but there are still a few structures having much lower energy close to that of the native state. Second, short peptides do not move smoothly during the folding process. The conformations in the trajectory aggregate to a few clusters or, in other words, intermediates. These results in turn validate the practicability of DED simulation. DED may provide an alternative approach to simulate the folding of peptides and proteins.

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
This work was supported by the National Natural Science Foundation of China under grant No. 90103031 and the Foundation of the Ministry of Education.

Submitted on June 1, 2004; accepted for publication January 31, 2005.

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