Coarse-Graining of Protein Structures for the Normal Mode Studies KILHO EOM,1 SEUNG-CHUL BAEK,2 JUNG-HEE AHN,2 SUNGSOO NA2 1

Microsystem Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea 2 Department of Mechanical Engineering, Korea University, Seoul 136-701, Korea Received 11 August 2006; Revised 7 December 2006; Accepted 8 December 2006 DOI 10.1002/jcc.20672 Published online 1 March 2007 in Wiley InterScience (www.interscience.wiley.com).

Abstract: The coarse-grained structural model such as Gaussian network has played a vital role in the normal mode studies for understanding protein dynamics related to biological functions. However, for the large proteins, the Gaussian network model is computationally unfavorable for diagonalization of Hessian (stiffness) matrix for the normal mode studies. In this article, we provide the coarse-graining method, referred to as ‘‘dynamic model condensation,’’ which enables the further coarse-graining of protein structures consisting of small number of residues. It is shown that the coarser-grained structures reconstructed by dynamic model condensation exhibit the dynamic characteristics, such as low-frequency normal modes, qualitatively comparable to original structures. This sheds light on that dynamic model condensation and may enable one to study the large protein dynamics for gaining insight into biological functions of proteins. q 2007 Wiley Periodicals, Inc.

J Comput Chem 28: 1400–1410, 2007

Key words: coarse-graining; normal mode analysis; protein dynamics; low-frequency normal modes; Gaussian network model

Introduction Proteins perform the biological functions through the structural change between two equilibrium states such as open and close forms.1–3 It is essential to understand the protein dynamics that provides the information about the structural changes of proteins. In general, molecular dynamics (MD) simulation has played a significant role in insight into protein dynamics.1,2 However, MD simulation is computationally prohibited for large proteins because the accessible time scale of MD simulation is currently at most 10 ns whereas proteins perform the functions at longer time scale.4 As a consequence, for large protein dynamics requiring large spatial and temporal scales, the normal mode analysis (NMA) has been an alternative to MD simulation. NMA has enabled one to gain insight into protein dynamics based on low-frequency normal modes.5–8 Nevertheless, for large proteins, NMA is often computationally restricted because of computational inefficiency to find the global equilibrium position by minimization of anharmonic potential energy, which possesses multiple local minima, and the Hessian (stiffness) matrix at the global equilibrium position. It has been recently reported that harmonic dynamics is sufficient for understanding the collective dynamic behavior of proteins.9–11 This enabled Tirion to develop a simple elastic model for protein’s conformational fluctuations.12 Tirion’s elastic

model provided an insight to Haililoglu et al., who introduced a Gaussian network model (GNM) on the basis of Gaussian dynamics of folded proteins.13 GNM regards the protein structure as a simple one-dimensional entropic spring network for residues (
carbons). GNM employs only two parameters such as a universal force constant and a cut-off radius for describing a onedimensional harmonic entropic spring network for protein structures. GNM, a simple one-dimensional elastic model, achieves the fast computing on the normal modes and thermal fluctuations. With computational efficiency, GNM enables one to predict the normal modes and their corresponding dynamics qualitatively comparable to an original molecular model and/or experiments. Specifically, GNM allowed several researchers to study the folding core predictions,14 structural changes of viruses,15–18 dynamical behavior of GroEL-GroES complex upon ATP binding,19–21 structural change upon ligand binding,22,23 and structural changes of motor proteins.24–27

Correspondence to: K. Eom; e-mail: [email protected] or S. Na; e-mail: [email protected] Contract/grant sponsor: Microsystem Research Center, Korea Institute of Science and Technology Contract/grant sponsor: Department of Mechanical Engineering, Korea University (Brain Korea 21 project)

q 2007 Wiley Periodicals, Inc.

Coarse-Graining of Protein Structures

Despite its capability of fast computation on the normal modes of proteins, GNM exhibits the computational inefficiency for diagonalization of large stiffness matrix for large proteins. Specifically, GNM possesses the computational cost proportional to O(N3), where N is the number of residues for a protein. In recent years, there have been attempts on model reduction that allows one to construct the stiffness matrix in lower-dimensional space. For instance, Doruker et al. reported the hierarchical model reduction method, which allowed them to construct a coarse-grained protein structure composed of (N/n) residues (nodal points).28 This coarsegrained structural model employs the parameters (e.g. force constant, cut-off radius) that are empirically determined by curve-fitting to B-factor obtained by experiment. This coarsegrained model exhibits the very fast computation, proportional to O(N3/n3), on normal modes as well as dynamic behavior qualitatively comparable to original protein structure and/or experiment. Recently, Chennubhotla and Bahar suggested the coarse-graining of protein structures by using a Markov statistical method.29 In their work, they introduced the transition matrix that maps the stiffness matrix of original structure to that of coarse-grained structure. Moreover, Cheng et al. developed the model reduction method to large physical/chemical system on the basis of low rank approximation to interaction matrix (e.g. stiffness matrix),30 even though they did not consider the protein structures. This model reduction method may be available for coarse-graining of protein structures. In this article, we provide the coarse-graining method referred to as ‘‘dynamic model condensation.’’ Dynamic model condensation allows the coarse-graining on the basis of elimination of entropic springs corresponding to residues not taken in the coarse-grained structure. For reconstruction of coarse-grained structures consisting of (N/n) residues, dynamic model condensation assumes that N(1
1/n) residues not retained in the coarsegrained structure exhibit the insignificant fluctuation, that is, they are in equilibrium. With the normal modes of coarsegrained structures reconstructed by dynamic model condensation, it is shown that a coarse-grained structure has dynamic behaviors, such as low-frequency normal modes and their collective motions, qualitatively and quantitatively comparable to the original structure. This may shed light on that dynamic model condensation and may enable one to gain insight into large protein dynamics related to biological functions.

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Table 1. Model Proteins.

Protein

PDB code

N

Nd

˚) rc (A

 (kcal/mol)

Hemoglobin in open form Hemoglobin in close form Creatine amidinogydrolase Citrate synthase in open form Citrate synthase in close form F0-ATPase motor protein F1-ATPase motor protein

1bbb 1a3n 1kp0 5csc 6csc 1c17 1e79

574 572 804 858 871 1090 3315

4 4 2 2 2 13 9

12 12 12 12 12 12 12

0.127 0.164 0.279 0.287 0.426 N/A 0.347

Here, N is the number of residues of model proteins, Nd is the number of domains of model proteins, rc is the cut-off radius for Gaussian network of model proteins, and  is the force constant for Gaussian network of model proteins.

to 3300, so that model proteins are suitable for coarse-graining studies. GNM (Elastic Network)

Since the conformational fluctuation of a protein is well described by harmonic dynamics,9,10 the elastic network model (ENM) considers the residues (
carbons) whose motions are well depicted by a harmonic elastic spring network.12,31 Specifically, the potential field E for Tirion’s elastic model (or ENM) is given by12 E¼

XX i

2

j

ðrij
rij0 Þ2 Hðrc
rij0 Þ

(1)

where rij is a distance between residues i and j, i.e. rij ¼ |ri–rj|,  is a force constant, rc is a cut-off radius typically set to ˚ , H(r) is a Heaviside unit step function defined rc ¼ 10*12 A as H(r) ¼ 1 if r  0; otherwise H(r) ¼ 0, and superscript 0 indicates the equilibrium state. The potential energy of Tirion’s elastic model, E, can be approximated by a harmonic expansion at a native conformation, resulting in a potential field of GNM.13 E¼

XX Cij ui uj 2 i j

(2)

Here, ui is the fluctuation of residue i, and Cij is the contact map representing the interaction between residue i and j, defined as

Method Model Proteins

We consider the model proteins suggested in Table1 for our studies on coarse-graining. Specifically, we take into account the model proteins such as hemoglobin in open and closed forms (pdb codes: 1a3n, 1bbb), citrate synthase in open and closed forms (pdb codes: 5csc, 6csc), creatine amidinogydrolase (pdb code: 1kp0), F0-ATPase motor protein (pdb code: 1c17), and F1ATPase motor protein (pdb code: 1e79). For these model proteins, the problem size N (number of residues) ranges from 500

Cij ¼
ð1
ij Þ Hðrc
rij0 Þ
ij

N X

Cik

(3)

k6¼i

where ij is the Kronecker delta, i.e. ij ¼ 1 if i ¼ j; otherwise ij ¼ 0. NMA of GNM provides the conformational fluctuation of proteins, whose motion is described by eigenvalue problem.

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Cij qj ¼ !2 qi DOI 10.1002/jcc

(4)

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where ! is the natural frequency, and qi is its corresponding normal mode. The equilibrium statistical mechanics theory32,33 allows one to construct the fluctuation matrix Qij given as Qij  hðri
hri iÞ  ðrj
hrj iÞi ¼

N X kT ðnÞ ðnÞ q qj 2 i ! n n¼2

(5)

Here, angle bracket hi indicates the ensemble (time) average, index n represents the mode index n, and summation excludes the one zero-mode corresponding to rigid body motion. The mean square fluctuation driven by thermal energy for a residue i about the equilibrium position is hð
ri Þ2 i ¼ Qii

142, 142 þ n, 142 þ 2n, . . . , 287, 287 þ n, 287 þ 2n, . . . , 428, 428 þ n, 428 þ 2n, . . .}. With the prescribed master and slave residues, one can express the potential energy described as eq. (2) into the form of

(6)

E¼

(7)

(11)

u¼

    Im u1 ¼ u1 u2
C12 C
1 22

(12)

(8)

The value of Cij close to 1 reflects the correlated motion between residues i and j, while the value of Cij is close to
1 represents the anti-correlated motion between residues i and j. When cross-correlation is close to zero, it indicates the uncorrelated motion or orthogonal motion between residues i and j. Furthermore, with the normal modes, one can compute the collectivity parameter defined as34 " #  2  N  X 1  ðiÞ   ðiÞ 2 i ¼ exp
qj  log qj  N j¼1

(10)

It can be easily shown, from eq. (11), that the fluctuation of residues is represented only by the fluctuation of master residues such as

The correlation of motions between residues i and j is well described by the cross-correlation defined as h
ri 
rj i Qij Cij  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Q ii Qjj hð
ri Þ ihð
rj Þ i

  u1 u2

@E ¼ C21 u1 þ C22 u2 ¼ 0 @u2

2

8 Qii 3

C12 C22

where asterisk indicates the transpose, subscripts 1 and 2 represent the master residues and slave residues, respectively. C11 indicates the pair-wise interactions between master residues, C22 provides the pair-wise interactions between slave residues, and C12 represents the pair-wise interactions between master residue and slave residue. It is assumed that the fluctuation of slave residues is not significant, that is, slave residues are in equilibrium.

and consequently, the Debye-Waller factor (B-factor) is given as Bi ¼

  C11    u Cu ¼ u1 u2 2 2 C21

Here, Im is the m  m identity matrix, where m is the number of master residues. From eqs. (4), (10), and (12), one can describe the fluctuation dynamics of coarse-grained structure composed of the master residues such as ~ ij qj ¼ !2 qj C

~ ij is the reconstructed Here, indices i and j run from 1 to m, and C contact map representing the pair-wise interactions between master residues in the coarse-grained structure, defined as

(9)

where i is in the range between 1/N and 1. For i-th mode, a small value of i close to 1/N reflects the localized motion, while a large value of i close to 1 represents the global (collective) motion. Dynamic Model Condensation (Coarse-Graining)

The dynamic model condensation enables the coarse-graining of protein structures with small number of residues. We take (N/n) residues, referred to as master residues, for the coarse-grained structure. We refer to N(1
1/n) residues, to be removed during the coarse-graining, as slave residues. The selection of master residues is uniformly implemented in every domain. For example, with a hemoglobin consisting of four domains such as A (residue: 1–141), B (residue: 142–286), C (residue: 287–427), and D (residue: 428–572), one can construct the coarse-grained structure retaining (N/n) residues such as {1, 1 þ n, 1 þ 2n, . . . ,

(13)

 ~  L½C ¼ Im C


C12 C
1 22

  C11 C21

C12 C22

  Im 0

(14)

where L[ ] is the transition operator mapping the original structure to the coarse-grained structure. The dynamic characteristics such as mean square fluctuation, cross-correlation, and collectivity parameter for the coarse-grained structure can be easily computed from eqs. (5)–(9), (13), and (14). For a coarse-grained structure by retaining (N/n) residues, direct computation on the transition operator L is computationally unfavorable because the computing expense is proportional to O(N3(1
1/n)3). Consequently, the coarse-graining process is implemented in the hierarchical manner by retaining N/2, N/4, N/8, N/16 residues (i.e. n ¼ 2, 4, 8, 16) as follows. i. Prescribe the (N/l) master residues (1 < l << n < N) and corresponding slave residues. ii. Partition the contact map C provided by eq. (10), so that the transition operator L is computed from eq. (14).

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~ for (N/l) master resiii. Reconstruct the effective contact map C idues. iv. Set the coarse-grained structure as initial structure for further ~ N / (N/l). coarse-graining, i.e. C C, v. Repeat the steps (i)–(iv) until one obtains the reconstructed ~ for a coarse-grained structure consisting of contact map C (N/n) residues.

Results Mean-Square Fluctuations

We consider the mean-square fluctuations of model proteins obtained by the original structure described by GNM and coarse-grained structures. In order to compare the mean-square fluctuation for every residue between original and coarse-grained structures, we consider all possible n coarse-grained structures for computing the mean-square fluctuation. The computational expense to compute the mean-square fluctuation for every residue by coarse-grained structures is proportional to O(N3/n2), while the original structure requires O(N3) computation. Even though coarse-graining allows us to reduce the computation on mean-square fluctuation by factor of n2, it is quite remarkable in that coarse-grained structures provide the mean-square fluctuation qualitatively comparable to original structure (see Fig. 1). Specifically, we take into account n ¼ 2*16 for coarsegraining of model proteins. In Figure 1, it is shown that the coarse-grained structures exhibit the qualitatively consistent mean-square fluctuation regardless of resolution of structures as long as n ranges from 2 to 16. The amplitude of fluctuations, however, becomes larger as we implement the further coarsegraining (i.e. larger n). This is quite rational because our coarsegraining is based on the elimination of entropic springs corresponding to the slave residues, and consequently, the coarse-grained structures are more flexible than the original structure. The difference of amplitude between coarse-grained structure and original structure may depend on the model proteins and their domains. Specifically, the increase in amplitude by coarse-graining is uniform for every domain for hemoglobin, whereas for F1-ATPase motor protein the increase in amplitude by coarse-graining is quite larger for H and I domains than that for other domains. This non-uniform increase in the amplitude for domains in F1-ATPase motor protein by coarse-graining may arise from the fact that we implement the coarse-graining uniformly for every domain, although the number of residues for H and I domains in F1-ATPase motor protein is quite smaller than that of other domains. This may indicate that one should implement the coarse-graining as long as the number of master residues for every domain is kept constant. For quantitative comparison of mean-square fluctuation predicted by coarse-grained structures and experimental data obtained by X-ray crystallography, the rescaling of a force constant  is mandatory. We consider the B factor for residues of model proteins predicted by coarse-grained structures that retain the N/16
carbons. Once we rescale the force constant for a coarse-grained structure, the B factor of model proteins are quantitatively comparable between coarse-grained structure and

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experimental data (See Fig. 2). This indicates that the fluctuation behavior of proteins can be predicted by coarse-grained structures as long as overall flexibility (or stiffness), responsible for fluctuation amplitudes, of proteins are preserved. This may shed light on that the fluctuation dynamics of proteins can be described by small number of degrees of freedom for rigid domains. It is very consistent with the modeling concept of RTB (Rotational and Translational Blocks) method35–37 and/or rigid cluster model,38,39 which consider the degrees of freedom for rigid blocks corresponding to rigid domains. Accordingly, one may replace the protein structures with rigid blocks that have small number of nodal points (master residues). Lowest-Frequency Normal Modes

We consider the primary low-frequency normal mode that is generally renowned to play a role in protein dynamics such as conformation change.19,21–24 In Figure 3, we take into account the primary low-frequency normal mode for both original structure and coarse-grained structures for model proteins. Remarkably, it is shown that the characteristics of lowest-frequency normal mode are well preserved during the coarse-graining. Specifically, for hemoglobin, both original structure and coarse-grained structures provide the collective motions described by lowestfrequency normal mode. In both original and coarse-grained structures, lowest-frequency normal mode suggests that two domains A and B move collectively, and that at two domains C and D the collective motion occurs. Moreover, the motion of a substructure consisting of domains A and B may be uncorrelated (or anti-correlated) with that of a substructure consisting of domains C and D (for details, read Section ‘‘Cross-correlation’’). The collective characteristics of domain motion described by lowest-frequency normal modes are well provided by both original and coarse-grained structures for all model proteins (see Fig. 3). This may suggest that coarse-grained structures possess the low-frequency dynamical behavior qualitatively comparable to an original structure. This may be ascribed to the fact that protein’s domains are relatively rigid, which may be responsible for collective motions. Collectivity

Low-frequency normal modes may play a role in collective motion of proteins.34 We consider the collectivity of normal modes for original and coarse-grained structures for model proteins. As shown in Figure 4, the large collectivity in normal modes is ascribed to the low-frequency normal modes, when both original and coarse-grained structures are considered for model proteins. It is quite remarkable that all coarse-grained structures predict the large collectivity of primary low-frequency normal mode quantitatively comparable to original structure. For instance, lowest-frequency normal mode of coarse-grained structures for hemoglobin provides the collectivity of 1 ¼ 0.8 that is exactly predicted by original structure. In the similar manner, for all model proteins, lowest-frequency normal modes for both original and coarse-grained structures have the same amount of collectivity 1. In Figure 4, it is shown that the collective lowfrequency normal modes are well predicted by coarse-grained

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Figure 1. The mean-square fluctuation of model proteins from the coarse-grained structures and the original structure described by Gaussian network: (a) hemoglobin in close form (pdb: 1a3n), (b) hemoglobin in open form (pdb: 1bbb), (c) creatine amidinogydrolase (pdb: 1kp0), (d) citrate synthase in open form (pdb: 5csc), (e) citrate synthase in closed form (pdb: 6csc), (f) F0-ATPase motor protein (pdb: 1c17), and (g) F1-ATPase motor protein (pdb: 1e79). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 2. Comparison of B factor obtained from the coarse-grained structure retaining N/16
carbons with experimental B factor by X-ray crystallography: (a) hemoglobin in close form, (b) hemoglobin in open form, (c) creatine amidinogydrolase, (d) citrate synthase in open form, (e) citrate synthase in close form, and (f) F1-ATPase motor protein. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

structures, whereas localized modes (high-frequency normal modes) cannot be obtained with coarse-grained structures. This indicates that collective modes (low-frequency normal modes) are sufficiently reproducible by coarse-grained structures, while localized modes (high-frequency normal modes) are not to be reproduced by coarse-grained structures. It suggests that localized modes are accurately predicted by a refined structural model (i.e. GNM, ENM, etc.) of proteins.40 Cross-Correlation

The cross-correlation map represents the collective behavior of domains as well as correlation of motions between domains. Figure 5 shows the cross-correlations obtained from both origi-

nal structure and coarse-grained structures that retains N/16 residues for model proteins. The red color in the cross-correlation map indicates the highly correlated motions, while the blue color represents the anticorrelated motions, and the white color shows the uncorrelated motions. Remarkably, the coarse-grained structures exhibit the cross-correlation qualitatively comparable to the original structure. For instance, both original structure and coarse-grained structure for hemoglobin predict that the residues within every domain move in a collective manner. The highly correlated motions between domains A and B (also, domains C and D) are well predicted by both original and coarse-grained structures. The coarse-grained structure of hemoglobin, however, overestimates the anticorrelated motion between a domain A (or domain B) and the domains C and D. This overestimation of

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Figure 3. Lowest-frequency normal mode obtained from the original structure and the coarse-grained structures for model proteins: (a) hemoglobin in close form, (b) hemoglobin in open form, (c) creatine amidinogydrolase, (d) citrate synthase in open form, (e) citrate synthase in close form, (f) F0-ATPase motor protein, and (g) F1-ATPase motor protein. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 4. Collectivity parameter obtained by original structure and coarse-grained structures for model proteins: (a) hemoglobin in close form, (b) hemoglobin in open form, (c) creatine amidinogydrolase, (d) citrate synthase in open form, (e) citrate synthase in close form, (f) F0-ATPase motor protein, and (g) F1-ATPase motor protein. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 5. (continued)

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Figure 5. Cross-correlation map for original and coarse-grained structures taking N/16
carbons for model proteins: (a) original structure for hemoglobin in close form, (b) coarse-grained structure for hemoglobin in close form, (c) original structure for hemoglobin in open form, (d) coarse-grained structure for hemoglobin in open form, (e) original structure for creatine amidinogydrolase, (f) coarsegrained structure for creatine amidinogydrolase, (g) original structure for citrate synthase in open form, (h) coarse-grained structure for citrate synthase in open form, (i) original structure for citrate synthase in close form, (j) coarse-grained structure for citrate synthase in close form, (k) original structure for F0-ATPase motor protein, (l) coarse-grained structure for F0-ATPase motor protein, (m) original structure for F1-ATPase motor protein, and (n) coarse-grained structure for F1-ATPase motor protein. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

anti-correlation may be contributed by the flexibility of coarsegrained structure. That is, the coarse-graining, based on the elimination of entropic springs corresponding to slave residues, makes the coarse-grained structure more flexible than the origi-

nal structure. Consequently, the more flexible coarse-grained structure may provide the larger value of anti-correlation between domains. In the same manner, for all model proteins, the correlated (collective) motion of domains is well predicted

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by coarse-grained structures, whereas the anticorrelation is overestimated when coarse-graining is implemented. As stated earlier, the rescaling of a force constant  is necessary in order for a coarse-grained structure to have the same amount of overall stiffness to protein structure, and then, the overestimation of anti-correlation between domains may not appear with rescaling of a force constant .

Conclusion In this paper, we demonstrated the coarse-graining method, referred to as ‘‘dynamic model condensation,’’ which enables one to reconstruct the contact (interaction) map for a coarsegrained structure. It is shown that the conformational motions of proteins, related to biological functions, are well delineated by coarse-grained structures. The success of a coarse-grained structure in predictions on conformational fluctuations may be attributed to the rigidity of domains, that is, relatively rigid domains can be represented by much less number of residues. This is consistent with the concept of RTB method35–37 and rigid cluster model,38,39 which take into account a protein structure with minimal degrees of freedom for rigid domains. Our coarse-graining method, ‘‘dynamic model condensation,’’ is robust for understanding large protein dynamics based on low-frequency normal modes. In conclusion, our coarse-graining method may allow one to gain insight into large protein dynamics and its related biological functions. Moreover, our coarse-graining method may be applicable to large physical/chemical system such as electrostatics, for which the short-range interaction is dominant so that the stiffness matrix is sparse (low-rank matrix).

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