Coupling between Normal Modes Drives Protein Conformational Dynamics: Illustrations Using Allosteric Transitions in Myosin II

Transcription

1 2128 Biophysical Journal Volume 96 March Coupling between Normal Moes Drives Protein Conformational Dynamics: Illustrations Using Allosteric Transitions in Myosin II Wenjun Zheng * an D. Thirumalai * Physics Department, University at Buffalo, Buffalo, New York; an Biophysics Program, Institute for Physical Science an Technology, University of Marylan, College Park, Marylan ABSTRACT Structure-base elastic network moels (ENMs) have been remarkably successful in escribing conformational transitions in a variety of biological systems. Low-frequency normal moes are usually calculate from the ENM that characterizes elastic interactions between resiues in contact in a given protein structure with a uniform force constant. To explore the ynamical effects of nonuniform elastic interactions, we calculate the robustness an coupling of the low-frequency moes in the presence of nonuniform variations in the ENM force constant. The variations in the elastic interactions, approximate here by Gaussian noise, approximately account for perturbation effects of heterogeneous resiue-resiue interactions or evolutionary sequence changes within a protein family. First-orer perturbation theory provies an efficient an qualitatively correct estimate of the moe robustness an moe coupling for finite perturbations to the ENM force constant. The moe coupling analysis an the moe robustness analysis ientify groups of strongly couple moes that encoe for protein functional motions. We illustrate the new concepts using myosin II motor protein as an example. The biological implications of moe coupling in tuning the allosteric couplings among the actin-bining site, the nucleotie-bining site, an the force-generating converter an lever arm in myosin isoforms are iscusse. We evaluate the robustness of the correlation functions that quantify the allosteric couplings among these three key structural motifs. INTRODUCTION It is important to obtain etails of the conformational changes in proteins to eluciate their molecular functions. To enable efficient simulations an analysis of protein conformational ynamics, coarse-graine moeling has been successfully evelope using simplifie structural representations an energy functions (1). A prime example of the structure-base coarse-graine moels is the elastic network moel (ENM), where the C a atoms of amino aci resiues that are within a cutoff istance are connecte by springs (2,3) with a uniform force constant (4). Normal moe analysis (NMA) base on the ENM has been extensively valiate (5,6) an employe to escribe the conformational ynamics in biomolecular structures (for reviews, see (7 10)). Remarkably, the global conformational changes (11 14), conformational transition pathways (15 20), an allosteric couplings (21 23) in complex systems are well escribe by few low-frequency normal moes. The ENM-base moes have also forme the basis of new computational techniques for protein structural moeling an refinements (24 27). The success of ENM-base lowresolution moeling has been attribute to the robust nature of the collective motions in multiomain proteins, which are apparently insensitive to the etails of microscopic interactions. Bolstere by the robustness argument, an ENM with a uniform force constant, k 0 an a uniform cutoff istance, R c, is typically use espite significant heterogeneity in the Submitte October 21, 2008, an accepte for publication December 5, *Corresponence: wjzheng@buffalo.eu; thirum@um.eu strength an range of physical interactions between the amino aci resiues. The use of R c values within the range 8 20 Å or other resiue-contact schemes (3,28 30) generally preserves the lowest few normal moes. In aition, the lowest moes of ENM were foun to be comparable with the lowest normal moes obtaine from the NMA of allatom force fiels (3). Another stuy suggeste that those few invariant or robust moes may be functionally important (31). The robustness in functionally important lowfrequency moes was also iscusse in the context of ribosome ynamics (32). Our recent stuies have also supporte the concept of robustness as a useful criterion for preicting functionally important moes (14,33). Despite earlier stuies in support of the robustness of ENM, the ynamic effects of nonuniform perturbations to the ENM force constant remain to be fully quantifie. In biomolecules, such perturbations may originate from several sources, such as the approximate use of uniform elastic interactions to account for heterogeneous physical interactions between resiues, an evolutionary sequence variations. To assess the effects of these complex factors, we will analyze how the normal moes are perturbe after a Gaussian ranom noise is ae to the uniform ENM force constant. The resulting simplification allows analytic treatment of the perturbation effects (see below), an it provies a reasonable starting point for further stuies of more complex perturbations. Using a brute-force approach, one can perform NMA for a large number of ENM parameter sets with heterogeneous force constants k ij ¼ k 0 þ k ij for resiue pairs hi, ji, whose equilibrium istance is 0 ij < R c; an then statistically Eitor: Nathan Anrew Baker. Ó 2009 by the Biophysical Society /09/03/2128/10 $2.00 oi: /j.bpj

2 Moe Coupling in Allosteric Transitions 2129 analyze the variations in the eigenvectors of the normal moes. Such an approach is computationally expensive for large structures typical of biological nanomachines. Therefore, it is useful to evelop an approximate metho that estimates the statistical variations of normal moes reliably an efficiently. The brute-force approach can be use to valiate the approximate metho (see below). Recently, we introuce the structural perturbation metho, base on first-orer perturbation theory, to estimate the robustness of normal moes in the presence of parameter changes ue to sequence variations (14,33). This metho is aopte here to investigate how nonuniform errors in the ENM force constant perturb the normal moes an the normal-moe-base correlation functions (see Methos). Unlike in our earlier stuies (14,33), we will a Gaussian noise to the force constant of all springs regarless of the conservation level of the resiues involve. This simplification will be justifie by showing that the results of moe robustness remain qualitatively unchange even when the perturbations are restricte to nonconserve resiue pairs only (see Results). Small perturbations in the ENM force constant will mix or couple those moes whose eigenvalues are close in the unperturbe spectrum. This is evient from first-orer perturbation theory, because the coupling coefficient between two moes is inversely proportional to the ifference between their eigenvalues (see Methos). Such moe mixing etermines the robustness of a normal moe: the more strongly a moe is couple to other moes, the less robust it is. However, it is unclear whether the moe coupling compute from the perturbation theory is relevant to realistic situations in which the perturbations to the ENM force constant are not small. For example, the anomalous coupling between nearegenerate moes might be suppresse in the presence of finite perturbations. This issue is critical to the applicability of the perturbation theory to the robustness assessment of normal moes. In this work, we will valiate the use of perturbation theory to estimate the moe robustness an the coupling between moes in the presence of finite perturbations to the ENM force constant. The moe coupling information is useful not only for estimating the robustness of iniviual moes, but also for eluciating the functional significance of seemingly nonrobust moes. A moe may appear to be nonrobust if it is strongly couple with other moes. However, superposition of the strongly couple moes may form a robust moe group whose members are weakly couple to those outsie the group. When functional motions of proteins are euce base on the NMA of an ENM with inaccurate force constant, it is important to analyze the moes not iniviually but in groups. We expect moes in the same group are strongly couple with one another, whereas moes from ifferent groups are only weakly couple. Structural motions escribe by moes in the same group are more likely to accompany each other. The variable combinations of motions escribe by a moe group allow sequence variations to fine-tune allosteric couplings in proteins. Another useful application of moe coupling analysis is to buil an invariant subspace spanne by a subset of low-frequency normal moes for enhance conformational sampling (34). The invariance can be ensure by the lack of strong coupling between moes inclue in an moes exclue from the subset. The concept of robustness is applicable not only to iniviual moes but also to quantities compute from all moes. We have recently explore the use of correlation functions to probe the allosteric coupling between a pair of functional sites in a protein structure (21). The correlation functions are compute as a weighte sum of contributions from all moes (up to a cutoff moe, see Methos). Therefore, the variations in normal moes will result in changes in the correlation functions. Here, we will investigate the robustness of correlation functions in the presence of nonuniform perturbations to the ENM force constant. A specific correlation is eeme significant or robust if its variation cause by perturbations is much smaller than its unperturbe value. This calculation will allow us to reliably preict couple motions between a pair of protein sites, which may enable allosteric communications between them. We illustrate the new analysis using myosin II as an example. Myosin II is a class of molecular motors that bin to an move along actin filaments by harnessing the chemical energy from ATP hyrolysis. The work cycle of myosin II consists of the following steps (35). In an ATPboun myosin etache from actin, ATP hyrolysis prouces ADP an inorganic phosphate, an it is accompanie by a large rotation of the lever arm to the prepowerstroke position (recovery stroke). Actin bining accelerates phosphate release from myosin, resulting in a force generation (powerstroke) as the lever arm rotates to the postpowerstroke position. Subsequent release of ADP is followe by the bining of a new ATP, which etaches myosin from actin an resets myosin for the subsequent cycle. Structural stuies have highlighte extensive communications in the myosin motor omain (36 40), which consists of four subomains (see Fig. 4 inset): the upper an lower 50 kda (U50 an L50) suboman, the N-terminal subomain, an the converter subomain. The nucleotie-bining site is locate at the intersubomain interface (incluing the P loop, switch I, an switch II three conserve loops involve in nucleotie bining an hyrolysis). Despite the strong structural conservation of the myosin motor omain, myosin isoforms from ifferent classes (41) vary significantly in their motor properties, pointing to variations in the functional couplings between the actin-bining site, the nucleotie-bining site, an the force-generating lever arm. In a series of recent stuies, the NMA has been employe to probe myosin s global conformational changes (12,42), local conformational changes at the nucleotie-bining site (43), allosteric coupling (21, 23), structural flexibility (42, 44),

3 2130 Zheng an Thirumalai an conformational transition (19,45). Base on the NMA using the ENM constructe from a prepowerstroke structure of myosin II, we have ientifie two functionally important moes (21,43). Moe 1 captures a large rotation of the converter accompanying the powerstroke. Moe 7 captures a rotation of the U50 subomain that simultaneously closes the actin-bining site an opens the nucleotie-bining site, which explains observations of a negative coupling between actin bining an nucleotie bining in myosin (38 40). Here, we assess the robustness of these moes an other low-frequency moes that are strongly couple to the functionally important moes. By analyzing the low-frequency moes in moe groups, we will iscuss how variations in myosin s functional motions can be attaine thorough moe coupling. Central to myosin motor function are allosteric couplings among actin bining, ATP bining, an prouct release, an the rotation of converter an lever arm. Structural comparisons have reveale two pairs of couple structural changes in myosin: 1), the closing/opening of the actinbining site an the opening/closing of switch I at the nucleotie-bining site (38 40); an 2), the upwar/ownwar rotation of the converter an lever arm an the closing/ opening of switch II at the nucleotie-bining site (37). By supplementing correlation analysis (21) with the robustness assessment, we will computationally valiate these structural coupling rules between the nucleotie-bining site (incluing switch I an switch II) an two other key sites (the actin-bining site an converter) in the presence of nonuniform errors in the ENM force constant. METHODS Elastic network moel In an ENM, a protein structure is represente as a network of beas, each corresponing to a C a atom. A harmonic potential with a uniform force constant k 0 accounts for pairwise interactions between all C a atoms that are within a cutoff istance, R c (set here to be 10 Å), of each other. The potential energy is (2,4) E ¼ 1 X 2; k 0 ij 2 ij 0 (1) ij 0<Rc where ij is the istance between C a atoms i an j, an ij 0 is the equilibrium istance between C a atoms i an j in the crystal structure. We expan the above potential energy function to the secon orer: Ez 1 2 XT H 0 X ¼ 1 X k 0 X T H ij X; (2) 2 ij 0<Rc where X ¼ X - X 0, X is a 3N-imensional vector representing the Cartesian coorinates of N C a atoms, X 0 gives the equilibrium C a coorinates in the crystal structure, H 0 ¼ P k 0 H ij is the Hessian matrix, where H ij 0 ij ¼ 1 2 V2 ½ð ij ij 0Þ2 Š: <Rc First-orer perturbation of normal moes We consier an ensemble of ENMs with a uniform cutoff istance R c an a hererogeneous set of force constants, k ij, that epens on resiue pair (i, j). The Hessian of each ENM is H ¼ X k ij H ij ; (3) ij 0<R c where k ij ¼ k 0 þ k ij. For simplicity, the istribution of k ij is assume to be a Gaussian function centere at 0 with a stanar eviation s k, an it is statistically inepenent between resiue pairs. The first-orer perturbation to the eigenvector V m of moe m is (14) where V m ¼ X nsm A mn V n ; (4) A mn ¼ VT m H V n ; (5) l m l n H ¼ X k ij H ij ; (6) ij 0<Rc an l m is the eigenvalue of moe m. The coupling strength C mn between moes m an n is given by the meansquare variation of A mn, which sums up contributions from all resiue pairs in contact: C mn ¼ X A 2 mn ¼ s 2 V T 2 m H ij V n k : (7) l ij 0 m l n <Rc To ientify resiues involve in the coupling between moes m an n, we first sort contributions from all resiue pairs (see Eq. 7) from high to low an keep the top 1% of them; the resiues involve in these selecte pairs are efine as those involve in coupling. The robustness of moe m is assesse using the score R m ¼ jv m j 2 ¼ X C mn : (8) nsm High (low) robustness is inicate by a low (high) value of the above score. Finite perturbation theory for normal moes Next, we test how well first-orer perturbation theory estimates the moe robustness score an the moe coupling strength in the presence of finite variations in the ENM force constant. For a perturbe Hessian matrix H ¼ P ðk 0 þ k ij ÞH ij ;a new set of moes (with eigenvectors represente ij 0 as Um) <Rc can be compute. The perturbe moes are numbere such that for a given m, jvn TU mj is maximal at n ¼ m. The analog of perturbational robustness score Rm (Eq. 8)) at finite perturbations is jum V m j 2 ¼ 2 1 V T m U X m z A 2 mn : (9) nsm The analog of the coupling strength, C mn, at finite perturbations is X 0:5, V T m U iv T n U 2 V T i z0:5, m V mv T n V 2 m i þðv T m V nv T n V 2 n ¼A 2 mn :ð10þ We compute the above two quantities for 100 samples of ENMs perturbe by Gaussian noise in the ENM force constant with 0 < s k % 1, an then compare with the corresponing quantities estimate by first-orer perturbation theory (Eqs. 7 an 8). First-orer perturbation of correlation function The correlation between two given irections of movement at site 1 (X 1 ) an site 2 (X 2 ) is given by the sum of contributions from the lowest M moes (21):

4 Moe Coupling in Allosteric Transitions 2131 C 12 ¼ X T 1 X m%m V m V T m l m X 2 ; (11) where M is the cutoff moe (with efault value 100). A positive value of C 12 inicates that the two movements are positively correlate. Some moes contribute positively an others contribute negatively. In the presence of perturbations to the Hessian matrix, H ¼ P k ij H ij ; ij 0<Rc first-orer perturbation theory preicts that the resulting change in C 12 is C 12 ¼ X "! X X k ij (X T Vm TH ijv n 1 V l ij 0<R c m%m nsm m ðl m l n Þ n V T m þ V m VT n ) VT m H ijv m V l 2 m V T m #X 2 ¼ X k ij B ij : ð12þ m ij 0<Rc robustness a perturbation σ k =0.1 σ k =0.3 σ k =0.5 σ k =0.7 σ k =1.0 Thus, the root mean-square variation of C 12 is qffiffiffiffiffiffiffiffiffiffiffiffiffi C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X s C12 ¼ 212 ¼ s k B 2 ij : (13) ij 0<Rc b moe The significance of a correlation function is assesse using the Z score: Z ¼ C 12 s C12 : (14) 10 0 Here, we set s k ¼ 1.0. If Z >>1, then the correlation is robust or significant; otherwise, it is insignificant. Dynamic omain partition robustness 10 1 To visualize the interomain motions escribe by a low-frequency moe, we ecompose the conformational change given by the eigenvector of a normal moe into several movements of ynamic omains that approximately move (rotate an translate) as rigi boies (for etails, see (22)). To allow a etaile comparison of the movements of four myosin subomains (see Fig. 4 inset) escribe by ifferent moes, four initial centrois for ynamic omains are chosen at resiues I581 (in L50), L429 (in U50), L106 (in the N-terminal suboman), an K743 (in the converter). RESULTS We will emonstrate the new analyses using a myosin II structure (PDB coe: 1VOM) as an example. Our main goal is to valiate the computational methos for assessing the robustness of an the coupling between low-frequency moes. We will also show the application of these methos to probing allosteric couplings within myosin II. First-orer perturbation theory for moe robustness an moe coupling We will evaluate the accuracy of first-orer perturbation theory in estimating moe robustness an moe coupling by comparing it with statistical estimations of these quantities for finite perturbations to the ENM force constant (see Methos). The perturbation theory is only accurate for infinitesimally small s k (stanar eviation of the perturbations to the ENM force constant). Here we will explore how its inaccuracy increases as s k graually increases from 0 to uniform cutoff=9 cutoff=8 cutoff= moe FIGURE 1 Robustness score, R m (in logarithmic scale), for the lowest 10 moes of the myosin II structure (PDB coe: 1VOM). (a) Soli lines show R m compute using first-orer perturbation theory, an otte lines correspon to results for finite perturbations at five s k values. (b) Results of first-orer perturbation theory in the presence of unrestricte an restricte Gaussian perturbations are shown as soli an otte lines, respectively (the latter are restricte to nonconserve resiue pairs with a given cutoff for conservation score). Note that a high (low) score means low (high) robustness. We have compute the robustness scores (see Methos) for the lowest 10 moes (see Fig. 1 a). The perturbation theory shows that moes 1 3 are the most robust (lowest robustness scores) (Fig. 1 a), whereas the least robust moes are moes 6 an 7. Comparison with the results of finite perturbations, with s k varying from 0.1 to 1.0, shows that the agreement is very goo for s k ¼ 0.1. As s k increases, the pronounce peak at moes 6 an 7 ecreases, whereas the robustness scores of the other moes increase (Fig. 1 a). For 0.5 % s k % 1, moes 1 an 2 remain relatively robust, whereas the other eight moes have higher robustness scores. Therefore, the estimation of perturbation theory for moe robustness remains quantitatively accurate as long as s k < 0.3. At higher

5 2132 Zheng an Thirumalai FIGURE 2 Coupling strength, C mn (in logarithmic scale), between the lowest 10 moes of the myosin II structure (PDB coe: 1VOM), which are compute using firstorer perturbation theory (a), an at finite perturbations of three ifferent magnitues: s k ¼ 0.3 (b), s k ¼ 0.5 (c), an s k ¼ 1.0 (). The gray scale is use to illustrate moe coupling strength. s k values, certain qualitative features (for example, moes 1 an 2 being the most robust) are still correctly capture. The fining of high robustness for moes 1 an 2 agrees with our recent fining that these two moes are most robust to sequence variations (14). The low robustness of moe 7 seems surprising given the previous suggestion (21) that it plays a functional role in coupling actin bining/release with nucleotie release/bining. We aress this apparent paraox below, after introucing moe coupling analysis. In our earlier stuies, we use first-orer perturbation theory to estimate the robustness of normal moes to sequence variations (14,33). In these stuies, we introuce sophisticate perturbations to resiue-resiue interactions, which are moulate by sequence variations. To investigate how the choices of ifferent perturbations affect the results of moe robustness, we have moifie the present perturbation protocol (see Methos) by restricting the Gaussian perturbations to those resiue pairs involving at least one nonconserve resiue. Here, a resiue position is conserve if the conservation score from ConSurf-HSSP (46) (score ranges from 1 to 9, where 1 is most variable an 9 most conserve) is greater than or equal to a cutoff value (set to be 7, 8, or 9). Then the robustness scores are compute for the lowest 10 moes of 1VOM in the presence of the restricte perturbations (see Fig. 1 b). Although the robustness scores are quantitatively reuce, the relative robustness of the lowest 10 moes is qualitatively preserve. In particular, moes 6 an 7 still have the highest scores, whereas moes 1 3 have the lowest scores. Therefore, the results of moe robustness are qualitatively insensitive to the choice of ifferent perturbations to the force constant. We have compute the strength of coupling (see Methos) between the lowest 10 moes, an the results are shown in Fig. 2. The perturbation theory preicts three strongly couple moe pairs, (6, 7), (4, 5), an (8, 9) (see Fig. 2 a), with the strongest coupling for moe pair (6, 7). These strong moe couplings account for the high robustness scores for moes 4~9 (see Eq. 8 in Methos), especially moes 6 an 7. In contrast, moes 1 3 are minimally couple with all the other moes, resulting in their low robustness scores. We compare the moe-coupling results of the perturbation theory with their counterparts for finite perturbations with s k varying from 0.3 to 1.0 (Fig. 2, b ). The agreement is very goo for s k ¼ 0.3 the same three strongly couple moe pairs are foun (Fig. 2 b). As s k further increases, stronger coupling arises between the three pairs (for example, between moes 6 an 7 an moe 5), but the couplings within these pairs remain higher than the couplings between them (Fig. 2, c an ). Therefore, perturbation theory gives a qualitatively correct preiction of the three strongly couple moe pairs that persists even for relatively large perturbations (for s k up to 1.0). In particular, the very high coupling between moes 6 an 7, also foun for finite perturbations, is not an artifact of perturbation theory cause by near-egeneracy of these two moes. The fining that moe 7 is strongly couple with moes 5 an 6 explains its low robustness. Nevertheless, this oes not rule out the functional relevance of moe 7. On the contrary, moe 7 is strongly couple to moes 5 an 6 to attain variations of functional motion between ifferent members of a protein family (see below). Inee, if we consier perturbations to the ENM force constant ue to sequence variations between myosin isoforms (33), it is very likely that the eigenvectors of the strongly couple moes woul be mixe in a perturbe normal moe spectrum. Therefore, to reliably euce functional motions an their variations in myosin, the strongly couple moes shoul be analyze together.

6 Moe Coupling in Allosteric Transitions 2133 FIGURE 3 Results of ynamic omain partition for moes #4 9. Dynamic omains are shown in four ifferent colors (see online version) (blue, cyan, green, yellow), an their irections of rotation with respect to a fixe omain (blue) are shown by arrows. The flexible parts not belonging to any omain are gray. The hea (stem) of an arrow is assigne the same color as the moving (fixe) omain. Biological implications of moe couplings in myosin Next we iscuss the potential roles of moe coupling in tuning the allosteric couplings among the actin-bining site, the nucleotie-bining site an the converter/lever arm in the myosin motor omain (see Fig. 4 inset). To this en, we will issect the motions involving the four subomains (see Fig. 4 inset), as escribe by the three pairs of strongly couple moes, by performing a ynamic omain partition analysis (see Methos). Moe pair (6, 7) Moe 7 escribes a rotation of the U50 subomain (Fig. 3,#7, cyan) accompanie by rotations of two other ynamic omains one consisting of the lower half of the relay helix, part of the converter, an part of the N-terminal subomain (green), an the other comprise of part of the converter (yellow). The first rotation of U50 appears to couple the closing of the actin-bining cleft with the opening of the nucleotiebining site (near switch I, see (21)). The other two rotations involve the relay helix, converter, an the N-terminal subomain. Therefore, moe 7 couples actin bining with istant movements in the N-terminal an converter subomains, which are potentially relevant to the powerstroke. In comparison, moe 6 escribes a rotation of the U50 subomain (Fig. 3, #6, cyan) similar to that of moe 7 (the rotational axes are pointe in similar irections (Fig. 3, #7)). However, the motions of the other subomains iffer significantly between these two moes, especially in the relay helix an the converter, which belong to ifferent ynamic omains (Fig. 3,#6, yellow, an Fig. 3, #7, green). Therefore, a variable mixing of these two moes can couple a similar rotation of the U50 subomain with a variety of movements in the relay helix an the converter. This may allow ifferent myosin isoforms, with varie resiue-resiue interactions, to couple actin bining to ifferent movements in the converter, leaing to variations in the force-generation process between myosin isoforms. Moe pair (4, 5) Unlike moe pair (6, 7), moes 4 an 5 escribe similar rotations in the N-terminal subomain (Fig. 3,#4 an #5, green), but ifferent rotations in the U50 subomain (Fig. 3, #4 an #5, cyan) an converter (Fig. 3, #4 an #5, yellow). Therefore, a combination of these two moes can couple a similar rotation of the N-terminal subomain with a range of movements in the U50 an converter subomains. Moe pair (8, 9) Moes 8 an 9 share similar rotations in the U50 subomain (Fig. 3,#8 an #9, cyan) an the N-terminal subomain (Fig. 3, #8 an #9, green), but they iffer in the converter (Fig. 3,#8 an #9, yellow). Therefore, similar to moe pair (6, 7), this coupling can facilitate variable allosteric coupling between the converter an the U50 subomain.

7 2134 Zheng an Thirumalai TABLE 1 myosin II Moe pair Resiues involve in the moeling coupling in Resiues involve in moe coupling 4, 5 7, 19, 34, 36, 37, 44, 45, 46, 47, 52, 60, 70, 71, 72, 77, 78, 79, 83, 95, 96, 97, 98, 104, 231, 233, 234, 368, 391, 392, 395, 396, 397, 398, 399, 400, 403, 404, 405, 406, 407, 408, 409, 410, 485, 508, 573, 593, 595, 646, 658, 659, 660, 662, 668, 669, 673, 674, 682, 683, 686, 723, 739 (62 total) 6, 7 7, 16, 18, 19, 21, 25, 28, 29, 36, 37, 44, 45, 46, 47, 72, 77, 78, 80, 82, 83, 84, 86, 97, 98, 104, 105, 112, 121, 397, 398, 399, 404, 407, 420, 485, 486, 488, 490, 493, 494, 496, 497, 501, 504, 505, 506, 509, 590, 591, 593, 629, 649, 650, 651, 652, 668, 669, 673, 674, 687, 690, 691, 692, 693, 695, 696, 697, 698, 699, 701, 702, 705, 706, 712, 713, 715, 720, 723, 734, 739, 741, 742, 744, 745 (84 total) 8, 9 7, 16, 18, 19, 20, 21, 25, 27, 28, 29, 32, 33, 34, 37, 45, 46, 51, 52, 53, 60, 72, 74, 76, 77, 78, 79, 80, 82, 83, 84, 86, 89, 94, 97, 99, 100, 112, 153, 194, 216, 397, 399, 404, 407, 494, 496, 499, 501, 575, 591, 629, 668, 673, 674, 677, 682, 683, 689, 690, 691, 692, 695, 699, 701, 712, 723, 731, 732, 733, 734, 735, 736, 737, 738, 739, 742, 746 (77 total) In summary, the strong couplings within the three moe pairs are likely to be functionally significant by allowing isoformepenent tuning of allosteric couplings among the actinbining site, the nucleotie-bining site, an the converter in myosin. The resiues involve in the coupling of these three moe pairs (see Methos) are shown in Table 1. For example, resiues involve in the coupling of moe pair (6, 7) are mostly istribute in the N-terminal subomain, relay helix, SH1 helix, an converter, inicating the importance of these regions in tuning the allosteric coupling between actin bining an converter movement. The biological significance of the preicte moe coupling can be teste by mutational experiments that perturb the resiues involve in the moe coupling. We propose that these targete perturbations moulate allosteric couplings, an alter the motor properties of myosin. Robustness of the allosteric couplings in myosin Finally, we revisit the allosteric couplings in the myosin motor omain in our previous stuy (21). Here, we assess the robustness of four correlation functions (see Methos) that quantify the observe couple structural changes between the nucleotie-bining site (incluing P loop, switch I, an switch II) an another two key sites, the actin-bining site (38 40) an the converter (37) (see Fig. 4 inset). The irections of movement in these sites are obtaine by aligning their conformations between a prepowerstroke structure of Dictyostelium myosin (PDB coe: 1VOM) an a postpowerstroke rigorlike structure of myosin V (PDB coe: 1W8J). The two myosins can be structurally aligne because their sequences are 41% ientical. Four movements in the above sites are observe from the prepowerstroke structure to the postpowerstroke structure (see Fig. 4): FIGURE 4 Conformational changes from the prepowerstroke myosin structure (1VOM; blue) to the rigorlike structure (1W8J; re). The two structures are aligne in the HLH motif. Local movements at the actin-bining site (HLH motif an HO helix), the nucleotie-bining site (P loop, switch I, an switch II), an the converter are highlighte by opaque cartoons. (Inset) Four subomains the upper an lower 50 kda (U50 an L50), the N-terminal subomain (N), an the converter (C) an key structural components of myosin (relay helix, SH1 helix, lever arm, an switches I an II). 1. Opening of switch I (resiues , using the resiue numbers of Dictyostelium myosin, same below) relative to the P loop (resiues ); 2. Opening of switch II (resiues ) relative to the P loop; 3. Closing of the actin-bining cleft, inicate by a relative rotation of the HO helix of the U50 subomain (resiues ) with respect to the helix-loop-helix (HLH) motif of the L50 subomain (resiues ); 4. Downwar rotation of the converter (resiues ) relative to the N-terminal subomain (resiues ). To test whether the above movements are couple by the normal moes calculate from the prepowerstroke structure, we compute four correlations (see Methos) between movement 1/2 an movement 3/4, as follows: 1. The correlation between the opening of switch I an the closing of the actin-bining cleft (Fig. 5 a) is foun to be positive an significant (C 12 ¼ 0.042, s C12 ¼ 0.015, so Z ~ 2.8). Its highest contribution is from moe 7, supporting the importance of this moe in allosterically coupling the actin-bining site an the nucleotie-bining site (21). 2. The correlation between the opening of switch II an the closing of the actin-bining cleft (Fig. 5 b) is weak an insignificant (C 12 ¼ , s C12 ¼ , so Z ~ 0.8). This weak correlation results from cancellation

8 Moe Coupling in Allosteric Transitions 2135 FIGURE 5 Correlation functions between (a) opening of switch I an closing of the actin-bining site (C 12 ¼ 0.042, s C12 ¼ 0.015); (b) opening of switch II an closing of the actin-bining site (C 12 ¼ , s C12 ¼ ); (c) opening of switch I an ownwar rotation of the converter (C 12 ¼ 0.038, s C12 ¼ 0.013); an () opening of switch II an ownwar rotation of the converter (C 12 ¼ 0.026, s C12 ¼ 0.011). The soli line shows the cumulative correlation as a function of the cutoff moe (see Methos), an the impulses show the contributions from each moe (see Methos). The moe number is shown in logarithmic scale to clearly illustrate the positions an contributions of lowfrequency moes. between positive an negative contributions from iniviual moes. Therefore, switch II is not irectly couple to actin bining. 3. The correlation between the opening of switch I an the ownwar rotation of the converter (Fig. 5 c) is positive an significant (C 12 ¼ 0.038, s C12 ¼ 0.013, so Z ~ 2.9). Its highest positive contribution is from moe 7, supporting the importance of this moe in allosterically coupling the converter an the nucleotie-bining site (21). We note that moe 2 gives a large negative contribution to this correlation that is cancelle by positive contributions from other moes, resulting in a positive net correlation. Therefore, it is important to analyze the net effects of all moes instea of focusing on a single moe. 4. The correlation between the opening of switch II an the ownwar rotation of the converter (Fig. 5 ) is also positive an significant (C 12 ¼ 0.026, s C12 ¼ 0.011, so Z ~ 2.4). Its highest contribution is from moe 8, supporting the importance of this moe in allosteric coupling. Therefore, the present correlation analysis has valiate in silico the structural coupling rules euce from structural comparisons (37 40). In particular, the observe couple motions from the prepowerstroke structure to the postpowerstroke structure are encoe in the former structure. Furthermore, we have uncovere a new coupling between the opening of switch I an the ownwar rotation of the converter. Therefore, actin bining may be inirectly couple to the ownwar rotation of the converter, with the signal being transmitte through switch I. DISCUSSION AND CONCLUSION In this stuy, we have focuse on the robustness of ENMbase normal moes to variations in the strength of elastic interactions instea of the network connectivity, because the former is amenable to quantitative comparison with first-orer perturbation theory. To further explore the moe robustness to variations in network connectivity, we have examine the effects of varying the cutoff istance, R c (with the aition of ranom Gaussian noise of s Rc ¼ 1or 2Å). The resulting robustness scores, R m, are qualitatively similar to the curve of s k ¼ 1.0 in Fig. 1 a the lowest two moes have low R m values, whereas the remaining eight moes have high R m values. Therefore, our first-orer perturbation theory makes qualitatively soun preictions for moe robustness to variations in both the interaction strength an the network connectivity. Our stuy shows that the moe robustness analysis shoul be combine with the moe coupling analysis to ientify groups of strongly couple moes, which must be analyze together to euce meaningful information about protein functional motions. Although some functionally relevant moes are foun to be robust iniviually (14,31), others are robust not by themselves but as a group (i.e., multiple collective motions can ominate protein functions). Moe coupling allows Nature to fine-tune protein conformational changes an thereby achieve functional iversity base on a common structural architecture. More case stuies along the lines propose here will offer a more complete unerstaning of both conservation an variation in protein conformational ynamics an the associate functions. Our results are consistent with the fining by Tama an co-workers (45) that a few low-frequency normal moes can escribe the observe conformational transitions, although they may be mixe by a changing coarseness of moel representation. In our formulation, the moes involve in the observe conformational transitions form a robust group, the members of which may be strongly couple to each other, leaing to mixing between them, but are weakly couple to outsiers, which explains the

9 2136 Zheng an Thirumalai robust escription of the observe conformational transitions. Therefore, it is important to use the moe coupling analysis propose here to assess the robustness of a group of couple moes for meaningful analysis of protein functional ynamics. By supplementing the ENM-base NMA with the assessments of moe robustness an moe coupling of the lowfrequency moes, we have foun a vali an efficient way to systematically explore the effects of parameter uncertainty in ENM-base moeling, an to evaluate the statistical significance of the results. This stuy lays a useful framework for using ENM-base moeling to preict combinations of moes that are important for functions. In aition, our stuy highlights the structural mechanism by which similar proteins (such as myosin II) in various organisms carry out functions with varying efficiency. W.Z. thanks the University at Buffalo for funing support. D.T. was supporte in part by grants from the National Science Founation (CHE ) an the Air Force Office of Scientific Research (FA ). REFERENCES 1. Tozzini, V Coarse-graine moels for proteins. Curr. Opin. Struct. Biol. 15: Hinsen, K Analysis of omain motions by approximate normal moe calculations. Proteins. 33: Atilgan, A. R., S. R. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin, et al Anisotropy of fluctuation ynamics of proteins with an elastic network moel. Biophys. J. 80: Tirion, M. M Large amplitue elastic motions in proteins from a single-parameter, atomic analysis. Phys. Rev. Lett. 77: Tama, F., an Y. H. Sanejouan Conformational change of proteins arising from normal moe calculations. Protein Eng. 14: Krebs, W. G., V. Alexanrov, C. A. Wilson, N. Echols, H. Yu, et al Normal moe analysis of macromolecular motions in a atabase framework: eveloping moe concentration as a useful classifying statistic. Proteins. 48: Bahar, I., an A. J. Raer Coarse-graine normal moe analysis in structural biology. Curr. Opin. Struct. Biol. 15: Ma, J. P Usefulness an limitations of normal moe analysis in moeling ynamics of biomolecular complexes. Structure. 13: Tama, F., an C. L. Brooks Symmetry, form, an shape: guiing principles for robustness in macromolecular machines. Annu. Rev. Biophys. Biomol. Struct. 35: Cui, Q., an I. Bahar Normal Moe Analysis. Theory an Applications to Biological an Chemical Systems... CRC press, Boca Raton, FL. 11. Delarue, M., an Y. H. Sanejouan Simplifie normal moe analysis of conformational transitions in DNA-epenent polymerases: the elastic network moel. J. Mol. Biol. 320: Zheng, W., an S. Doniach A comparative stuy of motor protein motions using a simple elastic network moel. Proc. Natl. Aca. Sci. USA. 100: Wang, Y., A. J. Raer, I. Bahar, an R. L. Jernigan Global ribosome motions reveale with elastic network moel. J. Struct. Biol. 147: Zheng, W., B. R. Brooks, an D. Thirumalai Allosteric transitions in the chaperonin GroEL are capture by a ominant normal moe that is most robust to sequence variations. Biophys. J. 93: Kim, M. K., G. S. Chirikjian, an R. L. Jernigan Elastic moels of conformational transitions in macromolecules. J. Mol. Graph. Moel. 21: Miyashita, O., J. N. Onuchic, an P. G. Wolynes Nonlinear elasticity, proteinquakes, an the energy lanscapes of functional transitions in proteins. Proc. Natl. Aca. Sci. USA. 100: Maragakis, P., an M. Karplus Large amplitue conformational change in proteins explore with a plastic network moel: aenylate kinase. J. Mol. Biol. 352: Franklin, J., P. Koehl, S. Doniach, an M. Delarue MinAction- Path: maximum likelihoo trajectory for large-scale structural transitions in a coarse-graine locally harmonic energy lanscape. Nucleic Acis Res. 35:W477 W Zheng, W., B. R. Brooks, an G. Hummer Protein conformational transitions explore by mixe elastic network moels. Proteins. 69: Chu, J. W., an G. A. Voth Coarse-graine free energy functions for stuying protein conformational changes: a ouble-well network moel. Biophys. J. 93: Zheng, W., an B. R. Brooks Ientification of ynamical correlations within the myosin motor omain by the normal moe analysis of an elastic network moel. J. Mol. Biol. 346: Zheng, W., J. C. Liao, B. R. Brooks, an S. Doniach Towar the mechanism of ynamical couplings an translocation in hepatitis C virus NS3 helicase using elastic network moel. Proteins. 67: Yu, H., L. Ma, Y. Yang, an Q. Cui Mechanochemical coupling in the myosin motor omain. II. Analysis of critical resiues. PLoS. Comput. Biol. 3: Tama, F., O. Miyashita, an C. L. Brooks III Flexible multi-scale fitting of atomic structures into low-resolution electron ensity maps with elastic network normal moe analysis. J. Mol. Biol. 337: Zheng, W., an B. R. Brooks Moeling protein conformational changes by iterative fitting of istance constraints using reoriente normal moes. Biophys. J. 90: Delarue, M., an P. Dumas On the use of low-frequency normal moes to enforce collective movements in refining macromolecular structural moels. Proc. Natl. Aca. Sci. USA. 101: Linahl, E., an M. Delarue Refinement of ocke protein-ligan an protein-dna structures using low frequency normal moe amplitue optimization. Nucleic Acis Res. 33: Jeong, J. I., Y. Jang, an M. K. Kim A connection rule for a-carbon coarse-graine elastic network moels using chemical bon information. J. Mol. Graph. Moel. 24: Konrashov, D. A., Q. Cui, an G. N. Phillips Jr Optimization an evaluation of a coarse-graine moel of protein motion using x-ray crystal ata. Biophys. J. 91: Eyal, E., L. W. Yang, an I. Bahar Anisotropic network moel: systematic evaluation an a new web interface. Bioinformatics. 22: Nicolay, S., an Y. H. Sanejouan Functional moes of proteins are among the most robust. Phys. Rev. Lett. 96: Tama, F., M. Valle, J. Frank, an C. L. Brooks Dynamic reorganization of the functionally active ribosome explore by normal moe analysis an cryo-electron microscopy. Proc. Natl. Aca. Sci. USA. 100: Zheng, W., B. R. Brooks, S. Doniach, an D. Thirumalai Low-frequency normal moes that escribe allosteric transitions in biological nanomachines are robust to sequence variations. Proc. Natl. Aca. Sci. USA. 103: Kitao, A., an N. Go Investigating protein ynamics in collective coorinate space. Curr. Opin. Struct. Biol. 9: De La Cruz, E. M., an E. M. Ostap Relating biochemistry an function in the myosin superfamily. Curr. Opin. Cell Biol. 16: Houusse, A., V. N. Kalabokis, D. Himmel, A. G. Szent-Gyorgyi, an C. Cohen Atomic structure of scallop myosin subfragment S1 complexe with MgADP: a novel conformation of the myosin hea. Cell. 97:

10 Moe Coupling in Allosteric Transitions Geeves, M. A., an K. C. Holmes Structural mechanism of muscle contraction. Annu. Rev. Biochem. 68: Coureux, P. D., A. L. Wells, J. Menetrey, C. M. Yengo, C. A. Morris, et al A structural state of the myosin V motor without boun nucleotie. Nature. 425: Coureux, P. D., H. L. Sweeney, an A. Houusse Three myosin V structures elineate essential features of chemo-mechanical transuction. EMBO J. 23: Reubol, T. F., S. Eschenburg, A. Becker, F. J. Kull, an D. J. Manstein A structural moel for actin-inuce nucleotie release in myosin. Nat. Struct. Biol. 10: Berg, J. S., B. C. Powell, an R. E. Cheney A millennial myosin census. Mol. Biol. Cell. 12: Li, G. H., an Q. Cui Analysis of functional motions in Brownian molecular machines with an efficient block normal moe approach: myosin-ii an Ca 2þ -ATPase. Biophys. J. 86: Zheng, W., an B. R. Brooks Probing the local ynamics of nucleotie bining pocket couple to the global ynamics: myosin versus kinesin. Biophys. J. 89: Navizet, I., R. Lavery, an R. L. Jernigan Myosin flexibility: structural omains an collective vibrations. Proteins. 54: Tama, F., M. Feig, J. Liu, C. L. Brooks, an K. A. Taylor The requirement for mechanical coupling between hea an S2 omains in smooth muscle myosin ATPase regulation an its implications for imeric motor function. J. Mol. Biol. 345: Glaser, F., Y. Rosenberg, A. Kessel, T. Pupko, an N. Ben-Tal Proteins. 58: , (URL: