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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Mol Biol. Author manuscript; available in PMC Oct 3, 2008.
Published in final edited form as:
PMCID: PMC2561047
NIHMSID: NIHMS18443

Conformational transitions of Adenylate Kinase: switching by cracking

Abstract

Conformational heterogeneity in proteins is known to often be the key to their function. We present a coarse grained model to explore the interplay between protein structure, folding and function which is applicable to allosteric or non-allosteric proteins. We employ the model to study the detailed mechanism of the reversible conformational transition of Adenylate Kinase (AKE) between the open to the closed conformation, a reaction that is crucial to the protein’s catalytic function. We directly observe high strain energy which appears to be correlated with localized unfolding during the functional transition. This work also demonstrates that competing native interactions from the open and closed form can account for the large conformational transitions in AKE. We further characterize the conformational transitions with a new measure ΦFunc, and demonstrate that local unfolding may be due, in part, to competing intra-protein interactions.

Keywords: Conformational Change, Energy Landscape Theory, Cracking, Strain Energy, Adenylate Kinase

Introduction

Flexibility and conformational changes are well acknowledged to be indispensable properties of proteins. New experiments using ultrafast laser technology and detailed computer simulations have begun to reveal the motions of these proteins, which encompass a rich repertoire of movements on various length and time scales. These motions, which complement the static three-dimensional structures provided by X-ray crystallography and NMR measurements, are essential to understand protein functions[1].

Protein flexibility and plasticity allow proteins to bind ligands, form oligomers, aggregate, and perform mechanical work. Therefore, the ability to alter protein dynamics may enable quantitative control of protein functionality. While this form of functional control is very important to biology, it is not well understood from either a theoretical or experimental basis. Thus, the question arises: How can we quantitatively connect conformational dynamics with biomolecular recognition and function? To address this question, we propose a new structure-based model to study the dynamical properties of proteins, specifically, conformational rearrangement.

Large conformational changes in proteins are important in many cellular signaling pathways, which can be generally described by the following steps. First, a signaling protein becomes activated, which then activates, or deactivates, signal transducing proteins, such as kinases. Signal transducing proteins are mobile and communicate with receptor proteins, which then produce specific reactions. The activity of many signal transducing proteins is associated with large conformational changes. For example, C-terminal Src Kinase protein[2], the Cyclin Dependent Kinase family [3], the Protein Kinase C family [4] and Adenylate Kinase (AKE)[5] have stable inactive conformations, in addition to active forms. Since the balance between conformations regulates protein activity, conformational transitions play important roles in the machinery of the cell[6].

Functional conformational transitions require a biomolecule to have at least a pair of conformational states of nearly equal free energy. The energy landscapes of these proteins have several basins of attraction and the transitions between basins dictates the conformational dynamics[7]. Despite the biological significance of these processes, the details of these processes are not fully understood. With a complete understanding of conformational changes we hope to predict which proteins have multiple conformations, predict these alternate conformations, determine the properties of the conformational transition ensemble, explain how proteins have evolved to have these properties and eventually design novel macromolecular machines which can execute any given biological function. To work towards these objectives, we explore the relationship between the structure, folding and function of AKE.

While many studies have investigated the relationship between protein structure and folding, fewer have focused on the relationship between structure and function, and even fewer have explored the interplay between protein structure, folding mechanism and function. Current experimental methods, including NMR, X-ray crystallography and fluorescence spectroscopy have been successful in describing the structural properties of individual states. These methods sometimes also manage to capture the chain flexibility[8,9]. Nonetheless, experimental techniques have not been able to provide the molecular details necessary to fully understand the mechanism of conformational changes. Due to these limitations, there has been significant effort to develop a theoretical framework for describing functional transitions in proteins[1015]. With a developed framework, one may study the energetic barriers associated with conformational transitions, their coupling to folding/unfolding (cracking), the role of ligands, and the role of energetic heterogeneity and frustration in conformational transitions[10,11]. In this work we propose a structure-based model that has a clear physical interpretation. Our model demonstrates that intraprotein contacts formed in the ligand bound structure of AKE can be responsible for the observed functional conformational changes. There has been success in applying simplified models to conformational changes, but our model provides a new physical interpretation that has not been proposed elsewhere.

The simplest model to describe functional transitions is based on landscape hopping and cracking between elastic networks[10,11]. To lowest order approximation, all interactions about a minimum are harmonic. Thus, this approach uses the most simplified approximation to the landscape about two energetic basins. From this model, the energetics of transitions are determined. This approach has been successful in demonstrating the physical relationship between protein fluctuations (low frequency normal modes) and protein function (conformational transition), and thus serves as a benchmark for further work.

To elucidate the relationship between protein structure, folding and function, functional transitions have been modeled as a result of ”hopping” [12,14] between structure-based energy surfaces. These structure-based potentials, which were inspired by the work of Gō[16], have had great success in explaining the interplay between protein structure and protein folding[17,18]. A limitation of these models is that the two structure-based energy surfaces have many nearly redundant contributions, since the conformations of interest have structural overlap. When applying these models to entire proteins, these near-redundancies may, or may not, contribute to the conformational changes. These redundancies add a degree of uncertainty to the physical interpretation of the system. Therefore, redundant interactions have been removed and replaced by single contacts for both structures.

Inspired by the successes of minimalist structure-based models in advancing our understanding of protein folding and molecular recognition[1726], our approach begins with the established theoretical framework of protein folding. As described below, we extend this framework to account for large conformational changes.

It is well established that protein folding is the result of a globally funneled, minimally frustrated energy landscape[2729]. The application of the principle of minimal frustration via structure-based potentials with single native basins has had considerable success in explaining the physics of protein folding. To now explain large functional transitions, there is a need for multiple basins. Thus, we generalize the minimally frustrated energy landscape of protein folding studies to incorporate biologically functional motions. We propose protein structure dominates functional behavior, as well as protein folding. Thus, we begin with a structure-based potential and add gradual perturbations, based on an alternate structure, to produce multiple minima. Our model implies, as does our previous model[10], that the transition ensemble can be determined from information of the conformations of interest. This is an implication of structure-based models in general (folding transition states can be determined by information of the native state). Using our model, we also show that multiple stable conformations may be due to amplified roughness in the global energetic landscape upon ligand binding.

Some proteins undergo large conformational changes without the aid of a cofactor. In allosteric proteins, however, such as Calmodulin and AKE, large conformational changes are associated with a co-factor, often an ion or a small biological molecule. Our model is general enough to be applied to both allosteric and non-allosteric conformational changes.

The model protein used in this study is E. Coli Adenylate Kinase. AKE is a 214 residue 3 domain protein (Figure 1) that catalyzes the reaction

Fig. 1
Functionally Relevant Conformations of AKE
ATP+AMPAKE2ADP
(1)

while undergoing large conformational changes which are believed to be the rate limiting steps of the reaction[30]. This protein was chosen mainly because it is well established that its multiple structures are catalytically relevant and because there is evidence that the conformational changes are rate limiting. Moreover, AKE is a good protein system to study the physics of conformational switching because there is a large amount experimental and theoretical data available on this process.

RESULTS

Hamiltonian Determination and Implications

First, we developed several potentials and determined which reproduces the structural properties of the open and closed forms of AKE. Second, we employed the superior potential to study the conformational transitions of AKE. Analogous to protein folding models where information of the native state is used to model the folding properties, this work uses information about two stable forms of AKE to infer conformational transition properties. To determine which potential most accurately accounts for the structural properties of AKE’s conformations, we compared conformational preference (i.e., open or closed), interresidue distance distributions and B-factors to experimental results. Four proposed Hamiltonians were compared: Hopen-Copen-D (open structure potential), Hclosed-Copen-D (open/closed mixed structure potential), Hopen-Cclosed-D (closed/open structure potential) and Hclosed-Cclosed-D (closed structure-based potential, see Models and Methods). Each potential stabilizes the contacts native to the open or closed form (denoted by C-open and C-closed) and the dihedral angles found in the open or closed form (D-open and D-closed). According to the above criteria, Hopen-Copen-D reproduces experimental results most accurately (explained below).

The first experimentally known property of AKE that our potential must reproduce is that the unligated protein must be predominantly in the open form. Since we later propose ligand binding can be represented by introducing contacts unique to the closed form (which are scaled by ε2; see Models and Methods), the simulated AKE without the contacts unique to the closed form (i.e. ε2 = 0.0) must also be in the open form. Hopen-Copen-D and Hopen-Cclosed-D have this property. Hclosed-Copen-D and Hclosed-Cclosed-D do not exhibit this property under any conditions (data not shown). Since the open state is not an energetic minimum (global or local) for Hclosed-Copen-D nor Hclosed-Cclosed-D these are not appropriate potentials for our investigation, and were not further considered. This result suggests that the open state is not purely a consequence of entropy, but energetic contributions are important as well.

Distance distributions P(r) of residues A55 and V169 (located in the NMP and LID domains, respectively) have been determined experimentally for unligated and ligated AKE[31] and were compared to the values obtained for the remaining two Hamiltonians: Hopen-Copen-D and Hopen-Cclosed-D. Rmax is the value of r at which P(r) is a maximum. In simulations, Rmax does not vary significantly for T < Tf. The resulting Rmax values from simulations and experiments are summarized in Table 1. Rmax for Hopen-Copen-D with ε2 = 0 (unligated) agrees very well with experiments. For Hopen-Cclosed-D residues A55 and V169 are closer than in experiments, indicating the relative locations of the LID and NMP domains do not reflect in-solution dynamics. Rmax for the closed form (ε2 = 1.6 maintains a closed conformation) for both potentials agrees equally well with the crystal structure distances but not as well with the fluorescence results. These results support Hopen-Copen-D as the more appropriate potential for our studies.

Table 1
Summary of CαCα distance distributions for A55 and V169 from experiments[31] and simulations. Rmax is value for which the CαCα distance distribution is peaked. This data suggests Hopen-Copen-D is ...

B-factors for each conformation, simulated under the different potentials, were compared to crystal structure B-factors. Correlation coefficients between simulated B-factors and experimental B-factors were computed. The correlation coefficient for the B-factors from the open crystal structure and Hopen-Copen-D is 0.68. To correlation coefficient between B-factors from the crystal structure and Hopen-Cclosed-D is 0.56. Both potentials’ B-factors were poorly correlated for the closed conformation (r<0.5). B-factors for all simulations were significantly larger than crystal structure B-factors, which is expected when comparing simulated to crystallographic B-factors[32]. Large B-factors have also been observed in all-atom, explicit solvent, simulations and were attributed to not including crystal contacts[33]. To validate this claim, we simulated the dimeric form (PDB entry 4AKE:chains A and B) of the open conformation using a structure-based potential. The native interactions were determined with CSU[34]. The average B-factor was reduced from the monomer value of 273 Å2 to 78 Å2 for the dimer, and the correlation with experimental B-factors decreased to 0.44. The experimental average B-factor is around 38 Å2. Since including a fraction of the crystal contacts (via a dimer) significantly alters the dynamics, crystallographic B-factors may not accurately describe in-solution multi-domain protein dynamics. Though, the discrepancy may also imply that our model is too flexible. Since the simulated dimeric B-factors are two times larger than the crystal B-factors, it is possible that the energetic barriers in our model are too small by up to a factor of two.

The analysis of conformational preference for the four designed potentials shows that Hopen-Copen-D most accurately represents the open and closed conformations of AKE and is thus used to study the conformational transitions. This decision has several implications. First, it suggests that the energy landscape has a single minimum corresponding to the open form, and additional minima are the result of perturbations from ligand binding. Second, the open conformation is the only stable state for some non-zero values of ε2 (Figure 2). This demonstrates that contacts that are not native to the open form can exist and aid in conformational changes while not destabilizing the open form. Finally, ligand binding is well represented by increasing ε2, therefore, to first approximation the effects of ligand binding are manifested in contacts found in the closed form.

Fig. 2
Contacts Native to the Closed Conformation Can Account for Large Conformational Changes

Energetic Barriers of Conformational Transitions

Free energy as a function of RMSD from the closed form is shown in Figure 3(a). TSE I and TSE II correspond predominantly to LID domain closure, and NMP domain closure, respectively (see Figure 4(d)). Figures 3(b) and 3(c) illustrate the energetic properties of TSE I and TSE II (see Functional Φ-values below). This finding is surprising, since there are more interactions between the NMP domain and CORE domain than there are between the LID domain and CORE domain. The barrier for LID closure is 0.9 kBT and the barrier for NMP closure is 3.0 kBT. Thus, if structural contributions dominate functional kinetics, then NMP domain closure should be rate limiting in AKE catalysis. While this prediction has not been made previously, it agrees with previous results from elastic network models where the lowest frequency normal mode corresponds to LID domain motion and the second lowest mode corresponds to NMP domain motion[35]. Thus, it is reasonable to expect the curvature of the NMP closure barrier will be greater than that of LID closure. While elastic network models have not explored the possibility of the intermediate we observe here (LID closed, NMP open), they have predicted a steeper strain energy barrier when opening the NMP domain than closing the LID domain[10], also in agreement with our results.

Fig. 3
Multiple Transitions Seen in Conformational Rearrangement of AKE
Fig. 4
Proposed Hamiltonian Captures Dynamics of AMP, ATP and Ap5A Binding

Localized Strain Energy and Unfolding Govern Conformational Changes

We previously proposed a cracking mechanism for allosteric proteins, based on normal mode analysis of AKE[10]. Upon translation along the lowest frequency modes, insurmountably high strain energy accumulated in very localized regions of the protein. This strain was enough to unfold the entire protein (>20kcal/mol), thus we predicted that localized regions of the protein unfold during conformational transitions, as a mechanism to reduce strain and enhance catalytic efficiency. The simulations reported here support the high strain energy and unfolding hypothesis (Figure 5).

Fig. 5
High Strain Energy Gives Rise to Local Unfolding

Figure 5 (top left) shows the average strain energy (defined as the total potential energy) by residue. There are clear peaks near residues 60–70, 120–125 and to a lesser extent residues 10–20, 30–35, 80–90 and 170–180. These finding are in excellent agreement with normal mode predictions of high strain in residues 10, 110–125, 150 and 160–170[10]. In normal mode studies, residues 30, 60 and 80 have only weak peaks at the later stage of the conformational transition. This is likely due to the previous data mainly reporting the strain associated with LID closure, and not NMP closure [10].

We believe that the high strain energy in AKE is the result of competing energetic contributions. Since competing energetic terms can not be satisfied simultaneously, internal strain must result. Some regions of strain drive the protein’s opening transition and other regions of strain drive the closing transition (see Functional Φ-values below). Thus, the balance between competing strains energies is very important for the function of AKE.

Local unfolding was measured by calculating the average deviation in dihedral angle i from the closed and open conformations <Δφiopen/closed> as a function of RMSD from the closed conformation. The inset in Figure 5 shows <Δφ63open> and <Δφ63closed> (residues 63–66). The fact that <Δφ63open> and <Δφ63closed> both exceed zero indicates an unfolding event is occuring. To quantify the unfolding we calculated the D-value, defined as

D2=|<Δφiopen>|+|<Δφiclosed>|-|<Δφiopen>-<Δφiclosed>|
(2)

as a function of RMSD from the closed form. By definition, D is greater than zero unless [var phi]closed < [var phi] < [var phi]open, for which values D is zero (if 0 < [var phi]open[var phi]closed < π). In other words, if a given dihedral angle is between the corresponding angles of the open and closed forms, D equals zero, otherwise D > 0 as it deviate from the boundary. The maximum value of D for each TSE, Dmax, was calculated for each dihedral angle and averaged by residue1. < Dmax > for the two energetic barriers is shown in Figure 5.

There is excellent agreement between regions of high strain energy and local unfolding (Figure 5). To no surprise the hinge region of the LID domain is under significant strain. More surprisingly, and in excellent agreement with normal mode predictions, is the strain experienced by residues 60–63. The different regions of strain correspond to strain that drives the protein opening and strain that drives the protein closing. Thus, large conformational changes in AKE are, in part, the result of competing intra-protein interactions. These interaction give rise to large strain energies, which are reduced through local unfolding.

Functional Φ-values

To determine which interactions are responsible for conformational changes, high strain energy and local unfolding, we calculated functional Φ-values for each residue[1012], defined as

ΦFunc=ΔΔG-XΔΔGY-X,
(3)

where ΔΔG‡−X = ΔG − ΔGX, ΔΔGY X = ΔGY − ΔGX and ‡ is the transition state between states X and Y2. The ΦFunc values for LID closure (TSE I) and NMP closure (TSE II) are mapped onto the structure of the closed form (Figures 3(b) and (c)). Before discussing the results, it is important to clarify the similarities and differences in the definition and interpretation of ΦFunc-values and protein folding Φ-values.

Protein folding Φ-values measure the amount of native content present in a residue at the folding transition state ensemble[36]. Owing to the funneled global landscape[28] and the principle of minimal frustration[37] ΔΔGY X ≥ ΔΔG‡ − X ≥ 0, where Y = native state and X = denatured state (i.e. non-local interactions stabilize the transition state more than the denatured state, and the native state more than the transition state). Thus, for protein folding, negative Φ-values and Φ-values greater than 1 are not easily interpreted.

As with folding Φ-values, ΦFunc measures the amount of state Y content present in the transition state ensemble. Though, due to the complexity of conformational changes, the same restrictions on ΔΔG and ΦFunc are not applicable. Since conformational changes arise from perturbations to the global landscape, |ΔΔGY X| can be less than |ΔΔG‡ − X| and ΔΔG‡ − X can be negative. In order to obtain a complete picture of the dynamics, we must study Φfunc in conjunction with ΔΔG.

A positive ΦFunc-value indicates ΔΔGY X ≥ ΔΔG‡ − X ≥ 0 or ΔΔGY X ≤ ΔΔG‡ − X ≤ 0. i.e., the residue stabilizes both the transition state and state Y (relative to state X), or destabilizes the transition state and state Y. Like-wise, negative ΦFunc-values (ΔΔGY X ≥ 0 ≥ ΔΔG‡ − X or ΔΔGY X ≤ 0 ≤ ΔΔG‡ − X) indicate the residue’s energetic effect on the transition state is opposite of that on state Y. The ambiguity is resolved by observing ΔΔGY X.

An additional deviation from Φ-values is ΦFunc is not calculated if |ΔΔGY X| < 0.75 kBT AND |ΔΔG‡ − X| < 0.75 kBT, rather than just using a cut-off for |ΔΔGY X|. Since we can’t assume |ΔΔGY X| > |ΔΔG‡ − X|, residues may strongly affect the transition state and not state X nor Y. We do not want to filter this feature out accidentally.

Our model indicates, as shown in Figure 3(b), that residues 131, 135 and 143 (blue residues in LID domain) stabilize the closed LID domain (ΔΔGY X > 0) and contribute to closure of the LID domain (ΦFunc > 0). Our model also predicts that residues 118 and 121 (red residues at hinge region between the LID and CORE domains) resist LID domain closure (ΔΔGY X < 0) and this resistance is accumulated during the closing transition state (ΦFunc > 0), in agreement with high strain energies observed during the closing transition in normal mode calculations[10]. Additionally, the large ΦFunc-values for ΔΔGY X > 0 that span the LID-NMP interface in Figure 3(c) and the lack of ΦFunc-values at the interface in Figure 3(b) indicate that NMP closure is stabilized substantially by the closed LID domain while LID closure is not highly influenced by the NMP domain. To further illustrate the NMP domain’s dependence on the LID domain and the LID domain’s independence of the NMP domain, we excluded the LID-NMP contacts and observed strong inhibition of NMP closure, with little effect on LID closure (not shown). Finally, our model shows that NMP intra-domain interactions resist conformational changes. Therefore, we predict that mutations in the core of the NMP domain will disrupt the interhelical interactions, reduce the energetic barrier to change conformation and/or stabilize the closed NMP domain.

These results support the claim that strain energy can be characterized as strain due to opening (ΔΔGY X > 0) or strain due to closing (ΔΔGY X < 0).

High Strain and Cracking are Robust Features of Conformational Changes

Using subsets of our ligand binding interactions, QLigand, we were able to simulate the unligated (Figure 4(a)), ATP bound (4(b)), AMP bound (4(c)) and Ap5A bound (4(d)) states. Since Ap5A is a bisubstrate analogue for AMP and ATP, which are biologically relevant, we have focused our attention there, thus far.

Strain energy and <Dmax> values were also computed for ATP binding and AMP binding (not shown). We observe that each peak in strain energy and <Dmax> in Figure 5 corresponds exclusively to either LID closure or NMP closure (not shown). This suggests the coupling between high strain and local unfolding (cracking) is a property common to many multi-domain proteins, where the number of domains is inconsequential.

CONCLUSIONS

Using a coarse-grained model, we have shown that large conformational changes in AKE can be accounted for by the intra-protein contacts that are formed upon ligand binding. This work demonstrates that the energy landscape of AKE follows the principle of minimal frustration, with the addition of contacts of two competing native states. Analysis of the structural properties of AKE has been performed via this model and has yielded several novel findings. One finding is that energetically competing native interactions can exist in AKE and contribute to its functional dynamics. This model provides the first direct measurements of cracking (which was proposed based on the results from normal mode analysis of AKE[10]). We have further demonstrated that this local unfolding is the result of competing strain energies in the protein and that this phenomenon applies to the motion of individual domains, suggesting it is not limited to three-domain proteins.

While this energetically heterogenous structure-based model has had considerable success, it is yet to be established whether or not it is common for intra-protein interactions to produce competing strain energies that give rise to local unfolding during functional transitions, or if AKE is somehow unique. Additionally, it will be interesting to see to what extent non-local interactions contribute to conformational changes. These answers, in addition to a more detailed understanding of conformational changes, will hopefully become clear as theoretical models become more widely applied and refined.

Models and Methods

Construction of the Energy Function

We study the conformational transitions of AKE by employing a structure-based Hamiltonian[21] with a modified contact map (described below). Structure-based potentials account for native interactions which are usually given the same energy weighting and produce a single funneled energy landscape. In a Cα-model each residue is represented by a single bead centered at the Cα position. Interaction energies of adjacent beads are harmonic in bond length and angle, with the geometry of the native state included through a dihedral term and non-local bead-bead interaction terms. Non-local contacts are included via

Econtacts=εn(5(σijr)12-6(σijr)10)
(4)

which has a minimum of depth εn at r = σij, with σij being the native distance between the Cα (i) and Cα (j) atoms in the crystal structure. In homogenous structure-based models, there is a single value for εn. To model conformational changes, different sets of contacts n are given different values of εn (see below). Temperature and energy values reported in this work are in units of ε1 (interaction strength of contacts in Qopen or Qclosed, see below), and distances are reported in Angstroms. A detailed description of a structure-based potential can be found elsewhere[21].

A simple structure-based potential has a single minimum. Making such a model for each form of AKE is straightforward. In this work, however, we construct a single potential that has multiple minima. Thus, to extend a structure-based model to systems with multiple minima, we modified the contact map and the dihedral angles. To determine whether dihedral angles of non-local interactions govern conformational change, two sets of non-local interactions were determined, one based on the open conformation and the other based on the closed conformation. In addition, two sets of dihedral values were considered, one from the open conformation and the other based on the closed conformation. All four combinations of contact map and dihedral values were simulated and compared to experimental data.

Native contact maps Qopen and Qclosed were generated using the CSU software package[34], and are assigned interaction distances from their respective structures. Contacts that are unique to the closed form, and are over 50% further apart when in the open form (QLigand in Figure 6) are deleted from Qclosed. Both contact sets, Qopen and Qclosed, were used in different simulations to model the open (unligated) conformation and were given an energetic weighting of ε1 = 1.0. These two contact maps were simulated to determine if non-local interactions govern the open conformational dynamics (see Results). To simulate the closed (ligand bound) conformation, QLigand contacts are added to each simulation with an energetic weight of ε2, which is varied from 0.0 to 2.0. QLigand is defined as the contacts when in the closed conformation with CαCα distances that are over 50% further apart in the open conformation. In summary, Qopen and Qclosed are contact maps based on the open and closed structures where additional contacts (QLigand) are added to represent ligand binding.

Fig. 6
Contacts Unique to Closed Form

Modeling ligand binding by including QLigand interactions is a warranted approximation. QLigand represents interactions that, inarguably, stabilize the ligand bound form. When ε2 > 0 there is an energetic bias to the closed form. Inspection of the ligand-protein interactions in AKE shows that many ligand mediated interactions are included in QLigand through direct residue-residue interactions. Additionally, some missing ligand mediated interactions are accounted for by local (in sequence) contacts. For example, the ligand mediated interaction between residues L58 and R167 is not included in QLigand, though several contacts between L58 and residues 170–175 are included.

The mixing of multiple structure-based potentials has been successful in representing conformational tranistions[12,14]. Previous studies mixed energetic terms at the global level (conformational switching is the result of all energetic contributions switching simultaneously between the two minima), whereas our potential is local (one interaction can stabilize the open conformation while a different interaction stabilizes the closed conformation). Thus, we must ask: Are dihedral angles or non-local contacts responsible for conformational transitions? To explore this question, Qopen and Qclosed were permuted with the open and closed values for dihedral angles, resulting in 4 possible Hamiltonians H(Qopen, [var phi]open), H(Qclosed, [var phi]open), H(Qopen, [var phi]closed) and H(Qclosed, [var phi]closed) (denoted by Hopen-Copen-D, Hclosed-Copen-D, Hopen-Cclosed-D and Hclosed-Cclosed-D, where C and D signify contacts and dihedral angles). The properties of AKE were examined with each potential. Several properties (conformational preference, interresidue distance distributions and B-factors) were compared to experimental values in order to determine the potential that most accurately captures AKE’s dynamics about the two known structures. As discussed in the Results section, we find the potential Hopen-Copen-D (single structure-based potential, based on the open structure, plus QLigand contacts) is the most appropriate to study the conformational transitions of AKE.

Calculating Thermodynamic Properties

We used Molecular Dynamics (MD) to simulate the conformational changes of AKE. We developed our own software to simulate constant temperature runs. Temperature was maintained using the Berendsen algorithm to couple the system to a thermal bath[38]. Our code was tested extensively to ensure correct calculations of energy and force through systematic debugging and simulated folding of CI2 and α-spectrin SH3 domain proteins. The folding mechanisms and thermodynamics of these proteins are in excellent agreement with previously reported simulations using AMBER[20].

Free energy profiles were obtained by simulating several constant temperature runs near room temperature and combining them by using the WHAM algorithm[39,40].

The folding temperature, Tf, was approximated, via kinetic unfolding simulations to be between, 1.15 and 1.25 (in reduced units). All results in this paper are at T=0.9≈ 0.8Tf, which corresponds approximately to room temperature.

RX-CORECM and RMSD calculations

Allosteric conformational changes involve major domain motion. We measure this motion via the spatial distance between the centers of mass of the domains and the RMSD from the closed conformation. RLID-CORECM and RNMP-CORECM are the distances between the centers of mass of the LID and the CORE domains, and the distance between the centers of mass of the NMP and the CORE domains, respectively. RLID-CORECM is 30.1 Å and 21.0 Å in the open and closed form. RNMP-CORECM is 22.0 Å and 18.4 Å in the open and closed form. Domain definitions are given in Table 2. RMSD was calculated using the McLachlan algorithm[41] in the PROFIT software package.

Table 2
Summary of symbols and names.

Definition of Open and Closed states

The open and closed structures were obtained from PDB entry 4AKE, chain A[42] and 1AKE, chain A[5]. The system is ”open” when F(RLID-CORECM) and F(RNMP-CORECM) have only one minimum each located near 30.1 Å and 22.0 Å. Figure 2 shows F(RLID-CORECM) for several values of ε2

Supplementary Material

Acknowledgments

We are are particularly grateful to Peter Wolynes for many discussions, in particular on the concept of ΦFunc. We also thank S. Takada and K. Okazaki for insightful comments on the manuscript. This work was funded by the NSF-sponsored Center for Theoretical Biological Physics (Grants PHY-0216576 and 0225630) and the NSF Grant 0543906. PW is supported by the NIH Molecular Biophysics Training Grant at UCSD (Grant T32 GM08326).

Footnotes

1For each residue, there are two Dmax values for which that residue is one of the middle two residues constituting the angle. We average over these two Dmax values for each residue.

2ΔΔG is the calculated change in free energy upon removal of all non-local interactions via perturbation theory. Thus, positive ΔΔGY X values indicate that a residue stabilizes state Y more than state X.

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