Protein Structure and Function: Application of Bioinformatics Methods - John Rigden 2014

Protein Dynamics: From Structure to Function
Collective Variable Sampling Algorithms
TEE-REX

Advanced sampling Methods, such as ED (Amadei et al. 1996), achieve their efficiency (Amadei et al. 1996; de Groot et al. 1996a, b) primarily because protein configurational dynamics is dominated by only a small number of internal collective degrees of freedom. Moreover, systems simulated using such methods are invariably in a nonequilibrium state, which complicates the calculation of thermodynamic—i.e., equilibrium—properties. On the other hand, generalized-ensemble algorithms like REX not only enhance sampling but also yield the proper statistical ensembles required to compute equilibrium properties amenable to experimental verification. However, REX quickly becomes computationally prohibitive for systems comprising more than a few thousand atoms, restricting its current application to small Peptides (Pitera and Swope 2003; Cecchini et al. 2004; Nguyen et al. 2005; Liu et al. 2005; Seibert et al. 2005). The novel Temperature Enhanced Essential dynamics Replica Exchange (TEE-REX) algorithm (Kubitzki and de Groot 2007) combines the appealing features of REX with those arising from the individual excitation of functionally relevant modes, while avoiding the drawbacks of both approaches.

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Fig. 9.9. Comparison of two-dimensional Free energy surfaces (in kJ/mol) of dialanine calculated using MD (panel a) and TEE-REX (panel b). The deviation ∆G_ТЕЕ-REX - GMD is comparable to the statistical error of -0.1 k_βT

Figure 9.6 depicts a schematic comparison between the standard-temperature REX algorithm (left) and the TEE-REX algorithm (right). TEE-REX follows the same overall scheme as REX: a series of system replicas are simulated in parallel and independently, with exchange attempts periodically performed between neighboring replicas. Unlike REX, however, all replicas except the reference one thermally excite only those degrees of freedom that contribute significantly to the overall fluctuations of the system (the collective subspace {es}). This allows several advantages to be combined while eliminating shortcomings. In contrast to standard REX, the individual excitation of collective coordinates promotes sampling along these functionally important motion modes—thereby leveraging the advantages of ED. To compensate for the drawbacks associated with such individual excitation (i.e., ensemble distortion), the scheme is embedded within the REX protocol. Consequently, the resulting ensembles exhibit an approximately Boltzmann distribution, benefiting from the enhanced sampling properties of REX. The exchange probability (Equation 9.1) between two replicas depends critically on the number of excited degrees of freedom in the system. Because these degrees of freedom constitute only a minor fraction of the total system degrees of freedom, the bottleneck of low exchange probabilities in full-atom REX simulations is circumvented. Thus, for given exchange probabilities, a temperature difference ∆T large enough to require only a few replicas can be employed.

Figure 9.9 illustrates the two-dimensional free energy landscape projection of dialanine calculated via MD (panel A) and TEE-REX (panel B). The thermodynamic behavior of a system can be considered fully determined if any thermodynamic potential, such as the relative Gibbs free energy ∆G, is known. Comparing free energies thus allows us to assess the degree of concordance between ensembles generated by different computational methods. For this, it is crucial that the ensembles have an identical composition. This requirement is met in the dialanine test case. A detailed Analysis of the free energy surface shapes obtained from MD and TEE-REX demonstrates that the maximum absolute deviation is 1.5 kJ/mol ≈ 0.6 k_BT from the ideal case ∆G_TEE-REX - G_MD = 0, which is commensurate with the maximum statistical error of 0.15 k_BT for each method. The minor discrepancies detected in the TEE-REX ensemble are likely attributable to the exchange of nonequilibrium structures within the reference ensemble.

For guanylin, a small 13-amino-acid peptide hormone, the sampling efficiency of the TEE-REX algorithm was evaluated in comparison with MD (Currie et al. 1992). Trajectories generated by both methods under identical computational resource expenditures were projected onto the (φ, ψ) space as well as onto various two-dimensional subspaces defined by principal components derived from the MD ensemble of guanylin structures. The temporal Evolution of the sampled configurational space volume was measured using these projections. Ultimately, the sampling performance of MD proved quite limited compared to TEE-REX, which outperformed MD by an average factor of 2.5, depending on the subspace used for projection.

9.3.2.1. Applications: Transition Pathway Search in Adenylate Kinase

Elucidating the functional basis of many Proteins (Gerstein et al. 1994; Berg et al. 2002; Karplus and Gao 2004; Xu et al. 1997) requires detailed knowledge of the transitions between functionally significant Conformations. In recent years, X-ray crystallography and NMR spectroscopy have predominantly provided static snapshots of various protein conformational states, leaving questions regarding the transitions between these states unanswered. For atomic-resolution MD simulations, clarifying the pathways and Mechanisms of Protein conformational dynamics remains a non-trivial task due to the long characteristic timescales involved. A prime example in this regard is adenylate kinase from E. coli. This monomeric enzyme plays a pivotal role in cellular energy Homeostasis by regulating intracellular ATP levels through the catalyzed reaction (Mg²⁺:ATP) + AMP ⇌ (Mg²⁺:ADP) + ADP. The enzyme's structure comprises three domains (Fig. 9.10): a large central “CORE” domain (shown in light gray), an AMP-binding domain designated as “AMPbd” (shown in black), and a lid-like ATP-binding domain termed “LID” (shown in dark gray), which shields the phosphate groups in the Active Site (Müller et al. 1996). In the absence of a Ligand, the LID and AMPbd domains adopt an open conformation, assuming that the closed conformation is observed in the structure crystallized with the transition-state inhibitor Ap5A (Müller and Schulz 1992). Here, the ligands reside in the highly specific environment required for catalysis. Recent ¹⁵N nuclear spin relaxation NMR studies (Shapiro and Meirovitch 2006) revealed nanosecond-range motions of the catalytic domain within the flexible AMPbd and LID domains, whereas CORE domain relaxation occurs on a picosecond timescale (Tugarinov et al. 2002; Shapiro et al. 2002). The conformational mobility of adenylate kinase has been explored in several computational studies (Temiz et al. 2004; Maragakis and Karplus 2005; Lou and Cukier 2006; Whitford et al. 2007; Snow et al. 2007). However, due to the large amplitude of the motions and the long timescales on which they occur, all-atom MD has thus far failed to capture spontaneous transitions between the open and closed conformations. Applying the TEE-REX algorithm facilitates the observation of such transitions and enables a full-atom Description of the transition pathway and its underlying mechanisms (Kubitzki and de Groot 2008). To construct this description, collective subspaces {es} were formulated from short MD simulations of an arbitrary conformation, incorporating structures in both open and closed states. In the latter case, {es} modes—including the difference mode linking the open and closed experimental structures—were excited.

Fig. 9.10. Closed (left) and open (right) crystallographic structures of adenylate kinase from E. coli, alongside intermediate structures depicting the two Phases of the closed-to-open transition. Adenylate kinase consists of the CORE domain (light gray), the AMPbd domain (black), and the LID domain (dark gray). In the closed structure (left), the Ap5A transition-state inhibitor has been removed

The observed transition pathway can be conceptualized in two phases. Starting from the closed state (Fig. 9.10, left), the LID domain remains closed, while the AMPbd domain, harboring helices α2 and α3, adopts a semi-open conformation. During this process, helix α2 bends toward helix α4 of the CORE domain by 15 degrees relative to helix α3. Such opening of the AMP-binding site could facilitate the efficient release of the generated product. In the second phase, opening of the LID domain is observed, which is partially correlated with the opening of the AMPbd domain. Compared to coarse-grained approaches, all-atom TEE-REX simulations allow for a detailed examination of inter-residue interactions. In the case of adenylate kinase, this revealed The formation of a stable salt bridge between residues Asp118 and Lys136 During the first phase, linking the LID and CORE domains. An evaluation of all non-bonded interactions between these domains demonstrated that this salt bridge makes a substantial contribution to interdomain binding. Consequently, abolishing the bridge via mutation—such as Asp118Ala—should diminish the Stability of the open state. A comparison of fourteen adenylate kinase PDB structures from Yeast, maize, humans, and Bacteria revealed that eleven of them feature such a salt bridge at the LID-CORE interface. An alternative transition pathway appears feasible, but analysis of all TEE-REX simulations suggests the existence of a high-energy barrier that impedes the full opening of the AMPbd domain once the LID domain opens. Coupled with the observed prominent fluctuations in Secondary structure elements—indicative of high internal strain energy—these enthalpic constraints along this route likely render it unsuitable as the transition pathway in adenylate kinase.