Using the experimentally determined components of the overall rotational diffusion tensor to restrain molecular shape and size in NMR structure determination of globular proteins and protein-protein complexes

Analysis of conformational exchange processes using methyl-TROSY-based Hahn echo measurements of quadruple-quantum relaxation

Transverse nuclear spin relaxation is a sensitive probe of chemical exchange on timescales on the order of microseconds to milliseconds. Here we present an experiment for the simultaneous measurement of the relaxation rates of two quadruple-quantum transitions in 13 CH3 -labelled methyl groups. These coherences are protected against relaxation by intra-methyl dipolar interactions and so have unexpectedly long lifetimes within perdeuterated biomacromolecules. However, these coherences also have an order of magnitude higher sensitivity to chemical exchange broadening than lower order coherences and therefore provide ideal probes of dynamic processes. We show that analysis of the static magnetic field dependence of zero-, double- and quadruple-quantum Hahn echo relaxation rates provides a robust indication of chemical exchange and can determine the signed relative magnitudes of proton and carbon chemical shift differences between ground and excited states. We also demonstrate that this analysis can be combined with established Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion measurements, providing improved precision in parameter estimates, particularly in the determination of 1 H chemical shift differences.

The structure of a minimum amyloid fibril core formed by necroptosis-mediating RHIM of human RIPK3

Receptor-interacting protein kinases 3 (RIPK3), a central node in necroptosis, polymerizes in response to the upstream signals and then activates its downstream mediator to induce cell death. The active polymeric form of RIPK3 has been indicated as the form of amyloid fibrils assembled via its RIP homotypic interaction motif (RHIM). In this study, we combine cryogenic electron microscopy and solid-state NMR to determine the amyloid fibril structure of RIPK3 RHIM-containing C-terminal domain (CTD). The structure reveals a single protofilament composed of the RHIM domain. RHIM forms three β-strands (referred to as strands 1 through 3) folding into an S shape, a distinct fold from that in complex with RIPK1. The consensus tetrapeptide VQVG of RHIM forms strand 2, which zips up strands 1 and 3 via heterozipper-like interfaces. Notably, the RIPK3-CTD fibril, as a physiological fibril, exhibits distinctive assembly compared with pathological fibrils. It has an exceptionally small fibril core and twists in both handedness with the smallest pitch known so far. These traits may contribute to a favorable spatial arrangement of RIPK3 kinase domain for efficient phosphorylation.

The amyloid structure of mouse RIPK3 (receptor interacting protein kinase 3) in cell necroptosis

RIPK3 amyloid complex plays crucial roles during TNF-induced necroptosis and in response to immune defense in both human and mouse. Here, we have structurally characterized mouse RIPK3 homogeneous self-assembly using solid-state NMR, revealing a well-ordered N-shaped amyloid core structure featured with 3 parallel in-register β-sheets. This structure differs from previously published human RIPK1/RIPK3 hetero-amyloid complex structure, which adopted a serpentine fold. Functional studies indicate both RIPK1-RIPK3 binding and RIPK3 amyloid formation are essential but not sufficient for TNF-induced necroptosis. The structural integrity of RIPK3 fibril with three β-strands is necessary for signaling. Molecular dynamics simulations with a mouse RIPK1/RIPK3 model indicate that the hetero-amyloid is less stable when adopting the RIPK3 fibril conformation, suggesting a structural transformation of RIPK3 from RIPK1-RIPK3 binding to RIPK3 amyloid formation. This structural transformation would provide the missing link connecting RIPK1-RIPK3 binding to RIPK3 homo-oligomer formation in the signal transduction.

Receptor Interacting Protein Kinase 3 (RIPK3) has a key role in TNF-induced necroptosis. Here, the authors combine solid state NMR measurements, MD simulations and cell based assays to characterize mouse RIPK3 and they present the structure of the RIPK3 amyloid core.

Fast NMR method to probe solvent accessibility and disordered regions in proteins

Understanding protein structure and dynamics, which govern key cellular processes, is crucial for basic and applied research. Intrinsically disordered protein (IDP) regions display multifunctionality via alternative transient conformations, being key players in disease mechanisms. IDP regions are abundant, namely in small viruses, allowing a large number of functions out of a small proteome. The relation between protein function and structure is thus now seen from a different perspective: as IDP regions enable transient structural arrangements, each conformer can play different roles within the cell. However, as IDP regions are hard and time-consuming to study via classical techniques (optimized for globular proteins with unique conformations), new methods are required. Here, employing the dengue virus (DENV) capsid (C) protein and the immunoglobulin-binding domain of streptococcal protein G, we describe a straightforward NMR method to differentiate the solvent accessibility of single amino acid N-H groups in structured and IDP regions. We also gain insights into DENV C flexible fold region biological activity. The method, based on minimal pH changes, uses the well-established ¹H-¹⁵N HSQC pulse sequence and is easily implementable in current protein NMR routines. The data generated are simple to interpret, with this rapid approach being an useful first-choice IDPs characterization method.

Xplor-NIH for molecular structure determination from NMR and other data sources

Xplor-NIH is a popular software package for biomolecular structure determination from nuclear magnetic resonance (NMR) and other data sources. Here, some of Xplor-NIH's most useful data-associated energy terms are reviewed, including newer alternative options for using residual dipolar coupling data in structure calculations. Further, we discuss new developments in the implementation of strict symmetry for the calculation of symmetric homo-oligomers, and in the representation of the system as an ensemble of structures to account for motional effects. Finally, the different available force fields are presented, among other Xplor-NIH capabilities.

The phosphocarrier protein HPr of the bacterial phosphotransferase system globally regulates energy metabolism by directly interacting with multiple enzymes in Escherichia coli

The histidine-phosphorylatable phosphocarrier protein (HPr) is an essential component of the sugar-transporting phosphotransferase system (PTS) in many bacteria. Recent interactome findings suggested that HPr interacts with several carbohydrate-metabolizing enzymes, but whether HPr plays a regulatory role was unclear. Here, we provide evidence that HPr interacts with a large number of proteins in Escherichia coli We demonstrate HPr-dependent allosteric regulation of the activities of pyruvate kinase (PykF, but not PykA), phosphofructokinase (PfkB, but not PfkA), glucosamine-6-phosphate deaminase (NagB), and adenylate kinase (Adk). HPr is either phosphorylated on a histidyl residue (HPr-P) or non-phosphorylated (HPr). PykF is activated only by non-phosphorylated HPr, which decreases the PykF K_half for phosphoenolpyruvate by 10-fold (from 3.5 to 0.36 mm), thus influencing glycolysis. PfkB activation by HPr, but not by HPr-P, resulted from a decrease in the K_half for fructose-6-P, which likely influences both gluconeogenesis and glycolysis. Moreover, NagB activation by HPr was important for the utilization of amino sugars, and allosteric inhibition of Adk activity by HPr-P, but not by HPr, allows HPr to regulate the cellular energy charge coordinately with glycolysis. These observations suggest that HPr serves as a directly interacting global regulator of carbon and energy metabolism and probably of other physiological processes in enteric bacteria.

Dynamic equilibrium between closed and partially closed states of the bacterial Enzyme I unveiled by solution NMR and X-ray scattering

Enzyme I (EI) is the first component in the bacterial phosphotransferase system, a signal transduction pathway in which phosphoryl transfer through a series of bimolecular protein-protein interactions is coupled to sugar transport across the membrane. EI is a multidomain, 128-kDa homodimer that has been shown to exist in two conformational states related to one another by two large (50-90°) rigid body domain reorientations. The open conformation of apo EI allows phosphoryl transfer from His189 located in the N-terminal domain α/β (EIN(α/β)) subdomain to the downstream protein partner bound to the EIN(α) subdomain. The closed conformation, observed in a trapped phosphoryl transfer intermediate, brings the EIN(α/β) subdomain into close proximity to the C-terminal dimerization domain (EIC), thereby permitting in-line phosphoryl transfer from phosphoenolpyruvate (PEP) bound to EIC to His189. Here, we investigate the solution conformation of a complex of an active site mutant of EI (H189A) with PEP. Simulated annealing refinement driven simultaneously by solution small angle X-ray scattering and NMR residual dipolar coupling data demonstrates unambiguously that the EI(H189A)-PEP complex exists in a dynamic equilibrium between two approximately equally populated conformational states, one corresponding to the closed structure and the other to a partially closed species. The latter likely represents an intermediate in the open-to-closed transition.

Coupling between overall rotational diffusion and domain motions in proteins and its effect on dielectric spectra

In this work, we formulate a closed-form solution of the model of a semirigid molecule for the case of fluctuating and reorienting molecular electric dipole moment. We illustrate with numeric calculations the impact of protein domain motions on dielectric spectra using the example of the 128 kDa protein dimer of Enzyme I. We demonstrate that the most drastic effect occurs for situations when the characteristic time of protein domain dynamics is comparable to the time of overall molecular rotational diffusion. We suggest that protein domain motions could be a possible explanation for the high-frequency contribution that accompanies the major relaxation dispersion peak in the dielectric spectra of protein aqueous solutions. We propose that the presented computational methodology could be used for the simultaneous analysis of dielectric spectroscopy and nuclear magnetic resonance data. Proteins 2015; 83:1571-1581. © 2015 Wiley Periodicals, Inc.

Correlated inter-domain motions in adenylate kinase

Correlated inter-domain motions in proteins can mediate fundamental biochemical processes such as signal transduction and allostery. Here we characterize at structural level the inter-domain coupling in a multidomain enzyme, Adenylate Kinase (AK), using computational methods that exploit the shape information encoded in residual dipolar couplings (RDCs) measured under steric alignment by nuclear magnetic resonance (NMR). We find experimental evidence for a multi-state equilibrium distribution along the opening/closing pathway of Adenylate Kinase, previously proposed from computational work, in which inter-domain interactions disfavour states where only the AMP binding domain is closed. In summary, we provide a robust experimental technique for study of allosteric regulation in AK and other enzymes.

NMR approaches for structural analysis of multidomain proteins and complexes in solution

NMR spectroscopy is a key method for studying the structure and dynamics of (large) multidomain proteins and complexes in solution. It plays a unique role in integrated structural biology approaches as especially information about conformational dynamics can be readily obtained at residue resolution. Here, we review NMR techniques for such studies focusing on state-of-the-art tools and practical aspects. An efficient approach for determining the quaternary structure of multidomain complexes starts from the structures of individual domains or subunits. The arrangement of the domains/subunits within the complex is then defined based on NMR measurements that provide information about the domain interfaces combined with (long-range) distance and orientational restraints. Aspects discussed include sample preparation, specific isotope labeling and spin labeling; determination of binding interfaces and domain/subunit arrangements from chemical shift perturbations (CSP), nuclear Overhauser effects (NOEs), isotope editing/filtering, cross-saturation, and differential line broadening; and based on paramagnetic relaxation enhancements (PRE) using covalent and soluble spin labels. Finally, the utility of complementary methods such as small-angle X-ray or neutron scattering (SAXS, SANS), electron paramagnetic resonance (EPR) or fluorescence spectroscopy techniques is discussed. The applications of NMR techniques are illustrated with studies of challenging (high molecular weight) protein complexes.

Equivalence between Euler angle conventions for the description of tensorial interactions in liquid NMR: application to different software programs

Long-range orientational restraints derived from alignment or rotational diffusion tensors have greatly contributed to the expansion of applications in biomolecular NMR. The orientation of the principal axis system of these tensors is usually described by the so-called Euler angles. However, no clear consensus has emerged concerning the convention of the associated orthogonal rotations. As a result, the different programs that derive or predict them have adopted different conventions, which make comparison between their results difficult. Moreover, the rotation schemes are seldom completely described. Here, we summarize the different conventions, determine which ones are adopted by commonly used software packages, and establish the formal equivalencies between the different calculated Euler angles.

Structure and dynamics of full-length HIV-1 capsid protein in solution

The HIV-1 capsid protein plays a crucial role in viral infectivity, assembling into a cone that encloses the viral RNA. In the mature virion, the N-terminal domain of the capsid protein forms hexameric and pentameric rings, while C-terminal domain homodimers connect adjacent N-terminal domain rings to one another. Structures of disulfide-linked hexamer and pentamer assemblies, as well as structures of the isolated domains, have been solved previously. The dimer configuration in C-terminal domain constructs differs in solution (residues 144-231) and crystal (residues 146-231) structures by ∼30°, and it has been postulated that the former connects the hexamers while the latter links pentamers to hexamers. Here we study the structure and dynamics of full-length capsid protein in solution, comprising a mixture of monomeric and dimeric forms in dynamic equilibrium, using ensemble simulated annealing driven by experimental NMR residual dipolar couplings and X-ray scattering data. The complexity of the system necessitated the development of a novel computational framework that should be generally applicable to many other challenging systems that currently escape structural characterization by standard application of mainstream techniques of structural biology. We show that the orientation of the C-terminal domains in dimeric full-length capsid and isolated C-terminal domain constructs is the same in solution, and we obtain a quantitative description of the conformational space sampled by the N-terminal domain relative to the C-terminal domain on the nano- to millisecond time scale. The positional distribution of the N-terminal domain relative to the C-terminal domain is large and modulated by the oligomerization state of the C-terminal domain. We also show that a model of the hexamer/pentamer assembly can be readily generated with a single configuration of the C-terminal domain dimer, and that capsid assembly likely proceeds via conformational selection of sparsely populated configurations of the N-terminal domain within the capsid protein dimer.

Determining interdomain structure and dynamics of a retroviral capsid protein in the presence of oligomerization: implication for structural transition in capsid assembly

Capsid (CA) proteins from all retroviruses, including HIV-1, are structurally homologous dual-domain helical proteins. They form a capsid lattice composed of unitary symmetric CA hexamers. X-ray crystallography has shown that within each hexamer a monomeric CA adopts a single conformation, where most helices are parallel to the symmetry axis. In solution, large differences in averaged NMR spin relaxation rates for the two domains were observed, suggesting they are dynamically independent. One relevant question for the capsid assembly remains: whether the interdomain conformer within a hexamer unit needs to be induced or pre-exists within the conformational space of a monomeric CA. The latter seems more consistent with the relaxation data. However, possible CA protein oligomerization and the structure of each domain will affect relaxation measurements and data interpretation. This study, using CA proteins from equine infectious anemia virus (EIAV) as an example, demonstrates a linear extrapolation approach to obtain backbone (15)N spin relaxation time ratios T1/T2 for a monomeric EIAV-CA in the presence of oligomerization equilibrium. The interdomain motion turns out to be limited. The large difference in the domain averaged <T1/T2> for a CA monomer is a consequence of the orthogonal distributions of helices in the two domains. The new monomeric interdomain conformation in solution is significantly different from that in CA hexamer. Therefore, if capsid assembly follows a nucleation-propagation process, the interdomain conformational change might be a key step during the nucleation, as the configuration in hexagonal assembly is never formed by diffusion of its two domains in solution.

Parameterization of solvent-protein interaction and its use on NMR protein structure determination

NMR structure determination is frequently hindered by an insufficient amount of distance information for determining the correct fold of the protein in its early stages. In response we introduce a simple and general structure-based metric that can be used to incorporate NMR-based restraints on protein surface accessibility. This metric is inversely proportional to the sum of the inverse square distances to neighboring heavy atoms. We demonstrate the use of this restraint using a dataset from the water to protein magnetization transfer experiment on the protein Bax and the solvent paramagnetic relaxation enhancement experiment on the protein ubiquitin and Qua1 homodimer. The calculated solvent accessibility values using the new empirical function are well correlated with the experimental data. By incorporating an associated energy term into Xplor-NIH, we show that structure calculation with a limited number of additional experimental restraints, improves both the precision and accuracy of the resulting structures. This new empirical energy term will have general applicability to other types of solvent accessibility data.

Protein fold determined by paramagnetic magic-angle spinning solid-state NMR spectroscopy

Biomacromolecules that are challenging for the usual structural techniques can be studied with atomic resolution by solid-state nuclear magnetic resonance. However, the paucity of >5 Å distance restraints, traditionally derived from measurements of magnetic dipole-dipole couplings between protein nuclei, is a major bottleneck that hampers such structure elucidation efforts. Here we describe a general approach that enables the rapid determination of global protein fold in the solid phase via measurements of nuclear paramagnetic relaxation enhancements (PREs) in several analogs of the protein of interest containing covalently-attached paramagnetic tags, without the use of conventional internuclear distance restraints. The method is demonstrated using six cysteine-EDTA-Cu²⁺ mutants of the 56-residue B1 immunoglobulin-binding domain of protein G, for which ~230 longitudinal backbone ¹⁵N PREs corresponding to ~10-20 Å distances were obtained. The mean protein fold determined in this manner agrees with the X-ray structure with a backbone atom root-mean-square deviation of 1.8 Å.

Water proton spin saturation affects measured protein backbone 15N spin relaxation rates

Protein backbone 15N NMR spin relaxation rates are useful in characterizing the protein dynamics and structures. To observe the protein nuclear-spin resonances a pulse sequence has to include a water suppression scheme. There are two commonly employed methods, saturating or dephasing the water spins with pulse field gradients and keeping them unperturbed with flip-back pulses. Here different water suppression methods were incorporated into pulse sequences to measure 15N longitudinal T1 and transversal rotating-frame T1ρ spin relaxation. Unexpectedly the 15N T1 relaxation time constants varied significantly with the choice of water suppression method. For a 25-kDa Escherichiacoli. glutamine binding protein (GlnBP) the T1 values acquired with the pulse sequence containing a water dephasing gradient are on average 20% longer than the ones obtained using a pulse sequence containing the water flip-back pulse. In contrast the two T1ρ data sets are correlated without an apparent offset. The average T1 difference was reduced to 12% when the experimental recycle delay was doubled, while the average T1 values from the flip-back measurements were nearly unchanged. Analysis of spectral signal to noise ratios (s/n) showed the apparent slower 15N relaxation obtained with the water dephasing experiment originated from the differences in 1HN recovery for each relaxation time point. This in turn offset signal reduction from 15N relaxation decay. The artifact becomes noticeable when the measured 15N relaxation time constant is comparable to recycle delay, e.g., the 15N T1 of medium to large proteins. The 15N relaxation rates measured with either water suppression schemes yield reasonable fits to the structure. However, data from the saturated scheme results in significantly lower Model-Free order parameters (<S2>=0.81) than the non-saturated ones (<S2>=0.88), indicating such order parameters may be previously underestimated.

Fast approximations of the rotational diffusion tensor and their application to structural assembly of molecular complexes

We present and evaluate a rigid-body, deterministic, molecular docking method, called ELMDOCK, that relies solely on the three-dimensional structure of the individual components and the overall rotational diffusion tensor of the complex, obtained from nuclear spin-relaxation measurements. We also introduce a docking method, called ELMPATIDOCK, derived from ELMDOCK and based on the new concept of combining the shape-related restraints from rotational diffusion with those from residual dipolar couplings, along with ambiguous contact/interface-related restraints obtained from chemical shift perturbations. ELMDOCK and ELMPATIDOCK use two novel approximations of the molecular rotational diffusion tensor that allow computationally efficient docking. We show that these approximations are accurate enough to properly dock the two components of a complex without the need to recompute the diffusion tensor at each iteration step. We analyze the accuracy, robustness, and efficiency of these methods using synthetic relaxation data for a large variety of protein-protein complexes. We also test our method on three protein systems for which the structure of the complex and experimental relaxation data are available, and analyze the effect of flexible unstructured tails on the outcome of docking. Additionally, we describe a method for integrating the new approximation methods into the existing docking approaches that use the rotational diffusion tensor as a restraint. The results show that the proposed docking method is robust against experimental errors in the relaxation data or structural rearrangements upon complex formation and is computationally more efficient than current methods. The developed approximations are accurate enough to be used in structure refinement protocols.

Impact of 15N R2/R1 relaxation restraints on molecular size, shape, and bond vector orientation for NMR protein structure determination with sparse distance restraints

(15)N R(2)/R(1) relaxation data contain information on molecular shape and size as well as on bond vector orientations relative to the diffusion tensor. Since the diffusion tensor can be directly calculated from the molecular coordinates, direct inclusion of (15)N R(2)/R(1) restraints in NMR structure calculations without any a priori assumptions is possible. Here we show that (15)N R(2)/R(1) restraints are particularly valuable when only sparse distance restraints are available. Using three examples of proteins of varying size, namely, GB3 (56 residues), ubiquitin (76 residues), and the N-terminal domain of enzyme I (EIN, 249 residues), we show that incorporation of (15)N R(2)/R(1) restraints results in large and significant increases in coordinate accuracy that can make the difference between being able or unable to determine an approximate global fold. For GB3 and ubiquitin, good coordinate accuracy was obtained using only backbone hydrogen-bond restraints supplemented by (15)N R(2)/R(1) relaxation restraints. For EIN, the global fold could be determined using sparse nuclear Overhauser enhancement (NOE) distance restraints involving only NH and methyl groups in conjunction with (15)N R(2)/R(1) restraints. These results are of practical significance in the study of larger and more complex systems, where the increasing spectral complexity and number of chemical shift degeneracies reduce the number of unambiguous NOE assignments that can be readily obtained, resulting in progressively reduced NOE coverage as the size of the protein increases.

New scenarios of protein folding can occur on the ribosome

Identifying and understanding the differences between protein folding in bulk solution and in the cell is a crucial challenge facing biology. Using Langevin dynamics, we have simulated intact ribosomes containing five different nascent chains arrested at different stages of their synthesis such that each nascent chain can fold and unfold at or near the exit tunnel vestibule. We find that the native state is destabilized close to the ribosome surface due to an increase in unfolded state entropy and a decrease in native state entropy; the former arises because the unfolded ensemble tends to behave as an expanded random coil near the ribosome and a semicompact globule in bulk solution. In addition, the unfolded ensemble of the nascent chain adopts a highly anisotropic shape near the ribosome surface and the cooperativity of the folding-unfolding transition is decreased due to the appearance of partially folded structures that are not populated in bulk solution. The results show, in light of these effects, that with increasing nascent chain length folding rates increase in a linear manner and unfolding rates decrease, with larger and topologically more complex folds being the most highly perturbed by the ribosome. Analysis of folding trajectories, initiated by temperature quench, reveals the transition state ensemble is driven toward compaction and greater native-like structure by interactions with the ribosome surface and exit vestibule. Furthermore, the diversity of folding pathways decreases and the probability increases of initiating folding via the N-terminus on the ribosome. We show that all of these findings are equally applicable to the situation in which protein folding occurs during continuous (non-arrested) translation provided that the time scales of folding and unfolding are much faster than the time scale of monomer addition to the growing nascent chain, which results in a quasi-equilibrium process. These substantial ribosome-induced perturbations to almost all aspects of protein folding indicate that folding scenarios that are distinct from those of bulk solution can occur on the ribosome.

Structural assembly of molecular complexes based on residual dipolar couplings

We present and evaluate a rigid-body molecular docking method, called PATIDOCK, that relies solely on the three-dimensional structure of the individual components and the experimentally derived residual dipolar couplings (RDCs) for the complex. We show that, given an accurate ab initio predictor of the alignment tensor from a protein structure, it is possible to accurately assemble a protein-protein complex by utilizing the RDCs' sensitivity to molecular shape to guide the docking. The proposed docking method is robust against experimental errors in the RDCs and computationally efficient. We analyze the accuracy and efficiency of this method using experimental or synthetic RDC data for several proteins, as well as synthetic data for a large variety of protein-protein complexes. We also test our method on two protein systems for which the structure of the complex and steric-alignment data are available (Lys48-linked diubiquitin and a complex of ubiquitin and a ubiquitin-associated domain) and analyze the effect of flexible unstructured tails on the outcome of docking. The results demonstrate that it is fundamentally possible to assemble a protein-protein complex solely on the basis of experimental RDC data and the prediction of the alignment tensor from 3D structures. Thus, despite the purely angular nature of RDCs, they can be converted into intermolecular distance/translational constraints. Additionally, we show a method for combining RDCs with other experimental data, such as ambiguous constraints from interface mapping, to further improve structure characterization of protein complexes.

Direct use of 15N relaxation rates as experimental restraints on molecular shape and orientation for docking of protein-protein complexes

(15)N relaxation rates contain information on overall molecular shape and size, as well as residue specific orientations of N-H bond vectors relative to the axes of the diffusion tensor. Here we describe a pseudopotential E(relax) that permits direct use of (15)N relaxation rates, in the form of R(2)/R(1) ratios, as experimental restraints in structure calculations without requiring prior information to be extracted from a known molecular structure. The elements of the rotational diffusion tensor are calculated from the atomic coordinates at each step of the structure calculation and then used together with the N-H bond vector orientations to compute the (15)N R(2)/R(1) ratios. We show that the E(relax) term can be reliably used for protein-protein docking of complexes and illustrate its applicability to the 40 kDa complex of the N-terminal domain of enzyme I and the histidine phosphocarrier protein HPr and to the symmetric HIV-1 protease dimer.