Numerous reviews (Clore et al., 1998; Zhou et al., 1999; Prestegard et al., 2000; Tolman, 2001; Al-Hashimi et al., 2002a; De Alba and Tjandra, 2002; Blackledge, 2005) highlight the utility of RDC data in a broad spectrum of applications to biological macromolecules. RDCs have been used in studies of carbohydrates (Tian et al., 2001a; Azurmendi and Bush, 2002; Azurmendi et al., 2002; Adeyeye et al., 2003), nucleic acids (Al-Hashimi et al., 2000a; Tjandra et al., 2000; Vermeulen et al., 2000; Al-Hashimi et al., 2002a; Al-Hashimi et al., 2002b) and proteins (Cornilescu et al., 1999; Fowler et al., 2000; Andrec et al., 2001; Tian et al., 2001b; Clore and Bewley, 2002; Assfalg et al., 2003; Bertini et al., 2003). Until recently, the role of RDCs in structure determination has generally been to provide supplemental restraints to a large number of distance-based NOE restraints. Recent developments (Tian et al., 2001b; Dosset et al., 2001; Prestegard et al., 2005; Valafar et al., 2005; Simin et al., 2014) have demonstrated the success of structure determination of macromolecules by using primarily or exclusively RDC data. The use of RDCs can lead to a significant reduction in data collection and analysis (Raman et al., 2010; Shealy et al., 2010; Lange et al., 2012; Valafar et al., 2012; Tang et al., 2015) while providing simultaneous resonance assignment, structure determination, and identification of dynamical regions (Tian et al., 2001b; Bernadó and Blackledge, 2004; Prestegard et al., 2005; Valafar et al., 2005; Shealy et al., 2011; Cole et al., 2016).

RDCs arise from the interaction of two magnetically active nuclei in the presence of the external magnetic field of an NMR instrument (Prestegard et al., 2000; Clore et al., 1998; Tjandra et al., 1996; Tolman et al., 1995). This interaction is normally reduced to zero, due to the isotropic tumbling of molecules in their aqueous environment. The introduction of partial order to the molecular alignment reintroduces dipolar interactions by minutely limiting isotropic tumbling. This partial order can be introduced in numerous ways (Prestegard and Kishore, 2001), including inherent magnetic anisotropy susceptibility of molecules (Prestegard et al., 2000), incorporation of artificial tags (such as lanthanides) that exhibit magnetic anisotropy (Nitz et al., 2004), or in a liquid crystal aqueous solution (Prestegard and Kishore, 2001). The RDC interaction phenomenon can be formulated in different ways (Tolman et al., 1995; Bax and Tjandra, 1997). In our work we utilize the matrix formulation of this interaction as shown in Eq. 1. The entity S shown in Eqs 1, 2 represents the Saupe order tensor matrix (Prestegard et al., 2000; Valafar and Prestegard, 2004; Saupe and Englert, 1963) (the ‘order tensor’) that can be described as a 3 × 3 symmetric and traceless matrix. D _max in Eq. 1 is a nucleus-specific collection of constants, r _ij is the separation distance between the two interacting nuclei (in units of Å), and v _ij is the corresponding normalized internuclear vector. The order tensor formulation of the RDC interaction provides a convenient mechanism of probing internal dynamics of proteins. Decomposition of the alignment tensor (Losonczi et al., 1999; Valafar and Prestegard, 2004) can reveal information regarding the level of order (Pomeranz and Gershenfeld, 2000; Tolman et al., 2001; Valafar and Prestegard, 2004) and the preferred direction of alignment (Tolman et al., 2001; Valafar and Prestegard, 2004). A careful comparison of order tensors obtained from different regions of a macromolecule can provide a diagnostic tool in identifying relative orientations between structural elements and/or the presence of internal dynamics (Tolman et al., 2001; Valafar and Prestegard, 2004; Bryson et al., 2008). D i j = ( D m a x r i j 3 ) v i j ∗ S ∗ v i j T (1) S = [ S xx S xy S xz S xy S yy S yz S xz S yz S zz ] ,   v ij =   ( cos ( θ x ) cos ( θ y ) cos ( θ z ) )     (2)

The collection of RDC data imposes additional steps in sample preparation and data acquisition when compared to the requisites of the traditional data acquisition by NMR spectroscopy. Despite the additional requirements, the use of RDCs may be justified based on several of their unique features. Our most recent work (Peti et al., 2002) illustrated the sensitivity of NOEs and RDCs as reporters of protein structures. Based on this work, NOEs tend to lose sensitivity as the search approaches the native structure, while RDCs become more sensitive. Therefore, the addition of RDCs has the potential of improving the structural resolution of proteins studies by NMR spectroscopy. RDCs can also report molecular motions on time-scales ranging from picoseconds to microseconds (Tolman et al., 1997; Meiler et al., 2001; Peti et al., 2002), during which many functionally important events occur. Indeed, in the 10 ns–1 s timescale window, RDCs are the most sensitive of NMR parameters (Tolman et al., 2001). Therefore, in instances of investigating internal dynamics of macromolecules, the use of RDCs can be very beneficial if not necessary. In summary, RDCs have the unique property of simultaneously reporting structural and dynamics information, which has not been fully explored. In this work, we extend our previous work by presenting the first instance of simultaneous characterization of structure and dynamics that include continuous and mixed-mode internal dynamics.

Previous works have described the theoretical aspects of the Suape Order Tensors (OTM) (Tolman et al., 1997; Shealy et al., 2011). Here we provide a more applied summary of this topic as it pertains to this report. Under purely theoretical and hypothetical conditions, a molecule that is absolutely devoid of any motion (internal or external tumbling) will achieve the highest level of order that is represented by the order tensor described in Eq. 3. Under realistic and unperturbed conditions, the isotropic tumbling of a macromolecule results in an order tensor that has been averaged to zero due to a uniform sampling of all possible molecular orientations. After inducing a tumbling anisotropy, a nonzero order tensor will be reintroduced based on the preferred orientation of the molecular tumbling, which is the origin of observing finite RDC data. In the absence of internal dynamics, the tumbling anisotropy is equally experienced by all portions of the molecule, and therefore OTMs reported by any portion of the molecule are equal to within the experimental error. The presence of internal dynamics will result in an OTM that is different than an OTM obtained from any other portion of the macromolecule. This is due to the fact that OTM from the dynamical region will consist of the effect of anisotropic molecular tumbling combined with the perturbation of internal dynamics. This is the primary principle that we employ in the development of our analysis. A systematic departure in OTMs reported from different portions of the protein are due to internal dynamics and can be used to identify dynamical regions, internally orchestrated motions, and be used in some instances to reconstruct the trajectory of motion (Cole et al., 2016). S = [ − 1 / 2 0 0 0 − 1 / 2 0 0 0 1 ] (3)

In this study we utilized dihydrofolate reductase enzyme (DHFR) that has been selected based on the substantial existing literature in support of major conformational changes when performing their enzymatic function (Bystroff and Kraut, 1991; Diez et al., 2001; Duan et al., 2001; Osborne et al., 2001).

Dihydrofolate reductase enzyme (DHFR) (Sawaya and Kraut, 1997) is a 159-residue long protein that has long been recognized for its central role in regulating tetrahydrofolate level in the cell, which directly aids in the synthesis of nucleic acid precursors. DHFR has been extensively studied and paramount evidence has confirmed its conformational changes as it binds to different intermediates (Fierke et al., 1987; Rod and Brooks, 2003; Antikainen et al., 2005; Mauldin and Lee, 2010). DHFR is a single-domain, monomeric molecule; the structure of which is divided into two subdomains: the adenosine binding subdomain and the loop subdomain. The gap separating the two subdomains is occupied by a nicotinamide ring, and the pteridine ring is located in the cleft between helices B and C. Four known states have been identified for this protein: open, closed, and occluded states depending on whether the active site is open, closed, or occluded by the loop. Due to internal dynamic, sometimes it becomes crystallographically unclear or invisible, hence the last state, known as the disordered state (Bystroff and Kraut, 1991). Although there exists ample evidence of the existence of internal dynamics, little is known regarding the exact nature of the structural rearrangement of this protein.

In this study we use DHFR to test the ability of our approach in concurrent characterization of structure and dynamics of proteins. To that end, we perform a fictitious, mixed-mode molecular dynamics simulation on DHFR (PDB-ID: 1RX2) in order to simulate RDC data and explore the possibility of identifying different dynamical regions of this protein by REDCRAFT, while providing atomic resolution structures for each dynamical domain. It is important to note that the imposed MDS is for illustration purposes only and it servers no useful information in recovering the actual dynamics of this protein in its native form.

A fictitious, molecular dynamics simulation was implemented for DHFR based on some of the information available in the literature. More specifically, the structure PDB-ID 1RX2 was fractionated and subjected to various models of internal dynamics to better test our approach. The overall model of dynamics consisted of four fixed regions, two segments that underwent rigid-body dynamics, and one unstructured region. These segments were connected by hinge regions as shown in Figure 1 and Figure 2. As the first step in our MD simulation, the protein structure was minimized in order to arrive at a more equilibrated state. In the next step, A mixed-mode constrained molecular dynamics simulation was performed in XPLOR-NIH (Schwieters et al., 2003; Schwieters et al., 2006) (version 3.3) by keeping segments 1 (residue 1–11), 3 (residue42-60), 5 (residue 92–115), and 7 (residue 137–159) fixed in space. Segment 2 (residue 15–28) and segment 4 (residue 64–88) were constrained to experience rigid body dynamics by permitting the hinge regions (regions connecting each segment) to fluctuate freely in space. Segment 6 (residue 116–136) was allowed to freely move in space without any additional constraints and therefore experienced a melting of that domain. The simulation was conducted for 100,000 steps with step size of 0.0001 psec in a 2,000 K bath temperature. A total of 102 uniformly sampled frames were produced during the course of the molecular trajectory to be used during the calculation of ensemble RDC data.

Using the trajectory produced from the MD simulation, 102 frames were generated uniformly to span the entire course of the dynamics. Auxiliary tools were used to separate each of these frames in a PDB format and to generate a corresponding REDCAT file. The software package REDCAT (Valafar and Prestegard, 2004) was used to calculate the RDCs values for backbone C′-N, N-H^N, and C′-H^N for each frame of the trajectory using the order tensors shown in Table 1 in two alignment media. REDCAT’s internal utility functions were used to create the observable RDCs by averaging the individual RDCs (for the three vectors) across the entire course of the dynamics (defined by 102 frames). To simulate a more realistic set of data, uniformly distributed noise in the range of ±0.5 Hz was added to all RDC data. These averaged RDCs were used for reconstruction of structure and study of the internal dynamics by REDCRAFT in a procedure highlighted in the following sections. It is important to comment on our choice of RDC data. Although a variety of highly informative RDC data (e.g., C_α-H_α, H_α-H^N, etc.) can be collected from smaller proteins, we have not used them in our studies since they may not be available in larger systems. To extend the applicable range of NMR spectroscopy to larger proteins, protons are exchanged with deuterons to improve spectral quality. Therefore in our study, we have confined the use of RDC data to what can be obtained from small or large and perdeuterated proteins. Finally, due to the existence of prolines, in general, the average number of RDCs is usually less than three per residue since only backbone N-H^N RDCs can be acquired. In the case of DHFR, the effective and average number of RDCs pre residue was reduced to 2.5 in each alignment medium.

PDBMine (Cole et al., 2019a) is a newly developed tool (https://ifestos.cse.sc.edu/PDBMine/) that performs an exhaustive search of the dihedral angles for a protein in the Protein Data Bank (Berman et al., 2000). As the first step, PDBMine creates a number of subsequences from the primary sequence of the query protein using a rolling window of size W. Therefore, for a protein of size N and a rolling window of size W, PDBMine creates N-W+1 subsequences. In the case of DHFR (159 residue protein) and a window size of 7, a total of 153 subsequences (residues 1–7, 2–8, 3–9 … 153–159) are created. As a second step, PDBMine gathers and aggregates an exhaustive list of all the observed dihedral angles for every residue in every subsequence present in the PDB. During the final step of its analysis, all the returned dihedral angles for all the subsequences are assembled into a final dihedral restraints for each residue of the query protein. In theory, a window size of one will reproduce the known Ramachandran dihedral space. Selection of a larger window size can be viewed as a context-sensitive Ramachandran space. Previous work (Cole et al., 2019b) has illustrated the differences between the dihedral spaces for a proline that precedes a glycine, versus a proline that succeeds a glycine. Therefore, having context specific estimations of dihedrals can be very useful in accelerating the task of structure determination. Another unique feature of PDBMine is its responsiveness; an exhaustive search of the PDB for a 159-residue protein will be completed in less than 10 min.

Under pragmatic conditions, use of the largest window size that produces a set of dihedrals is recommended. However, under testing conditions, it is important to exercise the necessary precautions to remove biases in the creation of the dihedral restraints. To that end, the primary objective is to avoid creation of the dihedral sets that are heavily populated with instance of 1RX2 or other homologous proteins. Therefore, any process that ensure diverse representation of dihedral angles will test the ability of REDCRAFT in identifying the correct dihedral angles among a large list of decoys. In this exercise, we explored window sizes of 3, 5, 7, and 9 after removing all instances of 1RX2 dihedrals. The window sizes of 3 and 5 produced an intractable number of hits, while the window size of 9 produced results that converged to the dihedrals of 1RX2 for some residues. The window size of 7 produced manageable results with at least 100 dihedrals that were separated from the actual dihedral of 1RX2 by more than 10° (some examples shown in the results section). REDCRAFT incorporates the results of PDBMine to improve its computation time by using the confined dihedral search space of the protein under investigation (in this case 1RX2). It is important to note that REDCRAFT can proceed in successful determination of protein structures in the absence of any dihedral constraints as demonstrated previously (Simin et al., 2014; Cole et al., 2021).

During the past decade, several approaches and programs for structure determination from RDC data have been described (Saupe and Englert, 1963; Cornilescu et al., 1999; Clore and Bewley, 2002; Assfalg et al., 2003; Bernadó and Blackledge, 2004; Nitz et al., 2004; Bouvignies et al., 2005; Shealy et al., 2011). Each of these programs has different advantages and disadvantages. REDCRAFT (Clore and Schwieters, 2004; Bouvignies et al., 2005; Prestegard et al., 2005; Valafar et al., 2005; Shealy et al., 2010), sets itself apart from other existing software packages by deploying a more efficient and effective search mechanism. As a result, REDCRAFT can achieve the same structure determination outcome as other methods with less data (Cole et al., 2021). REDCRAFT also allows simultaneous study of structure and dynamics of proteins (Bryson et al., 2008; Simin et al., 2014; Cole et al., 2016). Applications of REDCRAFT in structure calculation have been demonstrated using aqueous (Bryson et al., 2008; Simin et al., 2014; Cole et al., 2015) and membrane (Shealy et al., 2010) proteins with as little as two RDCs per residue (Shen et al., 2009; Shealy et al., 2010; Shen and Bax, 2015) (in two alignment media).

REDCRAFT has introduced a novel approach to structure determination of proteins from RDC data (Cole et al., 2021). Aside from an unorthodox search method that is robust and fast (Cole et al., 2021), REDCRAFT employs an incremental strategy to structure determination in contrast to the all-at-once approach that is adopted by other existing methods. REDCRAFT’s incremental structure determination strategy has certain advantages and starts with a search for the optimal torsion angles that join two neighboring peptide planes. This seed dipeptide plane is recursively extended by one residue at a time by exploring a directed and extensive combinatorial search of the dihedral angles that extend the seed structure by one peptide plane (or amino acid) that optimally satisfies the RDC constraints. This process can start from the N-terminus of the protein and continue until the C-terminal end, or traverse the structure of the protein in the reverse order (C to N-terminus). The structural fitness that is produced by REDCRAFT during the course of fragment extension (from dipeptide to the entire protein) is termed the “Dynamic Profile” (or DP), which plays an instrumental role in a number of analyses including assessing the quality of the final structure or elucidation of internal dynamics. Using the Dynamic Profile, we have defined a process that allows for simultaneous identification and characterization of structure and internal dynamics. This process consists of three functional steps: standard structure determination, identification of internal dynamics (hinge regions), a grouping of the structural domains (coordinated dynamics), followed by reconstruction of the atomic resolution dynamics when possible. While the last step in the reconstruction of atomic-resolution of dynamics has been discussed in our previous work (Bryson et al., 2008; Shealy et al., 2010; Cole et al., 2015; Cole et al., 2021), the former steps have not been fully described in the literature. In addition, our previous work has been applied to the cases of finite and discrete state dynamics. In this work, we will define and test a more rigorous method of studying continuous and mixed mode dynamics. The four comprehensive steps are as follows:

PDBMine was used as the first step to structure determination of DHFR by performing a search with a window size of 7. Figure 3 illustrates the number of hits that were identified by PDBMine with window size of 7 for each residue of DHFR. In average each residue received 5,923 possible dihedral angles with residues 37 and 57 receiving the least and the most (525 and 6,813 respectively) number of dihedral angles.

Figure 4 illustrates the aggregated dihedral angles for residues G14 (panel A) and G85 (panel B). In this figure all the dihedral angles reported by PDBMine are illustrated in blue and the corresponding dihedral angles obtained from the PDB (1RX2) is illustrated in red. Several noteworthy observations can be stated. First, the results of PDBMine in principle converge to a Ramachandran space as a reducing window size. However, due to the context-specific nature of the search, a more restricted dihedral space is reported by PDBMine. The second notable observation further expands on the context specific nature of the PDBMine search and is illustrated in Figure 4. Both of the results correspond to a glycine, but they differ substantially due to the context in which the two glycines appear in the primary sequence. The third important point is to confirm the proper precautions that we have deployed to remove any unintended biases in our evaluations. It is clear from these figures that there are significant number of decoy dihedrals among which, REDCRAFT successfully selects the correct dihedral angle.

It is important to quantify two aspects of internal dynamics. The first relates to capturing the magnitude of dynamics, and the second relates to the duration of time that was spent in different states. We first report the magnitude of dynamics for the rigid-body domains as an orientational departure from frame₀ as the point of reference. Figure 5 illustrates the descriptive statistics regarding the movement of two Rigid-Body domains. Panel (A) of this figure displays the angular departure of each domain (F2 and F4) with respect to the fixed domains (F1, F3, F5, F7) measured between frame_i and frame₀. Based on this information, Fragment 4 undergoes orientational rearrangement of as high as 32°, while Fragment 2 exhibits a much smaller motion of less than 15°. In addition to the magnitude of motion, it is important to assess the amount of time (or the number of frames) that each fragment spends in each orientational state during its trajectory. The frequency (or likelihood) of existing in a continuum of the orientational repositioning is illustrated in panel (B) of Figure 5. Based on this information, Fragment 2 spends a very small portion of its trajectory away from frame₀, while spending most of the trajectory in the vicinity of the original state (less than 5°). Fragment 4 on the other hand, spends more than 50% of the time in an orientation more than 10° away from the original state. The general summary is that Fragment 2 undergoes small amount of structural rearrangement, while Fragment 4 exhibits a larger motion with respect to the fixed domains of the protein. It is important to state that the MD simulation of DHFR is purely engineered with the primary intention of exploring the sensitivity of our approach in detection of motion.

As the first logical step, the structure of DHFR was determined in its entirety using REDCRAFT using Ramachandran dihedral restraints. As expected, this attempt at structure determination produced unsatisfactory results as indicated by the unacceptable fitness to the RDC data (1.14 Hz), and therefore are succinctly summarized here. The additional details are provided in Supplementary material in Supplementary Table S1 and Supplementary Figure S3. In summary, the overall structure exhibited 29 Å of bb-rmsd with respect to 1RX2 over the entire length of the protein with a fitness score of 1.14 Hz to the RDC data. The bb-rmsd computed over each of the fragments exhibited an average of 4.8 Å with localized similarities ranging from 0.8 to 9.7 Å.

As a more interesting case, the structure of DHFR was computed by REDCRAFT using the context specific dihedral restraints produced by PDBMine. The examination of the REDCRAFT’s DP will be crucial in assessing its success in the structure determination of this protein. The DP generated by REDCRAFT (shown in Figure 6) exhibits two indicators of the internal dynamics and therefore, a poor structure determination session. First, the final value of the fitness to the RDC data (1.2 Hz) compared to the expected value of 0.6 Hz (corresponding to the simulated error) indicates a failed attempt at structure determination. Second, the existence of sudden and anomalous increases in the DP in various places (e.g., at residues 12–14) is a potential indicator of internal dynamics that requires further examination. It is important to note the close correlation between the sudden increases in the DP and the location of hinge regions of our simulation (denoted by red markers in Figure 6).

Figure 7 illustrates the superimposed structure of DHFR (1RX2 shown in red) and the REDCRAFT recovered structure (shown in blue) by disregarding the existence of internal dynamics. Table 2 highlights the detailed results of comparing the structure of REDCRAFT to 1RX2. As a summary, the two structures exhibit a bb-rmsd of 21 Å and the comparison of fragments exhibit structural similarity in the range of 0.7 to 3 Å. Based on this information, in addition to the divergence in the overall structure, the structural error is also manifested in local fragments. It is important to note that the improved localized structural similarity is due to the effective restraining of the dihedrals accomplished by PDBMine. It is also important to note while the inclusion of PDBMine constraints improved the structural quality of our analysis, there is still substantial room for improvement.

Fragment 1: Residue 1–11—In consideration of the results shown in the previous section, fragmented study of the protein was conducted. The results of REDCRAFT for the region consisting of residues 1–11 exhibits an acceptable fitness score (around 0.5 Hz), and is devoid of any sudden increase. Therefore, the structure is deemed acceptable as the first fragment of this protein. Implementing steps 1 and 2 listed in the Methods section, the fragmented study continues from residue 16 (after skipping ahead 5 residues).

Fragment 2: Residue 17–38—Structure calculation of DHFR can proceed by investigating a new fragment. The start of the new fragment is based on skipping a fixed number of residues (i.e., 5 residues) from the onset of dynamics to pass the hinge region. The start of a new fragment essentially resets the calculation of an order tensor and therefore removes any inconsistency in the reported order tensors from two dynamically distinct domains of the protein. Therefore, structure calculation can proceed if a well-behaved DP is exhibited. Figure 8 illustrates the DP of the REDCRAFT for the new fragments starting at residue 17 and as expected, the REDCRAFT score increases at the beginning of the run due to lack of RDC data. Once stabilized, the general pattern is conserved until residue 38, at which point, the DP exhibits a distinct and anomalous increase in the REDCRAFT score. Indeed, residue 39 marks the beginning of the hinge regions and adjoins fragments 2 and 3 of this protein. Hence, we group residues 17–38 as the second Fragment in our investigation.

Fragments 3, 4, 5, 6, and 7—After completion of Fragment 2, a new structure calculation session was started from residue 44. As it can be observed in the DP for this segment (shown in Figure 8), the same general pattern as the previous two fragments is observed with an anomalous and notable increase in the REDCRAFT score at residue 61. This concluded the analysis of the third fragment that consisted of residues 44–60. The process of fragmented analysis was continued with the corresponding DP illustrated in Figure 8. The final completion of this process yielded four additional fragments F3 (44–60), F4 (65–88), F5 (97–116), and F7 (138–159). The range of the recovered fragments remarkably agree with the simulated MD. The DP of the only aberrant fragment, Fragment 6, is shown in Figure 8 as multiple attempts in structure recovery. Our first attempt at structure determination of this fragments started from residue 120 after skipping 5 residues from the end of the previous fragment. This attempt at structure determination was unsuccessful since the DP exhibited monotonically increasing score that exceeded the acceptable threshold of 0.6 Hz. The process of skipping forward by 5 residues was repeated with the objective of arriving at a well-behaved region of the protein. Each attempt at structure determination after skipping 5 residues is shown in Figure 8. This portion of the protein, unlike all other portions, never resulted in a well behaving DP due to the nature of its internal dynamics. Since the structure of this fragment was consistently modified in each frame, there is no conserved structure to recover, explaining the failure of structure calculation by REDCRAFT. This example also serves as a demonstration of cases where a gap region is larger than 5 residues.

The complete assessment of REDCRAFT’s results should consist of two parts. First, to evaluate the success of REDCRAFT in delineating different dynamical regions of the protein as described above. The second portion consist of assessing the structural accuracy of the recovered regions by REDCRAFT. Table 3 shows the results of the fragmented structure determination of DHFR by REDCRAFT while Figure 9 provides an illustration of the fragments (shown in blue) superposed on the corresponding regions of DHFR (shown in green). In Figure 9, we have omitted the REDCRAFT calculated structure of F6 due to the absence of a meaningful structure to compare. REDCRAFT was able to accurately recover the fragments of DHFR from three RDC data with an accuracy of less than 1 Å. It is important to note that these results are based on unrefined structures in order to expose and exhibit the raw capabilities of REDCRAFT. In practice however, these structure will benefit from refinement in platforms such as Xplor-NIH (Berman et al., 2000; Cole et al., 2019b), CNS (Brünger et al., 1998), or CYANA (Güntert and Downing, 2004) to name a few.

Fragment Assembly—Following the structure determination of the individual fragments, the assembly process can proceed based on the procedure described in the Methods section. We start the assembly process by transforming all the fragments into their Principal Alignment Frame (denoted at PAF₁) of the first medium and perform an initial investigation of their order tensor (OTM₁). The OTM for each fragment in the second alignment medium is also established using the PAF₁ as the common frame of comparison. Once the order tensors from all both alignment media have been canonicalized properly, a simple comparison of the order tensors will be sufficient to establish the relativly large motions between two fragments. In this case, F6 clearly was excluded based on the dissimilarity of its order tensors from the OTMs of any other fragment (due to one order of magnitude difference). However, since F2 and F4 were subjected to relatively small magnitudes of motion, the simple comparison of OTMs was inconclusive. A more sensitive discrimination of internal dynamics can be performed by assembling the fragments after examining all the inversion possibilities of each fragment. Table 4 provides a summary of the progressive fragment assembly using Q-Factor as a metric of fitness computed by REDCAT. The first column in this table indicates the progressively growing fragment during the course of the assembly. The nomenclature used in this column consists of the fragment number followed by subscript indicator of the fragment inversion examined in each evaluation. The second column indicates the fitness of the assembly to the combined RDC data in the first alignment medium. The following four columns signify the fitness of the assembly to the combined RDC data from the second alignment medium, after applying the indicated inversion to the last addition to the sequence. In these columns, I, R _x , R _y , and R _z indicate no rotation (Identity or as is), rotation about x, y, and z axes respectively. The fragment assembly starts with the first fragment and as noted in the first row of this table. Note that there is no effect in the rotation of this fragment from the perspective of the second alignment medium. Using the first fragment in its original orientation, fragment 3 has been appended and Q-Factors have been computed for all of 4 possible orientations of F3 (not F1). Since the rotation about y yielded an acceptable score, its extension by the fragment 5 will be based on the y-rotated fragment 3. As an empirically accepted practice in the community, Q-Factor scores with values less than 0.2 reflect a high-quality structure and are deemed acceptable. (Cornilescu et al., 1998; Cole et al., 2021). Using this practice of evaluation, it is clear that fragments 1, 3, 5, and 7 can successfully be assembled as one unit (the fixed core), while fragments 2 and 4 cannot be successfully accepted as part of the fixed domain of the protein.