Protein Dynamics to Define and Refine Disordered Protein Ensembles

Abstract

Intrinsically disordered proteins and unfolded proteins have fluctuating conformational ensembles that are fundamental to their biological function and impact protein folding, stability and misfolding. Despite the importance of protein dynamics and conformational sampling, time-dependent data types are not fully exploited when defining and refining disordered protein ensembles. Here we introduce a computational framework using an elastic network model and normal mode displacements to generate a dynamic disordered ensemble consistent with NMR-derived dynamics parameters, including transverse ${\text{R}}_2$ relaxation rates and Lipari-Szabo order parameters (${\text{S}}^2$ values). We illustrate our approach using the unfolded state of the drkN SH3 domain to show that the dynamical ensembles give better agreement than a static ensemble for a wide range of experimental validation data including NMR chemical shifts, J-couplings, nuclear Overhauser effects, paramagnetic relaxation enhancements, residual dipolar couplings, hydrodynamic radii, single-molecule fluorescence Förster resonance energy transfer, and small-angle X-ray scattering.

INTRODUCTION

It has long been perceived that proteins need to adopt a well-defined three-dimensional structure to carry out their function, although it is increasingly clear that all proteomes encode proteins that do not adopt stable three-dimensional structure that are nevertheless important for cellular function.^1–10 Proteins that do not possess stable structural domains, known as intrinsically disordered proteins (IDPs), or contain intrinsically disordered regions (IDRs), play a central role in signaling, transcription, and regulation of cell cycle, as well as in the formation of biomolecular condensates in the cell.^11–22 In addition, unfolded states of folded proteins are increasingly appreciated as critical for understanding protein folding and stability^23–25. Finally, both IDPs/IDRs and unfolded proteins can participate in pathological aggregation or fiber formation, with strong dependencies on the conformational equilibria in the disordered ensembles.^26–27

Because IDPs/IDRs and unfolded states have fluctuating heterogeneous conformations under physiological conditions, it necessitates detailed investigation into the underlying conformational ensembles to enable structural correlations with function.^{14, 28–29} There are a number of solution experimental techniques that can guide the construction of structural ensembles, many of which encompass key features of disordered states: fractional local structural propensities as measured by NMR chemical shifts (CSs)^{28, 30–32}, J-couplings (JCs)³³, and residual dipolar couplings (RDCs) ^34–37; global properties as measured through NMR hydrodynamic radii values³⁸ and small angle X-ray scattering (SAXS) data^39–42; and tertiary contact information as measured through NMR paramagnetic relaxation enhancements (PREs)^43–44, nuclear Overhauser effects (NOEs)²⁸ and single-molecule fluorescence Förster resonance energy transfer (smFRET)^{29, 45}.

However, due to the fast dynamics in the disordered state, the measured NMR, SAXS and smFRET observables are highly averaged, resulting in significant challenges to bridge the connection between experimental observables and structural ensembles.^{5, 45–46} Thus computational models play a critical role in constructing structural ensembles of disordered states by first creating an initial library of putative conformations, and then a selected subset is chosen based on improving the agreement with the available experimental solution data. Most computational efforts have focused on the ensemble subset selection process, such as the ENSEMBLE program⁴⁷ that uses a Monte Carlo algorithm or the ASTEROIDS approach with its evolutionary algorithm⁴⁸, to select a set of conformations for which the back-calculated data fit the available experimental data. Head-Gordon and co-workers have introduced the extended Experimental Inferential Structure Determination (X-EISD) procedure that uses a variety of experimental observables, and their known errors and variances, to determine the most probable structural ensemble from candidate structures for disordered states.³³ This Bayesian statistical method calculates log-likelihood scores of the selected ensemble corresponding to a set of experimental data and thus can be used to refine a structural ensemble of disordered states.³³ Often these approaches have relied on a somewhat arbitrary formulation of the underlying structural ensemble before optimization, such as use of the TraDES conformer generator.⁴⁹

In addition to structural characterization, protein dynamics is fundamental in understanding function of both ordered and disordered states of proteins⁵⁰, and NMR techniques have been used to investigate dynamics of proteins through measurement of spin relaxation properties.⁵¹ Heteronuclear spin relaxation studies typically measure internuclear bond vector dynamics (¹H-¹⁵N) by measuring ¹⁵N longitudinal ${\text{R}}_1$ and transverse ${\text{R}}_2$ relaxation rates along with the heteronuclear ¹H-¹⁵N NOE, potentially augmented by other relaxation experiments focused on ¹⁵N, ¹³C or ²H nuclei. These relaxation data can also be expressed in terms of spectral density functions and are often translated into model free Lipari-Szabo order parameters (${\text{S}}^2$ values).^51–52 The magnitude of ${\text{S}}^2$ values can vary from 0 to 1, corresponding to completely isotropic large amplitude internal motions or complete rigidity, respectively. The model free formalism assumes that the timescales of internal motion and overall rotation are very different⁵³ and provide a reasonable expectation for folded states but not for disordered proteins. Thus, most NMR relaxation studies of disordered proteins use raw rates or spectral density function mapping.^{52, 54–57}

While protein dynamics can potentially be of great importance for an accurate description of disordered protein conformational ensembles, there has been little effort to include dynamics data in computational tools for creating and/or refining disordered state ensembles. Molecular simulations such as molecular dynamics (MD) or Monte Carlo (MC) methods can elucidate information on protein motions.^{37, 58–60} Molecular simulations such as molecular dynamics (MD) or Monte Carlo (MC) methods can elucidate information on protein motions^{37, 58–60}, however fixed charge force fields have been shown to be unreliable for IDPs and better polarizable force fields are costly¹⁰. Coarse-grained elastic network models (ENMs), on the other hand, offer simple and computationally efficient alternatives to MD or MC simulations for sampling the intrinsic motions accessible to a protein, and have been widely used to study the intrinsic dynamics of folded systems from small globular proteins to large biomolecular assemblies.^61–72 The underlying principal is that dynamic properties of globular proteins are determined mainly by the protein topology, that can be captured as a collection of harmonic springs between native contacts, such that the most important global motions are captured.^73–75 It has been shown that the normal modes generated through ENM models can accurately predict the crystallographic B-factors as well as NMR dynamics data for folded proteins.^{56, 61} However, to the best of our knowledge, ENMs have not been used to explore the intrinsic dynamics of disordered proteins.

In this work, we introduce a framework for using an ENM and subsequent normal mode displacements for the creation and optimized selection of a disordered protein ensemble consistent with ${\text{R}}_2$ rates and ${\text{S}}^2$ values for the unfolded state of the N-terminal SH3 domain of the Drosophila drk protein (drkN SH3).^{57, 76} We show that dynamic ensembles created to satisfy the ${\text{R}}_2$ relaxation or ${\text{S}}^2$ order parameter restraints better agree with more commonly collected experimental solution NMR and SAXS data relative to statically generated ensembles. Furthermore, we observe that ${\text{R}}_2$ rates combined with NOEs alone can yield ensembles with the smallest RMSD with respect to all experimental data types for the unfolded state of the drkN SH3 domain. We conclude that incorporation of dynamic information in defining starting pools of conformers for disordered protein structure calculations is valuable for computational approaches to best represent the disordered protein ensemble.

METHODS

Structural ensembles of the unfolded states of the drkN SH3 domain.

To evaluate the suitability of ${\text{R}}_2$ data and ${\text{S}}^2$ values for selecting a disordered ensemble, we consider the unfolded state of the drkN SH3 domain.^{57, 76} The drkN SH3 domain has been extensively studied by the Forman-Kay and Gradinaru groups, yielding a wide variety of structural and dynamic experimental data types.^{33, 76–77} The domain exists in approximately 1:1 equilibrium between folded and unfolded states under typical buffer conditions⁵⁴, and the NMR data were acquired under conditions of the 50:50 mixture with slow exchange on the NMR timescale so that data for the unfolded state could be extracted separate from data for the folded state. NOE data were carefully analyzed to avoid transfer NOEs occurring due to exchange between folded and unfolded states during the mixing time.^{57, 76}

We use three initial structural ensemble pools previously investigated with the X-EISD method.³³ (i) The RANDOM pool is a collection of 100,000 structures including 1000 folded structures and 999,000 increasingly unfolded structures. The RANDOM pool is not optimized with respect to the experimental data. (ii) The ENSEMBLE pool is optimized with respect to the available NMR and SAXS experimental data and contains 1700 structures.²⁴ (iii) The MIXED pool combines the ENSEMBLE pool structures and optimized RANDOM pool structures. There are underlying structural differences in the RANDOM, ENSEMBLE, and MIXED pools, such as the percentage of the secondary structure for each residue (local structure) and global characteristics as measured by the radius of gyration (${\text{R}}_{\text{g}}$).³³

Anisotropic Network Model.

We use the Anisotropic Network Model (ANM) introduced by Bahar and co-workers⁷⁸, implemented in the Prody software program⁷⁹, to define the three lowest normal modes of each conformation of the original ENSEMBLE, RANDOM, and MIXED static pools. Since the frequencies of the normal modes are directly proportional to the energy required for the movement, high frequency modes describe local motions and low frequency modes represent collective conformational changes. Many studies have shown that the lowest three frequency normal modes often agree with experimentally observed functional motions for folded proteins.^{68, 80} Figure 1 shows a schematic of one of the conformers within the disordered ensembles modeled as an ANM and its three lowest frequency normal modes.

Figure 1. The dynamical description of the unfolded state of the drkN SH3 domain using an anisotropic network model.

Three lowest frequency normal modes for a single structure out of the disordered ensemble where green arrows, blue arrows, and black arrows represent the direction and amplitudes of the three lowest frequency motional modes, super-imposed on a schematic representation of the protein backbone in the unfolded state.

In ANM, the network nodes are the positions of the $\text{α}$-carbons, and uniform elastic springs with force constants $k$ connect the nodes located within a cutoff distance of ${\text{R}}_{\text{c}}$. The generalized form of the entire network potential of the ANM is given by:⁸¹

$$v=\frac{k}{2}\sum _{i<j}{\left(r_{ij}-r_{ij}^o\right)}^2$$ (1)

where $r_{ij}$ is the distance between atoms $\text{i}$ and $\text{j}$ and $r_{ij}^o$ is the distance between the atoms in the reference structure. The normal mode analysis using the ANM requires the diagonalization of the following generalized eigenvalue problem:

$$\mathit{H}\mathit{U}=\mathit{\lambda }\mathit{T}\mathit{U}where\mathit{U}=\left(u_1,u_2,\dots ,u_N\right)$$ (2)

where $\mathit{H}$ is the Hessian matrix of the partial second derivatives of the potential energy, $\mathit{T}$ the kinetic energy matrix, $\mathit{U}$ are the normal modes, and $\mathit{\lambda }$ is a diagonal matrix with the ${\lambda }_{\text{k}}$ eigenvalues associated to the kth normal mode $u_k$. A detailed description of the derivation of the Hessian matrix and the normal modes can be found elsewhere.^{78, 81} In this study we model the unfolded state of the drkN SH3 domain using an ANM model that has a force constant of $k=1$ kcal mol⁻¹ and a cutoff of ${\text{R}}_{\text{c}}=15\text{Å}$.

Normal Mode Displacements.

We also further extend the ANM ensembles to generate alternate conformations along the normal mode displacement vectors. Procedurally we used the Prody software to extend normal modes calculated for the $\text{Cα}$ model into an all-atom model. The extend-Model function in the Prody software⁷⁹ takes part of the normal modes for $\text{Cα}$ atoms and extends it to create all other atoms in the same residue. Then we use these all-atom models to generate an ensemble of randomly sampled conformations along the three lowest frequency normal modes. Then we use the all-atom model and normal mode displacements to generate alternate conformations along the three lowest frequency normal modes. For a given normal mode, $u_i$, and their eigenvalues, ${\lambda }_i$, new conformations are sampled using the relationship in Eq. 3, where $R_0$ is the active coordinate set of atoms and $\left[r_1^k{r}_2^k\dots r_m^k\right]$ are normally distributed random numbers generated for conformation $j$.

$$R_k=R_0+s\sum _{i=1}^m{r}_i^j{\lambda }_i^{-0.5}{u}_i$$ (3)

We use this procedure to create the ${\text{R}}_2$-dynamic and ${\text{S}}^2$-dynamic pools as our putative starting ensembles which we compare to the original “static” ensemble pools.

Sub-ensemble selection using ${\text{R}}_2$ rates and ${\text{S}}^2$ values.

Within the ENSEMBLE program⁴⁷, the ${\text{R}}_2$ restraint is defined as the correlation of ${\text{R}}_2$ values with the number of contacts of the amide group.^{24, 47} We use this structural definition as a dynamic proxy of the transverse ${\text{R}}_2$ relaxation rate to select sub-ensembles of structures from the original ENSEMBLE, RANDOM, and MIXED pools. We also chose to generate sub-ensembles selected using ${\text{S}}^2$ order parameters since they are conceptually easy to compare to dynamic modes that are computationally sampled. To select a sub-ensemble of structures that agrees with the ${\text{S}}^2$ values, we first compute the normalized mean square fluctuations (MSFs) of the lowest three frequency modes for each protein structure using an anisotropic network model (see below). Then, we select individual protein structures whose MSFs agree well with the ${\text{S}}^2$ order parameters.⁸² The underlying hypothesis is that the higher the ${\text{S}}^2$ values for a given segment, the lower its calculated MSFs should be. To measure the agreement between MSFs and ${\text{S}}^2$, we calculate Pearson correlation coefficients for six different residue ranges as provided in Supplementary Figure S1. The ${\text{R}}_2$ rate and ${\text{S}}^2$ order parameter sub-ensemble structure pools selected via these procedures are seen to have very similar profiles (Figure 2a and 2b). Therefore, we consider both ${\text{R}}_2$ and ${\text{S}}^2$ measurements in the creation of new dynamical ensembles for disordered states.

Figure 2. ${\text{R}}_2$ and ${\text{S}}^2$ profiles for the unfolded state of the drkN SH3 domain.

(a) ¹⁵N ${\text{R}}_2$ relaxation rates and (b) ${\text{S}}^2$ order parameters as a function of residue number.^{54, 82} Units in (a) are in 1/sec.

Back-calculations and scoring of disordered ensembles using the X-EISD algorithm.

The validity of the disordered conformational ensembles refined and expanded using ${\text{R}}_2$ and ${\text{S}}^2$ criteria and the degree to which they can represent the structural information available from experimental data is measured using X-EISD.³³ The X-EISD framework uses a maximum likelihood estimator formalism to assign a log likelihood score of a simulated ensemble matching an input set of experimental data. The procedure accounts for the uncertainties in both back-calculation and experiment by optimizing over the set of “nuisance parameters” that are treated as Gaussian random variables. X-EISD can be applied to multiple data types simultaneously to generate an aggregated probabilistic score of the form as in Eq. (4).

$$\text{log p}(X,\xi \mid D,I)=\text{log p}(X\mid I)+{\sum }_{j=1}^M\log \left[p\left(d_j\mid X,{\xi }_j,I\right)p\left({\xi }_j\mid I\right)\right]+C$$ (4)

Moreover, the X-EISD framework can provide a probabilistic score of an ensemble in a Markov Chain Monte Carlo (MCMC) optimization as given in eq. (5).³³ In this work, we use a simple direct maximization and perform 1000 exchange attempts to replace one conformation with another from the total pool of $\text{N}=100$ starting structures where an exchange is accepted only when the new ensemble has a higher probabilistic X-EISD score than the previous. We perform optimization by using either a single experimental type or multiple experimental types at a given time.

$$acc(i\to j)=X-EIS{D}_j>X-EIS{D}_i$$ (5)

The back-calculations for the different experimental types are done as follows: the Karplus equation is used to back-calculate the J scalar couplings (JCs) as reported previously³³; chemical shift (CS) calculations are performed using SHIFTX2 calculator;⁸³ residual dipolar couplings (RDCs) are computed using the local RDC back-calculator from the Forman-Kay group;³⁵ the hydrodynamic radius (${\text{R}}_{\text{h}}$) is calculated using the program HYDROPRO;⁸⁴ and NOE, PRE and smFRET calculations are done using in-house codes as described previously.³³ Finally, we calculate the properties of the optimized ensemble such as the ${\text{R}}_{\text{g}}$ distribution using the Prody implementation,⁷⁹ and secondary structure content using the implementation of the DSSP algorithm⁸⁵ within the AmberTools program cpptraj.⁸⁶

RESULTS

While methods to develop ensembles composed of independent “static” conformational snapshots have proven very useful to the disordered protein structural community, approaches for building meaningful ensembles consistent with all experimental data, including those that are time-dependent (e.g., NOEs, relaxation rates), require further attention. Hence the purpose of this study is to evaluate the sub-ensembles created via ${\text{R}}_2$ and ${\text{S}}^2$ selections and the degree to which they can represent the structural information available from all available experimental data for the drk SH3 domain.

Our first approach uses the correlations between ${\text{R}}_2$ rates and NH contacts or ${\text{S}}^2$ values and the low frequency modes of the ANM to create sub-ensembles of each of the original static ENSEMBLE, RANDOM, and MIXED pools; the resulting sub-ensembles are then optimized across all the experimental data types using the X-EISD optimization to create the ${\text{R}}_2$-select and ${\text{S}}^2$-select pools. Supplementary Table S1 reports the X-EISD scores as well as the root mean square deviations (RMSD) with respect to all 8 experimental data types. The ${\text{R}}_2$ and ${\text{S}}^2$ filtering step and subsequent X-EISD optimization improves chemical shifts, SAXS, and ${\text{R}}_{\text{h}}$ values with little degradation in the JC, RDC, and smFRET experimental data types for the ENSEMBLE and MIXED pools. However, ${\text{R}}_2$ filtering significantly decreases agreement with the JC, NOE and PRE values for the RANDOM pool, which we attribute to diminishment of compact structures relative to the original unfiltered structural ensemble.

Importantly, improved ensembles are realized when we expand the ${\text{R}}_2$ and ${\text{S}}^2$ filtering selections by introducing dynamical motions of their conformations using the normal mode displacements from the ANM model, and then optimizing these dynamically expanded ensembles with all the experimental data using the X-EISD procedure. As seen in Figure 3, the resulting ${\text{R}}_2$-dynamic and the ${\text{S}}^2$-dynamic ensembles have even better RMSD values against all independent experimental data types compared to the ${\text{R}}_2$-select and the ${\text{S}}^2$-select ensembles. This we take as our final dynamical model approach to improve experimental agreement for the unfolded state of the drkN SH3 domain.

Figure 3. The change in RMSD error of the eight experimental data types (x-axis) for the ${\text{R}}_2$ and ${\text{S}}^2$-selected ensembles (red-bar) vs the ${\text{R}}_2$- and ${\text{S}}^2$-dynamic ensembles (blue-bar).

The panels (a-c) show the results for ${\text{R}}_2$ and panels (d-f) show the results for ${\text{S}}^2$ for the ENSEMBLE, RANDOM, and MIXED pool optimizations using X-EISD. There are negligible errors according to the Bayesian model for SAXS, ${\text{R}}_{\text{h}}$ and smFRET. RMSD units are different for each experimental data type and are found in Table 1.

Table 1 quantifies the overall improvement in X-EISD scores and RMSD when comparing the optimized ${\text{R}}_2$-dynamic and ${\text{S}}^2$-dynamic pools against the original and static ENSEMBLE, MIXED, and RANDOM structural ensembles. For the ENSEMBLE pool, the ${\text{R}}_2$-dynamic ensemble performs better for nearly all the experimental properties compared to the original static ENSEMBLE pool. Improvements in RMSD can be seen in local data types such as chemical shifts, J-couplings, and RDCs, long-range distance restraints such as NOEs and PREs, as well as measurements of global shape information such as ${\text{R}}_{\text{h}}$, with essentially equivalent performance for smFRET and SAXS. Similar improvement with respect to experimental data types can also be seen for the MIXED ${\text{R}}_2$-dynamic ensemble, although our approach is not as consistent if the underlying pool is highly non-optimal such as the RANDOM ${\text{R}}_2$-dynamic pool. It has been previously shown that the RANDOM ensemble does not have sufficient conformers to represent the drkN SH3 domain unfolded state and is outside the uncertainties for local experimental data types such as J-couplings and chemical shifts.³³ Hence the dynamical expansion can’t overcome the original deficiencies of the static RANDOM ensemble.

Table 1: Evaluation of the optimized R₂-dynamic and ${\text{S}}^2$-dynamic ensembles and original ensembles for ENSEMBLE, RANDOM, and MIXED pools with 8 experimental data types.

We report both the X-EISD score, and the root mean square deviations (RMSD) with experiments. The results shown here are optimized with all experimental data types. The experimental and back calculations errors for CSs (${\text{σ}}_{\text{exp}}=0.03-0.3$ ppm; ${\text{σ}}_{\text{q}}=0.3-0.5$ ppm (hydrogen), 1.2–1.4 ppm (carbon)); JCs (${\text{σ}}_{\exp }=0.5$; ${\text{σ}}_{\text{q},\text{A}}=0.14$, ${\text{σ}}_{\text{q},\text{B}}=0.03$,${\text{σ}}_{\text{q},\text{C}}=0.08$); RDCs (${\text{σ}}_{\text{exp}}=1.0$; ${\text{σ}}_{\text{q}}=0.9$); NOEs (${\text{σ}}_{\text{exp}}=4.0-5.0\text{Å}$; ${\text{σ}}_{\text{q}}=0.0001$); PREs (${\text{σ}}_{\exp }=1.25-3.625\text{Å}$; ${\text{σ}}_{\text{q}}=0.0001$); smFRET <E> (${\text{σ}}_{\exp }=0.02$; ${\text{σ}}_{\text{q}}=0.007$); $R_h$ (${\text{σ}}_{\exp }=0.3$; ${\text{σ}}_q=0.8$); SAXS (${\text{σ}}_{\exp }=0.0008-0.002$, ${\text{σ}}_{\text{q}}=0.0006$).

Experimental data type	$\text{R}2$-Dynamic Pool		${\text{S}}^2$-Dynamic Pool		Original Pool
	X-EISD Score	RMSD	X-EISD Score	RMSD	X-EISD Score	RMSD
ENSEMBLE
CS (ppm)	117.5 (0.4)	0.3 (0.005)	117.9 (0.3)	0.34 (0.004)	110.1 (0.4)	0.51 (0)
JC (Hz)	43.68 (0.41)	0.16 (0.008)	41.2 (0.4)	0.21 (0.007)	43.3 (0.5)	0.18 (0.01)
RDC (Hz)	−47.8 (0.003)	0.04 (0.005)	−50.1 (0.2)	0.54 (0.02)	−50.4 (0.2)	0.56 (0.03)
NOE ($\text{Å}$)	534.8 (0.4)	2.6 (0.03)	533.7 (0.4)	2.7 (0.03)	532.2 (0.5)	2.8 (0.03)
PRE ($\text{Å}$)	460.4 (2.1)	0.86 (0.06)	443.6 (1.0)	1.42 (0.03)	453.9 (1.7)	1.02 (0.11)
smFRET <E>	6.91 (0.05)	0.006 (0.003)	6.95 (0.03)	0.004 (0.002)	7.0 (0)	0 (0)
Rh ($\text{Å}$)	−0.42 (0)	0	−0.57 (0.02)	0.47 (0.03)	−0.8 (0)	0.71 (0.04)
SAXS (Intensity)	458.3 (0.2)	0.001 (0)	458.0 (0.2)	0.001 (0)	457.9 (0.2)	0.001 (0)
RANDOM
CS (ppm)	118.3 (0.03)	0.33 (0.002)	109.59 (0.66)	0.47 (0.006)	103.6 (0.7)	0.55 (0.01)
JC (Hz)	−47.6 (1.7)	0.78 (0.006)	−34.39 (3.03)	0.73 (0.01)	−25.7 (2.4)	0.7 (0.01)
RDC (Hz)	−48.4 (0.02)	0.2 (0.007)	52.2 (0.02)	0.2 (0.007)	−55.4 (0.6)	0.98 (0.04)
NOE ($\text{Å}$)	506.0 (1.2)	4.3 (0.05)	526.03 (1.77)	3.20 (0.10)	528.5 (1.5)	3.06 (0.10)
PRE ($\text{Å}$)	261.1 (4.0)	3.4 (0.03)	429.53 (5.61)	1.69 (0.11)	450.0 (4.4)	1.24 (0.12)
smFRET <E>	3.9 (0.7)	0.05 (0.006)	6.84 (0.15)	0.009 (0.006)	6.9 (0.1)	0.01 (0)
Rh ($\text{Å}$)	−0.42 (0.1)	0	−0.47 (0.1)	0	−0.4 (0)	0.14 (0.10)
SAXS (Intensity)	452.4 (0.9)	0.003 (0)	455.03 (0.81)	0.002 (0)	456.3 (0.4)	0.002 (0)
MIXED
CS (ppm)	118.6 (0.45)	0.34 (0.006)	119.6 (0.3)	0.32 (0.005)	115.5 (0.5)	0.49 (0.01)
JC (Hz)	41.2 (0.60)	0.21 (0.01)	39.59 (0.66)	0.23 (0.01)	41.4 (0.7)	0.21 (0.01)
RDC (Hz)	−51.17 (0.35)	0.64 (0.03)	−51.16 (0.3)	0.62 (0.03)	−50.7 (0.3)	0.60 (0.03)
NOE ($\text{Å}$)	535.8 (0.66)	2.6 (0.04)	533.02 (0.53)	2.81 (0.04)	539.4 (0.7)	2.28 (0.07)
PRE ($\text{Å}$)	459.7 (2.3)	0.89 (0.06)	446.4 (1.8)	1.33 (0.03)	458.4 (4.3)	0.92 (0.11)
smFRET <E>	6.9 (0.08)	0.007 (0.004)	6.9 (0.05)	0.005 (0.004)	6.9 (0)	0.01 (0)
Rh ($\text{Å}$)	0.54 (0.05)	0.41 (0.09)	−0.46 (0.02)	0.21 (0.08)	−0.7 (0)	0.69 (0.05)
SAXS (Intensity)	457.5 (0.33)	0.002 (0)	457.9 (0.2)	0.001 (0)	458.0 (0.2)	0.001 (0)

Similar to the ${\text{R}}_2$-dynamic ensembles, the ${\text{S}}^2$-dynamic pools also have improved X-EISD scores and RMSD values with respect to the 8 data types, but quantitatively the ${\text{R}}_2$ relaxation rates seem to yield more consistent results than the ${\text{S}}^2$ order parameter. A T-test with a 95% confidence interval and a comparison of distribution of values indicate that the two dynamical selection methods are in fact statistically different from each other (Supplementary Table S1). The origin of this difference in the two dynamical measures could arise from a number of sources: (i) the ${\text{R}}_2$ rates are distinct from ${\text{S}}^2$ values, which include ${\text{R}}_1$ rates and NOE contributions, as well as assumptions about separation of motional timescales; (ii) the approach for selection is different, comparing the correlation between ${\text{R}}_2$ rates and numbers of HN contacts vs. the correlation between ${\text{S}}^2$ values and normalized mean square fluctuations of the lowest three frequency modes using ANM; and (iii) the selection method is on a per-residue basis for ${\text{R}}_2$ rates vs. on a segmental basis for ${\text{S}}^2$ values (see Supplementary Figure S2). Summarizing, we find that using dynamic information from ${\text{R}}_2$ relaxation data, and to a lesser extent ${\text{S}}^2$ values, as a prior to select a sub-ensemble improves the agreement with experiment for the unfolded state of the drkN SH3 domain.

Lincoff et al. have shown previously that certain NMR data types, such as J-couplings or NOEs, are very valuable in refining a structural ensemble since optimization of these single data types can help improve the other experimental data types such as SAXS or PREs.³³ Figure 4 provides the single optimizations with one data type (the diagonal entries) and its influence on the RMSDs of unoptimized data types (off-diagonal entries) compared with the unoptimized scores for all experimental data types in the last row for the ${\text{R}}_2$-dynamic structural ensembles. While the single optimizations of a given experimental data type using the ${\text{R}}_2$-dynamic ensemble improve RMSD errors of all the other experimental data types with few exceptions, the most dramatic improvement is the direct optimization of NOEs that improves the RMSD significantly for JC, PRE, RDC and smFRET for the MIXED ${\text{R}}_2$-dynamic ensemble (Figure 4c). Corresponding plots are shown using the ${\text{S}}^2$-derived ensembles in Supplementary Figure S3.

Figure 4. RMSD errors by optimizing the X-EISD score with a single experimental data type for $R_2$-dynamic ensembles derived from (a) ENSEMBLE, (b) RANDOM, and (c) MIXED pools.

The values are averages over 1000 ensembles of 100 structures each, and the numbers in parenthesis are standard deviations. The last row refers to the unoptimized ${\text{R}}_2$-dynamic pool of structures. Units are different for each experimental data type and are found in Table 1.

It is insightful to analyze the resulting refined ${\text{R}}_2$-dynamic ensemble from the perspective of structural signatures such as the radius of gyration (${\text{R}}_{\text{g}}$) (Figure 5). We find that the change in ${\text{R}}_{\text{g}}$ between the original and the ${\text{R}}_2$-dynamic pools is minimal for the ENSEMBLE structural ensemble, but that is not surprising since it had already been optimized by the same experimental data types as previously reported.³³ By contrast the MIXED ${\text{R}}_2$-dynamic ensemble has shifted to having more expanded conformers as evident in the ${\text{R}}_{\text{g}}$ distribution. For the RANDOM and MIXED pools, the refinement based on ${\text{R}}_2$ relaxation rates decreases the collapsed peak at ~12 $\text{Å}$ in the ${\text{R}}_{\text{g}}$ distribution, corresponding to the folded state conformers of the drkN SH3 domain and increases the importance of more extended conformations. Similar conclusions are reached for the ${\text{S}}_2$-dynamic ensembles as reported in the Supplementary Figure S2. This suggests that the selection and expansion based on dynamic properties enhances relevant structural properties within these disordered ensembles.

Figure 5. Radius of gyration distributions of the original (red) and ${\text{R}}_2$-dynamic (blue) ensembles for the unfolded state of drkN SH3 domain.

Shown for the (a) ENSEMBLE, (b) RANDOM, and (c) MIXED pools.

DISCUSSION AND CONCLUSIONS

In this study, we presented a computational method that uses NMR ${\text{R}}_2$ relaxation data, or ${\text{S}}^2$ order parameters in conjunction with an anisotropic network model, to define an underlying ensemble of conformations for a disordered protein that is more suitable for optimization against a range of experimental data that measure local, long range as well as global shape restraint information. Furthermore, we show that by using an anisotropic network model to generate alternate conformations along the normal modes, one can further improve the agreement of the selected ensemble with experiment.

It is understood that the ${\text{S}}^2$ value representing the internal dynamics of a protein derived from the Lipari-Szabo model free approach is not well formulated for the motions of an IDP or an unfolded protein.^{53, 82} This is because the ${\text{S}}^2$ value is based on the assumption that the correlation function that describes the dynamics of the amide bond vector can be written as a product of two exponentially decaying correlation functions: the overall tumbling motion of the protein molecule and the internal motion of the bond vector. The use of a single correlation time to characterize the overall tumbling is inappropriate for disordered proteins, in general, including the unfolded state of the drkN SH3 domain.⁸² Hence the use of ${\text{S}}^2$ order parameters for disordered proteins is not rigorous due to this lack of separation of motional timescales, and we would also conclude that it was not as fully effective as the ${\text{R}}_2$ relaxation rate, even with a crude structural estimate based on structural contacts.

Our previous work in developing the X-EISD method considered the full uncertainty in the experimental and back-calculation errors to enable meaningful comparisons between ensembles and their optimization. From that work, we show that J-couplings and NOEs can be highly useful data types that can simultaneously improve other data types such as smFRET, SAXS, and chemical shifts. However, it also matters that the underlying pool have meaningful conformers, and here we emphasize that the transverse ${\text{R}}_2$ relaxation rate is another useful measurement that can aid in selection of conformers to facilitate ensemble optimization. This contributes to the important goal of defining disordered state ensembles in order to better understand the relationships between the experimental data types that are most useful for characterizing IDP and unfolded protein conformational landscapes.