Simulations of protein folding transition states using ψ-values

Abstract

ψ-analysis has been used to identify inter-residue contacts in the transition state ensemble (TSE) of ubiquitin and other proteins. The magnitude of ψ depends on the degree to which an inserted bi-Histidine (biHis) metal ion binding site is formed in the TSE. A ψ equal to zero or one indicates that the biHis site is absent or fully native-like, respectively, while a fractional ψ implies that in the TSE, the biHis site recovers only part of the binding-induced stabilization of the native state. All-atom Langevin dynamics (LD) simulations of the TSE are performed with restrictions imposed only on the distances between the pairs of residues with experimentally determined ψ of unity. When a site with a fractional ψ lies adjacent to a site with ψ = 1, the fractional ψ generally signifies that the “fractional site” has a distorted geometry in the TS. When a fractional site is distal to the sites with ψ = 1, however, the histidines sample configurations in which the site is absent. The simulations indicate that the ψ = 1 sites by themselves can be used to generate a well-defined TSE having near native topology. φ-values calculated from the TS simulations exhibit mixed agreement with the experimental values. The origin and implication of the disparities are discussed.

The characterization of the TS is the goal of numerous protein folding studies. Because no population accumulates in the TS, kinetic methods such as φ- and ψ-analyses are used to probe the energetics and structure of the TS. Whereas mutational φ-analysis focuses on the energetic perturbation due to the alteration of a side chain, ψ-analysis focuses on identifying inter-residue contacts that define the structural topology of the TS.^{1; 2; 3} ψ-analysis proceeds by introducing relatively benign biHis metal ion binding sites one at a time on the protein surface. The addition of metal ions stabilizes the interaction between the two histidine partners which in turn stabilizes the corresponding secondary or tertiary structure. The metal-induced stabilization of the TSE relative to the native state, as represented by the extrapolation of ψ to vanishing metal ion concentration, reports on the proximity of the two partners in the TSE prior to any perturbation.

Values of ψ from zero to unity are observed for ubiquitin (Ub),⁴ dimeric and cross-linked versions of the GCN4 coiled coil,⁵ acyl phosphatase (AcP),⁶ and the B domain of protein A (BdpA).⁷ The limiting values of zero and unity are accepted as indicators that the probed region of the protein is either unfolded-like or native-like, respectively. However, the interpretation of fractional values^{1; 8; 9; 10; 11} and their use in modeling the TS¹² is more complicated, as has been discussed by a number of groups. A fractional ψ, i.e., 0 < ψ < 1, indicates that the TS is stabilized by ion binding to a lesser extent than the native state.¹⁰ This situation can occur when a native-like site is formed in only a subpopulation of the TSE (TS^present) with a fraction equal to the ψ-value, or when the site has binding affinity weaker than in the native state, or a combination thereof according to

$${\psi }_o=\frac{K_N}{K_{TS}^{\mathit{present}}}\frac{K_U-K_{TS}^{\mathit{present}}}{K_U-K_N}{F}^{\mathit{present}}$$

where F^present is the fraction of the TSE with the site formed, K_N, $K_{TS}^{\mathit{present}}$, and K_U are the respective metal binding affinities, and where the binding in the TS with the unformed biHis site, TS^absent, is assumed to have the same affinity as the unfolded state.

The ψ-values observed for the dimeric GCN4 coiled coil quantitatively reproduce the known degree of TS heterogeneity under the assumption that sites are either formed with native-like or unfolded-like binding affinity.^{5; 13} In addition, ψ for sites at one end of the coiled coil increased upon the introduction of destabilizing mutations at the far end as a result of a decrease in the probability of nucleation at the destabilized end.

All measured sites in Acp⁶ have ψ = 0 or 1, except for a site on helix H2 that lies next to a site with ψ = 1. The single fractional site yields ψ ~ 0.4, independent of whether the measurement uses either Ni²⁺, Zn²⁺, or Co²⁺ ions that have different coordination preferences¹⁴ and that stabilize the native protein by different amounts (1.5, 0.9 and 0.7 kcal·mol⁻¹, respectively). The invariance of ψ to metal ion supports the view that a fractional ψ reflects TS heterogeneity, because the same fractional binding affinity in the TS is unlikely to be maintained if the fractional ψ arises due to a single distorted site. Metals with different coordination geometries are expected to stabilize distorted sites in the TS to different extents relative to the stability the metals impart to the native state. Thus, the different metals should return differing ψ. Because ψ for Acp is independent of metal ion, the fractional ψ probably indicates that the biHis site lies in a region of the helix that is partially frayed in the TS.

On the other hand, numerous fractional ψ are observed for BdpA.⁷ They are best interpreted in the context of distorted sites based on the following observations. Kinetic amide isotope effects indicate that 2/3 of the helical hydrogen bonds are formed in the TS. Nevertheless, many ψ are fractional and metal-dependent for sites on portions of the helices that are formed in the TSE. Additionally, a fractional ψ (0.2) on Helix l is invariant to a destabilizing mutation introduced at the other edge of the structured portion of the TS. Hence, the fraction of the TSE in which this biHis site is formed remains invariant. This behavior is in contrast to the dimeric coiled coil where the populations of the biHis sites in the heterogeneous TS are subject to manipulation by destabilizing mutations far from the biHis site. Thus, we conclude that the packing of the three partially formed helices in the TS of BdpA is optimal, but the packing produces slightly non-native binding site geometries and fractional ψ.

These three proteins illustrate that a fractional ψ can either originate from a distorted site or a site that is formed in a subpopulation of the TSE. Here, we employ all-atom LD simulations to investigate the presence of structural diversity in the TS and the potential origins of fractional ψ. A minimalist approach is invoked in the simulations. Because sites with ψ = 1 found experimentally have the same ion binding affinity in the TSE as in the native state, the protein backbone at these sites is taken to adopt a native-like geometry throughout the simulations. Accordingly, the inter-residue C_α and C_β distances are constrained to their native values for each pair of residues that comprise the five native-like biHis sites. Similarly, the regions of the protein containing the sites with ψ = 0 are taken as disordered in the starting configurations for the simulations. With the constraints only applied to the sites with ψ = 1, we analyze the structural diversity in the LD trajectories for the regions of the protein with fractional ψ. We find that the separation between the pair of residues forming a site with fractional ψ either executes limited or large amplitude motions depending on the site’s distance from the native-like sites.

LD simulations of the TSE

Experimental data for ψ are combined with the native Ub sequence and structure (PDB 1UBQ),¹⁵ to create two initial TS models, TS^min and TS^max, with the minimum and maximum amount of possible structure, respectively (Fig. 1). Five of the fourteen measured ψ are unity (sites a, b, d, g, l), three are zero (f, i, j), and the remainder are fractional (c, e, h, k, m, n). The eight sites with the readily interpretable, unambiguous ψ = 0 or 1 identify the regions that are modeled in both initial TS models as unfolded or are left in their native geometry, respectively. The TS^max model also initially constrains the fractional sites to their native geometry, while these sites are permitted to become disrupted in the initial TS^min model. The TS kernel is represented by TS^min and contains a four-stranded sheet network docked against the carboxyl-terminal portion of the α-helix. The initial structures for the LD simulations are generated by selectively disrupting those regions of the protein that are assigned as unfolded in such a way that the global fold is undisturbed and the unfolded regions retain physically plausible backbone dihedral angles.

Fig. 1. ψ-analysis applied to Ub and the two TS models.

ψ has been measured at 14 sites.⁴ Five of these sites yield ψ = 1 and are taken as present in the entire TSE. Three sites in yellow produce ψ ~ 0 and are taken as absent from the TSE. Six other sites have intermediate ψ (“fractional ψ”). The maximum model is constructed such that both the sites with ψ equal unity and ψ fractional are constrained to remain in their native configurations (75% of the residues), with the other regions unfolded. The minimum TS model is the obligate kernel containing only the sites with ψ = 1 constrained to their native configurations (54% of the residues). Renderings created in PyMol (www.pymol.org).

LD simulations use the implicit solvent model developed in the Freed group^{16; 17} and implemented in a modified version¹⁸ of the TINKER dynamics package.¹⁹ To minimize the potential biases of different force fields, both the Garcia and Sanbonmatsu’s modified Amber94 (G-S Amber94)²⁰ and the OPLS/AA-L^{21; 22} force fields are utilized in separate simulations. Prior to any dynamics, the energy of the initial structures is minimized with the added constraint that backbone torsional angles of portions with native secondary structure remain close to their initial values. This constraint is accomplished using a harmonic force constant of 1 kcal·mol⁻¹·deg⁻². Ten to twenty 10–40 ns LD trajectories are performed for each TS model as well as for the native state. The native sequence (i.e., without the biHis substitutions) is used in all simulations because the experimental ψ-values are extrapolated to zero metal concentration and are corrected for the change in stability due solely to the biHis substitution.

We introduce inter-residue distance constraints only at the five sites with ψ = 1 for simulations of the transition and native states. The native C_β-C_β and C_α-C_α separations for these five residue pairs are obtained from the crystal structure, and a relatively soft harmonic spring (k = 2.4 kcal·mol⁻¹·Å⁻²) is used to maintain crystal structure distances for these residue pairs in the initial energy minimizations and in the LD simulations. Without the constraints, the TS structures fall apart, an unsurprising result given that these models are designed to represent intrinsically unstable TS structures. The average RMSD of the native state simulations is within 2Å of the crystal structure.

Distributions of the inter-residue C_α-C_α and C_β-C_β distances are calculated for each residue pair comprising a biHis site, first for individual trajectories and then for the average over all trajectories. Both force fields yield similar results, so only data related to the OPLS/AA-L force field are presented. The individual trajectories display a variety of behaviors, and comparing the TS^min and TS^max models helps overcome any sampling issues and provides an indication of the magnitude of structural diversity at the positions where ψ are fractional.

Fractional ψ

The distributions of distances between the residue pairs at the sites where ψ = 1 are peaked near the native separation, as expected given the harmonic constraints (Fig 2). The distance distributions for the pairs at the fractional sites are more diverse. Given the possible ambiguities in the interpretation of fractional ψ, we expect the distributions generally to fall into two classes. When a fractional ψ reflects TS heterogeneity, the distance distribution should exhibit one peak near the native distance, while the remainder of the distribution should encompass longer distances reflecting a “site absent” or unfolded-like condition (for example, beyond a 7.5 Å cutoff distance, Fig. 2). On the other hand, when a fractional ψ reflects the presence in the TS of a single distorted site with binding affinity weaker than in the native state, the distance distribution should be centered near the native separation.

Fig. 2. Distributions of C_β-C_β separations for N, TS^max, and TS^min states.

After an initial 2 ns equilibration period, distributions for the C_β-C_β separations are calculated during the next 8 ns for 10–20 trajectories per model using the OPLS/AA-L force field. Inset: C_α-C_α distributions. The distributions for sites with ψ = 1 are centered near their native state separations from the crystal structure (solid black line) due to the imposed harmonic constraints. Some of the distributions at the sites with ψ = 1, e.g. site l, broaden due to departures from the native structure (see Fig. 1). Fractional sites that represent distorted geometries (sites c, e, h, m) have native-like C_α-C_α and C_β-C_β distributions with centroids below 7.5 Å (dashed line). Sites n and k exhibit distributions that sample both native-like and unfolded-like distributions for both the C_α-C_α and C_β-C_β distances.

The sites with fractional ψ generate distance distributions that fall into the two anticipated general classes depending largely on the pairs’ proximity to the constrained ψ =1 positions. Sites c, e, h, and m, which have fractional ψ and are located next to sites with ψ = 1, yield distance distributions that are clustered near their respective native state distributions in simulations beginning from either TS^max or TS^min. These sites’ C_β;-C_β; distance distribution exhibit some fine structure, but most of the distribution is below the 7.5Å cutoff, and their C_α-C_α distributions are peaked near the native value. Evidently, the possible separations between residue pairs at these fractional positions are hindered during the simulations by the constraints imposed at the adjacent positions with ψ = 1.

Sites n and k are located further from the TS kernel and thus display broader distributions in both their C_α-C_α and C_β-C_β distributions. Site n is situated at the distal end of the amino-terminal hairpin, and site k is located on amino-terminal region of the α-helix. During the LD trajectories, the separation between these two sites is indicative of whether a contact between these two sites is present or not. Site n is formed initially in TS^max, and the distribution of separations during the dynamics slightly exceeds the native distance (Fig. 2). Site n is absent in the beginning TS^min structure, and the resulting distance distributions are very broad with minimal probability at the peak observed in simulations with TS^max. These two results suggest the presence of two stable wells, one native-like and the other unfolded-like. Site k appears as helical in the starting TS^max structure but as unfolded in TS^min and is the only fractional site whose distribution of separations in the TS^max trajectories clearly suggests a shuttling of this site between native-like and unfolded-like conformations (Fig 2). The site k distributions for TS^max and TS^min are broad and extend from native-like to unfolded distances, with TS^max favoring the former and TS^min the latter. The breadth of the distributions at larger distances is consistent with an unfolded-like ensemble rather than a single conformation. Figure 3 displays examples of the diverse behaviors observed in individual trajectories for sites n and k. Compared to the other four sites with fractional ψ, sites k and n, the two sites least affected by the constraints imposed on the five sites with ψ = 1, undergo fluctuations that are best interpreted as TS heterogeneity.

Fig. 3. Illustrative trajectories of C_β-C_β separations and hydrogen bond formation.

The C_β-C_β separation (black curve, distance in crystal structure denoted with dotted line) is presented along with the H-O distance for the hydrogen bond that is most associated with the C_β-C_β separation (red curve, 2.5 Å threshold separation is designated by black dotted line). The blue curve indicates whether the hydrogen bond is formed (1) or not formed (0). The C_β-C_β separation of sites with ψ = 1 remains close to their distance in the crystal structure. Sites with fractional ψ that emerge due to site distortion, e.g. sites c and e, experience some long-time fluctuations in which hydrogen bonds break and reform. These fluctuations occur primarily for trajectories where the sites begin in non-native conformations (TS^min). The two sites (k and n), whose behavior is indicative of structural heterogeneity, exhibit the most long-time fluctuations. A hydrogen bond is considered formed if the H-O distance is less than or equal to 2.5Å and the N-H-O angle is greater than 120º

This interpretation is further supported by an evaluation of the average number of native hydrogen bonds formed (Fig. 4A). The C_β-C_β separation at a site is an indicator of whether a hydrogen bond is formed between the two residues. Typically, a native-like separation implies hydrogen bond formation, while deviations from a native-like distance implicates the loss of the corresponding hydrogen bond (Fig. 3). Although fluctuations between native-like and unfolded-like separations appear in trajectories for the distorted sites c and e, the fluctuations are only observed in trajectories where the site is absent initially. The separations in sites k and n fluctuate in more trajectories and in simulations for both TS^max and TS^min.

Fig. 4. Hydrogen bond formation and computed φ.

A) The average native hydrogen bond fraction is computed for the donor amide residue listed for the sites with unity and fractional φ. B) Average φ-values are computed using either total or native side chain contacts and computed relative to either the static PDB structure or the average of the native simulations. A contact is defined by two side chain heavy atoms within 5.5Å of one another and the corresponding residues are at least two residues away from each other. The experimentally measured φ¹⁰; ²³ are depicted by short horizontal bars. An alternative calculation uses the average solvent accessible solvent area (SASA) of the whole residue relative to the average SASA in an unfolded state ensemble constructed from a statistical coil library.²⁵ The data presented are computed using the OPLS/AA-L force field.

This assessment presumes that the geometry of a distorted site remains hydrogen bonded, while non-native separations imply the lack of hydrogen bonds. The fractional sites located across two β-strands are associated with 1–2 hydrogen bonds. If at least one of these hydrogen bonds is often formed and native-like, we believe that the fractional ψ of this site is largely due to site distortion. Although the two different force fields differ quantitatively, the overall trends are the same. A site with a fractional ψ situated near a site with ψ = 1 is more likely to remain hydrogen bonded rather than fluctuate between native-like and unfolded-like states. Overall, the simulations indicate that a fractional ψ may arise from either a distorted site geometry, fluctuations between hydrogen bond present and absent configurations, or some combination thereof.

Varnai et al. ¹² also performed constrained simulations of Ub’s TSE. However, they conclude that the ψ-values do not adequately constrain the structure of the TSE, contrary to the situation described by our simulations. They incorporate the data for ψ into their simulations by constraining the TSE with a residue-residue potential constructed from observed His-His distances in various ion binding sites. The resulting potential is constant out to a separation of 9Å, which is far beyond the native biHis separations observed in Ub (4–8 Å). Thus, their implementation contrasts with our work where a native-like binding affinity in the TS (i.e., ψ = 1) is taken as evidence that the corresponding residue pair adopts a native-like separation in the TS. The broad potential of Varnai et al. allows residues to become unfolded-like even when they have a unity ψ. Not surprisingly, this implementation results in the prediction of a less structured TS. In particular, the α-helix is completely absent in their simulated TSE using only ψ as constraints. As a result, Varnai et al. conclude that the measured φ-values do not adequately constrain the structure of the TS and that the resulting structure would under-predict experimental φ values. However, these conclusions are erroneous because the experimentally determined ψ of 0.90 ± 0.14 and 0.48 ± 0.04 for helical sites l and k, respectively, mandate that the helix is present in the TS, whereas it is absent in Varnai et al.’s simulations. In fact, the model they construct using both ψ and φ data is remarkably similar to our model obtained using only the five ψ = 1 sites as constraints. We conclude that a proper incorporation of the unity ψ-values alone does suffice to generate a well-defined TSE exhibiting a near native topology.

Predicting φ from the TSE

With the TS structure independently determined from the data for ψ, we are well positioned to investigate the ability of simulations to accurately predict experimental φ-values. We have calculated φ from the number of side chain heavy atom contacts in three different ways. The customary manner is to compare the average number of native contacts in a simulated TSE relative to the number of (native) contacts in the original (static) PDB structure.¹² Two appealing alternatives are to calculate φ from the number of contacts observed in simulations of the native and transition states, either using the total or only the native contacts.

Burial levels and φ for both TS^min and TS^max have been evaluated from the simulations for five hydrophobic residues whose experimental φ have been measured (Fig. 4B).^{10; 23} Each residue is highly buried in the TSE (>75%). The three methods for calculating φ produce similar values across the five sites although the level varies between the three methods from 0.3 to 0.7. All three methods yield significant deviations from experiment, in particular over-predicting ${\phi }_{exp}^{L67A}$ which is observed experimentally to vanish. Calculations of φ using the total number of native contacts observed in simulations both of the native and transition state agree with φ^exp at I3, L15 and I30, but over-predict the value at L43. The customary method of using the number of native contacts relative to those in the static PDB structure leads to an underestimate of φ at the first three sites, while the third option of using the total number of contacts observed in simulations of both the native and transition states, always produces an overestimate of φ.

Based on these results, we believe that calculations of φ based on the number of side chain contacts should not always agree with the experimental data. The most glaring issue for the example of Ub is the significant over-prediction of φ^exp = 0 for the L67A substitution. This discrepancy is not necessarily an error in the simulation per se, but in the mode of calculating φ. Unlike the simulated values for φ, the experimental φ^exp reflect free energies. The observed lack of an energetic perturbation in the TSE for L67A can emerge due to multiple factors, including relaxation and energy minimization of the structure^{7; 10; 24} or a change in intrinsic secondary structure propensities of the substitution. Conversely, a vanishing experimental φ should not be taken as an absolute indication that a site is devoid of structure in the TSE.

Conclusion

We have simulated models of Ub’s TSE by constraining the separations between the five residue pairs for which experiments determine ψ as equal to unity. The constrained LD trajectories indicate that the TSE is composed of an obligate kernel consisting of portions of β1-β4 and the carboxyl-terminus of the α-helix. The periphery of the kernel relaxes to an energy minimum structure, while the tails of the β1-β2 hairpin and the amino-terminus of the α-helix are frayed. An analysis of the distribution of inter-residue separations indicates that fractional ψ can reflect either a distorted binding site with weakened ion binding affinity or a site that fluctuates between the hydrogen bond being present and being absent, especially for the two sites distal to the sites with ψ equal to unity. Calculations of φ using side-chain contacts for residues in the core of the simulated TS often differ from those observed experimentally. Generally, one should not expect agreement between theory and experiment because the experimental values by definition reflect free energies rather than contacts. This discrepancy should be considered in other studies that compare simulations to experimental data, particularly when the experimentally determined TSE appears to be small and polarized.

Abbreviations

biHis

bi-histidine

G-S Amber 94

Garcia and Sanbonmatsu’s modified version of Amber 94

$K_{eq}^D,K_{eq}^N,K_{eq}^{TS}$

metal ion binding affinity of the denatured, native and transition states, respectively

TSE

transition state ensemble

Ub

ubiquitin

LD

Langevin dynamics