Effect of a Variable Nonbonded Attractive Pair Interaction on the

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Effect of a Variable Nonbonded Attractive Pair Interaction on the Relaxation Dynamics of in Vacuo Unfolded Lysozyme Gustavo A. Arteca,*,†,‡ C. T. Reimann,§ and O. Tapia‡ De´ partement de chimie et biochimie, Laurentian UniVersity, Sudbury, Ontario, Canada P3E 2C6, Department of Physical Chemistry, Uppsala UniVersity, Box 532, S-751 21 Uppsala, Sweden, and Department of Analytical Chemistry, Chemical Center, UniVersity of Lund, Box 124, S-221 00 Lund, Sweden ReceiVed: May 18, 2000; In Final Form: September 12, 2000

Starting from a partly unfolded conformer of in vacuo lysozyme, we study the configurational transitions and molecular shape changes that accompany the relaxation (and eventual refolding) of the protein. In particular, we explore the effect of a variable monomer-monomer interaction on the folding dynamics within an ensemble of relaxation trajectories. We find that a strong attractive potential does not necessarily produce configurational “freezing,” but instead can be consistent with nativelike refolding. In contrast, a reduction in attraction below a critical value suppresses polymer collapse and eventually leads to complete unfolding. Our results suggest that, qualitatively, folding behavior may not be strongly dependent on the details of the model force field, but rather a feature associated with a range of potential energy functions.

Introduction The so-called “new view” of protein folding kinetics postulates the existence of multiple paths over a rugged energy landscape with a global bias toward the native fold.1-4 Over this surface, some paths produce nativelike conformers, whereas others can be temporarily locked (or “topologically frustrated”) in folded, but nonnative, structures.5-9 An important goal of protein modeling is to gain insights into how these pathways differ in terms of energetics and structure. In this work, we address some aspects of this issue. We explore how folding paths are affected by smooth changes in the model force field. To this end, we generate a large set of relaxation trajectories that start from the same unfolded protein conformer, but differ in the monomer-monomer interaction employed. In particular, we study the effect of a variation in the attractive part of the nonbonded pair potential. This approach serves as a simple model for the environmental conditions that favor either unfolding into noncompact structures or relaxation into compact conformers. Differences in relaxation behavior can be recognized by monitoring the evolution of large-scale molecular shape features in transient conformers. It is well established that molecular compactness, content of secondary structure, and the formation of tertiary contacts are key ingredients in a folding process.10-13 These three factors are associated with distinct aspects of molecular shape that, a priori, are not strongly correlated with each other. Results show that compactness can stabilize secondary structure if it is already present, but it does not necessarily lead to its formation.10-12 Moreover, simple protein models suggest that compactization (e.g., a polymer collapse transition) may eventually trigger growth in secondary structure but only for some specific strengths in the monomer-monomer interaction.13-16 Until now, these studies have used various forms * Corresponding author. Permanent address: Laurentian University. E-mail: [email protected]. † Laurentian University. ‡ Uppsala University. § University of Lund.

of lattice and off-lattice contact (Ising-like) Hamiltonians,5 where amino acids are represented as single beads with some form of excluded-volume (repulsive) and first-neighbor (contact) interactions.14-18 However, other potential energy terms play a role in forming a compact globule.19 In this work, we move a step forward by monitoring molecular shape transitions along folding pathways in the context of a more realistic protein model. We use an all-atom force field to account for the specific sequence effects. The model includes a Lennard-Jones (6-12) potential for the nonbonded pair interactions. It is by modulating this term that we explore the effect of monomer-monomer interactions on the formation of quasi-native and nonnative persistent conformers by relaxation. As a working system, we consider the relaxation dynamics of in vacuo unfolded proteins. Folding-unfolding transitions in vacuo provide an excellent laboratory to test the specific, intrinsic role of primary sequence and backbone-only interactions, as decoupled from solvent effects. Several conjectures on the possibility of folding transitions in vacuo20 have been tested experimentally21-25 and in computer simulations.25-30 Experiment and simulation indicate inequivocally that anhydrous proteins can coexist in both folded and partly unfolded states in gas phase.22-25,29-31 As well, data from both sources do not rule out the possibility that a near-native fold be stable (or at least metastable) even in absence of water.23-25 Indeed, experiments with proteins in various charged states suggest that refolding into these quasi-native compact states can take place from unfolded structures in vacuo.22,23 In addition, results show that rearrangements in vacuo occur in a shorter time scale when compared to processes in condensed phase (typically, over 1 ms).25 As a result, the quantitative exploration of the initial steps for in vacuo folding-unfolding transitions is now within reach of present-day nanosecond-long molecular dynamics (MD) simulations. Recently,29-31 we discussed the relaxation dynamics of a series of transient conformers of lysozyme derived via centrifugal unfolding.26 In ref 29, we showed the occurrence of a systematic pattern of large-scale molecular shape changes (i.e.,

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Relaxation Dynamics of in Vacuo Unfolded Lysozyme

J. Phys. Chem. B, Vol. 104, No. 47, 2000 11361

a folding process) leading to compact structures, both native and nonnative. In this work, we use in vacuo lysozyme to probe the effect of nonbonded interactions on relaxation dynamics. On a manifold of observed pathways, we monitor the relation between molecular size, compactness, and folding features. To this end, we map the dynamics trajectories onto a convenient space defined by two distinct large-scale molecular shape descriptors:29,32 the asphericity Ω, that measures the extent of compactization, and the mean number of oVercrossings N h , that conveys folding complexity in terms of chain self-entanglements. Below, we show that a pattern of shape transitions persists even after introducing significant changes to the attractive part of the monomer-monomer interaction potential. These variations in nonbonded pair interaction provide a crude, yet useful model to gauge the influence of screening effects, including those associated with a solvent. Computer Simulation and Molecular Shape Analysis. We use molecular dynamics to simulate the in vacuo relaxation of an unfolded conformer of lysozyme. To this end, we employ the GROMOS87 force field,33 with the D4 parameter set, with explicit polar hydrogens and electrostatic hydrogen bonding.34 We have studied the changes caused by variations in the attractive (nonbonded) pair potential, while keeping constant all other energy terms (i.e., stretching, bending, torsions, and electrostatics). The Lennard-Jones potential is modulated by scaling the attractive term with a constant f:

VLJ(rij) ) S(rij)

{

Aij

r12 ij

-f

}

Bij r6ij

,f>0

(1)

where rij is the distance between the ith and jth atom, and S(rij) is a cutoff function. The form of S(rij), as well as the A and B constants,33 do not change during the simulation. As initial seeds for relaxation, we use the ensemble of conformers generated by in vacuo centrifugal unfolding of neutral lysozyme.26 Unfolding begins from the crystal structure of lysozyme with intact disulfide bridges, PDB code 1hel, with 129 residues. Centrifugal unfolding leads to molecular sizes and a multistep transition consistent with available experimental data.23,35,36 If unfolding conditions are not applied,26 the initial conformer undergoes a slight compression in vacuo, but the native fold is essentially conserved over the length of the simulations. We refer to the resulting manifold of conformers as the “in vacuo native structure (IVNS)” of lysozyme, i.e., the in vacuo relaxation of the native structure as it appears in the crystal. Although it is not known whether this IVNS is the global freeenergy minimum in vacuo, it represents a dominant conformational attractor with the same structural features that characterize lysozyme in the crystal.31 Given that this marginally stable state becomes significantly populated prior to unfolding, it provides a proper reference structure to compare the outcome of relaxation trajectories starting from the in vacuo unfolded state. In this work, all relaxation trajectories start from a transient obtained at t ) 575 ps during the unfolding trajectory.26 This conformer exhibits nontrivial distortions from the IVNS. The unfolded transient has a noncompact backbone with reduced secondary structure and a significant rotation of the R-domains.26 The distortion amounts to a large root-mean-square deviation (rmsd) with respect to the backbone of the IVNS, σIVNS ) 8.8 Å. From this initial conformer, MD relaxation proceeds as in ref 29. (Partly unfolded initial transients with σIVNS e 5 Å relax immediately toward the IVNS,30,31 and thus are not considered here.) The essential details of the simulations are as follows: (i) the structure is assigned initial velocities from a Maxwell-

Boltzmann distribution at T ) 293 K; (ii) the temperature is maintained by a weak coupling to a Berendsen thermostat;37 (iii) for each Value of the f parameter in eq 1, we generate seVen MD trajectories (1 ns long each) by changing the initial distribution of velocities. In this way, we can test the variability associated with a nonequilibrium process,38 and obtain the mean conformational behavior associated with a range of values for the attractive monomer-monomer interaction. During relaxation, the fluctuations in large-scale molecular shape features are monitored. To this end, we project the MD trajectories onto a two-dimensional space of shape descriptors, defined by Ω, the asphericity, and N h , the mean overcrossing number.29,32 The asphericity Ω is defined as Ω ) {(I1 - I2)2 + (I1 - I3)2 + (I3 - I2)2)}/2(I1 + I2 + I3)2, in terms of the principal moments of inertia {Ii}.39 This parameter measures the compactness of a molecular chain as it changes from Ω ) 0 in a globule to Ω ) 1/4 in an elongated conformer. The mean overcrossing number N h is computed as the spatial average of the number of projected crossings between backbone bonds.40 It conveys how a molecular chain entangles with itself, and measures folding complexity in terms of both the geometry and the connectivity (or “topology”) of the backbone.41 In DNA knots, N h correlates with several observables.42 In proteins, N h characterizes folding features independently of how a secondary structure is defined. This is a desirable property when trying to establish if compactization helps in the formation of a tertiary fold.10 The descriptor increases from N h ) 0 in linear rods to N h ∼ 0.05n1.4 for the native states of globular n-residue proteins.43 For a given molecular size, folds with β-strands yield smaller N h values than folds rich in R-helices.44 The two-dimensional (N h,Ω)-map is a convenient tool to detect the pattern of molecular shapes that are conserved within an ensemble of folding pathways over the energy landscape. Pattern of Folding Transitions as a Function of the Pair Interaction Strength. We have varied the potential in eq 1 by giving the scaling parameter f ten different values from 0.2 to 2.0, in steps of 0.2. The depth of the potential well, Vmin, and the position of the minimum, rmin, depend on f as Vmin ∼ -cf 2, rmin ∼ c′ f -1/6, with two constants c and c′. When f > 1, the potential is more attractive. The case f ) 1 corresponds to the standard GROMOS87 potential, used in generating the initial structure for relaxation.26 For each f value, the seven independent MD trajectories differ only in the initial distribution of random velocities. However, the same series of initial seeds for the random number generator were used for all f values. In this manner, any observable differences relate specifically to changes in the nonbonded pair potential. Trajectories were generated with an integration step of 1 fs, and we extracted configurational snapshots every 5 ps during a total time of 1 ns. Below, each of the MD simulations is identified by the code “seed X,” where X ) 1, 2, ..., 7. For every conformational snapshot, N h and Ω are computed from the geometry and connectivity of the R-carbon backbone.32 The initial unfolded conformer (N h ≈ 39.5, Ω ≈ 0.158) has some residual secondary structure, but it is elongated and partly disentangled with respect to the IVNS. (The configurationally averaged descriptors for the IVNS are 〈N h 〉 ≈ 48.8 and 〈Ω〉 ≈ 0.035). These two structures have also very different mean sizes, with backbone radii of gyration Rg ≈ 19.8 Å and Rg ≈ 13.3 Å for the initial conformer and IVNS, respectively. (We do not use Rg for the rest of the analysis, since many trajectories produce transients with distinct folding features but similar mean sizes.) As f varies from 0.2 to 2.0, the MD relaxations show

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Arteca et al.

Figure 1. Variation in backbone entanglement during the relaxation of lysozyme. [The lines are results for single MD trajectories started with identical initial conditions but different scaling factors f for the Lennard-Jones attraction. The right-hand-side snapshots are conformers observed at the end of the runs. The snapshots on the left correspond to the initial conformer for relaxation (i.e., “unfolded transient”) and the in vacuo native structure (IVNS). The MD run with f ) 0.8 refolds to the IVNS, whereas the trajectory with f ) 0.2 shows complete unfolding.]

Figure 2. Variation in backbone asphericity during the relaxation of lysozyme. [Trajectories and symbols are those in Figure 1. Some conformers for the MD runs with f ) 0.2 and f ) 0.8 share the same Ω values, although they are distinguished in terms of folding complexity (cf., Figure 1).]

responses ranging from the reVersible refolding to the IVNS to the complete unfolding of the residual secondary structure. Figures 1 and 2 illustrate these opposite behaviors by displaying the variations in N h and Ω along the trajectory with

seed 1. Figure 1 shows the evolution of chain entanglements in terms of N h for the relaxations initiated with parameters f ) 0.8 and f ) 0.2. In the former case, the trajectory leads to quasinative entanglement complexity after 200 ps. In the latter case,



Figure 3. Molecular shape map for a single MD trajectory with f ) 0.4. The simulation indicates a transition from noncompact structures to compact conformers with nonnative fold.

large oscillations at smaller N h values are the signal of extensive disentanglement of the tertiary fold. Figure 2 complements these observations with the variation in asphericity. When relaxing with the more attractive potential (f ) 0.8), the chain refolds into a globule that resembles the IVNS. When using the leastattractive potential (f ) 0.2), the chain is in a “fluid” state, undergoing a series of continuous changes from spheroidal to elongated. The snapshots in Figure 1, taken at the end of the trajectories, are consistent with these observations. When f ) 0.8, the R-helical content of the initial transient is maintained and the structure regains a quasi-native fold. In contrast, when f ) 0.2, all residual R-helices disappear, and the transient conformers show fluctuating β-sheets. (Note that the elements of secondary structure, defined in terms of the hydrogen-bond pattern, provide simple visual cues to recognize differences among conformers. However, this information is provided only for illustration and it is not used in our quantitative analyses. Changes in tertiary structure are always described in terms of large-scale chain entanglements, i.e., by using N h , instead of the content of secondary structure.) Our results indicate a remarkable persistence of the local R-helical structure well into the regime of weak pair attractions. Throughout all trajectories, R-helices do not unfold for f g 0.4. There are, however, important variations in tertiary structure due to initial conditions, as discussed below. Figure 3 shows an example of this behavior in the molecular shape map of a trajectory for f ) 0.4. In this case, we find that the small f value does not impede compactization. After 1 ns, lysozyme reaches an ensemble of collapsed conformers with nonnative tertiary fold, yet an almost intact R-helical structure. This behavior can be contrasted with Figure 4, showing the evolution of an individual trajectory in the (N h , Ω)-shape space for f ) 0.2. Here, the initial local secondary structure is destroyed after 30 ps. For the remainder of the trajectory, the system fluctuates in a

continuum of partly disentangled structures with large oscillations in anisotropy. These differences in large-scale shape features reflect also variations in elements of secondary structure. For instance, the snapshots in Figure 4 show the occurrence of transient (short) β-sheets for t > 400 ps. Throughout our simulations, this type of behavior appears only for trajectories with f ) 0.2. Figure 5 illustrates these observations by displaying the snapshots corresponding to t ) 800 ps found in all seven MD runs with f ) 0.2. (As mentioned before, the elements of secondary structure are displayed only for visual inspection of the snapshots.) Figure 5 indicates the occurrence of a significant β-sheet content, especially in the MD runs corresponding to “seed X”, with X ) 2, 3, 4, and 5. Note as well the near absence of residual R-helical structure. Despite these similarities in local structural features, the f ) 0.2 transients differ largely in global folding features, as it is clearly recognized by their strong variations in compactness and entanglement. For this reason, we do not use the content of secondary structure as an indicator of backbone shape. Nevertheless, it is noteworthy that these conformers are unfolded, yet they are not random coils. Their characteristic (N h, Ω) values are qualitatively comparable to those found in noncompact proteins with the same chain length (e.g., viral capsid proteins).29,44 Aside from the above pattern, individual trajectories for f > 0.4 show a large variation in relaxation behavior. Depending on the initial velocities, a MD run with scaled potential can lead to a wide range of persistent (“stable”) conformers, including quasi-native and nonnative (both compact and noncompact) folds. Within a range of f values, we find that lysozyme can reach similar relaxed states whether: (1) we scale f while keeping the same distribution of initial velocities, or (2) we vary the initial conditions while keeping the same f value. The only systematic trend observed as a function of f is found in the initial response of the system to relaxation. Within the


Arteca et al.

Figure 4. Molecular shape map for a single MD trajectory with f ) 0.2. The simulation shows a fast “melting” of the initial local structure of the chain. At f ) 0.2, lysozyme is very flexible, and fluctuates between compact and noncompact structures. Note the formation of transient β-sheets.

first 5 ps, unfolded lysozyme reorganizes always toward a globule:

(∂Ω∂t )

tJ0

< 0, ∀ f

(2a)

whereas entanglements may increase or decrease depending on the scaling factor:

(∂N∂th )

tJ0

> 0, f > 1 <
0.6, relaxation pathways can lead to native refolding, compact (nonnative) conformers, as well as noncompact (“frozen” or “frustrated”) configurations. These compact structures are consistent with a variety of secondary structural features, in agreement with other results in the literature;51 (ii) Secondary structure persists for f g 0.4, but a weak pair attraction is not enough for refolding; (iii) The meltdown of all residual local structure takes place at a very weak potential, f ) 0.2. In this case, the protein behaves as a flexible polymer, switching between compact and noncompact chains. However, even at f ) 0.2, disulfide-intact lysozyme can still form β-strands that coalesce into short β-sheets. These conformers, though denatured, do not represent random structures. Their shapes are qualitatively comparable to those corresponding to the native states of noncompact proteins;29,44 (iV) Near-native refolding is not the most likely outcome within 1 ns of relaxation, although its probability does not appear to decrease as the potential is made more attractive. The results suggest that configurational frustration is perhaps weakly coupled to the form of the potential and more strongly dependent on the inital conditions, e.g., the initial distribution of velocities. Folding is possible in the regime of strong attractions (f ) 2), as much as topological frustration can still occur with a weakly attractive potential (f ) 0.6). Nevertheless, a minimum attraction may be needed to prevent the formation of a protein that is too flexible to be functional.53 These conclusions are a factor to take into account when attempting to redesign generic force fields for nonequilibrium molecular dynamics. Since the relaxation pattern observed in this work



Figure 7. Molecular shape map for all MD trajectories in the seed 3 series. The numbers in bold correspond to the f scaling parameters. [The lines alternate black and gray, starting with the MD run for f ) 0.2 in black. Most seed 3 relaxations lead to “topological” frustration.8 Cf., Figure 7.]

Figure 8. Molecular shape map for all MD trajectories in the seed 5 series. The numbers in bold correspond to the f scaling parameters. [The lines alternate black and gray, starting with f ) 0.2 in black. The seed 5 relaxations produce both compact and “frozen” conformers; cf., Figure 7.]

persists over large variations in the nonbonded pair interaction, it is possible that any fine-tuning of other potential energy terms may have an even smaller effect. The implication appears to

be that the folding transition in vacuo can be simulated by primarily controlling global factors (e.g., monomer sequence, type of monomer interaction, and the initial unfolded state),


Arteca et al.

Figure 9. Molecular shape map for all MD trajectories in the weak-attraction regime (with f ) 0.2, 0.4, and 0.6). [There is a gradual change in the accessible molecular shape features as a function of f. Rigid conformations dominate at f ) 0.6, persist at f ) 0.4, and disappear when f ) 0.2. In the latter case, full unfolding has taken place.]

present when studying folding in solution. It is conceivable that the relaxation dynamics of in vacuo unfolded proteins contains already all the essential elements needed to explain the folding transition in solvated proteins. Within this view, the solvent would only modulate the details of the free-energy surface by providing specific interactions, as well as slow the time scale of folding events by acting as a viscous medium causing strong frictional dissipation.

Figure 10. Root-mean-square deviation with respect to the IVNS, σIVNS, averaged over conformers with t g 400 ps. Each box gives σIVNS for a given MD run and f value. [For clarity, the σIVNS boxes have been slightly displaced around their f values. The bar labeled “Initial” is the σIVNS value averaged during the first 5 ps of relaxation. The symbols U, FF, PF, and N stand for “unfolded”, “configurationally frustrated/frozen”, “partly folded”, and “native fold”, respectively. Complete refolding events are infrequent, but observable for trajectories with f g 0.8. The different U, FF, PF, and N species are distinguished by their σIVNS values. More importantly, these structures occur in different regions of the two-dimensional molecular shape space.]

rather than by manipulating the details of the potential energy function. Some of these observations may reflect a fundamental characteristic of the folding transition. Note that a weaker internal attractive potential can be taken as a crude model for denaturing environmental effects. Accordingly, a similar insensitivity in global folding features with respect to the local properties of the potential energy landscape might also be

Acknowledgment. We thank I. Vela´zquez for his collaboration with part of the research and N. D. Grant for her comments on the manuscript. We thank Koradi et al. for use of their protein visualization program Molmol.54 G.A.A. acknowledges support from the Natural Sciences and Engineering Research Council of Canada. C.T.R. thanks the Swedish Technical Research Council (TFR), and O.T. is grateful to the Swedish Natural Sciences Council (NFR) for financial support. References and Notes (1) Dobson, C. M.; Sˇ ali, A.; Karplus, M. Angew. Chem., Int. Ed. Engl. 1998, 37, 868. (2) Socci, N. D.; Onuchic, J. N.; Wolynes, P. G. Proteins 1998, 32, 136. (3) Shakhnovich, E.; Fersht, A. R. Curr. Opin. Struct. Biol. 1998, 8, 65. (4) Dill, K. A.; Chan, H. S. Nat. Struct. Biol. 1997, 4, 10. (5) Bryngelson, J. D.; Wolynes, P. G. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 7524. (6) Garel, T.; Orland, H. Europhys. Lett. 1998, 6, 307. (7) Sˇ ali, A.; Shakhnovich, E.; Karplus, M. J. Mol. Biol. 1994, 235, 1614. (8) Shea, J.-E.; Onuchic, J. N.; Brooks, C., III. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 12512. (9) Guo, Z.; Thirumalai, D. Folding Des. 1997, 2, 377.

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Effect of a Variable Nonbonded Attractive Pair Interaction on the

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